contrast sensitization: function, theory, and …
TRANSCRIPT
CONTRAST SENSITIZATION: FUNCTION, THEORY, AND
MECHANISM OF A NOVEL RETINAL COMPUTATION
A DISSERTATION
SUBMITTED TO THE PROGRAM IN NEUROSCIENCE
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
David B. Kastner
June 2013
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/pt514dd8480
© 2013 by David Barak Kastner. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Stephen Baccus, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Denis Baylor
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Richard Tsien
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Brian Wandell
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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iv
Abstract
Adaptation provides a ubiquitous strategy for neural circuits to encode their inputs
using their limited dynamic range within the variety of sensory environments that
they encounter. However, because of the inherent timescale necessary to optimize the
response properties of a cell to its environment, any form of adaptive plasticity can
cause a neuron to fail to encode the stimulus when the environment changes. Many
ganglion cells, the output neurons of the retina, adapt so as to lower their sensitivity
in an environment of high contrast, but if the contrast subsequently decreases the cell
will fall below threshold and fail to signal. I have found a distinct form of plasticity
within the retina that acts in coordination with the process of adaptation. Cells using
this new form of plasticity elevate their sensitivity after a transition to low contrast.
This process, called sensitization, occurs in retinas from multiple species. Multielec-
trode recordings from sensitizing and adapting cells indicate that both populations
encode the same visual signals. The complementary action of the two populations
helps the retina encode its input over a broader range of signals and environmen-
tal changes, with one population continuing to respond when the other fails. The
threshold placement of these two cell types further enhances their coordination be-
cause sensitizing cells maintain lower thresholds, while adapting cells maintain higher
thresholds. Using a theoretical model, I was able to show that this behavior maxi-
mized the amount of information that the two populations can provide about their
input. I have further studied the spatiotemporal region that controlled the sensitivity
of a cell–the adaptive field. Just as retinal circuitry uses excitation and inhibition
to form biphasic center-surround receptive fields, the retina can also use adaptation
and sensitization to form biphasic adaptive fields in both the spatial and temporal
v
domains. Since visual statistics are correlated across time and space, center-surround
biphasic receptive fields more e�ciently encode the input by subtracting a prediction
of the stimulus so as to just encode the deviation from that prediction. Biphasic adap-
tive fields appear to perform an opposite function, transmitting a prediction of the
stimulus at the transition of a stimulus environment to weaker signals. This assists in
the encoding of an uncertain environment by storing features of a predictable input.
A model indicates that sensitization within the adaptive field can be produced by
adapting inhibition, a form of plasticity whose function was previously unknown. Us-
ing pharmacology, I confirmed this prediction, showing that GABAergic inhibition is
necessary for sensitization. Using simultaneous intracellular recording from inhibitory
amacrine cells and multielectrode recording from ganglion cells, I show that trans-
mission from a single amacrine cell is su�cient to cause sensitization. Using a novel
approach to analyze a circuit, I quantitatively describe the changes in amacrine cell
transmission that underlie sensitization thus elucidating how the retina performs this
sophisticated computation.
vi
Acknowledgements
I have had the good fortune to work with many wonderful people during my graduate
studies. First and foremost among them is my advisor Stephen Baccus. Steve has
been a spectacular mentor. No issue was too small for him to provide suggestions
and advice. Whatever rigor and quality my work has is largely due to my interaction
with Steve. The members of the Baccus lab have provided me with wonderful col-
leagues over the years. I would like to specifically acknowledge Pablo Jadzinsky for
his companionship and advice, and Mike Menz for his unceasing helpfulness.
I would like to thank all of the members of my committee: Denis Baylor, Brian
Wandell, and Richard Tsien. Each of them was always willing to take time out of
their busy schedules to discuss my work. I benefitted greatly from the interaction.
The Stanford neuroscience community has provided a very rich environment for my
development. I would like to thank John Huguenard for running a wonderful graduate
program. I would like to thank Tom Clandinin for his teaching me to competently
present science. I would like to thank Greg Barsh, Seung Kim, PJ Utz, and the
Stanford MSTP for providing a fantastic educational environment. I would also like
to thank Jay McClelland for establishing and running the Center for Mind Brain and
Computation. The MBC has not only provided me with funding, but it has created a
vibrant interdisciplinary community in which I have been glad to participate. Daniel
Fisher and Tatyana Sharpee co-mentored me during di↵erent points of my MBC
training. I gained broadening insight from both of them on di↵erent aspects of my
work. Andrew Huberman, Saskia de Vries, Georgia Panagiotakos, and Andrew Olson
were very considerate and helpful with technical assistance.
Many people have allowed me to focus on science without having to worry too
vii
much about ancillary things. Ross Colvin, Katie Johnson, Lorie Langdon, Moira
Louca, and Laura Hope have all been incredibly helpful. I would not have been able
to accomplish all that I have without their assistance.
Many people contributed to my being able to come to Stanford. First and fore-
most I have to thank my parents, Beverly and Michael Kastner. They provided me
with every opportunity I needed to succeed, I could not have asked for better par-
ents. I would like to thank my brother, Eitan Kastner, for always being available for
useful and interesting conversations. I must thank my sisters, Jennifer Newman and
Ayelet Hoenig, who always respected and supported my pursuits. I am thrilled to
acknowledge and thank my grandmother, Francis Kastner. I greatly appreciated her
unceasing and, often times, unwarranted support.
I have had many excellent teachers, formal and informal, over the years. Joan
Haahr, Bruce Hrnjez, Carl Feit, Jeremy Wieder, among others at Yeshiva University,
provided me with excellent intellectual training. I would particularly like to acknowl-
edge Joshua Gottlieb, who introducing me to the “ineluctable modality of the visible.”
I would like to acknowledge Aaron Cypess, who introduced me to the possibility of
an MD-PhD, and helped and encouraged me along the way, and Daniel Feldman,
who, for most of my youth, provided me with an excellent role model. I would like
to acknowledge Murray Goldberg, who gave me my first labaroty exposure, making
me realize how much I enjoyed science. And I would like to thank Deborah Fass and
Thomas Sakmar, in whose labs I first experienced the fertile and fun world of scientific
research.
While at Stanford, I have had the good fortune of being a part of a wonderful
community outside of acadmia. I would like to thank Larry and Marlene Marton, Ari
Tuchman, Maya Bernstein and Noam Silverman, Jeremy and Sara Goldhaber-Fiebert,
David Singer, Emily Schoenfeld and Binyamin Blum, Avital Livny, and Amos Bitzan
and Marina Zilbergerts. They have all opened their houses to me for wonderful meals,
great conversations, and, far more importantly, valued companionship.
viii
Contents
Abstract v
Acknowledgements vii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Summary of thesis work . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Coordinated dynamic encoding 5
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Adaptation and Sensitization in retinal ganglion cells . . . . . 7
2.3.2 Adapting and sensitizing populations encode the same signals 11
2.3.3 Sensitizing cells preserve weak signals, adapting cells preserve
strong signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.4 Ideal normalization and contrast estimation . . . . . . . . . . 18
2.3.5 Estimation of contrast in an uncertain environment . . . . . . 18
2.3.6 Variability and threshold correspond in the two populations . 19
2.3.7 Sensitizing cells decrease activity but convey more information 20
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.1 Experimental preparation . . . . . . . . . . . . . . . . . . . . 26
2.5.2 Linear-Nonlinear models . . . . . . . . . . . . . . . . . . . . . 26
ix
2.5.3 Adaptive index . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.4 Receptive fields . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.5 Discriminability . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.6 Models of contrast normalization . . . . . . . . . . . . . . . . 29
2.5.7 Information theory . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.8 Sensitization model . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Predictive sensitization 34
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Center-surround adaptive fields . . . . . . . . . . . . . . . . . 37
3.3.2 A model unifies the three adaptive fields . . . . . . . . . . . . 40
3.3.3 Subcellular sensitizing and adapting subunits . . . . . . . . . 43
3.3.4 Adaptation and sensitization in a rapidly changing environment 44
3.3.5 Feature detection in Fast O↵ cells . . . . . . . . . . . . . . . . 48
3.3.6 Encoding a signal in a noisy environment . . . . . . . . . . . . 48
3.3.7 Sensitization maintains the location of an object . . . . . . . . 55
3.3.8 Inhibition is necessary for sensitization and the establishment
of the adaptive field . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.1 Adaptive and receptive fields . . . . . . . . . . . . . . . . . . . 64
3.4.2 Integration of inhibition and excitation . . . . . . . . . . . . . 65
3.4.3 A functional role for adapting inhibition . . . . . . . . . . . . 65
3.4.4 Di↵erent levels of sensitization in di↵erent cell types . . . . . . 66
3.4.5 Updating the prior probability of a stimulus . . . . . . . . . . 67
3.4.6 Integrating information at the bipolar cell synaptic terminal . 67
3.4.7 The retinal neural code and the statistics of objects . . . . . . 69
3.5 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.2 Cell classification . . . . . . . . . . . . . . . . . . . . . . . . . 71
x
3.5.3 Receptive fields and sensitivity . . . . . . . . . . . . . . . . . 72
3.5.4 Adaptive field model . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.5 Temporal adaptive field . . . . . . . . . . . . . . . . . . . . . 77
3.5.6 Signal detection model . . . . . . . . . . . . . . . . . . . . . . 77
3.5.7 Duration of sensitization . . . . . . . . . . . . . . . . . . . . . 78
4 Optimal dynamic range placement 79
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Histogram equalization does not predict distinct thresholds . . 83
4.3.2 Binary response model for maximizing information . . . . . . 83
4.3.3 Low threshold cells should have less noise . . . . . . . . . . . . 85
4.3.4 Di↵erent amounts of noise produce di↵erent optimal coding
strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.5 Fast O↵ cells optimally space their response functions . . . . . 88
4.3.6 Model fits data across contrasts . . . . . . . . . . . . . . . . . 90
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.1 Limitations of the model . . . . . . . . . . . . . . . . . . . . . 93
4.4.2 Redundant versus distributed encoding . . . . . . . . . . . . . 93
4.4.3 Information maximization after signal detection . . . . . . . . 94
4.4.4 Optimal population encoding . . . . . . . . . . . . . . . . . . 95
4.5 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5.1 Experimental preparation . . . . . . . . . . . . . . . . . . . . 96
4.5.2 Linear-Nonlinear models . . . . . . . . . . . . . . . . . . . . . 96
4.5.3 Histogram normalization for multiple neurons . . . . . . . . . 97
4.5.4 Threshold model for contrast processing . . . . . . . . . . . . 98
4.5.5 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Mechanism of sensitization 100
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xi
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5.1 Experimental preparation . . . . . . . . . . . . . . . . . . . . 112
5.5.2 Receptive fields and nonlinearities . . . . . . . . . . . . . . . . 113
5.5.3 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Conclusions 115
6.1 Generalization of work . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A Publications 121
A.1 Journal Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Refereed conference articles and abstracts . . . . . . . . . . . . . . . 121
References 123
xii
List of Figures
2.1 Adaptation and sensitization in separate neural populations . . . . . 8
2.2 Linear-Nonlinear (LN) model of ganglion cell firing rate . . . . . . . . 9
2.3 Fraction of sensitizing cells correlates across species with loss of sensi-
tivity in adapting cells . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Sensitization occurs during a range of stimulus conditions . . . . . . . 10
2.5 Adaptation and sensitization to changes in luminance . . . . . . . . . 11
2.6 Sensitizing and adapting populations encode common stimulus features 12
2.7 Cell classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.8 Mammalian sensitizing cells are composed of various cell classes . . . 14
2.9 Sigmoid fits for nonlinearities . . . . . . . . . . . . . . . . . . . . . . 14
2.10 Improvement of discriminability in a combined population of sensitiz-
ing and adapting cells . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.11 Correlations between adapting and sensitizing cells . . . . . . . . . . 16
2.12 Sensitizing cells specialize to encode weak signals; adapting cells encode
strong signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.13 Noise and placement of threshold . . . . . . . . . . . . . . . . . . . . 20
2.14 Sensitizing and adapting cells increase information transmission using
opposing changes in firing rate . . . . . . . . . . . . . . . . . . . . . . 21
2.15 Information transmission in sensitizing cells increases when firing rate
decreases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.16 Model of sensitization . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Three di↵erent adaptive fields in the retina . . . . . . . . . . . . . . . 38
3.2 Sensitivity changes underlie activity changes in the adaptive field . . 40
xiii
3.3 Amount of adapting inhibition can determine the type of adaptive field 42
3.4 Changes in sensitivity within the receptive field center . . . . . . . . . 45
3.5 Temporal adaptive fields during rapidly changing contrast . . . . . . 46
3.6 Distinct cell types for object motion sensitivity and global sensitization 49
3.7 Fast-o↵ adapting cells are object motion sensitive . . . . . . . . . . . 49
3.8 Sensitization reflects an increased prior expectation of a signal . . . . 51
3.9 Variability in O↵ bipolar cells and changes in ganglion cell responses
during sensitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.10 Sensitization predicts future object location . . . . . . . . . . . . . . 56
3.11 Sensitization requires GABAergic transmission . . . . . . . . . . . . . 59
3.12 Sensitization does not require Glycine receptors or the On pathway . 60
3.13 Depolarization of bipolar cells during sensitization . . . . . . . . . . . 61
4.1 Adapting and sensitizing fast O↵ cells coordinate the encoding of the
input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Binary model for information maximization . . . . . . . . . . . . . . 84
4.3 Lower threshold response function optimally has less noise . . . . . . 86
4.4 Distinct population coding regimes . . . . . . . . . . . . . . . . . . . 87
4.5 Fast o↵ cells optimally space their response functions . . . . . . . . . 89
4.6 Model fits to the data . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.7 Detecting a signal in the presence of noise . . . . . . . . . . . . . . . 92
5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Adaptation in sustained O↵ amacrine cells . . . . . . . . . . . . . . . 105
5.3 Amacrine transmission is depressed at the transition to low contrast . 107
5.4 Amacrine transmission changes during low contrast . . . . . . . . . . 108
5.5 High contrast current in a single amacrine cell is su�cient to cause
sensitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6 Simplified model for sensitization . . . . . . . . . . . . . . . . . . . . 111
5.7 A diverse set of amacrine cells . . . . . . . . . . . . . . . . . . . . . . 112
xiv
Chapter 1
Introduction
1.1 Background
The computational perspective provides a useful framework for understanding the
nervous system. It focuses on the way in which the brain transforms an input to
an output, providing multiple clear avenues of inquiry. The brain is composed of
many complicated dynamic and nonlinear parts. To understand relevant functions of
such a complicated system an understanding of its native inputs is critical. Along
similar lines, such a complicated system can produce many di↵erent types of output;
it therefore also becomes critical to carefully choose outputs relevant to the system.
Then once the input and output are defined many tools and techniques exist, but in
no way guarantee success, to decipher the algorithm that converts the inputs to the
outputs.
The retina provides an excellent substrate for understanding neural computations.
A major strength of studying the visual system is our knowledge, at least in the
absolute sense, of the input to the system. Photons provide the natural input, and
the visual system maintains exquisite sensitivity to photons (Hecht et al., 1942; Baylor
et al., 1984). Furthermore, much work has been done to understand the structure of
natural visual scenes (Field, 1987; Ruderman and Bialek, 1994; van Hateren and
Ruderman, 1998; Simoncelli and Olshausen, 2001; Geisler, 2008), and the way in
which an animal experiences that visual input (Yarbus, 1967; Martinez-Conde and
1
2 CHAPTER 1. INTRODUCTION
Macknik, 2008; Rucci et al., 2007; Rucci, 2008).
The retina contains five di↵erent classes of cells. Photoreceptors transform light
into electrical activity. Bipolar cells are excitatory interneurons that receive input from
photoreceptors and provide a direct pathway to ganglion cells, the output neurons
of the retina. Horizontal cells are inhibitory interneurons in the outer retina that
inhibit the photoreceptor signal. And amacrine cells are a diverse class of inhibitory
interneurons in the inner retina that receive input from bipolar cells and inhibit
bipolar and ganglion cells (Dowling, 1987; Masland, 2001).
By removing the retina from an animal we can record from many ganglion cell
at once while presenting a visual stimulus (Meister et al., 1994; Segev et al., 2004).
This is a perfect set up for computational studies since we precisely control the input
and measure the output of a practically intact neural circuit. Additionally, in recent
years, this extracellular recording set-up was combined with intracellular recording
to allow for perturbations of the di↵erent cells in the retina; thereby, enabling an
understanding of the way in which the retina performs a given computation (Ge↵en
et al., 2007; Baccus et al., 2008; de Vries et al., 2011; Manu and Baccus, 2011).
In my studies I have focused on adaptation. Compared to the range of their
inputs neurons have a limited dynamic range for their outputs. This mismatch is
quite apparent in the visual system where we can detect ten orders of magnitude by
using neurons that can only provide outputs over two to three orders of magnitude
(Rieke and Rudd, 2009). One strategy that neurons use to handle this problem is
that they change their dynamic range to match the range of the inputs, becoming
less sensitive when there is a large range of inputs, and more sensitive when there is a
smaller range of inputs. Because this is such a general problem, neurons throughout
the brain will adapt (Smirnakis et al., 1997; Kohn and Movshon, 2003; Nagel and
Doupe, 2006; Maravall et al., 2007; Kobayashi et al., 2010).
1.2 Summary of thesis work
In this thesis, I introduce the phenomenon of contrast sensitization, show its functional
relevance for retinal visual processing, derive a theoretical basis to justify sensitization
1.2. SUMMARY OF THESIS WORK 3
and some of its functional consequences, and elucidate the mechanism by which the
retina performs sensitization.
Chapter 2 characterizes the discovery of sensitization in the retina of multiple
species. The discovery of sensitization sheds light on an interesting encoding scheme—
coordinated dynamic encoding—whereby the retina distributes the encoding of it
inputs between two populations of neurons. These two populations coordinate their
encoding by representing very similar features of the visual world, but then specialize
in encoding di↵erent components of those features. One population sensitizes and
maintains a lower threshold to encode weaker parts of the input, while the other
population adapts and maintains a higher threshold to encode stronger parts of the
input. Both of these populations alone fall victim to saturating parts of their nonlinear
response function, but combined the two populations can encode the input over a far
broader range because they coordinate the placement of their response functions such
that when one saturates the other does not.
Chapter 3 discusses the adaptive field, which is the spatio-temporal structure that
cause a neuron to adapt. Once we saw in Chapter 2 that the retina has two opposing
forms of plasticity, namely adaptation and sensitization, I drew a parallel to a more
well know opponency in the retina: the On and O↵ pathway. On and O↵ channels are
combined within the receptive field of ganglion cells to enable biphasic spatio-temporal
processing. I report that the retina also combines adaptation and sensitization into
biphasic adaptive fields. Further drawing upon the comparison to the receptive field,
where inhibition is necessary for the biphasic filtering, I created a model that used
adapting inhibition to create the biphasic adaptive field. Using the adaptive field,
Chapter 3 shows that sensitization can function to store information. Whereas the
receptive field removes correlations in the input through predictive encoding, the
adaptive field uses those correlations to forms predictions about the world to encode
during a time of uncertainty.
Chapter 4 studies the way in which a population of neurons should place their
dynamic ranges to maximally transmit information about their input. I develop a
simple model to determine the optimal placement of dynamic ranges that maximize
the information about the input. I then go on to show that the fast O↵ populations in
4 CHAPTER 1. INTRODUCTION
the retina optimally space their response functions, given the noise in their responses
and an overall metabolic constraint to enforce sparse encoding. Furthermore, I show
that there are two regimes for optimal population encoding, distributed encoding and
redundant encoding. The On population has responses consistent with their optimally
choosing redundant encoding, a finding that has the potential to explain the greater
homogeneity in the On population.
Chapter 5 focuses on the way in which the retina performs the computations
of sensitization. Using pharmacology, and combined intracellular and extracellular
recording, I show that inhibition is both necessary and su�cient for sensitization.
Furthermore, I go on to show that inhibitory dynamics underly the role of inhibition
in sensitization.
In Chapter 6 I conclude by showing the seeds of similar computations in other
parts of the brain, and highlight some implications of this work for future research
directions.
Chapter 2
Coordinated dynamic encoding in
the retina using opposing forms of
plasticity
This chapter has been published in Nature Neuroscience as “Coordinated Dynamic
Encoding in the Retina Using Opposing Forms of Plasticity,” with author list: Kastner
DB, Baccus SA.
2.1 Summary
The range of natural inputs encoded by a neuron often exceeds its dynamic range. To
overcome this limitation, neural populations divide their inputs among di↵erent cell
classes, as with rod and cone photoreceptors, and adapt by shifting their dynamic
range. We report that the dynamic behavior of retinal ganglion cells in salaman-
ders, mice and rabbits is divided into two opposing forms of short-term plasticity in
di↵erent cell classes. One population of cells exhibited sensitization—a persistent ele-
vated sensitivity following a strong stimulus. This newly observed dynamic behavior
compensates for the information loss caused by the known process of adaptation oc-
curring in a separate cell population. The two populations divide the dynamic range
of inputs, with sensitizing cells encoding weak signals and adapting cells encoding
5
6 CHAPTER 2. COORDINATED DYNAMIC ENCODING
strong signals. In the two populations, the linear, threshold and adaptive proper-
ties are linked to preserve responsiveness when stimulus statistics change, with one
population maintaining the ability to respond when the other fails.
2.2 Introduction
Adaptive systems adjust their response properties to the statistics of the recent in-
put (Laughlin, 1981). However, a fundamental tradeo↵ exists between optimizing for
the current environment, and being able to respond reliably when the environment
changes. Due to statistical limitations of how long it takes to estimate the recent
stimulus distribution (DeWeese and Zador, 1998; Wark et al., 2009), the timescale of
adaptation greatly exceeds the integration time of the response in many sensory sys-
tems (Laughlin, 1981; Smirnakis et al., 1997; Fairhall et al., 2001; Nagel and Doupe,
2006; Maravall et al., 2007). As a consequence, when stimulus statistics change sud-
denly, as often occurs in natural scenes (Frazor and Geisler, 2006), sensory neurons
often fall below threshold or saturate, until they successfully measure and adapt to
the statistics of the new environment.
In the retina, a transition from a high to a low contrast environment reveals this
tradeo↵, when the decreased sensitivity caused by high contrast prevents the neuron
from firing for some time after the contrast decreases (Smirnakis et al., 1997; Baccus
and Meister, 2002; Rieke and Rudd, 2009). Adapting primate retinal ganglion cells
are known to recover their activity after high contrast with a prolonged time constant
of ⇠ 6 s (Solomon et al., 2004). However, human psychophysical performance recovers
faster at early timescales (<1 s), matching an ideal observer model, indicating that
some adapting neural pathway can signal quickly even after exposure to high contrast
(Snippe and van Hateren, 2003). We recorded from retinal ganglion cells in amphib-
ian and mammalian retina during sudden changes in the statistics of the stimulus
to examine how neural populations maintain responsiveness when the environment
changes.
2.3. RESULTS 7
2.3 Results
2.3.1 Adaptation and Sensitization in retinal ganglion cells
We measured the average firing rate response of salamander, mouse, and rabbit gan-
glion cells to a contrast transition by presenting a spatially uniform visual stimulus.
The intensity was drawn from a Gaussian white noise distribution with a constant
mean and a standard deviation that alternated between high and low temporal con-
trasts (Figure 2.1a). Even after a short high contrast presentation, many ganglion
cells failed to respond for seconds after the transition to low contrast as their fir-
ing rate slowly recovered, consistent with previously reported properties of contrast
adaptation (Smirnakis et al., 1997; Brown and Masland, 2001; Fairhall et al., 2001;
Baccus and Meister, 2002; Maravall et al., 2007) (Figure 2.1a,b).
We found, however, that some neurons responded rapidly after a transition to low
contrast (Figure 2.1a), even after a long high contrast presentation (Figure 2.1b).
These cells exhibited an elevated response following high contrast that persisted for
several seconds, gradually decreasing during low contrast. This decay had an average
(± standard deviation) time constant of 2.4±1.1 s in salamanders, 1.3±0.3 s in mice,
and 4.1± 2.7 s in rabbits.
To measure how the sensitivity of the two populations changed during low con-
trast, we computed a linear-nonlinear (LN) model of each neuron’s firing rate (Baccus
and Meister, 2002) (see methods) (Figure 2.2). We compared the nonlinearities com-
puted early (Learly
) and late (Llate
) after the transition to low contrast (bars in Figure
2.1a). For the two populations of ganglion cells, the change in firing rate arose from
a change in average sensitivity, defined as the average slope of the nonlinearity (Fig-
ure 2.1c). For salamanders, cells that elevated their activity at the transition to low
contrast doubled their average sensitivity (2.1± 0.3) during Learly
relative to Llate
. In
part, a change in threshold underlaid this change in average sensitivity. Because the
presence of a strong stimulus elevated the sensitivity to a subsequent weak stimulus,
we term this property sensitization, by analogy to behavioral sensitization (Pinsker
et al., 1973).
Sensitizing cells were found in salamanders (Figure 2.1a,c) (32%, 80 out of 250
8 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.1: Adaptation and sensitization in separate neural populations. (a) Stimulus inten-sity alternating between high and low contrast during a single trial (top), for salamander(left) and mouse (right). Firing rate response for adapting (middle) and sensitizing (bot-tom) cells, averaged over all trials, each with a di↵erent stimulus sequence. Color indicatesresponse to low contrast. (b) Average time to first spike after a transition from high tolow contrast (n = 2 � 12 cells). (c) Nonlinearities of an LN model (see methods) for cellsin (a) calculated during intervals indicated by bars in (a) for salamander (left) and mouse(right). The interval L
early
was defined as 0.5 � 2 s after the transition to low contrast,and L
late
was 10� 16 s for salamander and 10� 15 s for mouse. (d) Adaptive indices (seemethods) for 190 ganglion cells from 16 salamander retinas. The distribution is significantlybimodal (Hartigans dip test, p < 0.05). (e) High contrast (35%) was presented for 1, 2 or 5s, followed by low contrast (3%) for 15 s. The average change in firing rate between L
early
and Llate
is shown normalized by the average rate for low contrast in all conditions (n = 5cells). Black line is an exponential fit to the data. (f) For the same cells, the adaptive indexwas computed separately for changing contrast at a fixed luminance, and compared to theadaptive index when changing the mean luminance a factor of 16 at a fixed contrast of 10%(see Figure 2.5).
2.3. RESULTS 9
Figure 2.2: Linear-Nonlinear (LN) model of ganglion cell firing rate. For a fast O↵-typeganglion cell, the stimulus, s(t), was convolved with a linear filter, F (t), to yield the filteredstimulus, g(t). The filtered stimulus was then transformed by the nonlinearity, N(g), to yieldthe predicted response, r0(t). The linear filter in all conditions was biphasic, and lasted lessthan 0.5 s (Baccus and Meister, 2002).
cells), mice (Figure 2.1a,c) (12%, 5 out of 41 cells), and rabbits (21%, 8 out of 39
cells) (Figure 2.3a,b). A similar ratio of salamander ganglion cells has been reported in
abstract form to respond to contrast decrements (C.A. Burlingame, A.Y. Dymarsky,
M.J.Berry II, Soc Neurosci. Abstr. 506.11, 2007). Recording from many cell types
in the salamander, we found that adapting and sensitizing cells formed two distinct
classes (Figure 2.1d). For each species, we used the nonlinearities during Learly
and Llate
to compute the average loss of sensitivity. The sensitivity loss in adapting cells during
Learly
correlated with the fraction of sensitizing cells in the species (Figure 2.3c),
suggesting that sensitizing cells compensate for the sensitivity loss due to adaptation.
Sensitization occurred over a broad range of spatial frequencies and stimulus sizes
(Figure 2.4). By measuring sensitivity after di↵erent high contrast durations, we found
that after 0.55 s of high contrast, a cell reached 63% of its peak sensitization (⌧ = 0.55
s) (Figure 2.1e). Thus, significant sensitization is expected even during brief fixations.
After the transition to low contrast, increased activity was not instantaneous, but
reached a peak in 0.98 ± 0.03 s. This delay may reflect the statistical limitation
necessitating su�cient temporal integration for any system to adapt to a contrast
decrement (DeWeese and Zador, 1998; Fairhall et al., 2001; Snippe and van Hateren,
2003).
We tested whether the two forms of plasticity generalized to statistics other than
10 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.3: Fraction of sensitizing cells correlates across species with loss of sensitivity inadapting cells. (a) Example response of a sensitizing cell from rabbit. High contrast (25%)and low contrast (4%) were both presented for 30 s over 27 trials. (b) Nonlinearities for cellin (a) calculated during the times indicated by the corresponding colored bars in (a). L
early
,0.5� 4 s after the transition to low contrast; L
late
, 15� 30 s. (c) The fraction of sensitizingcells in each species plotted against the average decrease in sensitivity during L
early
relativeto L
late
.
Figure 2.4: Sensitization occurs during a rangeof stimulus conditions. (a) The response of asalamander sensitizing cell to stimuli with dif-ferent spatial frequencies. Stimuli alternated be-tween high contrast (35%) and low contrast(5%) and were composed of 12 µm bars (left), 50µm bars (center), or a uniform field (right). Col-ored regions indicate responses to low contrast.Average adaptive indices for the three condi-tions were 0.45±0.01, 0.27±0.01 and 0.23±0.01,respectively (n = 4 cells). (b) The response of asensitizing cell to di↵erent size stimuli. Stimuliwere composed of 50 µm bars that covered thewhole retina, >2mm (left), or a 200 µm square.Adaptive indices for the two conditions were0.27± 0.07 and 0.49± 0.03, respectively (n = 3cells).
2.3. RESULTS 11
contrast by changing the mean luminance while keeping the contrast fixed. Each cell
type showed consistent sensitizing or adapting behavior for changes in both stimulus
parameters (Figure 2.5, and Figure 2.1f).
Figure 2.5: Adaptation and sensitization tochanges in luminance. (a) Stimulus during a sin-gle trial (top), and average response over 64 tri-als for example salamander adapting (middle)and sensitizing (bottom) cells at a fixed con-trast of 10% during high and low luminance,which di↵ered by a factor of 16. (b) Nonlinear-ities for cells in (a) calculated during the timesindicated by the corresponding colored bars in(a). L
early
, 0.5 � 3 s after the transition to lowluminance; L
late
, 10� 20 s.
2.3.2 Adapting and sensitizing populations encode the same
signals
Although adaptation and sensitization slowly modulated the average firing rate, reti-
nal ganglion cells encode visual information on a much finer timescale using repro-
ducible firing events—intervals of high firing probability lasting <0.1 s in duration
(Baccus and Meister, 2002). We compared firing events for adapting and sensitizing
cells recorded simultaneously by repeating an identical stimulus sequence during Learly
and Llate
. During Llate
, 94% of adapting cell firing events occurred synchronously with
a sensitizing cell firing event (Figure 2.6a,b). Consistent with the changing nonlin-
earities, during these individual common firing events the activity of adapting cells
during Learly
decreased by 41± 3% relative to Llate
(n = 28), whereas the activity of
sensitizing cells increased by 93 ± 8% (n = 12). Thus, the two populations coordi-
nated their encoding such that they responded to the same visual stimuli, with the
representation shifting more to the sensitizing population during Learly
.
To examine the specific messages encoded by sensitizing and adapting cells, we
12 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.6: Sensitizing and adapting populations encode common stimulus features. (a)Average response of salamander adapting and sensitizing cells to 26 trials of the samestimulus repeated during L
early
and Llate
after 4 s of high contrast (35%). Low contrast was3� 5%. Firing rate binned at 10 ms. (b) Absolute di↵erence in time between events in allpairs of fast O↵-type adapting cells (n = 28) and sensitizing cells (n = 12). Events definedas times when a cell’s firing rate, binned at 10 ms, exceeded 20 Hz. (c) Average temporal(top) and spatial (bottom) filters for adapting (n = 142), and sensitizing (n = 48) fastO↵ cells, mapped in one dimension. Curves obscure the error bars located at the peak andtrough of the temporal filters and along the spatial filters. Spatial filters normalized to theirpeaks. (d) Fractions of adapting and sensitizing cells of di↵erent cell types, as classifiedby a cell’s temporal filter (n = 209 fast O↵, 16 medium O↵, 20 slow O↵, 9 On) (Figure2.7). (e) Spatial receptive field centers of fast O↵ adapting and sensitizing cells recordedsimultaneously. Receptive fields displayed at one standard deviation of a 2-D Gaussian fit.(f) Histogram of spacing (see methods) between nearest neighbors of fast O↵ adapting(n = 615) and sensitizing (n = 171) cells.
2.3. RESULTS 13
Figure 2.7: Cell classification. (a)Temporal filters obtained from a sin-gle salamander retina. (b) Filtersfrom (a) projected onto the first andsecond principal components of allfilters. (c) Filters from (a) projectedonto the second principal componentplotted against each cell’s adaptiveindex. Cells that did not respondduring low contrast do not appearon the plot because their adaptiveindex is undefined. Cells were clas-sified based upon their grouping inplots (b) and (c). There were 4 broadcategories of cells as classified bytheir filter alone. Within the fast O↵population most analyses for adapt-ing cells were performed on the firstgroup (red).
measured how the plasticity of a cell corresponded to its linear spatio-temporal re-
ceptive field (Figure 2.6c,d). For all salamander O↵-type cells—⇠90% of the cells
in the salamander retina (Segev et al., 2006)—the adaptive index divided each cell
type into two groups, composed of both adapting and sensitizing cells. Within a cell
class, the spatial receptive fields of adapting and sensitizing cells overlapped (Figure
2.6e), but maintained a minimum spacing between members of the same class (Fig-
ure 2.6f) (Huberman et al., 2008). This indicates that a mixed group of cells with
highly similar linear receptive fields (Segev et al., 2006), splits into two classes with
di↵erent short-term plasticity, each of which appears to tile the retina. Thus, adapt-
ing and sensitizing populations represent the same stimuli. In mice, sensitizing cells
also comprised di↵erent cell types, including both On and O↵ classes (Figure 2.8a).
In addition, some adapting and sensitizing cells in mice and rabbits had very similar
temporal properties (Figure 2.8b).
Because sensitizing cells compensate for the loss of sensitivity in the adapting
population during low contrast, we tested whether the reverse was true during high
14 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.8: Mammalian sensitizing cells are composed of various cell classes. (a) Temporalfilters from four sensitizing mouse ganglion cells shown in di↵erent colors. (b) Example tem-poral filters from mouse (left) and rabbit (right) with similar properties between adaptingand sensitizing cells. In mice, 3 out of 18 On cells, and 2 out of 18 O↵ cells sensitized. Inrabbits 1 out of 10 On cells, and 7 out of 18 O↵ cells sensitized. Temporal kernels could notbe classified for three of the rabbit cells.
contrast. During Hearly
, 0.5�5 s after a transition to high contrast, the nonlinearity of
sensitizing cells saturated (Figure 2.9), reaching 98± 1% of their estimated maximal
firing rate, and their sensitivity dropped to 10± 4% of the peak sensitivity (n = 3).
Adapting cells did not saturate, however, and only reached 79± 4% of their maximal
rate while retaining 63±8% of their peak sensitivity (n = 11). Thus, adapting cells
compensated for saturation in the sensitizing population at the transition to high
contrast.
Figure 2.9: Nonlinearities and sigmoid fits for exam-ple adapting (top) and sensitizing (bottom) cells. Usingthis sigmoid, we estimated maximal firing rate and thepeak sensitivity, which is the slope at the midpoint.
2.3. RESULTS 15
2.3.3 Sensitizing cells preserve weak signals, adapting cells
preserve strong signals
To measure the functional benefit of having the two opposing forms of plasticity,
we quantified the discriminability, d0, in the combined population of sensitizing and
adapting cells after a decrease in contrast (see methods). This measure derives from
the Fisher information, an upper bound on the information available by any unbiased
decoding scheme (Dayan and Abbott, 2009). Discriminability, and Fisher informa-
tion, increases with the slope of the nonlinearities at each input (Figure 2.10a), but
decreases with the variability of the response at that input. It also depends on corre-
lations between cells, which can either increase or decrease information (Abbott and
Dayan, 1999). We used simultaneously recorded populations of adapting and sensitiz-
ing cells to account for the nonlinearities, variability, and covariance as a function of
distance between cells (Figure 2.11) (see methods). Discriminability in the adapting
population alone decreased 44.2±1.9% during Learly
relative to Llate
. However, for the
combined population of sensitizing and adapting cells, discriminability only decreased
16.8±2.3% during Learly
. Thus, the addition of sensitizing cells to the population sub-
stantially reduced the loss of discriminability when the contrast of the environment
changed.
Figure 2.10: Improvement of discriminabilityin a combined population of sensitizing andadapting cells. (a) Nonlinearities for adapting(n = 21) and sensitizing (n = 13) cells duringLearly
(left) and Llate
(right). (b) Discriminabil-ity between nearby stimuli, d0(g), as a functionof the stimulus (see methods) in the full popu-lation minus d0(g) for the adapting populationalone (blue) or minus d
0(g) for the sensitizingpopulation alone (red) during L
early
(left) andLlate
(right). All values were normalized by thearea of the total d0 in the full population duringLlate
.
16 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.11: Correlations betweenadapting and sensitizing cells. (a)The covariance of an example pairof sensitizing cells is shown as afunction of the filtered stimulus g.In addition, a model of this co-variance is shown, computed as thegeometric mean of the two vari-ances weighted by the correlationcoe�cient between the two cells(see methods). (b) Correlation co-e�cient as a function of distanceduring L
late
within the adaptingand sensitizing populations, andbetween populations. Lines are ex-ponential fits to the data.
We then examined this improvement in discriminability in the full population at
each separate stimulus, and found that the addition of sensitizing cells to a popula-
tion of adapting cells enhanced the discriminability of weak signals (Figure 2.10b).
The improvement produced by including sensitizing cells during Learly
was 1.8 times
the improvement during Llate
. Discriminability improved most in the region of the
reduced threshold of the nonlinearities of sensitizing cells, indicating that this re-
duction during Learly
further enhanced the encoding of weak signals. Conversely, the
addition of adapting cells to a population of sensitizing cells enhanced discriminabil-
ity of strong signals (Figure 2.10b). As expected, this contribution of adapting cells
increased during Llate
as their threshold decreased and sensitivity increased.
The dynamics of adapting and sensitizing cells decayed towards a steady-state
response that depended on the contrast. To understand the endpoint of this adap-
tive process, we measured the steady-state nonlinear response curve from LN models
computed across a ten-fold range of contrasts (Figure 2.12a). Compared to adapt-
ing cells, sensitizing cells had a threshold closer to the mean (Figure 2.12a,b). Thus,
across all contrasts the two populations divided the range of inputs, with sensitizing
cells encoding weak signals, and adapting cells encoding strong signals.
2.3. RESULTS 17
Figure 2.12: Sensitizing cells specialize to encode weak signals; adapting cells encode strongsignals. (a) Twelve di↵erent contrast levels (3 � 36%) were randomly interleaved for atleast 110 s and three repeats, and the first 10 s of data in each contrast was discarded.Nonlinearities are shown for an adapting (top) and sensitizing (bottom) cell for the di↵erentcontrasts. Each row is a di↵erent nonlinearity, displayed in a color scale. Black dots indicateone standard deviation above the mean for each contrast level. Nonlinearities calculated fromthe data (left), and as predicted using a model described in panel (c) (right). (b) Normalizednonlinearities from cells in panel (a). For each contrast, the nonlinearity was scaled alongthe abscissa by the input standard deviation (top) or shifted by a common factor (↵) andthen scaled along the abscissa by the contrast (bottom). (c) Model M
↵
. Input values werepassed through a threshold function, which shifted the mean value by a factor, ↵, thenwere rescaled by the contrast (�), and then passed through a secondary nonlinearity withthreshold ✓ to recreate the range of nonlinearities shown in (a). The secondary nonlinearity isthe average nonlinearity for a cell after shifting by ↵ and rescaling. (d) Nonlinearities N
i
(g)were computed for each 3 s bin. For each bin, an estimate of the contrast was determined asthe contrast, �, for which the steady-state nonlinearity of the model M
↵
(�) had the smallestmean-squared di↵erence to N
i
(g). Low contrast (5%) followed 40 s of high contrast (35%).
18 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Consistent with this division of labor, sensitizing cells had a larger center and
weaker surround than did adapting cells (Figure 2.6c). This di↵erence likely enables
sensitizing cells to improve their signal to noise ratio for weak inputs by spatial
averaging, as occurs for ganglion cells during low luminance conditions (Enroth-Cugell
and Robson, 1966; Srinivasan et al., 1982).
2.3.4 Ideal normalization and contrast estimation
To explain the relationship between contrast and the steady-state dynamic range of
adapting and sensitizing cells, we considered that an ideal encoder that maximizes
information from a stimulus distribution should change its sensitivity inversely with
the contrast (Laughlin, 1981). This ideal normalization is thought not to occur in
the retina because ganglion cells reduce their sensitivity by a fraction less than the
change in contrast. This can be seen by comparing nonlinearities whose input has
been normalized by the contrast (Figure 2.12b top) (Smirnakis et al., 1997; Chander
and Chichilnisky, 2001).
We found, however, that a model, M↵
(see methods), using ideal normalization
does account for steady-state adaptation, causing the normalized curves to nearly
overlay, if one considers that the rescaling occurs after a threshold (Figure 2.12b
bottom). This type of normalization could occur if the stimulus passes through a
threshold, such as from voltage-dependent Ca channels in bipolar cell presynaptic
terminals (Mennerick and Matthews, 1996), and then rescaling occurs about that
threshold (Figure 2.12c).
2.3.5 Estimation of contrast in an uncertain environment
A change in stimulus statistics, as has recently occurred during Learly
, necessarily
brings uncertainty as to the new range of inputs (DeWeese and Zador, 1998; Fairhall
et al., 2001; Wark et al., 2009). As seen in the di↵erent dynamics of their firing rates
(Figure 2.1a) and nonlinearities (Figure 2.1c), the two populations make di↵erent
choices during that time of uncertainty, and then adjust their response to the new
contrast. Thus, we can view the initial placement of the nonlinearity as corresponding
2.3. RESULTS 19
to an initial estimate of the contrast.
The model M↵
represents an idealized relationship between contrast and the op-
timized response of a cell to that contrast. We therefore used the model as a lookup
table to identify the contrast estimate given the nonlinearity of a cell at di↵erent times
during low contrast (Figure 2.12d). We mapped nonlinearities for each cell at di↵erent
time intervals to a given estimated contrast by finding the most similar nonlinearity
in the steady-state model M↵
. During Learly
, adapting cells overestimated the contrast
at 1.6± 0.1 times the actual value (n = 12), and sensitizing cells underestimated the
contrast at 0.5± 0.1 times the actual value (n = 6).
2.3.6 Variability and threshold correspond in the two popu-
lations
We next sought to explain why sensitizing cells raised their threshold during prolonged
exposure to the low contrast environment, rather than maintaining a continued higher
firing rate during low contrast. For optimal encoding of an input, the level of noise
can influence the placement of threshold, with higher noise necessitating a higher
threshold (Field and Rieke, 2002). Sensitizing cells had lower variability than adapting
cells as measured by the Fano factor, or variance to mean ratio, by a factor of 1.86±0.17 (Figure 2.13a). This may occur in part due to their di↵erent receptive field sizes,
which would predict, assuming independent noise from photoreceptors, that their
variability would di↵er by the ratio of the receptive field areas, which was 2.07± 0.06
(26 sensitizing and 74 adapting cells).
We then examined the parameters of the model M↵
, which resembles an ideal ob-
server model of human perception having ideal contrast normalization with a thresh-
old set by internal noise (Snippe and van Hateren, 2003). Compared to adapting cells,
sensitizing cells had a lower initial threshold, ↵ (by a factor of 1.96) and a lower final
threshold, ✓, (by a factor of 3.6) (Figure 2.13b), possibly constrained by the di↵erent
variability in the two populations. Because of this connection between variability and
threshold, and the defined relationship of the steady-state threshold with contrast
(Figure 2.12ac), we considered that after a change in contrast, the threshold might
20 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.13: Noise and placement ofthreshold. (a) Variance and mean ofthe spike count during L
late
from re-peated stimuli (Figure 2.6a) calcu-lated every 100 ms. Black lines arelinear fits to the data. Results arefrom 28 adapting cells and 12 sen-sitizing cells recorded from 3 reti-nas. The Fano factor is 1.43 ± 0.08for adapting and 0.77± 0.06 for sen-sitizing cells. (b) Value of ↵ plot-ted against the threshold ✓ from themodel in Figure 2.12c.
then become optimized in the steady state to convey greater information about the
current stimulus.
2.3.7 Sensitizing cells decrease activity but convey more in-
formation
These observations led us to propose that during low contrast, from Learly
to Llate
, in-
formation transmission improved for both adapting and sensitizing cells, even though
the firing rates of the two populations moved in opposite directions. We thus mea-
sured the mutual information during Learly
and Llate
for adapting and sensitizing cells
by presenting pulses of eight di↵erent intensities during Learly
and Llate
(Figure 2.14a).
As expected, adapting cells conveyed less information during Learly
than sensitizing
cells, and increased their information transmission between Learly
and Llate
. Remark-
ably, we found that sensitizing cells also conveyed more information during Llate
than
Learly
(Figure 2.14b) even though their activity decreased during Llate
(Figure 2.15a).
Thus, the increase in threshold for sensitizing cells from Learly
and Llate
improves in-
formation transmission. This increase in mutual information was consistent with the
population measurement of discriminability (Figure 2.10b), in that the sensitizing
population alone lost 8.4 ± 4.0% of its discriminability during Learly
. Thus although
both sensitizing and adapting cells lose information at the transition to low contrast,
2.3. RESULTS 21
sensitizing cells lose much less.
This loss of information in the sensitizing population despite the increase in firing
rate can be explained by comparing the variability during Learly
and Llate
. A lower
threshold during Learly
exposed an increase in noise at the weakest stimuli for sensi-
tizing cells (Figure 2.14c), but not for adapting cells (Figure 2.15c), confirming that
subthreshold noise limits the steady-state placement of threshold. Previously, it has
been shown that higher firing rate correlates with greater information transmission
(Wessel et al., 1996; Reinagel and Reid, 2000). Here, however, the decay in activity
in sensitizing cells actually improves the encoding of the low contrast stimulus.
Figure 2.14: Sensitizing and adapting cells increase information transmission using opposingchanges in firing rate. (a) Stimulus used in the calculation of mutual information and thestimulus specific information (SSI) for low contrast. 20 s of identical high contrast pulseswere followed by L
early
, which was 2 s of 8 randomly presented low contrast pulses. ForLlate
, every 180 s, 44s seconds of continuous, randomly organized, low contrast pulses waspresented. (b) Mutual information during L
late
versus Learly
. Llate
occurred from 22� 44 safter high contrast, and L
early
occurred from 0.5 � 2 s after high contrast. All sensitizingcells had a higher firing rate during L
early
than Llate
(Figure 2.15a). A bin size of 150 mswas used, but the increase of information during low contrast is independent of bin size(Figure 2.15b). (c) Average mean and variance during L
early
(lighter colors, thicker lines)and L
late
(darker colors, thinner lines) for the sensitizing cells in (b), shown as a functionof the stimulus pulse amplitude. (d) Stimulus specific information, I
SSI
, for each of the 8di↵erent low contrast stimuli. In (c) and (d), flash amplitude is the Michelson contrast,(I
max
� I
min
)/(Imax
+ I
min
), of the 8 brief flashes in the low contrast stimulus.
22 CHAPTER 2. COORDINATED DYNAMIC ENCODING
Figure 2.15: Information transmission in sensitizing cells increases when firing rate decreases.(a) Firing rates during L
early
and Llate
for the sensitizing cells from Figure 2.14b. Lowcontrast stimuli were eight randomly presented low contrast pulses of di↵erent amplitudespresented either after high contrast pulses (L
early
) or in a prolonged presentation (Llate
) asdescribed in the legend of Figure 2.14. (b) Average mutual information during L
early
andLlate
for the sensitizing cells in Figure 2.14b calculated across a range of bin sizes. (c) Meanand variance of firing rate averaged across adapting cells from Figure 2.14b during L
early
(lighter colors, thicker lines) and Llate
(darker colors, thinner lines). For Learly
, the curvefor variance obscures that of the mean rate. Flash amplitude is the Michelson contrast. (d)Examples of the bias correction for finite data Strong et al. (1998) (see methods) for mutualinformation (left) and stimulus specific information measurements (right). The informationmeasurements were computed on all of the data, and the data divided into x parts, wherex is the value on the abscissa. The average value is plotted for all fractions of the data,and a second-order polynomial fit was used to extrapolate the curve to the case of infinitedata. For the stimulus specific information, I
SSI
, during Llate
is shown for all of the di↵erentintensity values in di↵erent colors.
2.4. DISCUSSION 23
To further examine how encoding changed for individual stimuli, we computed
the stimulus-specific information (Butts, 2003) (see methods), during Learly
and Llate
.
This measure reflects the contribution of each specific stimulus to the mutual infor-
mation. During both Learly
and Llate
, adapting and sensitizing cells favored di↵erent
ends of the input signals (Figure 2.14d), with sensitizing cells conveying the greatest
amount of information about the weakest stimuli during Learly
. This was consistent
with the measure of discriminability, which showed that the additional discriminabil-
ity conveyed by the two populations separated during Learly
(Figure 2.10b). However,
across all stimuli, information transmission improved from Learly
to Llate
. Thus, after
the initial opposing thresholds chosen by sensitizing and adapting cells, both popu-
lations improved their information transmission with more prolonged exposure to a
steady environment.
2.4 Discussion
These results give an explanation for the opposing dynamics of sensitizing and adapt-
ing cells. A decrease in contrast creates the greatest ambiguity as to the statistics of
the stimulus, because the new range of inputs only contains weak signals that fall
within the most probable values of the previous distribution (DeWeese and Zador,
1998; Fairhall et al., 2001). The addition of sensitizing cells to the population im-
proves the encoding of weak signals, with the greatest improvement being after the
contrast decreases (Figure 2.10b). However, neither population perfectly encodes the
new distribution after the contrast decreases (Figure 2.14b,d), a condition compelled
by the uncertainty that accompanies a transition to an environment of weak signals
(DeWeese and Zador, 1998; Fairhall et al., 2001). But by positioning their sensitivity
at di↵erent sides of the steady-state value, sensitizing and adapting cells bracket the
target sensitivity by underestimating or overestimating, respectively, the steady-state
sensitivity (Figure 2.12d). Thus, during the time of greatest statistical uncertainty
the two populations span the range of inputs. Because this initial position deviates
from optimal, both sensitizing and adapting cells then increase their information
transmission by adopting their steady-state positions (Figure 2.14b,d). Therefore,
24 CHAPTER 2. COORDINATED DYNAMIC ENCODING
the coordinated dynamics of adapting and sensitizing cells (Figure 2.1a) represent a
tradeo↵ between the immediate encoding of an uncertain distribution and the delayed
optimization for that distribution.
Dynamic changes within the circuitry of the inner retina underlie contrast adap-
tation (Rieke, 2001; Baccus and Meister, 2002; Kim and Rieke, 2003; Manookin and
Demb, 2006). Two adapting pathways, one excitatory and one inhibitory could com-
bine to produce sensitization (Figure 2.16). In this scheme, high contrast causes in-
hibitory transmission to adapt. Then, at the transition to low contrast, the residual
lowered inhibition raises sensitivity (Figure 2.16b,c). This model of sensitization in-
dicates that sensitizing cells receive a negative version of an adapting cell’s response.
This causes the two populations to encode di↵erent signals, in particular during the
time when each population has the highest likelihood of failing to encode the stimulus.
The model also indicates that the source of increased variability during sensitization
lies prior to the initial threshold in the excitatory pathway, as decreased inhibition
prior to this threshold could result in greater transmission of noise.
A neuron with a response curve that spans its distribution of inputs will encode
those inputs e�ciently (Laughlin, 1981). However, to perform this task dynamically
would require that the neuron maintain its threshold to encode the weakest signals,
and its maximal response to encode the strongest, making both ends of the response
curve vulnerable to saturation should the stimulus distribution change. Here, we have
shown that the retina divides this problem in two, with linear filtering, threshold
placement, and dynamic plasticity combining to encode a specific range of inputs.
Low threshold cells with weaker surrounds sensitize to reliably encode weak signals.
High threshold cells with stronger surrounds adapt to reliably encode strong signals.
When one population saturates, the other compensates. The ability to coordinate
opposing forms of dynamic encoding allows a neural population to avoid the inherent
losses of any single type of plasticity.
2.4. DISCUSSION 25
Figure 2.16: Model of sensitization. (a) Sensitization results from the di↵erence betweentwo adapting pathways, one excitatory and one inhibitory. In each pathway, the stimulusis passed through a linear filter, L, a threshold, N , and then an adapting block, A. Theadapting block is a feedforward module. In the inhibitory pathway, the input, u(t), is con-volved with an exponential filter, F
A
, yielding F
A
⇤u (see methods). The input, u(t), is thendivided by the filtered input, F
A
⇤u, such that the output of the adapting block, v(t), has asmaller amplitude than the input, u(t). A temporal filter, L
Q
, and saturating function, NQ
,is applied to the inhibitory pathway before the two pathways are combined. (b) Response ofthe model to an input that repeated, and was identical during L
early
and Llate
. (c) Averageresponses over many white noise sequences, shown at di↵erent stages in the model. (v) Inthe inhibitory pathway, the response decreases during high contrast, and recovers duringlow contrast. (w) The synaptic functions decrease the response modulation during highcontrast. (y) The decrease in inhibition at the transition to low contrast elevates activity inthe excitatory pathway. (z) The final adapting block, A
E
, in the excitatory pathway yieldsadaptation during high contrast, and preserves sensitization during low contrast.
26 CHAPTER 2. COORDINATED DYNAMIC ENCODING
2.5 Materials and Methods
2.5.1 Experimental preparation
We recorded from retinal ganglion cells of larval tiger salamanders, mice, and rabbits
using an array of 60 electrodes (Multichannel Systems) as described (Baccus and
Meister, 2002). Ringer solution (124 mM NaCl, 2.6 mM KCl, 2 mM CaCl2, 2 mM
MgCl2, 1.25 mM NaH2PO4, 26 mM NaHCO3, 22.2 mM glucose) perfused the mouse
retina at 32� 35�C and the solution maintained a pH of 7.35� 7.4 by aeration with
95/5% O2
/CO2
. Ames medium perfused the rabbit retina at 37�C.
A video monitor projected the visual stimuli at 30 Hz using Psychophysics Toolbox
(Brainard, 1997; Pelli, 1997) in Matlab (Mathworks). Stimuli were uniform field with
a constant mean intensity, M , of 8 � 10 mW/m2 and were drawn from a Gaussian
distribution unless otherwise noted. Contrast is defined as � = W/M , where W is the
standard deviation of the intensity distribution, unless otherwise noted. To measure
changes in firing rate for adapting and sensitizing cells (Figure 2.1a), for salamander,
80 trials were presented, alternating between 4 s high (35%) and 16 s low contrast
(5%). For mouse, 104 trials were presented of 15 s high (30%) and 15 s low contrast
(9%). For the measurement of the average time to the first spike after the transition
to low contrast (Figure 2.1b), results were pooled over 5 experiments, with >50 trials
for each cell. To measure the development of sensitization (Figure 2.1e), conditions
were interleaved in blocks of 17 trials for a total of 102 trials in each condition.
2.5.2 Linear-Nonlinear models
LN models (Figure 2.2) consisted of the light intensity passed through a linear tem-
poral filter, which describes the average response to a brief flash of light, followed
by a static nonlinearity, which describes the threshold and sensitivity of the cell. To
compute the model, the stimulus, s(t), was convolved with a linear temporal filter,
F (t), which was computed as the time reverse of the spike triggered average stimulus,
such that
2.5. MATERIALS AND METHODS 27
g(t) =
ZF (t� ⌧)s(⌧)d⌧. (2.1)
A static nonlinearity, N(g), was computed by comparing all values of the firing
rate, r(t), with g(t) and then computing the average value of r(t) over bins of g(t). The
filter, F (t), was normalized in amplitude such that it did not amplify the stimulus,
i.e. the variance of s and g were equal (Baccus and Meister, 2002). Thus, the linear
filter contained only relative temporal sensitivity, and the nonlinearity represented the
overall sensitivity of the transformation. Adapting and sensitizing cells were equally
well fit by an LN model. Model and data had a correlation coe�cient of 56± 2% for
adapting cells, and 61 ± 3% for sensitizing cells. The sigmoidal function used to fit
the nonlinearities was
y = x0
+m
1 + exp⇣
x
1
/
2
�x
r
⌘ , (2.2)
where x0
is the basal firing rate, m is the maximal firing rate, x1
/
2
is the x value at
50% of maximal firing, and r controls the maximal slope.
2.5.3 Adaptive index
The adaptive index was r
early
�r
late
r
early
+r
late
, where rearly
and rlate
are the firing rates during a
3 s window beginning during Learly
and Llate
, respectively. Only cells that responded
during rearly
and rlate
were included.
2.5.4 Receptive fields
Spatio-temporal receptive fields were measured in one or two dimensions by the stan-
dard method of reverse correlation (Chichilnisky, 2001) of the spiking response with
a visual stimulus consisting of either lines or squares. The spatio-temporal recep-
tive field was approximated as the product of a spatial profile and a temporal filter
(Olveczky et al., 2003). The normalized distance between receptive fields (Figure 2.6f)
was the spacing, S = d/(r1
+ r2
), where d is the distance between the center of the
28 CHAPTER 2. COORDINATED DYNAMIC ENCODING
two cells, and r1
and r2
are the radii of the two cells along the line connecting their
centers.
2.5.5 Discriminability
The average discriminability between nearby stimuli (Dayan and Abbott, 2009) as a
function of the stimulus was estimated as:
d0(g) =p
IF
(g), (2.3)
where IF
, the Fisher information, was computed as,
IF
(g) = N0(g)TQ(g)�1N0(g) +1
2Tr[Q0(g)Q�1(g)Q0(g)Q�1(g)]. (2.4)
Total discriminability was computed as d0 =Rd0(g)dg. The vector N0(g) is the deriva-
tive of the nonlinearity for a population of cells with respect to the filtered stimulus
g, Q(g) is the covariance matrix as a function g, and the function Tr is the trace of
a matrix (Abbott and Dayan, 1999). Nonlinearities, N(g), were sigmoidal fits to the
measured nonlinearities. The diagonal terms of Q(g), which were the variance of each
cell as a function of the stimulus g, were empirically well fit by a combination of mul-
tiplicative noise � that depended on g, and additive noise � that was independent of
g. This relationship was fit to the data (Figure 2.14c and Figure 2.15c) for sensitizing
and adapting cells during Learly
and Llate
,
Qii
(g) = �Ni
(g) + �. (2.5)
Only sensitizing cells had significant additive noise. The o↵-diagonal terms of Q(g),
the covariance between cells, were well fit by the geometric mean of the two variances
weighted by distance (Figure 2.11a),
Qij
= c(dij
ii
(g)Qjj
(g). (2.6)
2.5. MATERIALS AND METHODS 29
The correlation coe�cient c(dij
) decayed exponentially as a function of distance be-
tween two cells (Figure 2.11b). Di↵erent functions c(dij
) were fit during Learly
and
Llate
for pairing within and between adapting and sensitizing cells. The fractional loss
in discriminability during Learly
was 1� (d0early
/d0late
). Distance values dij
were taken
from a complete population of sensitizing and adapting cells spanning ⇠1 mm (Figure
2.6e). Error was computed by multiple random draws from a set of 21 adapting and
13 sensitizing cells.
2.5.6 Models of contrast normalization
Nonlinearities, N�
(g), were computed across 12 steady-state contrasts ranging from
3� 36%. The basic model of normalization by the contrast, M�
, was computed as:
M�
= N(g
�), (2.7)
where � is the contrast, and a single function N() was chosen to minimize the error
between model and data, E�
, defined as the average rms di↵erence between the model
and the set of nonlinearities, N�
. The model of normalization following a threshold,
M�
, was computed as:
M↵
= N
✓U↵
(g)
�
◆, (2.8)
where U↵
is a threshold function. In practice, because the threshold of U↵
was nearly
always lower than that of N , U↵
could be substituted with a simpler form having a
single parameter, ↵,
M↵
= N
✓g � ↵
�
◆. (2.9)
The nonlinearity N() and ↵ were similarly chosen to minimize the error, E↵
. The
model normalizing each curve separately, Mfull
, was computed as:
Mfull
= N
✓g � ↵(�)
c(�)
◆, (2.10)
30 CHAPTER 2. COORDINATED DYNAMIC ENCODING
where in addition to a single N(), a separate ↵ and c were chosen for each N�
to
minimize the error, Efull
. For all models sigmoid fits to the data were used for N�
.
We compared the relative performance of M↵
to M�
and Mfull
by computing (E�
�E
↵
)/(E�
�Efull
). Relative toM�
, the single parameter modelM↵
captured 92.6±1.0%
for adapting cells (n = 40) and 85.6± 1.8% for sensitizing cells (n = 12), of the error
reduction produced by Mfull
, which contained 22 parameters.
Thresholds were computed from a fit to a nonlinearity using the equation:
N(x) =
(0 if x < ✓
ax� ✓ if x � ✓, (2.11)
where ✓ is the threshold, and a is the slope above threshold. The line above threshold
was fit below saturating levels of the nonlinearity.
2.5.7 Information theory
To gather su�cient data to compute the mutual information after a transition to low
contrast, Learly
intervals occurred in di↵erent periods than Llate
. To gather data for
Learly
, the stimulus alternated between 20 s of identical high contrast pulses and 2
s of low contrast containing 8 randomly chosen stimulus intensities. To gather data
for Llate
, the stimulus consisted of a continuous 44 s sequence of random low contrast
pulses. Learly
and Llate
conditions were alternated every 180 s. The response to stimulus
pulses separated by 0.5 s was defined as a series of spike counts in bins of duration
150 ms (Figure 2.14a), or in durations ranging from 10�150 ms (Figure 2.15b). Each
response spanned a window of 150 ms, which included all spikes from a given pulse
of the stimulus. Mutual information was computed by taking the di↵erence between
the total response entropy, H(R), and the noise entropy, H(R|S), where the entropy
(H) is
H(X) = �X
x
p(x)log2
p(x). (2.12)
2.5. MATERIALS AND METHODS 31
The stimulus specific information (ISSI
) was computed as:
ISSI
(s) =X
r
p(r|s){H(S)�H(S|r)}. (2.13)
This measure is the average reduction of uncertainty gained from a measurement of
the set of responses given a particular stimulus s (Butts, 2003). The weighted average
ISSI
over all stimuli is the mutual information between stimulus and response. All
information measurements were corrected for limited data by computing the infor-
mation for fractions of the data and then extrapolating the result to infinite data
(Strong et al., 1998; Gollisch and Meister, 2008) (see Figure 2.15d).
2.5.8 Sensitization model
The model for sensitization was generated to reproduce the qualitative behavior of
sensitizing cells. Excitatory and inhibitory adapting pathways were linked by a synap-
tic pathway. A prime candidate for the proposed inhibitory pathway could be the sig-
nal passed through amacrine cells, inhibitory interneurons in the retina (Baccus and
Meister, 2002). Variables correspond to symbols in Figure 2.16. The biphasic linear
filter for both pathways matched the time to peak of sensitizing cells,
g(t) =
ZL(t� ⌧)s(⌧)d⌧. (2.14)
For the inhibitory pathway, a linear-threshold function contained a threshold set at
the mean of the input g(t),
u(t) = NI
(g) =
(0 if g < hgig(t)� hgi if g � hgi
, (2.15)
Brackets, h...i, denote the average quantity. Adaptation occurred through a feedfor-
ward divisive operation. This adaptation could either occur at the level of a bipolar
or amacrine cells (Baccus and Meister, 2002). The input u(t) was convolved with an
exponential filter, FA
, and then u(t) was divided by the result. A constant term in
32 CHAPTER 2. COORDINATED DYNAMIC ENCODING
the denominator set the magnitude of adaptation,
v(t) = AI
(u(t)) =u(t)
2 +RFA
(t� ⌧)u(⌧)d⌧. (2.16)
The time constant of FA
, representing the timescale of integration of the contrast,
was 2 s,
FA
(t) = 0.5e�t
2s (2.17)
The connection between the two pathways contained a temporal filter LQ
defined as
an alpha function with a time to peak of 150 ms,
LQ
(x) = 2x
.15
exp��x�.15
.15
�
wL
(t) =RLQ
(t� ⌧)v(⌧)d⌧. (2.18)
This delay could result from the action of metabotropic GABA receptors (Maguire
et al., 1989). The connection between the two pathways also contained a saturating
nonlinearity NQ
,
w(t) = NQ
(wL
(t)) =0.3
1 + exp⇣
0.03�w
L
(t)
0.01
⌘ , (2.19)
the e↵ect of which was to diminish the modulation of inhibitory transmission at
high contrast, and amplify the modulation at low contrast. This saturation could
arise from either synaptic depression or receptor desensitization (Li et al., 2007).
An alternative source of the inhibitory pathway could be that adaptation is in the
bipolar cell, and the delay and saturation (LQ
and NQ
) are produced by the filtering
and membrane properties of an intervening amacrine cell (Baccus and Meister, 2002).
The two pathways combined linearly,
x(t) = g(t)� w(t), (2.20)
The pathways combined prior to the excitatory pathway threshold indicating that the
amacrine cell might synapse presynaptically onto a bipolar cell terminal. The nonlin-
earity, NE
, in the excitatory pathway was a linear-threshold function with threshold
2.5. MATERIALS AND METHODS 33
↵, representing the initial threshold ↵ in the model M↵
in Figure 2.10,
z(t) = NE
(x(t)) =
(0 if x(t) < hx(t)i+ ↵
x(t)� (hx(t)i+ ↵) if x(t) � hx(t)i+ ↵, (2.21)
with ↵ set to 0.025. Finally, the excitatory input y(t) was passed through another
stage of divisive adaptation, and the result thresholded by an output nonlinearity NF
.
The threshold ✓ represented the final threshold ✓ of the model M↵
from Figure 2.10,
z(t) = NF
(AE
(y(t))) = NF
⇣y(t)
3+
RF
A
(t�⌧)y(⌧)d⌧
⌘
NF
(x) =
(0 if x < ✓
x if x � ✓
. (2.22)
✓ was set to 0.005. Error indicates standard error of the mean, computed across cells,
unless otherwise noted.
Chapter 3
Spatial segregation of adaptation
and predictive sensitization in
retinal ganglion cells
This chapter has been accepted to Neuron as “Spatial segregation of adaptation and
predictive sensitization in retinal ganglion cells,” with author list: Kastner DB, Baccus
SA.
3.1 Summary
Sensory systems change their sensitivity based upon recent stimuli to adjust their
response range to the range of inputs, and to predict future sensory input. Here we
report the presence of retinal ganglion cells that have antagonistic plasticity, showing
central adaptation and peripheral sensitization. Ganglion cell responses were captured
by a spatiotemporal model with independently adapting excitatory and inhibitory
subunits, and sensitization requires GABAergic inhibition. Using signal detection
theory we show that the sensitizing surround conforms to an optimal inference model
that continually updates the prior signal probability. This indicates that small recep-
tive field regions have dual functionality—to adapt to the local range of signals, but
34
3.2. INTRODUCTION 35
sensitize based upon the probability of the presence of that signal. Within this frame-
work, we show that sensitization predicts the location of a nearby object, revealing
prediction as a new functional role for adapting inhibition in the nervous system.
3.2 Introduction
Visual scenes have correlations over space and time that arise from the properties
of environmental conditions, objects, eye movements, and self motion (Field, 1987;
Ruderman and Bialek, 1994; Frazor and Geisler, 2006). Because of this statistical
regularity, it has long been thought that the visual system might improve its e�ciency
and performance by adjusting its response properties to the recent history of visual
input (Barlow et al., 1957; Blakemore and Campbell, 1969; Laughlin, 1981).
In early stages of sensory systems, studies of how stimulus statistics influence
the neural code have focused mainly on adaptation, which also occurs throughout
the brain (Smirnakis et al., 1997; Nagel and Doupe, 2006; Maravall et al., 2007;
Tobler et al., 2005; Kohn and Movshon, 2003). Given the recent stimulus distribution,
response properties change over multiple time scales to encode more information and
remove predictable parts of the stimulus (Fairhall et al., 2001; Hosoya et al., 2005;
Wark et al., 2009; Ozuysal and Baccus, 2012). A primary concept underlying studies
of adaptation is that early sensory systems should transmit as much information as
possible for further processing in the higher brain (Atick, 1992; van Hateren, 1997).
Studies in the higher brain and in behavior often take a di↵erent point of view:
the goal is to generate a behavior given a stimulus (Kersten et al., 2004; Yuille and
Kersten, 2006; Kording and Wolpert, 2006; Schwartz et al., 2007b). Accordingly, such
studies highlight the benefit of predicting future stimuli, using inference about the
prior probability of future sensory input for choosing appropriate actions.
Recent results indicate that many ganglion cells encode specific features with a
sharp threshold, implying that these ganglion cells make a decision as to the presence
of a feature (Olveczky et al., 2003; Zhang et al., 2012). If so, one might expect that
plasticity in the retina also take advantage of the principles of signal detection and op-
timal inference. At the photoreceptor to bipolar synapse, even though at the dimmest
36 CHAPTER 3. PREDICTIVE SENSITIZATION
light level the threshold of the synapse is close to the optimal level to perform signal
detection, it does not appear that any adjustment due to the prior probability of a
signal occurs (Field and Rieke, 2002). This problem, however, has not been explored
at the level of ganglion cells. Given the complex circuitry of the inner retina (Roska
and Werblin, 2003) and the di↵erent types of ganglion cell plasticity (Hosoya et al.,
2005; Olveczky et al., 2007; Kastner and Baccus, 2011), we examined this plasticity
in the context of both signal transmission and signal detection.
Here we systematically mapped the spatial arrangement of plasticity in retinal
ganglion cells, finding that many ganglion cells adapted to a localized stimulus, but
sensitized in the surrounding region. The spatiotemporal properties of this plasticity
were captured by a computational model having independently adapting excitatory
subunits, producing localized adaptation, and larger adapting inhibitory subunits,
producing sensitization.
Using knowledge of the detailed computation, we then combined signal detection
theory and optimal inference to qualitatively account for several properties of sensiti-
zation. This indicated that sensitization creates a regional prediction of future input
based upon prior information of local signal correlations in space and time. We then
test this theory in a more natural context by showing that object motion sensitive
ganglion cells use sensitization to predict the location of a camouflaged object.
Finally, we show that sensitization requires GABAergic inhibition, and that di↵er-
ent levels of inhibition can account for di↵erences in sensitization between ganglion
cell types. Together these results show how two functional roles of plasticity are
combined in a single cell—to adapt to the range of signals, and predict when those
signals are more likely to occur. Furthermore, these results establish a functional role
for adapting inhibition in predicting the likelihood of future sensory input based upon
the recent stimulus history.
3.3 Results
We first measured the spatiotemporal region whose statistics control the sensitiv-
ity of a cell—the adaptive field. Previous measurements of spatial properties of the
3.3. RESULTS 37
adaptive field focused primarily on fast adaptation—changes in sensitivity occurring
within the integration time of a cell. These fast, suppressive e↵ects in the retina and
lateral geniculate nucleus extend beyond the receptive field center (Victor and Shap-
ley, 1979; Werblin, 1972; Olveczky et al., 2003; Bonin et al., 2005; Solomon et al.,
2002). Considerably less e↵ort has been devoted to measurements of the adaptive
field with changes in sensitivity lasting longer than the integration time of the cell.
Recent results have shown that delayed changes in sensitivity in salamanders, mice,
and rabbits have two opposing signs, adaptation and sensitization (Kastner and Bac-
cus, 2011). The spatial extent of adaptation has been measured as it pertains to
mechanisms of adaptation within the circuit, indicating that small regions of the gan-
glion cell receptive field adapt somewhat independently (Brown and Masland, 2001).
Spatial properties of sensitization have not been measured.
To measure prolonged changes in sensitivity at di↵erent spatial locations, we pre-
sented a low contrast flickering checkerboard. Every 20 s, one region of space changed
to high contrast for 4 s (Figure 3.1a,b). The high contrast stimulus was presented at
di↵erent retinal locations, allowing for the creation of a spatial map of slow changes
in sensitivity. We compared the firing rate during two time intervals after the high
contrast spot disappeared: Learly
, 0.5 to 3 s after the transition to all low contrast, and
Llate
, 13.5 to 16 s after the transition to all low contrast, a time that approximated
the steady state.
3.3.1 Center-surround adaptive fields
Fast O↵-type adapting and sensitizing cells are two defined cell types that each form
an independent mosaic in the salamander retina (Kastner and Baccus, 2011). In
response to a spatially global transition between high and low contrast, defined as
the standard deviation of the intensity computed over time, adapting cells decrease
their sensitivity following a high contrast stimulus, whereas sensitizing cells increase
their sensitivity.
Fast O↵ cells that adapted to a global contrast change also adapted when the high
contrast spot was directly over their receptive field center. However, when the high
38 CHAPTER 3. PREDICTIVE SENSITIZATION
contrast spot neighbored their receptive field center they sensitized, increasing their
response during Learly
relative to Llate
(Figure 3.1c,d). Thus, the adaptive field of this
type of cell exhibited spatial antagonism, showing central adaptation but peripheral
sensitization.
Sensitizing cells also had a spatially varied response to a local high contrast spot.
During Learly
, these cells sensitized both in their central and surround region (Figure
3.1c,d). However, when the firing rate was examined at an even earlier time, from
0�0.5 s after the transition from high contrast (L0�0.5
), sensitizing cells also adapted
in their center. Thus both cell types had an adapting center and sensitizing surround,
although with apparently di↵erent dynamics in their adaptation. In comparison, On-
type cells had a spatially monophasic adaptive field, adapting both in the central and
surround regions. This was the case for all On cells examined (Figure 3.1c,d).
Although a change in firing rate reveals an overall change in the response of a
Figure 3.1 (facing page): Three di↵erent adaptive fields in the retina. (a) A single frame ofthe stimulus when a high contrast square was presented. Boxes were either 300 µm or 200µm on each side. (b) Temporal sequence of the binary stimulus in the di↵erent regions. Highcontrast was 100% and lasted 4 s in a single region, indicated by the black box in panel (a).Low contrast was 5%. Contrast was defined to be: C = I
max
�I
min
I
max
+I
min
, with I
max
and I
min
beingthe maximum and minimum intensity values. (c) Response of an adapting On (left), fast-O↵adapting (middle), and fast-O↵ sensitizing (right) cell to the adaptive field stimulus. PSTHsare shown with the high contrast region (square) located at two positions relative to thereceptive field center (circle), and show the average response for >60 stimulus sequences.Data binned at 0.5 s. Colored responses indicate when all regions were low contrast. Blackindicates the time of the local high contrast. (d) Adaptive indices for all cells. The color ofeach point in the polar plot indicates the cell’s adaptive index when the adapting square wasat that location. The origin indicates the cell’s receptive field center. The adaptive index is:A =
r
early
�r
late
r
early
+r
late
, where r
early
is the average firing rate during L
early
, and r
late
is the average
firing rate during L
late
. Data comes from 9 adapting On, 21 sensitizing, and 74 adapting O↵cells. For sensitizing cells, adaptive fields are shown during L
early
(top), and during L
0�0.5
, 0 � 0.5 s after high contrast (bottom) (e) Average adaptive index for each cell type as afunction of distance from the cell’s center. Results were averaged across angles in (d), andcolors correspond to (c). Solid black lines are single or di↵erence of Gaussian fits to thedata. For sensitizing cells, the larger standard deviation (s.d.) was 0.41 mm. For adaptingO↵-cells, the smaller s.d. was 0.11 mm and the larger s.d. was 0.30 mm. For adapting Oncells, a single Gaussian was used with an s.d. of 0.32 mm. In (e), error values are s.e.m. andwere computed across cells contributing to each point.
3.3. RESULTS 39
c
a
Inte
nsity
4 s
Learly Llateb
300 µmd
e 0.50
-0.5
Adap
tive
Inde
x
0.50Distance (mm)
Learly
L0 - 0.5
40 CHAPTER 3. PREDICTIVE SENSITIZATION
cell, we tested whether changes in firing rate observed following a local high contrast
stimulus were accompanied by spatial changes in visual sensitivity. We computed the
sensitivity at each spatial location during Learly
and Llate
using a spatiotemporal filter
representing the receptive field (see methods). In all cell types, a prolonged adaptive
change in sensitivity underlied the changes in activity as measured using a spatiotem-
poral linear-nonlinear (LN) model (Figure 3.2a,b). Therefore, three di↵erent popu-
lations of cells—fast-O↵ adapting cells, fast-O↵ sensitizing cells and On cells—had
distinct spatiotemporal adaptive properties, with O↵ cells exhibiting center-surround
adaptive fields.
Adapting Off!Sensitizing!
b
-0.6-0.3
00.3
∆ se
nsiti
vity
0.50Distance (mm)
a Adapting On!
Figure 3.2: Sensitivity changes underlie activity changes in the adaptive field. (a) Sensitivitywas measured in each spatial region using a spatiotemporal linear-nonlinear (LN) model (seemethods). Average di↵erence in sensitivity at each location for each cell type, normalizedby the average sensitivity at each location during L
late
. The map of � sensitivity for eachcell was centered on the location of the high contrast region. Data comes from 7 adaptingOn, 43 adapting O↵, and 10 sensitizing cells. (b) Average di↵erence in sensitivity betweenL
early
and L
late
for each cell type as a function of distance from the cell’s center. Datafrom panel (a) were averaged across angles. Colors correspond to labels above figures in(a). Error values (s.e.m.) are obscured by the data points, and were computed across cellscontributing to each point.
3.3.2 A model unifies the three adaptive fields
To gain insight into both the computation and potential mechanisms that underlied
the construction of the adaptive field, we modeled the center-surround adaptive field
3.3. RESULTS 41
by extending a previous model that produced sensitization using an adapting in-
hibitory pathway (Kastner and Baccus, 2011). In this model, adapting excitation and
inhibition combine, such that a high contrast stimulus causes inhibitory transmission
to adapt, thus reducing inhibition and generating a residual sensitization after the
high contrast ceases.
To extend the previous model, we added adapting spatial subunits for both excita-
tory and inhibitory pathways (Figure 3.3a). Excitatory subunits, representing bipolar
cells, had receptive fields smaller than that of the ganglion cell, and inhibitory sub-
units were three times larger than excitatory subunits (Figure 3.3a). This size ratio
was taken from a di↵erence of Gaussians fit to the center-surround adaptive field (Fig-
ure 3.1e). Based upon the earlier model, inhibition was delivered prior to a strong
threshold, as occurs at the presynaptic terminal of the bipolar cell (Heidelberger and
Matthews, 1992). The ganglion cell then received a spatially weighted average of the
excitatory subunit outputs (Figure 3.3b). Most parameters of the model were taken
from previous full-field experiments with fast-O↵ sensitizing cells (Kastner and Bac-
cus, 2011). Two spatial parameters, representing the scale of excitatory and inhibitory
subunits, were calculated from the spatial adaptive field averaged across a popula-
tion of only fast-O↵ adapting cells, which had center-surround adaptive fields (Figure
3.1e).
Using a stimulus similar to that shown in Figure 3.1a,b, the model produces
an output that either adapts or sensitizes depending upon the location of the high
contrast (Figure 3.3c), consistent with the responses of cells with center-surround
adaptive fields. Thus, a di↵erent spatial scale of adapting excitation and inhibition
yield a center-surround adaptive field, just as a di↵erent scale for simple excitatory
and inhibitory inputs yield a center-surround receptive field (Thoreson and Mangel,
2012). Because the three types of adaptive fields had distinct properties, one might
expect that di↵erent circuitry would be required to generate the di↵erent adaptive
fields. However, we tested whether we could generate the di↵erent adaptive fields
simply by changing the parameters of the model, but not the circuitry. We were
able to reproduce all three adaptive fields by simply changing the strength of the
inhibitory weighting onto the excitatory subunits (Figure 3.3d). The adaptive fields
42 CHAPTER 3. PREDICTIVE SENSITIZATION
Excitatory
Inhibitory
Ganglion
cell
subunits
subunits
a b
c
d
−
wI
Inhibitory!
Excitatory!
wE
-0.5
0
0.5
Adap
tive
inde
x
wmax
Adapting!Sensitizing!
Figure 3.3: Amount of adapting inhibition can determine the type of adaptive field. (a)Spatial scale of the subunits in a model of a ganglion cell with a center-surround adaptivefield (see methods). Colored bars show the di↵erent locations used for the high contrast.Line thickness indicates the weight of a given subunit onto the ganglion cell. (b) Boththe inhibitory and excitatory subunits are composed of spatio-temporal receptive fields,threshold nonlinearities, and adaptive blocks (arrow in circle). The inhibitory populationwas spatially weighted (w
I
) before inhibiting an excitatory subunit, with a weighting equalto the spatial overlap between each excitatory and inhibitory subunit. The excitatory pop-ulation was likewise spatially weighted (w
E
) and then passed through a threshold to createthe output of the model. The average responses for two excitatory subunits are shown inresponse to a spot centered over the ganglion cell receptive field (colored bar in panel (a)matching the low contrast response). Markers in the top right corner of the two subunitresponses correspond to their weighting, w
E
, in the model output and their spatial locationin (a). (c) Output of the model (top) for three di↵erent locations of high contrast corre-sponding to the colored bars in (a) that match the low contrast responses. Example dataPSTHs from fast-O↵ adapting cells are shown below for comparison. (d) Adaptive indicesfrom models with di↵erent maximal inhibitory weighting (w
max
) (see methods). The valueof the adaptive index is plotted for when the high contrast spot was located directly aboveor just neighboring the ganglion cell receptive field. Line colors correspond to the bars in(a). Green and purple icons represent the type of adaptive fields measured during L
early
that correspond to that range of wmax
.
3.3. RESULTS 43
of sensitizing cells resulted from the strongest adapting inhibition, center-surround
adaptive fields resulted from intermediate adapting inhibition, and a monophasic
adaptive field, showing adaptation, resulted from the weakest adapting inhibition.
Thus, all the three adaptive fields, as well as intermediate examples that we did not
encounter experimentally, could arise solely by changing the strength of inhibition
from a single population of amacrine cells onto di↵erent bipolar cells.
This adaptive field (AF) model predicts several distinct features of the data. Sen-
sitizing cells produce less sensitization when they are directly centered under a high
contrast spot than when the spot is slightly o↵set from the receptive field center
(Figure 3.1e, 3.2a,b, and 3.3d). The model also predicts that when the high contrast
region is further removed from the receptive field center, the cell had a larger steady
state response at low contrast than high contrast, but an elevated response at the
transition to both low and high contrast (Figure 3.3c). This occurs because in the
periphery of the receptive field center the weight of inhibition is greater than exci-
tation by virtue of the greater spatial spread of inhibition (Figure 3.3a). However, a
delay in inhibitory transmission causes excitation to be transiently greater than inhi-
bition at the transition to high contrast. Thus, a model with independently adapting
excitation and inhibition predicts multiple distinct spatiotemporal properties of the
adaptive field.
3.3.3 Subcellular sensitizing and adapting subunits
The AF model contains subunits that independently undergo plasticity, with the
final response exhibiting the summed adaptive behavior of each subunit. Because
these subunits are smaller than the receptive field center, the model predicts that
individual regions of the response of the cell may sensitize, even when the overall
firing rate response adapts (Figure 3.3b). We tested whether the AF model, fit to a
coarser spatial stimulus (Figure 3.1), would predict changes in sensitivity at a high
spatial resolution within a single cell without reoptimizing the model. We stimulated
the retina with a low contrast white noise stimulus composed of concentric flickering
annuli centered on a single ganglion cell (Figure 3.4a,b). In the central 200 µm, every
44 CHAPTER 3. PREDICTIVE SENSITIZATION
20 s the stimulus turned into a uniform circle that flickered with high contrast for 4
s. The diameter of the high contrast spot was smaller than the receptive field center
of a cell.
We measured the subcellular changes in sensitivity following high contrast using
a spatiotemporal LN model of the receptive field similar to Figure 3.2a, except that
each spatial region represented an annulus. This allowed for a higher resolution mea-
surement with respect to distance. We then compared the sensitivity during Learly
and
Llate
at each spatial location (Figure 3.4c). Cells with a center-surround adaptive field
showed negative changes in sensitivity within the receptive field center below the high
contrast, and positive changes in sensitivity next to the location of the high contrast,
just as predicted by the AF model (Figure 3.4c). Thus, even though the AF model
parameters were fit using di↵erent experimental data (full field and checkerboard
changes in contrast), the model predicted subcellular adaptation and sensitization
using concentric annuli. Previously, it was shown that adaptation occurs at a subcel-
lular scale (Brown and Masland, 2001). The present result shows that interneurons
contribute spatially localized plasticity both for adaptation and sensitization within
the adaptive field.
3.3.4 Adaptation and sensitization in a rapidly changing en-
vironment
In the AF model, excitatory and inhibitory subunits adapted independently. To fur-
ther explore this property, we tested whether the model fit to the localized step change
in contrast (Figure 3.1a,b) predicted the response when all regions were activated to-
gether by a uniform field stimulus whose contrast changed with a broad temporal
bandwidth. Such rapid changes in contrast are found during natural viewing condi-
tions where the eye is moved frequently (Frazor and Geisler, 2006). We presented a
uniform field Gaussian stimulus where the temporal contrast (standard deviation of
intensity) changed randomly every 0.5 s (Figure 3.5a). The value of the contrast was
drawn from a uniform distribution. We then computed a temporal filter representing
the average e↵ect of a brief increase in contrast by correlating ganglion cell spiking
3.3. RESULTS 45
a b
c
1
0.5
0
Nor
mal
ized
sens
itivi
ty/a
rea
0.20Distance (mm)
Figure 3.4: Changes in sensitivity within the receptive field center. (a) A single frame of thestimulus used to map changes in sensitivity at a high resolution, composed of concentricannuli with radii increasing by 50 µm that were modulated independently with Gaussianwhite-noise having a 5% contrast. In the central 200 µm, the stimulus alternated between16 s of the 5% low contrast stimulus, and 4 s of a uniform circle that flickered with a 100%Michelson contrast. (b) Normalized spatial sensitivity of an adapting O↵ cell during L
late
,computed as the rms value of the spatiotemporal receptive field at each distance for thestimulus in (a). Because annuli had a di↵erent area unlike a checkerboard stimulus, thesensitivity at each distance was normalized by the area of the annulus that contributed tothat point in the receptive field. The vertical dotted line shows the point of zero crossing,which was used to define the center of the receptive field. (c) Average di↵erence between thespatial sensitivity during L
early
and L
late
for adapting O↵ cells (left) (n = 7) and the modelfrom Figure 3.3 (right). The spatial sensitivities during L
early
and L
late
were normalizedby the maximum sensitivity during L
late
. Positive values indicate regions that were moresensitive during L
early
compared to L
late
, while negative values indicate regions that wereless sensitive during L
early
compared to L
late
. The solid vertical line shows the extent of thecentral circle, which was the only location that experienced the high contrast. The dottedvertical line shows the average point where the total sensitivity within the receptive fieldcrossed zero, indicating the border between receptive field center and surround, as shownfor a single cell in (b).
46 CHAPTER 3. PREDICTIVE SENSITIZATION
with the random sequence of contrast (Figure 3.5b). This temporal filter represented
the temporal adaptive field, which is the spatial average of the spatiotemporal adap-
tive field. Note that this computation measures the average contribution of both
increases and decreases in contrast, analogous to how the linear receptive field aver-
ages over both increases and decreases in intensity. All of these functions had a large
peak in the first time bin, from 0 � 0.5 s, something expected since higher contrast
invariably produces a higher firing rate. To examine the temporal adaptive field, we
therefore focused on the component of the temporal filter outside of the first 0.5 s
bin, representing how the recent history of contrast outside the integration time of
the cell influenced the firing rate.
-1
0
Filte
r (s-1
)
42Delay (s)
42
Data AF Modelc
Inte
nsity
Con
trast
1 s
b
4
2
0
-2
Filte
r (s-1
)
420Delay (s)
SensitizingAdapting OffAdapting On
a
Figure 3.5: Temporal adaptive fields during rapidly changing contrast. (a) Experimentaldesign. The spiking response of ganglion cells was recorded in response to a Gaussian whitenoise stimulus, where the contrast, �(t), changed randomly every 0.5 s drawn from a uniformdistribution between 0 and 35% contrast. (b) Example temporal adaptive fields, whichwere calculated as the spike triggered average of the contrast, �(t). (c) Average temporaladaptive fields for ganglion cells (left), and the output of the AF model from Figure 3.3(right), computed with no refitting of AF model parameters for the rapidly changing contraststimulus. In the data figure, the width of the lines indicates the s.e.m. For the model, threeinhibitory weightings (Figure 3.3d) were chosen to represent the three di↵erent types ofspatial adaptive fields. The x-axis begins at 0.5 s to highlight the slower changes due tochanging contrast. All filters are normalized in amplitude to have the same rms value.
3.3. RESULTS 47
The three types of cells had distinct temporal adaptive fields (Figure 3.5b,c).
Adapting On cells had a slow negative monophasic filter, indicating that a brief in-
crease in contrast decreased activity between 0.5�3 s. Sensitizing cells had a biphasic
filter, such that elevations of contrast initially decreased activity, but only for a du-
ration of up to 1 s. With a delay of ⇠1 s, contrast on average increased activity, and
the e↵ect then decayed away after ⇠3 s.
Adapting O↵ cells had a temporal adaptive field that was negative and monopha-
sic, but with a more rapid decay than that of adapting On cells, indicating that an
increase in contrast caused a decrease in activity between ⇠0.5 and 1.5 s (Figure 3.5c).
After ⇠1 s the temporal filter of the adaptive O↵ cell was between those of the other
two cells. Just as with the spatial adaptive field, where adapting O↵ cells showed a
mixture of adaptation and sensitization, the temporal adaptive field of adapting O↵
cells appeared to be a mixture of the time courses of the two extremes. Although
sensitization did not completely cancel adaptation, adaptation was reduced at later
times.
We then evaluated whether the AF model could reproduce the di↵erent tempo-
ral adaptive fields using the same stimulus that changed in contrast with a broad
bandwidth (Figure 3.5a). For each of the three types of cells, we used a model with
a di↵erent strength of adapting inhibition. Even though the model parameters were
adjusted only using the spatial map of the adaptive field (Figure 3.1), we found that
the model reproduced the di↵erent average temporal adaptive fields in the three pop-
ulations, indicating that the same circuitry that underlies the spatial adaptive field
can su�ciently account for the temporal adaptive field. In particular, for adapting
O↵ cells with center-surround adaptive fields, the temporal filter in the model was
approximately an average of the filters of adapting On and sensitizing cells. Thus, a
model of the adaptive field fit to responses from a step change in contrast, captures
the adaptive dynamics during rapidly changing contrast.
Although the full spatio-temporal model (Figure 3.3) produces more complex be-
havior, such as asymmetric responses at increases and decreases in contrast, the com-
bined e↵ects of the subunits in the spatiotemporal model predict the response to
dynamically varying contrast. Because the model with independent subunits fit to
48 CHAPTER 3. PREDICTIVE SENSITIZATION
local adaptation predicts the sum total adaptation for spatially global stimuli, we
conclude qualitatively that excitatory and inhibitory subunits within the adaptive
field adapt independently.
3.3.5 Feature detection in Fast O↵ cells
Having characterized the combined spatiotemporal computation of adaptation and
sensitization, we then considered the functional relevance of the sensitizing region of
the adaptive field. Many sensory neurons encode specific visual features using a high,
sharp threshold, and signal when the stimulus matches that feature (Ringach and
Malone, 2007). In the retina, for example, object motion sensitive (OMS) (Olveczky
et al., 2003) and W3 cells (Zhang et al., 2012) selectively report the presence of
di↵erential motion.
We measured both di↵erential motion sensitivity and sensitization in the same
cells. We found that fast-O↵ adapting cells were object motion sensitive (OMS) cells,
suppressed by motion in the distant periphery, whereas fast-O↵ sensitizing cells were
not (Figure 3.6 and 3.7). Although they receive di↵erent levels of peripheral inhibi-
tion, the two cell types fire synchronously in response to a stimulus with no spatial
structure, and thus respond to the same local stimulus features (Kastner and Baccus,
2011). Consistent with a role as a feature detector, the nonlinearity of O↵-cells was
more strongly rectified than that of On cells using a previously described index of
rectification. This index measures the logarithm of the ratio of the maximum slope
of the nonlinearity to the slope at zero input (Chichilnisky and Kalmar, 2002). O↵
cells had an index of 2.2 ± 0.1 (n = 80), whereas On cells had an index of 1.3 ± 0.2
(n = 9), meaning that relative to the slope at an input of zero (the average input),
O↵ cells increased their slope ⇠8 times more than On cells.
3.3.6 Encoding a signal in a noisy environment
To better understand the function of sensitization, we formalized the apparent role
of fast O↵ cells as feature detectors using a simple model of optimal signal detection
that changes with stimulus history. In a signal detection problem, the position of
3.3. RESULTS 49
a b Global!
Background!Object!
Differential!
Figure 3.6: Distinct cell types for object motion sensitivity and global sensitization. (a)Schematic diagram of the stimulus to test for object motion sensitivity. A central objectregion was shifted either together with the background (global motion), or at a di↵erent time(di↵erential motion). (b) Histogram of the response ratio between global and di↵erentialmotion for adapting (n = 59) and sensitizing (n = 16) fast-O↵ cells.
a b c
Figure 3.7: Fast-o↵ adapting cells are object motion sensitive. (a) The average response of anadapting O↵ (top) and sensitizing (bottom) cell to a stimulus that changed back and forthbetween high (35%) and low (5%) contrast placed over the receptive field center. PSTHsbinned at 0.5 s show the average over 50 trials, each with a di↵erent stimulus sequence.(b) Spatial receptive fields of the cells in (a) relative to the location of the object region(box). (c) Average response to 10 repeats of alternating global and di↵erential motion (seemethods) for the same cells in (a).
50 CHAPTER 3. PREDICTIVE SENSITIZATION
the optimal threshold function depends upon the distributions of signal and noise,
as has been examined at the photoreceptor to bipolar cell synapse (Field and Rieke,
2002). Although the threshold at the photoreceptor to bipolar cell synapse does not
appear to change according to prior information about the probability of photons, we
considered that changes in the response function of ganglion cells reflects the changing
likelihood of a signal.
By recording intracellularly from O↵-type bipolar cells in response to a repeated
5% Gaussian white noise stimulus, we found that the noise was 0.44 ± 0.12 (n = 5,
mean ± s.d.) times the standard deviation of the recorded membrane potential fluc-
tuations (Figure 3.8a and 3.9). Thus, for weak, low contrast, signals the probability
distribution of an input, v, given the presence of a signal, p(v|s), greatly overlaps with
the probability distribution of that same input in the presence of only noise, p(v|⌘).This overlap creates a benefit from a careful threshold placement to discriminate
between the two conditions. Although both positive and negative signals are distin-
guishable from noise, because many ganglion cells have monotonic response curves,
we focused our analysis on positive signal deviations.
The probability that a particular voltage arises from the signal distribution de-
pends upon the prior probability, p(s), of the presence of a signal. Thus, when p(s)
increases, the optimal threshold decreases (Field and Rieke, 2002). What then would
lead to an increase in the prior signal probability? For the visual system, an important
source of prior information comes from the strong spatial and temporal correlations
present in natural visual stimuli (Geisler and Perry, 2009). Objects do not suddenly
disappear, therefore once they are detected they are highly likely to be present nearby
in space. We incorporated this natural visual prior into a spatiotemporal version of
an optimal inference model (Figure 3.8b), similar to that used previously (DeWeese
and Zador, 1998; Wark et al., 2009). The model has two steps. First, at each point in
time and space, a new measurement of intensity, vx,t
, combines with the prior prob-
ability of a signal, p(sx,t
), at that location to yield a new posterior estimate of signal
probability, p(s|vx,t
) (see methods). Second, at each point in time, the prior, p(sx,t+1
),
is updated from the posterior at the previous point in time p(s|vx,t
), smoothed by
a Gaussian function, h(x) (see methods), representing the di↵usion of an object or
3.3. RESULTS 51
edge due to the random walk movement of fixational eye movements. The integral of
h(x) was less than 1, reflecting the occasional possibility of saccadic eye movements
that redirect gaze to a di↵erent image location. When presented with a brief strong
stimulus (35% contrast) on a background of weak input (5% contrast), this optimal
model was biased in space and time to predict an increased probability that a signal
was present outside of the spatial range of the object, even after the object was no
longer detectable (Figure 3.8c). This optimal behavior was qualitatively similar to
the sensitizing field we observed in O↵-type ganglion cells (Figure 3.1).
We compared how the changes in the response function during sensitization corre-
spond with changes expected from the framework of ideal signal detection. The e↵ect
of a changing prior value, p(s), on the posterior probability, p(s|v), depends upon the
Figure 3.8 (facing page): Sensitization reflects an increased prior expectation of a signal.(a) Conditional probability of an input, v, given either signal, s, (5% contrast) or noise, ⌘(0% contrast). Thick line p(v|s) is the distribution of measured voltages from an O↵-typebipolar cell responding to a 5% contrast Gaussian white noise stimulus. Thick line p(v|⌘) isthe estimated noise measured from repeated presentations of the same stimulus. Thin linesare Gaussian fits to the data. (b) Schematic diagram of a recursive model whereby a one-dimensional spatiotemporal input, v
x,t
, illustrated as a bright stimulus at one point in space,x, is combined with a prior stimulus probability, p(s
x,t
), to yield a posterior, p(sx,t
|vx,t
).The colored curves indicate p(s
x,t
|vx,t
) as a function of vx,t
given di↵erent values of p(sx,t
).The posterior p(s
x,t
|vx,t
) is then smoothed by a spatial filter, h(k), to yield a new prior,p(s
x,t+1
). The integral of h(k) was 0.96, reflecting the fact that signals may disappear dueto saccadic eye movements. To convert the posterior probability to a firing rate, p(s
x,t
|vx,t
),is passed through a rectifying function, N(), which was normalized by the average ganglioncell firing rate. (c) Top, posterior probability, p(s
x,t
|vx,t
), at each point in space and timein the inference model in panel (b), in response to a stimulus that changed between 5%contrast and a 35% contrast bar applied at the region and time interval indicated by thethick black lines. Bottom, average time course of p(s
x,t
|vx,t
) at the spatial location indicatedby the arrow. The average is taken over 1000 trials of di↵erent intensity sequences but thesame change in contrast, analogous to experiments in Figure 3.1 where firing rate changesare computed over the same change in contrast but di↵erent intensity sequences. (d) Top,average posterior, hp(s|S)i, in the center of the object during L
early
and L
late
. Bottom, firingrate nonlinearities for a ganglion cells compared with the model firing rate output duringL
early
and L
late
. (e) Comparison of slope (top) and midpoint (bottom) of sigmoid fits todata and model nonlinearities during L
early
and L
late
. The abscissa is in units of s.d. at 5%contrast. Also compared are the change in nonlinearity slope and midpoint between L
early
and L
late
(�) for the data and model. The dotted line is the identity.
52 CHAPTER 3. PREDICTIVE SENSITIZATION
Data
210Model
321
Midpoint
Slope
LearlyLlateΔ
2
0Nor
mal
ized
firin
g ra
te
20Input v (σ)
1
0.5
0
< p(s
|v)>
LearlyLlate
DataModel
e c d
0.5
0
Con
ditio
nal
prob
abilit
y
210Input v (σ)
p(v|η)p(v|s)
a
2
0
Norm
alizedfiring rate
10.50p(s|v)
Nonlinearity Np(p(s|v))
0.10
-5 0 5Spatial offset k
Spatial filter h(k)
p(sx,t|vx,t)p(sx,t+1)
1
0.6
0.2
Posterior p(s|v)210
Input (v)
p(s)0.80.60.40.2
b
3.3. RESULTS 53
shapes of p(s|v) and p(v|⌘). For the case where p(v|s) and p(v|⌘) are both Gaussian
with a di↵erent width, when p(s) decreases, the slope decreases, the threshold de-
creases and the baseline increases, reflecting the increased bias towards the presence
of the signal (Figure 3.8b).
After a transition to low contrast, sensitization, by definition, consists of a decrease
in threshold (Kastner and Baccus, 2011). By intracellularly recording from sensitizing
ganglion cells we found that an increased baseline of the nonlinearity accompanied
the decreased threshold during Learly
(Figure 3.9b). This depolarization was 35±18%
of the membrane potential standard deviation during Llate
(n = 3). Finally, even
though sensitization decreases the threshold during Learly
, it also decreased the slope
in the spiking nonlinearity as measured from extracellular recordings (Figure 3.9c).
This indicates that sensitization di↵ers from changes in sensitivity due to adaptation,
where the slope increases when the threshold decreases (Baccus and Meister, 2002).
The decrease in slope occurs because of the bias conferred by an increased p(s). When
a signal is more likely, a greater influence on p(s|v) comes from the prior p(s), and a
smaller influence comes from the new input, v. In the extreme, when p(s) = 1, the
posterior will always be 1, and the cell should always fire, regardless of the input, v.
In fact, this reduced dependence on the current input is consistent with a decrease
in mutual information between stimulus and response during the higher firing rate of
Learly
reported previously for sensitizing cells (Kastner and Baccus, 2011).
During Learly
, sensitization displays all three properties—decreased threshold, in-
creased baseline observed in the subthreshold membrane potential, and decreased
slope—expected from an ideal model of signal detection. Thus, changes in the re-
sponse curve during sensitization are of the same type found in an ideal model of
signal detection when the probability of the signal increases.
We then compared the output of the optimal model more quantitatively to the
change in firing rate seen in the nonlinearities from Learly
and Llate
. Low values of input
should yield near zero firing rate in ganglion cells, owing to the apparent pressure to
convey information about the stimulus using few spikes (Pitkow and Meister, 2012).
To convert the prior probability, p(s|v), to a firing rate, we used a spiking nonlinearity
(Figure 3.8b) optimized to map p(s|v) to the firing rate averaged over all cells during
54 CHAPTER 3. PREDICTIVE SENSITIZATION
5 %!
35 %!
b 4
3
Slop
e L early
43Slope Llate
-65
-60
-55
Out
put (
mV)
-2 0 2Input (!)
LearlyLlate
a
c
Figure 3.9: Variability in O↵ bipolar cells andchanges in ganglion cell responses during sen-sitization. (a) Top, intracellular recording froman O↵-type bipolar cell responding to a Gaus-sian white noise contrast (5%). Di↵erent col-ored traces show two repeats of the same stimu-lus sequence. Bottom, response at 35% contrast.(b) Nonlinearity of the membrane potential of asensitizing ganglion cell recorded intracellularlyduring L
early
and L
late
for a switch between 35%and 5% contrast. (c) Slope of spiking nonlinear-ities during L
early
and L
late
for sensitizing gan-glion cells recording extracellularly.
both Learly
and Llate
conditions, i.e., only a single function was used for all cells and
all conditions. This function had a sharp threshold corresponding to approximately
a p(s|v) of ⇠0.5. Thus, a comparison of ganglion cell firing with the optimal signal
detection model allowed us to interpret that the cell fired when it was more likely
than not that a signal was present.
After finding the best single spiking nonlinearity we then chose a single parameter
corresponding to the standard deviation of the noise distribution, p(v|⌘), for each
cell. This noise level was chosen to match the model to the response curve of Llate
alone. We then examined how closely the model matched the nonlinearity during
Learly
. We found that the optimal signal detection model predicted the magnitude of
the change in both midpoint and slope of the nonlinearity between Learly
and Llate
(Figure 3.8d,e).
In the signal detection model, the time course that the signal probability increased
was faster than when it decayed, di↵ering by a factor of 3 (Figure 3.8c). This temporal
asymmetry reflects the fact that it is easier to detect an increase in contrast than a
decrease in contrast (DeWeese and Zador, 1998). After a decrease in contrast, most
intensity values lie in the center of a previous higher contrast distribution, whereas
after an increase in contrast, extreme intensity values are quickly encountered that
are inconsistent with a previous lower contrast value. This asymmetry corresponded
3.3. RESULTS 55
to our measurements, as sensitization decayed with a tau 4.4 times longer than sen-
sitization developed (2.4 versus 0.55 s) (Kastner and Baccus, 2011). Therefore, both
qualitatively and quantitatively, sensitization within the adaptive field conforms to
an optimal model of signal detection in the presence of background noise.
3.3.7 Sensitization maintains the location of an object
We thus propose that the sensitizing field provides a bias for the detection of a signal
based upon the prior probability of that signal conditioned on the stimulus history.
We tested this idea in a more natural context relating to the motion of objects,
which represents an important source of signals in the visual scene. In a natural
environment, objects do not suddenly disappear, therefore once detected they are
highly likely to be present nearby in space. We thus presented a stimulus where
changes in spatiotemporal contrast were generated by changes in the velocity of object
motion on a background in the presence of fixational eye movements. We chose the
spatial texture of the object and background to be identical, and thus the object
was camouflaged and could only be detected by its motion. A camouflaged object
presents a di�cult signal detection problem for the visual system (Cuthill et al., 2005;
Stevens and Cuthill, 2006). When the object moved, it stimulated the retina with both
di↵erential motion and an increase in spatio-temporal contrast; however, once the
object ceased its di↵erential motion relative to the background, it was undetectable
relative to the background, and thus any information about its location could only
arise as a prediction based upon prior measurements.
The stimulus was a textured background of vertical lines with an intensity drawn
randomly from a Gaussian distribution that jittered in one dimension to mimic fix-
ational eye movements (Olveczky et al., 2003; Martinez-Conde and Macknik, 2008)
(Figure 3.10a). Every 8 s, three neighboring bars, representing an object, moved co-
herently for 250 ms at a speed of 1.1 mm/s (for a total distance of 275 µm), 9 times
greater than the average speed of the background. This prolonged period was used
only to provide a steady baseline for the measurement, as experiments changing con-
trast every 0.5 s (Figure 3.5) show that sensitization occurs even in a rapidly changing
56 CHAPTER 3. PREDICTIVE SENSITIZATION
environment. Thus the object part of the stimulus changed its spatio-temporal con-
trast by virtue of its changing motion—with fast motion representing high contrast,
and background motion representing low contrast.
a
b
2
1
Norm
alize
d ra
te
-1 0 1-1 0 1 -1 0 1Distance from center (mm)
SensitizingAdapting On Adapting Off
Figure 3.10: Sensitization predicts future object location. (a) Camouflaged stimulus (top),shown in space (vertical axis) and time (horizontal axis), was composed of 50 µm bars, of15% contrast. The dashed vertical line indicates the time the object stopped moving rel-ative to the background. Average firing rate response of sensitizing cells (n = 7, bottom),normalized for each cell by the average steady state response for that cell during jitteringbackground motion, 5 � 8 s after the object stopped moving. (b) Average normalized re-sponse for adapting On-type (left) (n = 5), adapting O↵-type cells (OMS cells, middle)(n = 39) and sensitizing cells (right) (n = 7), where each cell experienced >1000 repeats ofthe stimulus. Firing rate in (a) and (b) is shown as a function of distance from the centerof the object’s trajectory during the time interval ‘Early’ in (a), 0.5� 1.25 s after the ob-ject stopped moving relative to the background. Positive and negative locations reflect cellswith receptive field centers on either side of the object. Error was s.e.m. and was computedacross the number of trials contributing to each spatial bin.
We measured the spiking responses of the di↵erent populations of ganglion cells
to the camouflaged object located at many di↵erent positions on the retina. We
computed the average firing rate of each population as a function of the distance
3.3. RESULTS 57
between the cell and the center of the object’s trajectory. As expected, when the
object moved, cells responded strongly in the location of the moving object (Figure
3.10a).
After the object stopped its di↵erential motion, disappearing into the background,
a population of On-type cells decreased their activity within 0.5 mm of the object,
consistent with their monophasic adaptive fields (Figure 3.10b). However, sensitizing
cells showed persistent elevated activity in the location where the object recently
moved (Figure 3.10a,b). This activity was significantly (p < 0.002) above the steady
state response for 2.8 s after the object stopped its motion relative to the background.
We compared the duration of this elevated activity to the duration of the immediate
response, defined as the time that cells under the moving object first fell below the
baseline firing rate, reflecting the end of the linear filter and the onset of brief, local
adaptation. Sensitizing cells showed elevated activity for 21 times longer than the
immediate response to the fast motion, which was 133 ms. Thus, sensitizing cells
functionally stored the location of the previously moving object with locally increased
activity.
Adapting O↵ cells had diminished activity in the immediate location where the
object had stopped, indicated by a distance of zero in Figure 3.10. However, adjacent
to the location of the moving object these cells increased their activity (Figure 3.10b).
Like sensitizing cells, this increased activity remained significantly (p < 0.005) above
the steady state response for 2.8 s after the object stopped moving, 12 times longer
than the immediate response to the fast motion, which was 233 ms. When one con-
siders the total magnitude of the peripheral increase in activity from sensitization,
as measured by the area under the curve (Figure 3.10b), this was at least as large
as (1.1 times) as the central decrease in activity caused by adaptation. These results
were consistent with the center-surround organization of their adaptive fields (Figure
3.1d,e). Therefore, following the motion of a camouflaged object, adapting O↵ cells
stored and transmitted a prediction of the location of the boundaries of the object
after it ceased to move.
58 CHAPTER 3. PREDICTIVE SENSITIZATION
3.3.8 Inhibition is necessary for sensitization and the estab-
lishment of the adaptive field
Guided by the presence of inhibitory pathways in the AF model (Figure 3.3), we tested
whether inhibitory neurotransmission was necessary for sensitization. We measured
the responses of sensitizing cells to a stimulus that changed globally between high
and low contrast before, during, and after the application of 100 µM picrotoxin,
a drug that inhibits ionotropic GABAergic receptors, one of the major inhibitory
pathways of the inner retina. Picrotoxin abolished the ability of these cells to respond
during Learly
(Figure 3.11a), and turned the sensitizing response into an adapting
response, as measured by the adaptive index (Figure 3.11b). The change of plasticity
during pharmacologic block was specific to picrotoxin because sensitization persisted
in the presence of strychnine, a glycinergic antagonist, and APB, which blocks the
On pathway in the retina (Figure 3.12). Thus, GABAergic transmission underlies
sensitization, and enables sensitizing ganglion cells to respond quickly after a decrease
in contrast.
In the AF model, inhibition combines with the excitatory pathway prior to the
threshold of the excitatory pathway (Figure 3.3b). This is necessitated because inhi-
bition delivered after the threshold would produce a vertical shift during sensitization
instead of a horizontal shift (Kastner and Baccus, 2011). Such connectivity is most
consistent with amacrine cells inhibiting bipolar cell terminals. Salamander retinal
bipolar cell terminals express GABAC
receptors that can be blocked by Picrotoxin,
but not by Bicuculline, which block GABAA
receptors found on amacrine and gan-
glion cells (Lukasiewicz et al., 1994). Therefore our model predicts that sensitization
should persist in the presence of Bicuculline, and indeed this was the case (Figure
3.11a,b).
Previous studies have shown that intracellular recordings of bipolar cells can reveal
e↵ects of inhibition at their synaptic terminals, in particular those bipolar cells that
are likely to convey input to OMS cells (Olveczky et al., 2007). If one interprets
the excitatory subunits of the AF model to be bipolar cells, the model predicts that
during Learly
, bipolar cell terminals receive less steady inhibition than during Llate
.
3.3. RESULTS 59
c
a b
0.40.20
Adap
tive
inde
x
0.60.30Distance (mm)
-0.5
0
75 µM200 µM
d
Figure 3.11: Sensitization requires GABAergic transmission. (a) The stimulus changed backand forth for 80 trials between a global high contrast (35%) for 4 s and a global low contrast(3%) for 16 s (top). Ganglion cells were continuously recorded and drug was added to thesolution after 30 min, and then washed out of the solution 1 h later. Top, example averageresponses for sensitizing cells during the 30 minutes prior to drug addition (left), and from 30min to 1h after the drug addition (right), for a cell in 100 µM picrotoxin. Bottom, same fora di↵erent sensitizing cell in 200 µM bicuculline methiodide. (b) Average adaptive indicesfor sensitizing cells in picrotoxin (top)(n = 4) or bicuculline (bottom)(n = 5). Error bars,which show the s.e.m., are obscured by the data. (c) The response of a cell with a centersurround adaptive field to the stimulus from Figure 3.1a,b before and during the addition of75 µM picrotoxin. High contrast stimulus was positioned in the sensitizing region of the cell.Each histogram comes from 60 repeats of the stimulus, with di↵erent intensity sequencespresent on each trial. (d) Average adaptive indices as a function of distance for fast-O↵adapting cells (top) (n = 68) and sensitizing cells (bottom) in control solution and 75 µM(n = 12, closed circles) or 200 µM (n = 6, open circles) picrotoxin. Colored circles show theresponses before the addition of the drug, and black circles show the responses after thedrug had been washed in for 20 min. Error bars are s.e.m.
60 CHAPTER 3. PREDICTIVE SENSITIZATION
b a
c
Figure 3.12: Sensitization does not require Glycine receptors or the On pathway. (a) Thestimulus changed back and forth between a global high contrast (35%) for 4 s and a globallow contrast (5%) for 16 s (top). Ganglion cells were continuously recorded and drug wasadded to the solution after 30 min, and then washed out of the solution 1 h later. Exampleresponses for sensitizing cells during the 30 minutes prior to drug addition (left), and from30 min to 1h after the drug addition (right) for cells in 20 µM strychnine (middle) and 15 µML-APB (bottom). Each histogram comes from >60 repeats of the stimulus, with di↵erentintensity sequences presented on each trial. (b) Average adaptive indices for sensitizing cellsin strychnine (top) (n = 5) and L-APB (bottom) (n = 4). Error bars (s.e.m.), are obscuredby the data. (c) Stimulus (top) and average response for >27 trials of a sensitizing cell(bottom) to two di↵erent changes in contrast, 35 to 5% (left) and 100 to 7% (right).
3.3. RESULTS 61
As previously reported (Rieke, 2001; Baccus and Meister, 2002) we found that some
bipolar cells had a hyperpolarized membrane potential during Learly
compared to Llate
.
However, we also found bipolar cells with a depolarized membrane potential during
Learly
compared to Llate
(Figure 3.13a,b).
Inte
nsity
5 s2 mV
100 Hz40 ms
Ganglion cell responsewhen bipolar cell:
depolarizedhyperpolarized
-1
0
Δ po
tent
ial (
mV)
100Flash amplitude
bipolar cells with:afterdepolarizationafterhyperpolarization
a b
c d
10
0
Rat
e (H
z)de
pola
rized
100Rate (Hz)
hyperpolarized
AdaptingSensitizing
0.4 s
…! …!
Figure 3.13: Depolarization of bipolar cells during sensitization. (a) Top. Stimulus that con-sisted of biphasic flashes that changed from high (100%, black) to low (7%, blue) contrast.The low contrast stimulus was composed of 9 randomly interleaved intensity flashes. Insetshows transition from high to low contrast, colors indicate di↵erent flash amplitudes. Eachflash amplitude was repeated a total of 3 times at each time point. Bottom. Average re-sponse of a bipolar cell. (b) Change in membrane potential between L
early
(0.8� 3.2 s afterhigh contrast) and L
late
(12 � 16 s after high contrast) at each flash amplitude averagedover bipolar cells that showed an afterdepolarization (n = 4 cells) or afterhyperpolarization(n = 3 cells) following high contrast. (c) Average response of a ganglion cell simultaneouslyrecorded with the bipolar cell from (a) to a low contrast flash while a +500 pA or -500 pApulse was injected into the bipolar cell. (d) Changes in the firing rate of all simultaneouslyrecorded adapting O↵ (n = 4) and sensitizing (n = 6) ganglion cells within 0.2 mm of thefour bipolar cells that showed an afterdepolarization.
Although the existence of bipolar cells that depolarize during Learly
is consistent
with the AF model, these bipolar cells must connect to fast O↵ adapting and sensitiz-
ing cells. Therefore, while recording intracellularly from these bipolar cells that display
62 CHAPTER 3. PREDICTIVE SENSITIZATION
a afterdepolarization we simultaneously recording extracellularly from ganglion cells
(Asari and Meister, 2012). Injecting depolarizing and hyperpolarizing current into
these bipolar cells changed the response of all neighboring adapting and sensitizing
O↵ ganglion cells (Figure 3.13c,d). The current injection changed the response of the
ganglion cells from 7.4± 1.5 Hz when the bipolar cell was depolarized to 4.3± 1.3 Hz
when the bipolar cell was hyperpolarized (p <0.0003), indicating that these bipolar
cells reside within the fast O↵ ganglion cell circuitry.
The AF model predicts that all three types of adaptive field can be generated using
the same circuitry by changing the strength of adapting inhibition (Figure 3.3d). We
therefore tested whether a lower concentration of picrotoxin would transform a center
surround adaptive field into a monophasic adapting adaptive field, and transform
the sensitizing adaptive field into a center-surround adaptive field. For cells with a
center-surround adaptive field, 75 µM picrotoxin caused the surround of a cell to
change from sensitizing to adapting (Figure 3.11c,d). Thus, GABAergic transmission
was also necessary for sensitization in cells with a center-surround adaptive field.
In addition, when the high contrast region was close to the receptive field center of
the cell, at an average distance of 100 µm, inhibition acted to oppose adaptation,
as the adaptive index was greater in the absence of inhibition (�0.47± 0.05 control,
�0.59± 0.06 picrotoxin, p < 0.0125).
We then examined the e↵ect of 75 µM picrotoxin on sensitizing cells. We found
that cells located closer to the high contrast region changed from sensitizing to adapt-
ing, whereas those further away from the high contrast region still sensitized, but to
a lesser degree (Figure 3.11d). Sensitization was completely abolished at all distances
by added 200 µM picrotoxin (Figure 3.11d). Thus, a partial block of GABAergic
transmission transformed the sensitizing adaptive field into a center-surround adap-
tive field (Figure 3.11d). This confirms that a combination of adapting excitation and
inhibition constructs the adaptive field. As predicted by the AF model (Figure 3.3d),
reductions in the strength of one broad class of inhibition changed the adaptive field
from sensitizing to center-surround and then to adapting.
One potential concern with experiments using picrotoxin is that an increased firing
rate might cause increased adaptation to mask intact sensitization. In picrotoxin, the
3.4. DISCUSSION 63
high contrast response increased on average by 38 ± 18%, and the steady state low
contrast response increased by 123± 14%. However, an increased firing rate can also
occur with stronger stimuli in control solution. Therefore we compared the response
of individual sensitizing cells (n = 8) to two di↵erent contrast transitions (35 � 5%
versus 100�7%) (Figure 3.12c). Sensitizing cells increased their high contrast response
by 61 ± 17% in 100% contrast compared to 35% contrast. They also increased their
steady state low contrast response by 153 ± 51% in 7% contrast compared to 5%
contrast. Even with a firing rate higher than in picrotoxin, sensitizing cells continued
to sensitize under the higher contrast condition, as the adaptive index was 0.36±0.06
for 35 to 5% contrast, and 0.21± 0.01 for 100 to 7% contrast (Figure 3.12c).
3.4 Discussion
Here we have studied multiple aspects of how adaptation and sensitization combine
in single ganglion cells. As to the general phenomenon, Fast O↵-type retinal ganglion
cells have center-surround adaptive fields, showing central adaptation but peripheral
sensitization (Figure 3.1). Furthermore, spatial antagonism of plasticity occurs at a
subcellular scale (Figure 3.4) and sensitization occurs in a rapidly changing contrast
environment (Figure 3.5). As to the computation, a model with independently adapt-
ing excitatory and inhibitory subunits explains the spatiotemporal plasticity within
the adaptive field (Figure 3.3�3.5). The model further shows that di↵erent types of
adaptive fields can be explained by di↵ering strength of inhibition. As to the un-
derlying mechanisms, a membrane potential depolarization underlied sensitization of
the firing rate (Figure 3.9b). Sensitization also requires GABAergic inhibition, but
not transmission through GABAA
receptors (Figure 3.11). Certain bipolar cells ex-
perience a depolarization following high contrast and also connect to ganglion cells
that show sensitization (Figure 3.13). Furthermore, partial blockade of GABAergic
transmission supports the idea that di↵erent levels of inhibition can produce di↵erent
types of adaptive fields. As to the functional relevance of sensitization, one popula-
tion of cells with center-surround adaptive fields are object motion sensitive cells, thus
conforming to the notion of a feature detector (Figure 3.6). Fast-O↵ sensitizing cells,
64 CHAPTER 3. PREDICTIVE SENSITIZATION
although they are not OMS cells, have a similarly sharp threshold and respond to the
same local features as fast-O↵ adapting cells (Kastner and Baccus, 2011). Finally, as
to a theoretical understanding of these results, the sensitizing e↵ect on nonlinearities
is consistent with a simple model showing that inhibition acts as a bias in the detec-
tion of an e↵ective stimulus (Figure 3.8). Furthermore, the spatiotemporal sensitizing
field conforms to a recursive inference model that updates the prior probability of
a signal, predicting a sensitizing surround larger than the immediate response to a
stimulus. Testing this idea with a stimulus representing a camouflaged object, we
showed that sensitization enables the prediction of a future object position (Figure
3.10).
3.4.1 Adaptive and receptive fields
The most basic stimulus we used to characterize the adaptive field (Figure 3.1) is sim-
ilar in nature to the stimulus first used to describe the receptive field (Ku✏er, 1953;
Barlow, 1953), except that in our case we used changing spots of contrast instead
of flashing spots. Even though the classical receptive field incompletely describes the
response of a cell, part of its usefulness comes from the fact that, to some extent,
di↵erent spatial regions provide an independent contribution to the response of the
cell. Similarly, our measurements of the adaptive field indicate that excitatory and
inhibitory subunits contribute, to some extent, independently towards adaptation
and sensitization, respectively. Toward that end, we confirmed that the adaptive field
could be used to explain and interpret responses to di↵erent (global stimuli) and
more ecological stimuli (moving objects). We thus expect that the basic model of the
adaptive field o↵ered here should prove useful in an analysis of other visual stimuli.
Recently it was shown that at the level of the ganglion cell membrane potential, all
adaptive properties for a uniform stimulus could be explained by a model of synaptic
adaptation (Ozuysal and Baccus, 2012). If local sites of adaptation contribute inde-
pendently, this implies that spatiotemporal plasticity may be explained substantially
by knowledge of the local adaptive properties of synapses and of anatomical circuitry.
3.4. DISCUSSION 65
3.4.2 Integration of inhibition and excitation
A number of papers have described the di↵erent spatiotemporal properties of local
inhibition integrated with local excitation (Munch et al., 2009; Russell and Werblin,
2010; Manu and Baccus, 2011; de Vries et al., 2011; Bolinger and Gollisch, 2012; Asari
and Meister, 2012; Garvert and Gollisch, 2013). The integration of inhibition and
excitation at di↵erent spatial and temporal scales thus appears to be a common theme
to generate diversity among ganglion cell types, depending upon the combination of
all properties, including linear spatiotemporal filtering, nonlinearities, and dynamic
adaptation. Recent studies of responses across either shorter timescales or in the
steady state (Bolinger and Gollisch, 2012; Garvert and Gollisch, 2013), likely study
populations overlapping with those that we have studied here; however it is di�cult
to make a direct correspondence as the previous studies did not examine plasticity
over the timescales studied here. Because our main results are confined to identified
cell types known to form two separate mosaics (fast O↵ cells) (Kastner and Baccus,
2011), future results can compare di↵erent phenomena as we have done here with
object motion sensitivity and sensitization.
3.4.3 A functional role for adapting inhibition
A strong parallel exists between the role of inhibition in the receptive field, and the role
of adapting inhibition in the adaptive field. Just as the receptive field surround relies
on inhibition with a wider spatial extent than excitation (Thoreson and Mangel, 2012),
our AF model (Figure 3.3) and pharmacological experiments (Figure 3.11) indicate
that di↵erent levels of adapting inhibition produce the various spatial adaptive fields.
Although adaptation in inhibitory amacrine cells was known to exist (Baccus and
Meister, 2002), it lacked any apparent role in the plasticity of ganglion cells (Brown
and Masland, 2001; Rieke, 2001; Manookin and Demb, 2006; Beaudoin et al., 2007).
Our results and model show that by opposing excitatory adaptation and producing
sensitization, inhibitory synaptic transmission plays a critical role in retinal plasticity.
When GABAergic transmission was blocked, sensitizing cells adapted and failed
66 CHAPTER 3. PREDICTIVE SENSITIZATION
to signal at the transition to low contrast. Thus at the transition to low contrast, ac-
tivity was lowered in the absence of inhibition. This confirms the previously proposed
active role of inhibition in preserving sensitivity at the transition to a low contrast
environment (Kastner and Baccus, 2011). Because GABAergic amacrine cells in the
salamander are known to be sustained cells, this e↵ect is also consistent with the
role of these amacrine cells in generating ganglion cell activity through disinhibition
(Manu and Baccus, 2011).
However, it is likely that the classical linear surround and sensitization arise from
di↵erent sources of inhibition. Fast-O↵ adapting cells have a stronger inhibitory sur-
round than sensitizing cells (Kastner and Baccus, 2011), yet sensitizing cells appear to
have stronger input from adapting inhibition (Figure 3.11). This suggests that some
part of the inhibition that produces the linear receptive field surround is distinct from
the GABAergic inhibition that produces sensitization. Accordingly, we found a min-
imal correlation between the strength of the linear surround and the adaptive index
within adapting O↵ (r2 = 0.051) and sensitizing (r2 = 0.009) cells.
3.4.4 Di↵erent levels of sensitization in di↵erent cell types
Three di↵erent cell types showed di↵erent levels of sensitization, with On cells show-
ing no sensitization, and OMS cells showing intermediate sensitization. One possible
functional explanation for this observation is that On cells have a more shallow re-
sponse curve than o↵ cells (Chichilnisky and Kalmar, 2002; Zaghloul et al., 2003). As
such, they act less as a feature detector and therefore may have less of a benefit from
sensitization. Although fast-O↵ adapting cells are OMS cells, and sensitizing cells are
not (Figure 3.6 and 3.7), the two cells do respond to the same local temporal features
(Kastner and Baccus, 2011). Because OMS cells receive information from the wider
surround, indicating whether a di↵erential motion signal is present, they may rely
less on prior information in the form of sensitization.
3.4. DISCUSSION 67
3.4.5 Updating the prior probability of a stimulus
Models that use ongoing inference to adjust the prior probability are consistent with
behavior (Kording and Wolpert, 2006; Schwartz et al., 2007b), but a similar question
has not been explored in early sensory systems. Furthermore, previous theoretical
work has suggested that an optimal model that updated its prior probability is in-
consistent with observed physiological data precisely because such a model would not
predict adaptation (‘repulsion’) of a tuning curve, but an opposite e↵ect (‘attrac-
tion’) (Stocker and Simoncelli, 2006). In fact, in the primate LGN and primary visual
cortex, stimulus-specific enhancement of sensitivity from peripheral stimuli has been
shown and has been explained by a model containing adaptation of an inhibitory
surround pathway analogous to what we have proposed (Camp et al., 2009; Wissig
and Kohn, 2012). As to whether this behavior might be consistent with updating of a
prior stimulus probability, it has been noted that during low contrast or noisy stim-
uli, prior information would become particularly important, but these conditions have
not been thoroughly explored (Schwartz et al., 2007b), most likely because conditions
of strong stimuli are often more amenable to experimentation. In fact, we observed
sensitization under conditions of weak stimuli, when prior information from nearby
or previous strong stimuli is most critical in detecting signals in a noisy environment.
3.4.6 Integrating information at the bipolar cell synaptic ter-
minal
Several lines of evidence suggest that sensitization first arises in the bipolar cell presy-
naptic terminal, although a definitive confirmation must come from more mechanistic
future experiments. Sensitization produces a horizontal shift on the ganglion cell non-
linearity (Kastner and Baccus, 2011). For such a shift to occur a steady change in
inhibition must be delivered prior to a strong threshold, as occurs at the bipolar
cell terminal (Heidelberger and Matthews, 1992). Furthermore, although GABAergic
transmission is required (Figure 3.11), transmission through GABAA
receptors is not.
This shows that GABAergic transmission directly onto ganglion cells is not required
for sensitization (Figure 3.11), and indicates a requirement for transmission through
68 CHAPTER 3. PREDICTIVE SENSITIZATION
GABAC
receptors on bipolar cell terminals. Although a more direct test would involve
a specific blocker of GABAC
receptors, these may not be specific in salamanders (Ichi-
nose and Lukasiewicz, 2005). Finally, recordings from a subset of bipolar cells show a
depolarization after high contrast. These bipolar cells also connect to fast O↵ adapt-
ing and sensitizing cells (Figure 3.13). Although one may consider the possibility that
depolarization in bipolar cells reflects an intrinsic change in conductance instead of
changing inhibitory input, the spatial extent of sensitization (space constant of 0.41
mm for sensitizing cells) appears too large to arise from bipolar cells alone.
The bipolar cell synaptic terminal is also thought to be the site of contrast adap-
tation (Manookin and Demb, 2006; Jarsky et al., 2011; Ozuysal and Baccus, 2012).
Accordingly, we have seen that adaptation and sensitization combine at a scale smaller
than the ganglion cell receptive field (Figure 3.4). Previous studies of the OMS circuit
likewise indicate that the synaptic terminal does not simply relay the local photore-
ceptor input, but performs an integration of local and distant signals (Olveczky et al.,
2007; Baccus et al., 2008).
The depolarizing o↵set observed in bipolar cells that connect to sensitizing gan-
glion cells could underlie the shift in threshold, and potentially the decrease in slope
seen during sensitization. Inactivation of voltage dependent ion channels or synaptic
depression at the bipolar cell terminal could potentially decrease the slope of the
nonlinearity when the bipolar cell is depolarized, although future studies must be
performed to identify the biophysical mechanisms underlying the observed changes
in sensitivity. Studies of GABAergic receptors on bipolar cell terminals indicate that
transmission through GABAC
receptors does indeed undergo depression, with a re-
covery time constants of seconds, somewhat longer than the time course of recovery
of depression of excitatory transmission at the terminal (Li et al., 2007; Sagdullaev
et al., 2011).
The threshold at the bipolar cell terminal plays a key role in establishing certain
ganglion cells as feature detectors. Taking the functional point of view that the steady
level of inhibition is related to the prior probability of a signal (Figure 3.8), the bipolar
cell terminal adapts to the range of local signals, and steady presynaptic inhibitory
input provides information about the likelihood of those signals. One may wonder
3.4. DISCUSSION 69
why the retina performs the inference produced by sensitization, as opposed to only
occurring in the higher brain. The sharp threshold of ganglion cells acting as feature
detectors again provides the answer. If a signal fails to cross this threshold, it cannot
be detected by a later brain region no matter what future computation is performed.
Consistent with this idea, previous results indicate that sensitization preserves signals
that would otherwise be lost in cells with less sensitization (Kastner and Baccus,
2011). Thus, for the brain to take the greatest advantage of prior knowledge about
simple spatiotemporal correlations, the sensitizing signal must be delivered prior to
this threshold.
3.4.7 The retinal neural code and the statistics of objects
The detection, classification, and representation of objects is a di�cult task that oc-
curs throughout the visual hierarchy (Logothetis and Sheinberg, 1996). In the retina,
the neural code takes advantage of the distinct statistics of objects to encode an
object’s location and trajectory. For example, the trajectory of an object necessar-
ily di↵ers from background motion due to eye movements, a property used by OMS
cells to detect the presence of objects (Olveczky et al., 2003). Object motion is often
smooth, a property the retina uses to anticipate the location of a moving object,
(Berry et al., 1999), and signal a deviation from a smooth trajectory (Schwartz et al.,
2007a).
An additional important statistical property of an object is that its identity re-
mains constant, a property that underlies the cognitive representation of object per-
manence (Bower, 1967; Shinskey and Munakata, 2003). Thus object constancy pro-
vides the basis for an inference about the source of a visual stimulus.
However, objects present the retina with signals of vastly di↵ering strengths de-
pending upon motion, ambient lighting, or context. Adaptation to these changing
properties provides one strategy to reduce the e↵ects of environmental conditions
(Olveczky et al., 2007; Wark et al., 2009). Nevertheless, in cases of very large changes
in visual input, such as when an object stops moving, cells will fall below threshold
notwithstanding adaptation (Olveczky et al., 2007; Kastner and Baccus, 2011).
70 CHAPTER 3. PREDICTIVE SENSITIZATION
With respect to the problem of maintaining a continuous representation of an
object, a camouflaged object presents a particularly di�cult stimulus to the visual
system. Motion reveals the object, causing it to pop out from its surroundings, a
property that may arise in the retina due to OMS cells (Olveczky et al., 2003). Yet,
once the object stops, it nearly disappears into its surroundings. In this case, the visual
system has no choice but to rely upon prior information to represent the object, as
occurs higher in the brain (Graziano et al., 1997). Because of the di↵erent histories
of the motion trajectories of the object and the background, and because of the
property of object constancy, sensitization helps preserve an object’s location across
changes in object motion, thus contributing to the stable representation of objects.
Because a saccadic eye movement will change the location of an object on the retina,
it is expected that this preservation of object location will function within a saccadic
fixation. Across a saccade, the spatial scale of prolonged plasticity will relate to both
the size of the object and the saccade. Further experiments will be necessary to
determine how the present image is influenced by adaptive plasticity from a previous
fixation.
Although a number of sophisticated computations have been described in the
retina, these properties are typically studied in isolation (Rieke and Rudd, 2009; Gol-
lisch and Meister, 2010; Schwartz and Rieke, 2011). Here we have shown that several
computations—adaptation, sensitization and object motion sensitivity—combine to
enable a prolonged representation of an object in the retina. The basic principles
of adaptation and prediction are common to all sensory regions of the brain. Similar
synaptic mechanisms have been proposed to accomplish adaptation both in the retina
and in the cortex (Jarsky et al., 2011; Ozuysal and Baccus, 2012; Chance et al., 2002).
Given the simple underlying mechanism of adaptation of inhibitory transmission that
we propose to generate predictive sensitization, one might expect that similar pro-
cesses underlie prediction elsewhere in the nervous system.
3.5. MATERIALS AND METHODS 71
3.5 Materials and Methods
3.5.1 Electrophysiology
Retinal ganglion cells of larval tiger salamanders of either sex were recorded using
an array of 60 electrodes (Multichannel Systems) as described (Kastner and Baccus,
2011). A video monitor, which had a frame rate of 60 Hz, projected the visual stimuli
at 30 Hz controlled by Matlab (Mathworks), using Psychophysics Toolbox (Brainard,
1997; Pelli, 1997). The video monitor was calibrated using a photodiode to ensure
the linearity of the display. Stimuli had a constant mean intensity of 10 mW/m2.
Contrast was defined as the standard deviation divided by the mean of the intensity
values, unless otherwise noted.
Simultaneous intracellular and multielectrode recordings were performed as de-
scribed (Manu and Baccus, 2011). Sensitizing ganglion cells were identified by their
level in the retina, spiking response and sensitizing behavior. O↵-type bipolar cells
were identified by their flash response, receptive field size, and level in the retina.
3.5.2 Cell classification
For all experiments, all ganglion cells (>250 cells in >45 retinas) were classified as pre-
dominantly On-type or O↵-type based upon their linear temporal filters computed by
reverse correlation to a spatial visual stimulus. They were further classified as adapt-
ing or sensitizing by whether a spatially global high contrast stimulus subsequently
decreased or elevated the firing rate response to low contrast, as previously described
(Kastner and Baccus, 2011). Sixty percent of the cells we recorded from were fast-
O↵ cells (adapting or sensitizing) or On cells. Other cell types could not be sampled
in large enough numbers to assign them to distinct classes, and therefore were not
included.
Additionally, for experiments in Figures 3.6 and 3.10, cells were classified as object
motion sensitive (OMS) cells using a stimulus similar to one previously described
(Olveczky et al., 2007). A striped grating shifted ⇠25 µm back and forth over the
retina with a frequency of 0.5 Hz. A central 800 µm object region moved coherently
72 CHAPTER 3. PREDICTIVE SENSITIZATION
with the background (global motion) or out of phase with the background (di↵erential
motion) with the condition changing every 6 s. The ratio of the average firing rate
during global vs. di↵erential motion was computed.
3.5.3 Receptive fields and sensitivity
To compute sensitivity at each spatial location, a spatiotemporal LN model was com-
puted by correlating the spiking response with a visual stimulus consisting of binary
squares (Figure 3.1) such that
F (x, y, ⌧) =1
T
ZT
0
s(x, y, t� ⌧)r(t)dt, (3.1)
where F (x, y, ⌧) is proportional to the linear response filter at position (x, y) and
delay ⌧ ; s(x, y, t) is the stimulus intensity at position (x, y) and time t, normalized to
zero mean; r(t) is the firing rate of a cell; and T is the duration of the recording. For
stimuli containing Gaussian white-noise annuli (Figure 3.4), the filter had one spatial
dimension and was normalized by the area of the annulus at each distance to account
for the di↵erence in area of di↵erent annuli. The filter had a significant amplitude
for a duration of ⇠0.5 s, shorter than the time intervals of Learly
and Llate
used to
calculate the LN model. Then the predicted linear response, g(t), was computed by
convolving the stimulus with the linear response filter such that
g(t) =
ZT
0
s(x, y, ⌧)F (x, y, t� ⌧)dxdyd⌧. (3.2)
The filter was initially normalized in amplitude so that the standard deviation of g(t)
and s(x, y, t) were equal (Baccus and Meister, 2002). A static nonlinearity, N(g), was
then computed as the average value of the response r(t) over bins of g(t). Model and
data had a correlation coe�cient of 0.52 ± 0.03 for the ganglion cells contributing
to Figure 3.4. This correlation was comparable to our previous work (Kastner and
Baccus, 2011).
To measure the sensitivity within each spatial region between Learly
and Llate
the
3.5. MATERIALS AND METHODS 73
nonlinearities were fit to a sigmoid function,
y = x0
+m
1 + exp⇣
x
1
/
2
�x
r
⌘ , (3.3)
where x0
is the basal firing rate, x0
+m is the maximal firing rate, x1
/
2
is the x value
at the midpoint of the range of firing rates, and r�1 controls the maximal slope.
This equation was also used to report parameters of nonlinearities in Figure 3.8e and
3.9c. For sensitivity measurements, the nonlinearity measured during Learly
was scaled
along the x-axis to match the nonlinearity from Llate
. The factor needed to scale the
two nonlinearities together was then multiplied into the linear filter computed during
Learly
(Kim and Rieke, 2001). This placed the sensitivity in the magnitude of the
linear filter at each spatial location. The spatial sensitivity was computed to be the
root-mean-squared value of the temporal filter in a given region in space. For Figure
3.2 the sensitivity di↵erence was the di↵erence between the spatial sensitivity between
Learly
and Llate
in all regions of space that had a temporal filter with a sensitivity
value that deviated by four standard deviations from the temporal filters in all regions
of space.
For all cells, a receptive field was independently measured using reverse correla-
tion to a binary checkerboard stimulus, and the spatio-temporal receptive field was
approximated as the product of a spatial profile and a temporal filter (Olveczky et al.,
2003).
3.5.4 Adaptive field model
The model of Kastner and Baccus (2011) was extended by replicating the model’s
structure as a set of excitatory and inhibitory subunits, along with the two new spa-
tial parameters that described the size of the subunits. The output of the inhibitory
subunits were linked by a synaptic pathway to the excitatory subunits. Then exci-
tatory subunits were spatially weighted and passed through a threshold to produce
the final output. A prime candidate for the proposed inhibitory pathway could be the
signal passed through amacrine cells, inhibitory interneurons in the retina (Baccus,
74 CHAPTER 3. PREDICTIVE SENSITIZATION
2007).
All subunits contained separable spatiotemporal filters, F (x, ⌧). For each subunit,
the stimulus was convolved with the spatio-temporal filter,
g(t) =
ZT
0
s(x, ⌧)F (x, t� ⌧)dxd⌧. (3.4)
The temporal component of all subunit filters was identical, and modeled as a biphasic
function (di↵erentiated Gaussian) that roughly matched the peak time of fast-O↵
adapting and sensitizing cells. The spatial component of each filter was modeled as a
Gaussian with standard deviation, �, and subunits within a type were separated by
2�, representing tiling of the subunits. Because the extent of sensitization was roughly
three times that of adaptation in center-surround cells (Figure 3.1e), the inhibitory
spatial filters were three times the size of the excitatory spatial filters.
In the inhibitory subunits gI
(t) was then passed through a linear-threshold func-
tion, which was the same for all inhibitory subunits,
hI
(t) = NS
(gI
) =
(0 if g
I
< hgI
i+ 0.025
gI
(t)� (hgI
i+ 0.025) if gI
� hgI
i+ 0.025. (3.5)
This same nonlinear function, NS
(), was also used for excitatory subunits as described
below. Brackets h...i denote the average quantity, which for all g was constant because
the stimulus mean did not change.
After the threshold in each subunit, adaptation occurred through a feedforward
divisive operation, such that the response of the subunit was divided by the mean
value of its recent history weighted by an exponential filter of two seconds. This was
chosen for simplicity, although more accurate models exist of adaptation (Jarsky et al.,
2011; Ozuysal and Baccus, 2012). This adaptation could either occur at the level of
a bipolar or amacrine cell synaptic terminals (Baccus and Meister, 2002; Sagdullaev
et al., 2011). For all excitatory and inhibitory subunits the same adaptive filtering
was used. The input, h(t), was convolved with an exponential filter, FA
, and then h(t)
was divided by the result. A constant term in the denominator set the magnitude of
3.5. MATERIALS AND METHODS 75
adaptation,
mI
(t) = A(h(t)) =h(t)
2 +RFA
(t� ⌧)(⌧)d⌧, (3.6)
where m(t) is the output of the subunit. The time constant of FA
, representing the
timescale of integration of the contrast, was 2 s,
FA
(t) = 0.5e�t
2s . (3.7)
For inhibitory subunits, the output mI
(t) passed to excitatory subunits through a
synaptic temporal filter LQ
defined as an alpha function with a time to peak of 150
ms,
LQ
(x) = 2x
0.15
exp⇣
�(⌧�0.15
0.15
⌘
n(t) =RLQ
(t� ⌧)mI
(⌧)d⌧. (3.8)
This synaptic delay could result from the action of metabotropic GABA receptors
(Maguire et al., 1989). The connection between the inhibitory subunits and an exci-
tatory subunit also contained a saturating nonlinearity, NQ
,
pI
(t) = NQ
(n(t)) =0.3
1 + exp⇣
0.03�n(t)
0.01
⌘ , (3.9)
the e↵ect of which was to diminish the modulation of inhibitory transmission at high
contrast, and amplify the modulation at low contrast. Thus, pI
(t) represented the
unweighted inhibitory synaptic input to an excitatory subunit. This saturation could
arise from either synaptic depression or receptor desensitization (Li et al., 2007). An
alternative source of the inhibitory pathway could be that adaptation is in the bipolar
cell synaptic terminal that forms the input to an amacrine cell, and the delay and
saturation (LQ
and NQ
) are produced by the filtering and membrane properties of
the intervening amacrine cell (Baccus and Meister, 2002).
In narrow field amacrine cells, neural processes produce both input and output.
Thus, the weighting wI,E
of the connection between each inhibitory and excitatory
76 CHAPTER 3. PREDICTIVE SENSITIZATION
subunit was equal to the overlap between the spatial filters of the subunits,
wI,E
=X
x
FI
(x)FE
(x), (3.10)
as shown in Figure 3.4b, where wI,E
computed as the dot product between the two
spatial filters.
For each excitatory subunit, all inhibitory subunits were linearly combined,
qE
(t) = gE
(t)� wmax
X
j
wI
pI
(t), (3.11)
with a scaling of wmax
to increase or decrease the magnitude of the inhibition in
the model (Figure 3.3d). The pathways combined prior to the excitatory pathway
threshold indicating that the amacrine cells might synapse presynaptically onto a
bipolar cell terminal.
Within the excitatory subunit, qE
(t), was then passed through the same linear-
threshold function NS
() and adapting function A() as in the inhibitory pathway.
The excitatory subunits were then combined linearly according to
r(t) =X
i
wE
A(NS
(qE
(t))) (3.12)
Each excitatory subunit was weighted by wE
according to the overlap between the
subunit’s spatial filter and the ganglion cell spatial filter, which was also three times
as large as an excitatory subunit. This size was chosen because for cells with a center-
surround adaptive field (Figure 3.1e), the receptive field center was approximately
the size of sensitizing surround of the adaptive field (Figure 3.4c). The output was
then passed through another linear threshold function
u(t) = NF
(r(t)) =
(0 if r(t) < ✓
r(t)� ✓ if r(t) � ✓. (3.13)
Where ✓ was a function of the total magnitude of the inhibition, used to provided a
comparable steady state response across all inhibitory magnitudes used. This ensured
3.5. MATERIALS AND METHODS 77
that the only di↵erences seen in the adaptive index were due to changes in the strength
of the plasticity and not due to changes in thresholds at steady state.
3.5.5 Temporal adaptive field
To measure the temporal adaptive field we presented a stimulus whose contrast was
drawn randomly from a uniform distribution of 0 � 35% contrast every 0.5 s. The
intensities presented for each contrast were randomly drawn from a Gaussian distri-
bution defined by the contrast of that time point. Since the intensities were randomly
drawn, as the input for the filter we computed the contrast from the mean, M , and
standard deviation, W , of the sequence of intensities that were actually presented,
which were 0� 66%. The contrast, �, was � = W/M .
3.5.6 Signal detection model
The probability, p(v|s), of an input, v, given a signal, s, was taken from a Gaussian
fit from the distribution of bipolar cell membrane potentials at 5% contrast. The
probability of an input, v, given that no signal was present, p(v|⌘), was estimated as
a Gaussian distribution from repeated presentation of the same 5% contrast stimuli.
For the model, the average ratio of the s.d. of a Gaussian fit to p(v|⌘) and p(v|s) wasthe only parameter taken from the data. For the recursive spatiotemporal inference
model at each time point the posterior probability, p(sx,t
|vx,t
) was computed from
Bayes’s rule as
p(sx,t
|vx,t
) =p(v
x,t
|sx,t
)p(sx,t
)
p(vx,t
|sx,t
)p(sx,t
) + p(vx,t
|⌘)(1� p(sx,t
)). (3.14)
The denominator, p(v), reflected the fact that p(s) + p(⌘) = 1 (either a signal is
present or it is not). The prior probability, p(sx,t
), was updated from the previous
posterior probability at each time point by convolving a Gaussian smoothing filter,
h(k), with p(sx�k,t�1
|vx�k,t�1
) according to
p(sx,t
) =
Zh(k)p(s
x�k,t�1
|vx�k,t�1
)dk. (3.15)
78 CHAPTER 3. PREDICTIVE SENSITIZATION
The average posterior, hp(s|v)i, during Learly
and Llate
was computed. To convert
the posterior probability to an average firing rate, a nonlinear function, N(hp(s|v)i),was computed to map hp(s|v)i in the center of the object to a firing rate nonlinear-
ity averaged over all ganglion cells at both Learly
and Llate
. This function captured
the average relationship between signal probability and firing rate for all cells and
conditions. Then to assess whether the optimal model could reproduce the changing
nonlinearities between Learly
and Llate
, a single parameter representing the standard
deviation of the noise distribution p(v|⌘) was fit. This parameter varied across cells
and ranged between 4.0 and 4.3% contrast, and was optimized only to make the
output of the model approximate the nonlinearity during Llate
. Then, we examined
whether the model’s firing rate output, N(hp(s|v)i), during the time interval Learly
matched the measured nonlinearity during Learly
. Nonlinearities for data and model
were compared using Equation 3.3.
3.5.7 Duration of sensitization
For experiments in Figure 3.10, the significance of the response of a population of cells
was assessed in the spatial region that gave the largest response. The number of trials
contributing to the calculation of significance for the sensitizing and adapting OMS
O↵ cells was 1000 and 8000, respectively, averaged across 7 retinas. Significance was
measured using Student’s t-test by comparing the distribution of values at a given
time point to the distribution of values from 5� 8 s after the object stopped moving.
Chapter 4
Optimal dynamic range placement
by coordinated populations of
ganglion cells
This chapter is in preparation as “Optimal dynamic range placement by coordinated
populations of ganglion cells,” with author list: Kastner DB, Baccus SA, Sharpee TO.
4.1 Summary
Neural circuits use populations of neurons to encode their inputs. Although there
have been various theoretical motivations for population encoding there have been
few comparisons of theory to data. Here we focus on fast O↵ retinal ganglion cells,
a set of cells made up of two distinct populations of neurons. These neurons have
been shown to encode very similar features of the visual scene, allowing us to focus
exclusively on their dynamic range placement. We develop a simple model, based
upon binary neural encoders, constrained by neural noise and metabolic e�ciency.
We show that these populations of neurons optimally place their dynamic ranges
to maximize the amount of information they encode about the input distribution.
We also find that there are two distinct regimes for optimal population encoding.
At low noise the optimal solution distributes the encoding with distinct thresholds,
79
80 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
while at high noise the optimal solution redundantly encodes the input. On cells have
response functions consistent with their using redundant encoding, explaining their
greater homogeneity.
4.2 Introduction
Neurons have a limited dynamic range, placing a premium on the way in which
a neuron encodes its inputs. In select circumstances it has been shown that neurons
approach the optimal positioning of their dynamic range to maximally transmit infor-
mation about their inputs (Laughlin, 1981; Brenner et al., 2000; Pitkow and Meister,
2012). Furthermore, neurons dynamically change their response range in ways con-
sistent with optimal behavior (DeWeese and Zador, 1998; Kastner and Baccus, 2011;
Sharpee et al., 2006; Wark et al., 2009). However, neural circuits use populations
of neurons to encode their inputs. In fact, both optimal information transmission
(Nikitin et al., 2009) and metabolic e�ciency (Laughlin et al., 1998) theoretically
motivate the use of population encoding in neural circuits. Yet, whether or not popu-
lations of neurons perform optimally within neural circuits remains poorly explored.
The salamander retina contains two populations of ganglion cells that encode
the same features of the visual world, but do so with distinct thresholds (Kastner
and Baccus, 2011). These two populations show distinct forms of plasticity upon
a change in the contrast distribution. Sensitizing cells maintain a lower threshold,
encoding weaker signals, and adapting cells maintain higher thresholds, encoding
stronger signals. We sought to determine whether this coordination of signal strengths
could be a consequence of maximizing the information these two populations can
provide about the input.
Toward that end we developed a simple model to determine the spacing of response
functions for two neurons that maximizes the amount of information they provide
about the input. Since ganglion cells produce sparse and noisy responses (Pitkow and
Meister, 2012), the model found the spacing of the response function that maximized
the information given constraints put on the overall rate of the response functions,
and the noise of the individual response functions. With those constraints in place,
4.3. RESULTS 81
we show that, across a large range of contrasts distributions, adapting and sensitizing
cells space their response functions very close to the optimal spacing that maximizes
information. Furthermore the model predicts two distinct regimes for population en-
coding, one using response functions with di↵erent thresholds, and one using identical
response functions, redundantly encoding the input. We show that whereas the O↵
population in salamanders conformed to the former option, the homogeneity within
the On population in salamanders behaves consistently with the prediction for redun-
dant encoding.
4.3 Results
To measure dynamic ranges we recorded the responses of ganglion cells to Gaussian
white noise across multiple contrast distributions (Figure 4.1a). That enabled the
fitting of a linear-nonlinear model to the response of each ganglion cell at each contrast
(see methods). The linear component of the model represents the feature to which the
cell has the greatest sensitivity, on average, while the static-nonlinearity of the model
captures the dynamic range of the cell in each contrast condition (Chichilnisky, 2001).
In salamanders, all O↵ cell types split into adapting and sensitizing populations, and
the two fast-O↵ populations tile the retina, indicating that they are distinct cell types
(Kastner and Baccus, 2011). Because they form distinct cell types, here we will focus
on the dynamic range placement of the fast-O↵ adapting and fast-O↵ sensitizing cells.
As previously reported (Kastner and Baccus, 2011), sensitizing cells maintain
lower thresholds across the entire range of recorded contrast distributions (Figure
4.1b,c). To gain further insight into encoding with populations of cells, we wished to
understand the positioning of response functions for two cells that would maximize
the amount of information transmitted about the input distribution.
82 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
c
a b 60300R
ate
(Hz)
0.20 0.40Input
0.60
Sensitizing!Adapting!
32
x 1/2
/ σ
0.30.2Contrast (σ)
Figure 4.1: Adapting and sensitizing fast O↵ cells coordinate the encoding of the input. (a)Stimulus (top) and average response of fast O↵ sensitizing (middle) and adapting (bottom)cells recorded in three di↵erent contrast distributions (12, 24, 36%). The cells were recordedsimultaneously. 9 di↵erent contrasts were presented from 12 � 36% in 3% intervals. Eachcontrast was presented for > 110 s. The contrasts were randomly interleaved and repeated.Each contrast was presented for > 600 s. (b) Nonlinearities for the adapting and sensitizingcells shown in (a). Black lines are sigmoid fits to the data. The first 10 s of each repeat wasnot included in the LN model calculation. (c) Average values for all sensitizing (n = 11)and adapting (n = 36) cells at each contrast, �. The x1
/
2
values were normalized by � tomake it easier to see the di↵erences across all contrasts. s.e.m. values are obscured by thedata points.
4.3. RESULTS 83
4.3.1 Histogram equalization does not predict distinct thresh-
olds
Histogram normalization has proven to be a successful framework for understanding
the optimal placement of a dynamic range in single neurons (Laughlin, 1981; Brenner
et al., 2000). By matching an equal area of the input to a comparable range of output
it ensures optimal usage of a limited dynamic range. Therefore we extended the
approach of Laughlin to the case of two response functions (see methods). However,
given the assumptions of equal Gaussian noise at each response level, the optimal
solution for this approach never splits the response ranges of the two neurons. Rather
histogram equalization extends to predict two response functions located at the same
location, and the only thing that can vary between the two responses functions is the
overall scaling.
Beyond the assumption of equal Gaussian noise at each response level, the straight-
forward solution of histogram normalization will never produce a response function
shifted away from the mean. Even in the single cell case the mean of the response
function must reside at the mean of the input distribution. This, too, provides a sig-
nificant limitation when trying to understand the sparse responses typically found in
vertebrate neural circuits, not to mention the retina.
4.3.2 Binary response model for maximizing information
Therefore we decided to take a di↵erent approach. Our approach turned out to be
similar to that in a recently published paper, which looked at the optimal response
curve for a single ganglion cell in the retina (Pitkow and Meister, 2012). We modeled
the responses of individual neurons as a binary output with a probability of spiking
defined by a sigmoid function,
y =1
1 + exp⇣
x
1
/
2
�x
b
⌘ , (4.1)
The sigmoid function had two parameters, x1
/
2
, which is the x value that leads to
a 50% spiking probability, and b, where the inverse of b corresponds to the slope of
84 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
a b
10.50p(
1|x)
-2 0 2Input (x)
0.40.20
p(x)
x1/2
0.40H
(r|x)
61
2 4 610
2
Slope (b-1)0.40.20
p(x)
-2 0 2Input (x)
10.50
p(1|x)
0.40.20
p(x)
-2 0 2Input (x)
10.50
p(1|x)
Input (x)!
c
Figure 4.2: Binary model for informationmaximization. (a) Example input distribu-tion (top), and response function (bottom).(b) Noise entropy as a function of the slopeof the sigmoid. (c) Optimal placement of thetwo response functions without (left) andwith (right) a rate constraint.
the sigmoid. Multiplying the response function by the input distribution provides the
instantaneous probability of spiking, p(1). Since there are only two possible outcomes
from a single response function, subtracting 1 from p(1), provides the probability that
the response function produced no spike, p(0). Because we used Gaussian distributions
to generate the stimuli for the data in this work, we will use Gaussian distributions
as the input distributions (Figure 4.2a); however, there is nothing in this model that
requires any specific form to the input distribution.
For two neurons, instead of there only being two possible outcomes there are now
four: when both spike, p(11), when neuron one spikes but neuron two does not, p(10),
when neuron one does not spike but neuron two does, p(01), and when neither neuron
spikes, p(00). This treats the neurons as independent without significant correlations
in their responses, a good first approximation for the fast O↵ adapting and sensitizing
cells (Kastner and Baccus, 2011).
With the ability to calculate all possible outputs for the two model response
functions, it is straightforward to compute the mutual information they provide about
the input. The mutual information is computed by subtracting the noise entropy from
the total entropy. Here the total entropy is computed by:
H(r) = �X
p(ri
)log2
p(ri
). (4.2)
The noise entropy is computed by:
H(r|x) = �Z
p(x)X
p(ri
|x)log2
p(ri
|x)dx. (4.3)
4.3. RESULTS 85
Within this framework the slope of the sigmoid introduces noise into the system
(Figure 4.2b). When the slope is infinite, the sigmoid turns into a step function with
no uncertainty as to the outcome, and the slope decreasing creates uncertainty as to
the outcome of producing a spike. When finding the optimal position of the response
functions we will impose slopes for the functions. This constraint requires there to be
a certain amount of noise in the system, a rather plausible constraint for a biological
system.
Unlike with histogram normalization, the response functions that provide the op-
timal amount of information with this binary model separate from each other (Figure
4.2c); however, the curves produced still surround the mean of the input distribution,
and are not sparse like the data (Figure 4.1b). Therefore to shift the curves away from
the mean we imposed a constraint on the average rate of the two response functions
(Figure 4.2c)
hri = p1
(1) + p2
(1). (4.4)
We can interpret this rate limitation as enforcing a metabolic constraint on the sys-
tem. Neural activity brings with it a large metabolic cost (Ames et al., 1992; At-
twell and Laughlin, 2001). This metabolic cost is so strong that it likely provided
a strong evolutionary limitation on brain size (Kotrschal et al., 2012). Furthermore,
retinal ganglion cells show responses consistent with e�ciently taking into account
that metabolic cost (Balasubramanian and Berry, 2002).
4.3.3 Low threshold cells should have less noise
Now that we have a model that has the potential to produce curves similar to the
data we wanted to understand the general behavior of the model. Toward that end
we simulated the model across a large range of noise values and average rates. For
each grouping of two slopes, which sets the noise levels in the two response functions,
and average rate there is a single spacing between the two curves that maximizes
the information that the two curves provide about the input. When we simulated the
optimal response functions in many di↵erent conditions we found that all solutions
placed the response function with the lower amount of noise, steeper slope, closer to
86 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
a b
3.53
2.5Slop
e / !
0.30.2Contrast (!)-4
-2
0
2
4
Spac
ing
( !x 1/2)
2 4 61
2 4
Slope Ratio
0.40.2
< r >
Sensitizing!Adapting!
Figure 4.3: Lower threshold response function optimally has less noise. (a) Many simulationsof the binary model. Each data point shows the spacing between the two response functionsthat maximizes the information they provide about the input. For each point two slopes anda rate were randomly chosen. Spacing is defined as the x1
/
2
value from response functionone minus the x1
/
2
value from response function two. The slope ratio was taken as the slopeof response function one over the slope of response function two. (b) Average slope valuesfor the same group of cells from Figure 4.1c. The slope values were normalized by � to makeit easier to see the di↵erences across all contrasts. s.e.m. values are obscured by the datapoints.
the mean (Figure 4.3a).
These simulations provide the first prediction of the model: the lower threshold
cell should have less noise. It has previously been shown that sensitizing cells have less
noise than adapting cells (Kastner and Baccus, 2011), but this model makes a finer
prediction, the less noise should also manifest as a steeper slope of the sensitizing
response function. We, therefore, compared the slopes of the response functions of
adapting and sensitizing cells across the range of recorded contrasts. Across the full
range of contrasts, sensitizing cells maintained a steeper slope than the adapting
cells (Figure 4.3b). Just as predicted by the model, sensitizing cells have the lower
threshold (Figure 4.2c), and have less noise, as manifested by their steeper slope.
4.3.4 Di↵erent amounts of noise produce di↵erent optimal
coding strategies
In understanding the general behavior of the model, we next sought to understand
the spread that occurs in the spacing as the slopes of the response functions become
4.3. RESULTS 87
a c b 1
0.5
0Spac
ing
( ! x 1
/2)
65432Slope (b-1)
0.40.2
< r >
20
0Rat
e (H
z)
-0.5 0 0.5Input
3
2.5
Estim
ated
Slo
pe
Fast On adapting
Fast Off adaptingFast Off sensitizing
Figure 4.4: Distinct population coding regimes. (a) Many simulations of the binary modelwith the slopes and rates chosen randomly; however, the slopes of the two response functionswere the same. Each point shows the spacing that maximizes the amount of informationconveyed about the input. (b) Example nonlinearities for a fast O↵ sensitizing, a fast O↵adapting, and an On cell recorded simultaneously. (c) Average slopes for fast O↵ sensitizing(n = 95), fast O↵ adapting (n = 388), and On (n = 58) cells. For (b) and (c) the stimuluswas over 500 s of a spatial Gaussian white noise stimulus, made up of independent 50 µmbars, where each bar had a contrast of 35%.
similar (Figure 4.3a). When the slopes were the same, slope ratio of 1 in Figure
4.3a, sometimes the optimal solution placed the curves in the same location, with no
spacing between the response functions, and sometimes the optimal solution spaced
the two curves apart. To better understand what contributed to that di↵erence we
reran the simulations with two response functions that had the same slope. We found
a very sharp transition between two di↵erent coding strategies for populations (Figure
4.4a). When the noise is high, as manifested by shallow slopes, the optimal solution
places the response functions in the same location, using redundant encoding. As the
slope becomes steeper, and the noise is low enough, the optimal solutions sharply
transitions to coordinated encoding, with curves spaced apart.
This provides an explanation for a previously perplexing result. In salamanders
all O↵ populations were heterogeneous, splitting into adapting and sensitizing types;
however, the On population was homogeneous, only displaying adaptation. A possible
explanation for the homogeneity of the On population now could be that the noise
in the On population is such that the optimal coding strategy would be redundant
encoding. That predicts that the On response functions should have a shallower slope
than the fast O↵ cells, which was the case (Figure 4.4b,c).
88 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
The data from Figure 4.4b,c was collected in response to a spatial stimulus, un-
fortunately, preventing a direct comparison of the slopes measured to those used in
the simulation of the model since we cannot know the actual contrast that the cell
experienced. However, the data collected in Figure 4.1, was in response to a uniform
stimulus, where there is no uncertainty as to the contrast. Since the slope divided by
the contrasts of the fast O↵ sensitizing cells was e↵ectively a constant (Figure 4.1b),
we found the contrast that would create the best match of the data in Figure 4.4c
to that in Figure 4.1b. That transformation allowed us to compare the values of the
slopes in the data to the axis of the simulation. The fast O↵ slopes were above the
sharp threshold where coordinated encoding was optimal, and the On slopes were
below that threshold, putting them within the regime of redundant encoding.
4.3.5 Fast O↵ cells optimally space their response functions
Up until this point we have considered the general behavior of optimal response func-
tions. This model allows us to go a step further, and directly compare the placement
of the response functions of the recorded ganglion cells to the optimally placed re-
sponse functions. By fitting the data to sigmoid functions we can extract the noise
and rates that are the constraints to the model. We can then calculate the informa-
tion transmitted by a range of spacing that are all consistent with the constraints
from the sigmoid fits to the data. This creates a function that relates the transmitted
information to the spacing between the response functions. This bi-lobed function has
a maximum at the spacing that provides the maximal amount of information, and a
second local maximum at spacing that has the lower sloped response function closer
to the mean (Figure 4.5a).
Across the full range of contrasts the spacing of the data was very close to the
spacing that maximizes the information about input (Figure 4.5a). On average fast
O↵ cells had a spacing that provided > 97% of the maximum amount of information
(Figure 4.5b).
4.3. RESULTS 89
1
0.9
0.8Rel
ativ
e in
form
atio
n
-1 0 1Spacing (Δx1/2)
302010
Contrast
a b 1
0.9
0.8R
elat
ive
info
rmat
ion
0.30.2Contrast (σ)
Figure 4.5: Fast o↵ cells optimally space their response functions. (a) Each curve shows theinformation for a given spacing given the noise and rate contraint at each contrast. Theslopes used were taken from sigmoid fits to the nonlinearities of a pair of fast O↵ sensitizingand adapting cells recorded simultaneously. The rate was also taken from the sigmoid fitsafter rescaling to make the maximum 1 for each fit. The black dots are the spacing seen inthe data at each contrast. Each curve was normalized by the maximum information at thatcontrast. The curves are o↵set from each other along the y-axis to be able to see each curveindependently. (b) The average percentage of the maximum information reached for all cellpairs (n = 7) at each contrast. Colors correspond to the colors from (a). The dotted line ispositioned at 97%. s.e.m. values are obscured by the data points.
90 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
4.3.6 Model fits data across contrasts
Finally, we sought to fit the measured response functions with the model. For a
given set of two slopes and the rate constraint a single pair of response functions
maximizes the information about the input. We searched through the space of optimal
response functions to find the set that best fit the data after being normalized to a
maximum firing rate of one (Figure 4.6a). To determine how well the optimal fits
approximated the data we compared the error associated with the optimal fits to two
di↵erent error values. The minimal error would come about if the model fit the data
as well as independent sigmoid fits to the two response functions. The underlying
principle of the model is that the spacing, and thereby the half maximal points, x1
/
2
,
of the response functions were optimized to maximally transmit information about
the input. Therefore at the opposite extreme we compared the model fits to sigmoid
fits to the data that could have independent slopes, but shared the same half maximal
point, x1
/
2
. Across all cell pairs and contrasts the model fit the data well, reaching
> 65% of the error associated with the independent sigmoid fits to the response
functions, which had an extra parameter for the fit (Figure 4.6b).
For the optimal fits, the x1
/
2
and the slopes changed in a stereotyped manner
with contrast (Figure 4.6c). Previously it was shown that each individual ganglion
cell could be viewed as rescaling their response function by the contrast after an initial
threshold (Figure 2.12c)(Kastner and Baccus, 2011). Here we see that this type of
rescaling is the optimal behavior for the pair of neurons.
Up until this point we have treated each input distribution as if they were all
fundamentally identical just with di↵erent scaling factors that led to the di↵erent
contrasts. However, for the optimal model fits the fact that the rate constraints, hri,increased with increasing contrast distributions (Figure 4.6d) highlights that these
paired ganglion cells do not behave identically at each contrast. To understand the
di↵erence between the di↵erent contrasts we considered that ganglion cells do not
encode a perfectly detectable signal. Rather ganglion cells encode their input within
the context of noise.
To properly function, information processing systems, like the brain, need to be
able to detect a signal in the presence of noise (Kording and Wolpert, 2006), and the
4.3. RESULTS 91
0.60.2x 1
/20.30.2
Contrast (σ)
0.10.05
b
10.50
Rel
ativ
e er
ror
0.30.2Contrast (σ)
a
b
c
d
10.50R
ate
(Hz)
0.20 0.40Input
0.60
Sensitizing!Adapting!
0.20.10<
r >
0.40.2Contrast (σ)
Figure 4.6: Model fits to the data. (a) Data (color circles) and optimal fits (black lines)for response functions for a pair of adapting and sensitizing cells at three di↵erent contrastdistributions (12, 24, 36%). (b) Average error for the seven pairs of adapting and sensitizingcells across the range of contrast distributions. (c) Average parameters for the fits to theadapting and sensitizing cells. Black lines show the fits for the model that scales by thecontrast after an initial threshold (see methods). (d) Average rate constraints for the pairs ofganglion cells across the range of contrast distributions. Black line shows the rate constraintextracted from the threshold model from (c).
92 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENTPr
obab
ility
0.20Input
NoiseLow contrastHigh contrast
a b
Det
ecta
bilit
y0.40.2
Contrast (σ)
Figure 4.7: Detecting a signal in the presence ofnoise. (a) Three sample distributions: noise (grey),low contrast (green), and high contrast (purple).(b) The relative detectability of signals as a func-tion of the contrast distribution in the presenceof a fixed noise distribution (see methods for thecalculation of detectability).
retina is no exception. Already at the first synapse in the retina a threshold separates
the reliable signal from perverting noise (Field and Rieke, 2002). Indeed, discriminat-
ing a signal from noise in a dynamic environment predicts sensitization itself (Figure
3.8) and responses similar to sensitization (Stocker and Simoncelli, 2006). Therefore
we found it quite useful to consider signal detection within the framework of optimal
information processing in the retina.
Within a low contrast distribution it is quite di�cult to distinguish signal from
noise because of the large overlap between the two distributions. However, as the con-
trast increases a far greater proportion of the distribution separates from the distri-
bution of the noise (Figure 4.7a). The increase that we observed in the rate constraint
with increasing contrast (Figure 4.6d) can reflect this increasing discriminability in
the signal (Figure 4.7b).
4.4 Discussion
We presented a model that determines the dynamic range placement for populations
of neurons that maximizes the information about the input distribution. Given noise
constraints for the individual neurons and an energy constraint for the pair of neu-
rons we found that the fast O↵ adapting and sensitizing cells position their response
functions to maximize the amount of information they encode about the input. These
two populations maintain that optimal placement across a broad range of contrast
distributions. Furthermore, we found two distinct regimes for optimal population en-
coding. A low noise regime distributes the encoding of the input between low and
high threshold response functions, just as seen with the coordinated encoding within
4.4. DISCUSSION 93
the fast O↵ populations. While a high noise regime forces the response functions to
overlap and redundantly encode the input. The On population within salamanders
appears to be within this higher noise regime, explaining why it is a more homogenous
set.
4.4.1 Limitations of the model
One of the powers of the model is its simplicity. Modeling neurons as noisy binary
encoders makes the calculation of mutual information quite straightforward. However,
because the model response functions range between zero and one there is no place
within the model for the di↵erent absolute firing rates seen within the data (see
Figure 4.1b). This complicates the interpretation of the energy constraint within the
model as a simple mapping onto the spiking of individual neurons since adapting and
sensitizing cells have di↵erent maxima for their response functions. Rather it requires
a more nuanced accounting of the energy that each population expends.
The simplicity of the model also allowed us to numerically solve for the optimal
response functions, using error minimization. However, an analytic solution to the
general problem could prove useful for verification of the error minimization proce-
dures. Additionally, having an analytic solution to the problem could also make a
more comprehensive model feasible, such as fitting a single optimal model across all
contrasts.
4.4.2 Redundant versus distributed encoding
Although redundant encoding can be thought to be suboptimal at the populations
level (Puchalla et al., 2005), here we have shown that it too can be an optimal
solution for information maximization at the population level. Our finding that the
level of noise creates distinct optimal encoding regimes is reminiscent of work on
optimal transcriptional regulation (Tkacik et al., 2009). The work on transcriptional
regulation also took advantage of a simple model to show the optimal behavior of a
biological system (Tkacik et al., 2008).
94 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
To understand why the On populations falls within the regime of redundant en-
coding it is helpful to consider what we generally know about the On population. The
fact that the On and O↵ populations di↵er is not new. The On population has been
shown to more linearly encode its input (Chichilnisky and Kalmar, 2002; Zaghloul
et al., 2003), On and O↵ populations show di↵ering levels of adaptation (Chander
and Chichilnisky, 2001; Ozuysal and Baccus, 2012), and there are di↵erent numbers
of On and O↵ cells within the retina (Ratli↵ et al., 2010). By more linearly encoding
its input the On population conveys a more complete representation of the visual
world. In so doing it has a dynamic range that encodes input values through the
mean, which are also the noisiest values of the input. Therefore it might be the case
that the On population has traded higher noise levels to maintain that more linear
encoding, but in so doing those noise levels force the population into the regime of
redundant encoding that we find here.
4.4.3 Information maximization after signal detection
To have the model produce curves relatable to response functions seen in the data it
was necessary to limit the overall rate of the model (Figure 4.2c). In so doing we, as
other before us (Pitkow and Meister, 2012), placed a constraint on the overall energy
used by the neurons. However, when we considered that energy constraint across
contrasts we found that it increased with contrast (Figure 4.6d). This result allowed
us to reconsider the function of the energy constraint. As opposed to it simply being
an accounting for each and every spike, rather it can be viewed as a weighting of
spikes given their likelihood to arise from the signal rather than noise (Figure 4.7).
At lower contrasts since there is far greater overlap between the noise distribution
and the signal distribution, it would be a waste of the limited energy resources to
devote a similar amount of energy as in high contrast, when the signal distribution is
far more easily separated from the noise.
This perspective raises the possibility that there are two things at play in the
determination of dynamic range placement in the retina. The first is signal detection,
and the second is information maximization. There is an initial threshold placed
4.4. DISCUSSION 95
to separate out signal from noise (Kastner and Baccus, 2011). This threshold also
functions to limit overall metabolic cost of the response, and it does so somewhat
rationally by distributing the resources to the more informative inputs. Then there is
an optimal placement of the dynamic range to encode that more informative signal.
4.4.4 Optimal population encoding
The fast O↵ population within the salamander retina proved to be a useful population
to test our theory of optimal population encoding since within this grouping there
are two populations that encode very similar features of the visual world (Kastner
and Baccus, 2011). That allowed us to focus exclusively on dynamic range placement
for the fast O↵ adapting and sensitizing populations. To broaden the theory to less
similar initial filtering, for instance to incorporate the slower O↵ populations or the
On populations, the model will need to be extended to incorporate multidimensional
nonlinearities. This comes about because each visual filter produces a dimension upon
which the stimulus is projected to convert the input into the output. There is nothing
preventing our simple model from being extended to multiple dimensions, but such a
multidimensional approach raises the question of how to choose the optimal filtering
dimensions. Such considerations will be necessary to fully understand the way in
which the retina encodes its input.
Of course the retina is not the only neural circuit to encode with populations of
neurons. The auditory cortex also seems to have neurons split into distinct forms
of plasticity (Watkins and Barbour, 2008). The somatosensory system also might
make use of di↵erent forms of plasticity (Ganmor et al., 2010), perhaps indicating the
possibility of similar distributed encoding. It will be interesting to see if these diverse
neural circuits can also be understood from the perspective of optimal information
transmission.
96 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
4.5 Materials and Methods
4.5.1 Experimental preparation
We recorded from retinal ganglion cells of larval tiger salamanders using an array of
60 electrodes (Multichannel Systems) as previously described (Kastner and Baccus,
2011). A video monitor projected the visual stimuli at 30 Hz controlled by Matlab
(Mathworks), using Psychophysics Toolbox (Brainard, 1997; Pelli, 1997). Stimuli were
uniform field with a constant mean intensity, M , of 10 mW/m2 and were drawn from
a Gaussian distribution. Contrast is defined as � = W/M , where W is the standard
deviation of the intensity distribution.
4.5.2 Linear-Nonlinear models
LN models consisted of the light intensity passed through a linear temporal filter,
which describes the average response to a brief flash of light, followed by a static
nonlinearity, which describes the threshold and sensitivity of the cell. To compute the
model, the stimulus, s(t), was convolved with a linear temporal filter, F (t), which
was computed as the time reverse of the spike triggered average stimulus, such that
g(t) =
ZF (t� ⌧)s(⌧)d⌧ . (4.5)
A static nonlinearity, N(g), was computed by comparing all values of the firing rate,
r(t) with g(t) and then computing the average value of r(t) over bins of g(t). The
filter, F (t), was normalized in amplitude such that it did not amplify the stimulus,
i.e. the variance of s and g were equal (Baccus and Meister, 2002). Thus, the linear
filter contained only relative temporal sensitivity, and the nonlinearity represented
the overall sensitivity of the transformation.
4.5. MATERIALS AND METHODS 97
4.5.3 Histogram normalization for multiple neurons
We would like to maximize the entropy of the neural response
~r = (r1
(x), r2
(x)), (4.6)
where r1
(x) is the firing rate of neuron 1 for stimulus x, and r2
(x) is the firing rate
of neuron 2 for stimulus x. The stimulus x varies between �1 and 1. Note that r1
and r2
cannot take arbitrary values because in the space of two neural responses they
form a one dimensional line obtained by varying stimulus value x.
To maximize the response entropy,
H = �Z
p(l)lnp(l)dl, (4.7)
where the variable l marks the position along the curve of ~r, we want
p(l)dl = p(x)dx, (4.8)
where the probability to be within a segment dl along the curve equals the probability
to be within interval dx on the stimulus axis. Since
dl =q
r02
1
(x) + r02
2
(x)dx, (4.9)
the entropy equations becomes,
H = �Z
p(x)lnp(x)dx+
Zp(x)ln
qr02
1
(x) + r02
2
(x)dx. (4.10)
Since the first part of Equation 4.10 does not depend upon either r1
(x) or r2
(x) we
can focus on the second part of the equation.
The variation in H when we change r1
(x) is:
�H =1
2
Zp(x)
2r01
(x)�r01
(x)
r02
1
(x) + r02
2
(x)dx, (4.11)
98 CHAPTER 4. OPTIMAL DYNAMIC RANGE PLACEMENT
which is equivalent to
�H = �Z
d
dx
✓p(x)r0
1
(x)
r02
1
(x) + r02
2
(x)
◆�r0
1
(x)dx. (4.12)
Since the variation is zero if the response is maximal, Equation 4.12 is set to zero.
Correspondingly there is an equivalent equation for r2
(x), which provide the equations
p(x)r01
(x) = c1
(r02
1
(x) + r02
2
(x)) (4.13)
p(x)r02
(x) = c2
(r02
1
(x) + r02
2
(x)), (4.14)
where c1
and c2
are constants. When we divide equation 4.13 and 4.14 we get
r01
(x) = c3
r02
(x), (4.15)
where c3
is another constant, which shows that the derivatives of the two response
functions are proportional to each other, and do not split apart as seen in the data
(Figure 4.1).
4.5.4 Threshold model for contrast processing
We have previously shown that a simple model that thresholds the input prior to
scaling by the contrast was capable of capturing the response of the adapting and
sensitizing fast O↵ cells (Kastner and Baccus, 2011). That model took the form
N�
= N
✓x� ↵
�
◆, (4.16)
where N�
is the response function at a given contrast, �, N is the generic response
function, and ↵ is the threshold. We can rewrite this, given that the generic response
function is the sigmoid function (Equation 4.1).
N�
=1
1 + exp
✓x
1
/
2
�(x�↵
�
)ˆ
b
◆ . (4.17)
4.5. MATERIALS AND METHODS 99
This then leads to how x1
/
2
and b should change as a function of contrast,
x1
/
2
(�) = �x1
/
2
+ ↵ (4.18)
b(�) = b�, (4.19)
which are the equations we used to fit the data in Figure 4.6c. We then used these
equations for the pair of cells to fit the hri data in Figure 4.6d.
4.5.5 Detectability
When there is a signal in the presence of noise the probability that any input, ⌫, is
from the signal distribution is
w(⌫) =p(s|⌫)
p(s|⌫) + p(⌘|⌫) , (4.20)
where p(s|⌫) is the probability of the signal given an input, ⌫, and p(⌘|⌫) is the
probability of the noise given an input, ⌫ (Field and Rieke, 2002). w(⌫) was calculated
for each contrast distribution with a fixed level of noise comparable to a 2% contrast
distribution. Since both the signal and noise distributions had the same mean, w(⌫)
had a minimum at the mean, which was arbitrarily placed at zero. Since we focused
on the O↵ population we only considered the function at positive input values. The
overall detectability was measured as the integral of w(⌫) for each contrast.
Chapter 5
Inhibitory dynamics underlying
contrast plasticity in the retina
This chapter is in preparation as “Inhibitory dynamics underlying contrast plasticity
in the retina,” with author list: Kastner DB, Jadzinsky PD, Panagiotakos G, Baccus
SA.
5.1 Summary
Understanding how neural circuits compute comprises a fundamental goal of neuro-
science. The more di↵erent circuit elements and cellular dynamics contribute to the
computation the more complicated it becomes to understand how the biology per-
forms the computation. Contrast plasticity in the retina provides a rich example, since
much work has been done to expose the cellular components of the computation. Yet,
the role of inhibitory dynamics in contrast plasticity remains unknown. Here we show
that sustained O↵ amacrine cells are su�cient for sensitization, a newly discovered
form of plasticity in the retina. Furthermore, we measure the changes in dynamics
at the level of the input and output of these amacrine cells to show how they create
sensitization.
100
5.2. INTRODUCTION 101
5.2 Introduction
Throughout the brain inhibition shapes neural computations (Baccus, 2007; Isaacson
and Scanziani, 2011). In the retina, inhibition suppresses and modulates excitatory
responses, as with the surround of the classical receptive field (Flores-Herr et al.,
2001; McMahon et al., 2004), the selectivity of ganglion cells to di↵erential motion
(Olveczky et al., 2003; Baccus et al., 2008), and direction selectivity (Euler et al.,
2002). However, neurons dynamically change their response properties dependent
upon the recent stimulus history. Mechanisms within excitatory neurons contribute
to these changes in the neural code. Photoreceptors and bipolar cells change their
sensitivity so as to produce luminance adaptation (Dunn et al., 2007). Bipolar and
ganglion cell dynamics provide the necessary components for contrast adaptation
within the retina (Rieke, 2001; Kim and Rieke, 2003; Manookin and Demb, 2006;
Beaudoin et al., 2007). Although inhibition is necessary for some of these dynamic
processes (Hosoya et al., 2005; Ge↵en et al., 2007; Olveczky et al., 2007)(Chapter 3),
the contribution of inhibitory dynamics remains largely unclear.
Pharmacologic experiments provide useful information for determining the neces-
sity of inhibition for retinal processing (Olveczky et al., 2003; Hosoya et al., 2005;
Bolinger and Gollisch, 2012); however they require caution when trying to decipher
mechanistic roles for that inhibition. Drug application a↵ects the entire retina. All
neurotransmission of the type a↵ected by the drug will be altered. If that neurotrans-
mitter plays diverse roles a pharmacologic blockage will alter all of them, and the
results will reflect a combination of all of those e↵ects. Furthermore, such a global
change manipulates the overall state of the retina, making it practically impossible
to tease apart whether inhibition mediates the computation of interest, or rather
just modulates the computation by generating an appropriate environment for the
true mediators of the computation of interest. Therefore, to mechanistically deter-
mine the role of inhibition in retinal processing we need more refined techniques for
manipulation of inhibitory circuitry.
Here we used paired recordings between amacrine and ganglion cells to selectively
manipulate inhibition in order to measure inhibitory dynamics and understand their
102 CHAPTER 5. MECHANISM OF SENSITIZATION
role in retinal contrast processing. Ganglion cells undergo two types of plasticity with
changes in contrast. They can show decreased or increased sensitivity after a change
from a high to a low contrast distribution, performing adaptation or sensitization
respectively. Even though amacrine cells also adapt to contrast (Baccus and Meister,
2002), and undergo synaptic depression (Sagdullaev et al., 2011), their role in retinal
contrast plasticity has remained opaque (Rieke, 2001; Brown and Masland, 2001;
Manookin and Demb, 2006; Beaudoin et al., 2007) until very recently (Chapter 3).
Here we show that amacrine cells are su�cient to cause sensitization, and we measure
the various stages of plasticity that underlie sensitization.
5.3 Results
Recently the retina was shown to process contrast with two opposing forms of plas-
ticity, adaptation and sensitization (Kastner and Baccus, 2011). Although synaptic
plasticity in excitatory neurons has been proposed to underlie adaptation (Manookin
and Demb, 2006; Jarsky et al., 2011), the circuit mechanisms underlying sensitization
remain far less understood. In response to a more local high contrast some cells vary
their form of plasticity from adaptation to sensitization based upon the location of
the cell relative to the high contrast (Figure 3.1). This suggests that intrinsic ganglion
cell properties are not solely responsible for sensitization. A membrane depolariza-
tion underlies sensitization even in cells that exclusively sensitize (Figure 3.9). Such
membrane dynamics typically indicate presynaptic computations (Carandini and Fer-
ster, 1997; Manookin and Demb, 2006). Also a computational model that combines
adapting inhibition and excitation captures the phenomenon of sensitization (Kastner
and Baccus, 2011), and suggests that di↵erences in the type of adaptive response can
be explained by di↵erences in the amount of adapting inhibition (Figure 3.3). These
results call for a further study into the role of amacrine cell dynamics in contrast
processing.
Although pharmacologic experiments place a necessary role on GABAergic inhi-
bition for sensitization (Figure 3.11) they cannot identify the way in which inhibitory
transmission shapes contrast plasticity. Blocking inhibition increases the overall firing
5.3. RESULTS 103
rates of the ganglion cells (Figure 3.11), making a comparison of the details of the
dynamics of the response of a ganglion cell between the control condition and in the
presence of drug inappropriate, since e↵ectively the strength of the overall stimulus
di↵ers between the two conditions.
Therefore we simultaneously recorded intracellularly from an amacrine cell and
extracellularly from multiple nearby ganglion cells, all while presenting a visual stim-
ulus to the retina (Figure 5.1a,b). We focused on sustained O↵ amacrine cells to see
if they could contribute the inhibition underlying the picrotoxin results (Figure 5.1c).
Sustained O↵ amacrine cells were good candidates for several reasons. First, we chose
to focus on amacrine cells in general as the source of the inhibition instead of horizon-
tal cells since the outer retina does not adapt to contrast (Baccus and Meister, 2002).
Second, sensitization persists with blockage of the On pathway, making On-O↵ cells
much less likely a candidate. Third, a model that reproduces sensitization contains a
nonlinearity between the inhibitory and excitatory pathways (Figure 2.16) that can
be understood as the inhibition working largely through disinhibition. Recently, sus-
tained O↵ amacrine cells were shown to function largely through disinhibition (Manu
and Baccus, 2011). Finally, sensitization even occurs with smaller and more local
high contrast transitions (Figure 2.4 and 3.1), making it less likely that the inhibitory
cell will have a large receptive or projective field. Sustained O↵ amacrine cells have
smaller receptive and projective fields (de Vries et al., 2011), especially when com-
pared to transient cells like the polyaxonal amacrine cell (Cook et al., 1998; Baccus
et al., 2008).
Since horizontal cells also have sustained O↵ flash responses we measured the
spatio-temporal receptive field of the interneurons (Figure 5.1d). By ensuring that an
interneuron had a surround we were able to separate amacrine from horizontal cells.
For some of the cells we confirmed that they were in fact amacrine cells by filling
and imaging them (Figure 5.1e). By verifying that an interneuron sent projections
into the inner plexiform layer we were able to distinguish an amacrine cell from a
horizontal cell. We correctly identified all of the amacrine cells that we filled (n = 4).
Some amacrine cells adapt to contrast while others do not (Baccus and Meister,
2002). Therefore we determined if sustained O↵ amacrine cells as a class adapted to
104 CHAPTER 5. MECHANISM OF SENSITIZATION
300 µm
Electrode array!
H!
G!
P! P!
A!B!B!
G!
Visual stimulus!
Mem
brane potential!
Current injection!
a b
c
d e
Ligh
t OnOff
10 mV
1 s
Figure 5.1: Experimental setup. (a) The retina is placed onto a multielectrode array, whichallows for extracellular recording from multiple ganglion cells. An interneuron is simultane-ously recorded using a sharp electrode with a resistance of 150�300 M⌦. A visual stimulusis focused onto the photoreceptors. (b) Spatial receptive fields of an amacrine (red) and gan-glion cells (black) recorded simultaneously. (c) Average response of a sustained O↵ amacrinecell to a 0.5 Hz flashing stimulus. (d) Spatio-temporal receptive field of an amacrine cell.Red indicates sensitivity below the mean, blue indicates sensitivity above the mean. (e)Image of the same amacrine cell from (d). INL is the inner nuclear layer, IPL is the innerplexiform layer, and GCL is the ganglion cell layer.
5.3. RESULTS 105
1 mV
0.4 sIn
tens
ity4 mV
5 s
LlateLearlya b c
-0.20
0.2
Δ m
embr
ane
pote
ntia
l (m
V)
100Flash amplitude
HorizontalAmacrine
Figure 5.2: Adaptation in sustained O↵ amacrine cells. (a) Visual stimulus (top) and theresponse of an example amacrine cell (bottom). High contrast was 100% Michelson contrast,and low contrast was composed of 9 di↵erent flashes randomly delivered. The high contrastwas only presented over the receptive field of the amacrine cell. The rest of space was lowcontrast. (b) Average membrane potential of the amacrine cell shown in (a) to the ninedi↵erent flashes delivered during low contrast. Times for L
early
and L
late
are shown abovethe stimulus in (a). Average di↵erence in membrane potential between L
late
and L
early
foramacrine (n = 21) and horizontal (n = 4) cells. Negative values indicate that the cell washyperpolarized during L
early
compared to L
late
.
contrast. First we created an online map of the receptive field of the amacrine cell using
a white-noise checkerboard stimulus (Figure 5.1b). This allowed us to place a local
high contrast spot over the amacrine cell and then measure its response to a stimulus
that changed from a high to a low contrast (Figure 5.2a). Sustained O↵ amacrine
cells adapted to contrast (Figure 5.2a). Their membrane potential hyperpolarized
immediately after the transition from a high to a low contrast, Learly
, compared to
later during low contrast, Llate
(Figure 5.2b,c). Consistent with a previous report
(Baccus and Meister, 2002), horizontal cells showed practically no adaptation with a,
relatively, minimal hyperpolarization (Figure 5.2c).
The adaptation seen in these amacrine cells could contribute to sensitization in
ganglion cells because this adaptation diminishes the amount of inhibition that a
ganglion cell will receive after a transition from high contrast leading to an e↵ective
increase in excitation. However, this adaptation only provides one half of the picture
of the amacrine cell dynamics. To fully implicate these sustained O↵ amacrine cells
in sensitization we need to not only characterize their input dynamics, comparable to
their receptive field, we also must characterize their output dynamics, comparable to
their projective field (Lehky and Sejnowski, 1988; de Vries et al., 2011).
106 CHAPTER 5. MECHANISM OF SENSITIZATION
To measure the output dynamics of these amacrine cells we presented a visual
stimulus to the retina that allowed for the rapid measurement of the ganglion cell
response function during Learly
and Llate
without current injected into the amacrine
cell, and while depolarizing or hyperpolarizing the amacrine cell (Figure 5.3a,b). The
slower temporal frequency of the response allowed for a rapid measurement of the
ganglion response function without having to also measure the ganglion cell filter
(Brenner et al., 2000). To be able to extract the di↵erence between amacrine cell
transmission during Learly
and Llate
we took advantage of the fact that ganglion cell
response functions are well fit by sigmoid equations (Kastner and Baccus, 2011). We
then assumed that the way in which current injected into the amacrine cell a↵ected
the ganglion cell only changed in magnitude between Learly
and Llate
. That means
that if the amacrine cell changes the threshold of a ganglion cell during Learly
, we
assume that it will also only change the threshold of the ganglion cell during Llate
,
but it can change the threshold more or less than during Learly
.
Using this assumption we were able to model the e↵ects of the hyperpolarizing
and depolarizing current during Learly
and Llate
(Figure 5.3c) (see methods). From the
model, we computed the change in the strength of transmission between Learly
and
Llate
. We found that transmission is greater during Llate
than Learly
both when we
hyperpolarized and depolarized the amacrine cell (Figure 5.3d). Therefore, not only
did the input to these amacrine cells adapt, but their output was also depressed.
In modeling the e↵ect of the current we were able to see the depressed transmission
in many amacrine-ganglion cell pairs, but it did require an assumption. Therefore we
also sought to verify the depressed transmission in a model independent way. In
most cases it is di�cult, if not impossible to just look at the response functions and
determine changes in transmission between Learly
and Llate
because of the changes
in the response functions in the control conditions (Figure 5.3b); however there are
situations when we can unambiguously say that the transmission is depressed by
high contrast. That occurs when during the control condition there is sensitization,
but during the current condition there is adaptation or no change in the response
between Learly
and Llate
. We were able to observe such e↵ects on multiple occasions
(Figure 5.4), confirming depressed transmission in a model independent way between
5.3. RESULTS 107
200
100
0
Rat
e (H
z)
100Flash amplitude
HyperpolarizationControl
LearlyLlate
200
100
0
Rat
e (H
z)
100Flash amplitude (%)
100
Learly Llate
Visu
alIn
tens
ity
Cur
rent
Inte
nsity
... ...
200 ms
...
... ...
200
100
0
Rat
e (H
z)
100Flash amplitude (%)
Depolarization!Control!Hyperpolarization!Model!
a b
c d 1
0.50Δ transmission
hyperpolarization
10.50-0.5Δ transmissiondepolarization
Flash amplitude!
Flash amplitude!
Figure 5.3: Amacrine transmission is depressed at the transition to low contrast. (a) Visualstimulus changed back and forth between high and low contrast. High contrast was 100%Michelson contrast, and low contrast was composed of 9 di↵erent flashes randomly delivered.Depolarizing (+500 pA) and hyperpolarizing (�500 pA) current was precisely delivered for200 ms starting when the visual stimulus flash went from On to o↵. Current was onlydelivered during the flashes that occurred 0.8 � 3.2 s (L
early
) and 16 � 20 s (Llate
) afterthe high contrast. The high contrast was only presented over the receptive field of theamacrine cell. The rest of space was low contrast. (b) Average response of an exampleganglion cell to the 9 di↵erent visual flashes during the 6 conditions of the experiment.(c) Same curves from (b) just with L
early
on the left and L
late
on the right for clarity.The grey curves show the output of the model, which capture the e↵ect of the current.(d) Change in transmission between L
early
and L
late
for when the current is depolarizingand hyperpolarizing (see methods). Positive values indicate that inhibitory transmission isgreater during L
late
.
108 CHAPTER 5. MECHANISM OF SENSITIZATION
200
100
0
Rat
e (H
z)
100Flash amplitude
HyperpolarizationControl
LearlyLlate
Figure 5.4: A model independent view of depressed amacrine transmission. Multiple exam-ples of response functions of ganglion cells from the experimental paradigm shown in Figure5.3.
sustained O↵ amacrine cells and ganglion cells following a transition from high to
low contrast. This increase in transmission from Learly
to Llate
is consistent with a
recovery from synaptic depression, in accordance with recent studies of amacrine cell
transmission (Sagdullaev et al., 2011).
Now that we have established that these sustained O↵ amacrine cells have both
an adapted input and a depressed output we wanted to verify that these cells were
su�cient for sensitization. To do that we designed an experiment such that only the
amacrine cell e↵ectively experiences high contrast, and we measured the response of
ganglion cells following the exposure of that single circuit element to high contrast.
The visual stimulus was comprised of exclusively low contrasts. Instead of presenting
a high contrast visual stimulus, we periodically injected a current designed to mimic
the voltage change in the amacrine cell recorded during high contrast (Figure 5.5a),
while simultaneously measuring the response of multiple ganglion cells.
5.4. DISCUSSION 109
We found that the high contrast current injected into the amacrine cell was suf-
ficient to cause sensitization in ganglion cells (Figure 5.5b-d), and this e↵ect was
specific to ganglion cells that had receptive fields within several hundred µm to the
amacrine cell. Whereas amacrine cells were su�cient for sensitization, horizontal cells
were not (Figure 5.5d). The sensitization measured here was likely due to the high
contrast current marshaling synaptic depression in the amacrine cells (Figure 5.3 and
5.4) since the high contrast current did not change the membrane potential like a
high contrast visual stimulus (Figure 5.5e). Therefore, our su�ciency measurement
here provides a minimal amount of sensitization that an individual amacrine cell can
create. We would expect individual amacrine cells to contribute even more to sensi-
tization than what we can measure here, since in response to a high contrast their
output would be further decreased by their input adaptation (Figure 5.2)
5.4 Discussion
We have found that sustained O↵ amacrine cells show adaptation at the level of their
input (Figure 5.2) and depression at the level of their output (Figure 5.3 and 5.4).
This combination of dynamics in the inhibitory signal is consistent with previous
models of sensitization (Figure 2.16 and 3.3), whereby after a high contrast stimulus
inhibition is decreased leading to an e↵ective increase in excitation, i.e. sensitization.
Furthermore, we have shown that these sustained O↵ amacrine cells are su�cient to
generate sensitization in ganglion cells.
To fully understand and explain the results in this paper we generated a sim-
plified model (Figure 5.6). We modeled the amacrine response function as a linear
transformation between its input and output. During Learly
the response function is
depolarized and less sensitive than during Llate
. This amacrine output then inhibits
the bipolar cells response more (Llate
) or less (Learly
), leading to more excitation
during Learly
. It is worth noting that it is necessary to incorporate the fact that these
sustained O↵ amacrine cells e↵ect ganglion cells largely through disinhibition (Manu
and Baccus, 2011), meaning that there is a nonlinearity between the amacrine cell and
ganglion cell output. That nonlinearity explains the larger amount of depression seen
110 CHAPTER 5. MECHANISM OF SENSITIZATION
0.020
-0.02Ad
aptiv
e in
dex
0.30.20.1Distance (mm)
AmacrineHorizontal
4
2
0
Rat
e (H
z)
100Flash amplitude
a
c
Cur
rent
inte
nsity
Visu
alin
tens
ity
4 s
Learly Llate
-0.20
0.2
Δ m
embr
ane
pote
ntia
l (m
V)
100Flash amplitude
CurrentVisual
b
d
e
Figure 5.5: High contrast current in a single amacrine cell is su�cient to cause sensitization.(a) The visual stimulus was a continuous low contrast composed of 9 di↵erent flashes ran-domly presented at 2.5 Hz (top), identical to the low contrast from Figure 5.3a. Current wasinjected into a single amacrine cell to mimic its high contrast response, which was previ-ously recorded (Figure 5.2a). The current had a maximum amplitude of 1 nA. A membranetime constant of 30 ms was assumed to convert the current into membrane voltage, but thistechnique is not particularly sensitive to the exact value of the membrane time constant(Manu and Baccus, 2011). (b) Average response of a ganglion cell to the stimulus protocolin (a). Black indicates the time of current injection, and the color indicates the time withoutcurrent injection. (c) Average response of the ganglion cell in (b) to the 9 di↵erent flashespresented in the visual stimulus during L
early
and L
late
, as indicated in (a). (d) Averageadaptive index of ganglion cells plotted relative to the distance between the ganglion cellsand amacrine cells or horizontal cells. Data comes from a total of 76 amacrine and ganglioncell pairs and 10 horizontal and ganglion cell pairs. (e) Average di↵erence in membranepotential between L
late
and L
early
for the amacrine cells (n = 21) for when they were eitherstimulated with a visual high contrast, or injected with current to mimic their high contrastresponse. Negative values indicate that the cell was hyperpolarized during L
early
comparedto L
late
. Error bars indicate s.e.m.
5.4. DISCUSSION 111
Amac
rine
outp
ut! Depolarization!
Hyperpolarization!
Learly!
Llate!
Amacrine input!
− Amacrine output!
Gan
glio
n ou
tput!
Figure 5.6: Simplified model for sensitization. Input adaptation is illustrated as an o↵-set in the amacrine input-output function between L
early
and L
late
. Synaptic depressionis illustrated as a decreased slope of the amacrine cell input-output function during L
early
compared to L
late
. The same current is delivered, but the amacrine output is less duringL
early
. Hyperpolarization shows a greater di↵erence in transmission because of the subse-quent threshold between amacrine output and ganglion cell response. This threshold can beat the bipolar cell synapse or in the ganglion cell itself.
with a hyperpolarizing than with a depolarizing current in the amacrine cell (Figure
5.3d) since hyperpolarization sends that ganglion cell into the expansive part of the
nonlinearity, whereas depolarization sends the ganglion cell into the compressive part
of the nonlinearity.
The disinhibitory nature of sustained O↵ amacrine cells also explains the increase
in response seen in the ganglion cell during the current injection during the experi-
ments to test su�ciency (Figure 5.5b). Consistent with previous results (Manu and
Baccus, 2011), a large variation current will increase the output of connected gan-
glion cells because of the intervening nonlinearity. Since hyperpolarizing current has
more of an e↵ect than depolarizing current, the nonlinearity e↵ectively causes such
a stimulus to remove inhibition. Even though the current injected into the amacrine
cell causes an increase in response in the ganglion cell during the current injection,
the increase in response in the ganglion cell alone is not su�cient to cause sensiti-
zation because some ganglion cells change their type of plasticity from adapting to
sensitizing based upon the location of a high contrast stimulus (Figure 3.1).
Throughout this work we have focused on the sustained O↵ amacrine cells as if
they were a single population of interneuron. Although the population as a whole
112 CHAPTER 5. MECHANISM OF SENSITIZATION
50 µm!
Figure 5.7: A diverse set of amacrine cells. Two fills of sustained oO↵ amacrine cells.
clearly shows the results described here it is unlikely that this is a single type of
amacrine cell. Rather it is more likely that we are dealing with multiple di↵erent types
of cells. Comparing the projection pattern of filled sustained O↵ amacrine clearly
makes the point that within the sustained O↵ population there are multiple di↵erent
amacrine cell types (Figure 5.7). It will have to be left to future work to determine the
nature of the di↵erent populations within the class of sustained O↵ amacrine cells.
5.5 Materials and Methods
5.5.1 Experimental preparation
Retinal ganglion cells of larval tiger salamanders of either sex were recorded using
an array of 60 electrodes (Multichannel Systems) as described(Kastner and Baccus,
2011). A video monitor projected the visual stimuli at 30 Hz controlled by Matlab
(Mathworks), using Psychophysics Toolbox (Brainard, 1997; Pelli, 1997). Stimuli had
a constant mean intensity of 10 mW/m2. Contrast was defined as either the the
standard deviation divided by the mean of the intensity values, or as the Michelson
contrast, which is C = I
max
�I
min
I
max
+I
min
, with Imax
and Imin
being the maximum and minimum
intensity values.
5.5. MATERIALS AND METHODS 113
5.5.2 Receptive fields and nonlinearities
Spatio-temporal receptive fields were measured in two dimensions by the standard
method of reverse correlation (Chichilnisky, 2001) of the spiking response with a
visual stimulus consisting of binary squares, such that
F (x, y, ⌧) =
ZT
0
s(x, y, t� ⌧)r(t)dt, (5.1)
where F (x, y, ⌧) is the linear response filter at position (x, y) and delay ⌧ , s(x, y, t) is
the stimulus intensity at position (x, y) and time t, normalized to zero mean, r(t) is the
firing rate of a cell, and T is the duration of the recording. Then the predicted linear
response, g(t), was computed by convolving the stimulus with the linear response
filter such that
g(t) =
ZT
0
s(x, y, t� ⌧)F (x, y, t� ⌧)dxdyd⌧. (5.2)
A static nonlinearity, N(g), was then computed as the average value of the response
r(t) over bins of g(t). For all experiments a cell’s receptive field was independently
measured using reverse correlation to a binary checkerboard stimulus, and the spatio-
temporal receptive field was approximated as the product of a spatial profile and a
temporal filter (Olveczky et al., 2003).
5.5.3 Transmission
To measure changes in transmission between Learly
and Llate
the ganglion cell response
function without current was fit to a sigmoid function to interpolate between the data
points. The sigmoid function used was,
y = x0
+m
1 + exp⇣
x
1
/
2
�x
r
⌘ , (5.3)
where x0
is the basal firing rate, x0
+m is the maximal firing rate, x1
/
2
is the x value
at the midpoint of the range of firing rates, and r�1 controls the maximal slope. Then,
114 CHAPTER 5. MECHANISM OF SENSITIZATION
the ganglion response during current injection was fit such that
NC
(x) = aNI
(↵x+ �) + b, (5.4)
where NC
is the ganglion response function in the control condition, NI
is the ganglion
response function during current injection, and ↵, �, a, and b are the shifting and
scaling parameters for the x and y-axes. The shifting and scaling parameters were fit
together for all current condition between Learly
and Llate
except that the parameters
could change together such that
↵E
= m↵L
+ 1
�E
= m�L
aE
= maL
+ 1
bE
= mbL
, (5.5)
where m is the di↵erence in transmission between Learly
and Llate
, and the subscript
indicates the time during low contrast, with E for Learly
and L for Llate
. The scaling
parameters (↵ and a) need to have the addition o↵set of one because if there is no
change in those parameters due to contrast, that corresponds to a multiplication by
1.
Chapter 6
Conclusions
6.1 Generalization of work
Sensitization and coordinated dynamic encoding highlight that adaptation, although
beneficial, comes with its own costs. Neurons have limited dynamic ranges. To be
able to encode the range of inputs that they encounter they have to manipulate that
dynamic range to match their input. However, in so doing they, inevitably, will not
be able to match the range of a di↵erent set of inputs. With coordinated dynamic
encoding the retina splits the problem between two distinct classes of cells that un-
dergo di↵erent forms of plasticity: the previously known contrast adaptation, and the
newly discovered contrast sensitization.
The notion that adaptive systems have di�culty during transitions of their inputs
is not a new one. We have all experienced the failure of light adaptation when we
leave a dark room for one with more light. Our visual system initially saturates in
the light, until it can adapt to the new environment. However, as we have seen with
coordinated dynamic encoding, there is no reason that every transition of inputs must
lead to such absolute failures in encoding.
In a now classic paper, Fairhall and colleagues highlighted the ambiguity inherent
to adaptation (Fairhall et al., 2001). They focused on the H1 neuron in the blowfly,
which encodes the velocity of visual inputs. The H1 neuron adapts to the variance
of the distribution of the velocities, a quantity comparable to contrast. In so doing
115
116 CHAPTER 6. CONCLUSIONS
it was expected that the H1 neuron should lose the ability to encode the variance of
the velocity since it was scaling that quantity out of its response function (Brenner
et al., 2000). However, they showed that the variance could be decoded from the spike
timing, instead of the firing rate (Lundstrom and Fairhall, 2006), an instance where
di↵erent features within a single neuron combine to encode more than a single feature
could.
The seeds of coordinated dynamic encoding exist in other areas of the nervous
system. In the auditory cortex of marmosets at least two di↵erent types of adaptation
exist (Watkins and Barbour, 2008). One group of cells always adapted to maintain
sensitivity around the most common intensities of the stimulus distribution. Another
group of neurons did the same except if the most common intensities were too high,
at which point these neurons raised their sensitivity to maintain the ability to encode
lower intensities. Had the second form of adaptation not existed those lower intensities
would have been poorly encoded and potentially even lost (Watkins and Barbour,
2008).
In the somatosensory cortex of rats, neurons adapt less than would be expected
based on simple models of synaptic depression (Ganmor et al., 2010). This study
looked at the potential outcome of using coordinated dynamic encoding, whereby
using distinct types of adaptation in the sensory periphery can enable a more sta-
ble response to a dynamic stimulus in the cortex. In fact, the same group recently
reported, in abstract form, that di↵erent subcortical structures use di↵erent types
of adaptation (B. Mohar, I. Lampl, Soc. Neurosci Abstr, 704.17, 2011), potentially
combining to form the more stable representation in the cortical neurons.
Within early sensory systems it is easier to relate the input to the output, making it
an ideal place to study dynamic encoding. However, adaptation and plasticity are not
exclusive to early sensory systems. Even higher processing centers have been shown to
use distinct forms of plasticity in encoding their inputs (Miller and Desimone, 1994).
Using a task that requires working memory for monkeys to select a matching item,
neurons in the inferotemporal cortex show two responses to the second presentation
of the item. Some neurons have a diminished response, while others show an enhanced
6.1. GENERALIZATION OF WORK 117
response. Although the functional relevance of such parallel processing remains un-
clear, even the inferotemporal cortex distributes its encoding between distinct forms
of plasticity.
In addition to the coordination between cell types with di↵erent plasticity, I have
also measured the adaptive field of retinal cell types. Surprisingly, I found that a
single cell type changes its form of plasticity from adaptation to sensitization based
upon the stimulus. These center-surround adaptive fields enable coordinated dynamic
encoding at the level of a single population of cells.
In the somatosensory cortex of the rat the e↵ective spatial adaptive field of neurons
was measured (Katz et al., 2006). Here the spatial component came from stimulating
di↵erent whiskers, which fed into di↵erent barrels. These neurons had a local adaptive
field, in that adapting to the stimulation of one whisker does not e↵ect the response
to stimulation of another whisker.
The adaptive field has also recently been measured in the auditory cortex of ferrets
(Rabinowitz et al., 2011). In audition, frequency comprises the spatial aspect of the
neuron’s responsiveness, and adaptation at a neuron’s best frequency was a↵ected
by stimulation at other frequencies. As was the case in somatosensory cortex, and
the previous measurement of slow changes of the adaptive field in the retina (Brown
and Masland, 2001), these measurements were made to understand the mechanisms
of adaptation. The functional significance of these other adaptive field remain unex-
plored.
Although the phenomenon of sensitization is new in the retina, it resembles plastic-
ity in other brain regions. In the somatosensory thalamus facilitating responses have
been reported, whereby a cell’s response will be sensitized by a previously strong
stimulation (Ganmor et al., 2010). This facilitation was also recently reported, in
abstract form, to exist in the barrel cortex as well (K. Cohen-Kashi, M. Jubran, I.
Lampl, Soc. Neurosci Abstr, 385.10, 2011). Interestingly, this group has suggested
that the facilitation is due to depressed inhibitory inputs, in a very similar fashion to
what is seen in the retina.
Because of the view that sensitization can function as a form of information stor-
age, it is not too surprising that people have tried to connect similar types of responses
118 CHAPTER 6. CONCLUSIONS
to memory storage in the hippocampus (Larimer and Strowbridge, 2010), and other
brain regions (Miller and Desimone, 1994). Additionally, it has been proposed the-
oretically, that information storage can come about by combining facilitating and
depressing excitatory inputs (Mongillo et al., 2008). Such a model is functionally
indistinguishable from the underlying mechanism of sensitization where depressing
inhibition plays the role of facilitating excitation.
Finally, sensitization also contains features similar to exogenous attention. When a
strong stimulus is flashed in a region of space, a subject will become more sensitive in
that region of space (Bisley and Goldberg, 2003; Carrasco et al., 2004; Ingle, 1975). I
have shown that sensitization can be evoked with a very brief strong stimulus. Perhaps
similar forms of plasticity underlie various aspects of attention as well.
6.2 Future directions
A major question that my work raised, but that I have not been able to address is
if and how the adapted and sensitized signals are combined. I have shown that the
retina splits the encoding of its input between two populations of cells. One takes
responsibility for the weak signals, while the other focuses on the strong signals.
It could be very interesting and informative to understand what the visual system
does with these two streams of information. The most appealing and straightforward
outcome would be that these two populations are recombined in the thalamus or
primary visual cortex. However, that is in no way necessary. The important thing is
that the animal has access to the two streams of information. It could be possible
that the animal uses the stronger signals for certain behaviors, while it uses the
weaker signals for di↵erent behaviors. That leaves the possibility open that these two
information streams go to very di↵erent brain regions, and then the information is
only recombined at the level of the behavior of the animal.
To address this question, the salamander would probably not be the ideal organ-
ism. Fortunately, I have shown sensitization to exist in mice as well. All of the new
mouse lines that have genetically labeled ganglion cells (Huberman et al., 2008, 2009;
Kim et al., 2008) could prove quite useful. In essence all that would be required is the
6.2. FUTURE DIRECTIONS 119
identification of the genetically labeled lines as either sensitizing or adapting because
the projection patterns of the labeled cell types is readily accessible, if not already
known.
Once the ganglion cells within a genetically labeled mouse line are identified as
sensitizing then another implication of my work can be tested. We have proposed
multiple di↵erent, yet linked, functions for sensitization. If the labeled ganglion cells
can be knocked out of the retinal circuit, what behavioral deficits will the mice exhibit?
Will they have deficiencies in tracking camouflaged objects? Will they be deficient in
exogenous attention? Such studies would be extremely interesting and informative.
Our understanding of the role of inhibition in neural computations is advancing
rapidly (Baccus, 2007; Isaacson and Scanziani, 2011; Moore et al., 2010). I have
elucidated a role for inhibitory dynamics in retinal contrast plasticity. However, it is
also clear that inhibition helps to set the steady state response of ganglion cells during
contrast plasticity as well. Why is the steady state response so tightly regulated? What
aspects of the visual input are better encoded due to the role of inhibition?
A way forward for these questions can come about by getting a richer understand-
ing of what ganglion cells encode even while they undergo contrast plasticity. I have
begun this process in the measurement of the temporal adaptive field. That measure-
ment has a direct parallel to the spike-triggered average of standard linear-nonlinear
modeling. Because of that correspondence, one could also measure a nonlinearity that
follows the temporal adaptive field. Such a nonlinearity would indicate the contrasts
which could be decoded from a cell’s response. Additionally, it could allow for better
characterization of a cell’s response, to see how that response changes with drugs
that block various forms of inhibition, or during current injection into amacrine cells.
Either manipulation could begin to give clues as to how inhibition shapes the retinal
response.
Finally, the methodology used to show that inhibitory transmission depresses over
the course of sensitization is quite general. It could be used to determine a role for
other dynamic processes in the retina, such as one proposed for inhibition in pattern
adaptation (Hosoya et al., 2005). But it need not be limited to the retina. Optogenetics
(Yizhar et al., 2011) and microstimulation (Clark et al., 2011) can also be used as
120 CHAPTER 6. CONCLUSIONS
the mode of perturbation of the system. As long as there was some other secondary
readout, these perturbations can be used to great e↵ects in deciphering the underlying
mechanisms of various neural computations.
Appendix A
Publications
A.1 Journal Articles
1. Kastner DB, Baccus SA. Coordinated dynamic encoding in the retina using
opposing forms of plasticity. Nature Neuroscience, 14(10):1317-1322.
This work appears as Chapter 2. I collected the data, designed and performed
all analyses with SAB, and wrote the manuscript with SAB.
2. Kastner DB, Baccus SA (accepted). Spatial segregation of adaptation and pre-
dictive sensitization in retinal ganglion cells. Neuron.
This work appears as Chapter 3. I collected the data, designed and performed
all analyses with SAB, and wrote the manuscript with SAB.
A.2 Refereed conference articles and abstracts
1. Kastner DB, Baccus SA (2009, poster/spotlight). Distinct adaptive modes for
weak and strong signals in a retinal populations Frontiers in Neuroscience. Con-
ference Abstract: Computational and Systems Neuroscience (COSYNE), Salt
Lake City, UT.
This work became Kastner et al. (2011).
121
122 APPENDIX A. PUBLICATIONS
2. Kastner DB, Baccus SA (2010, poster). The adaptive field and predictive object
representation in the retina Conference Abstract: Gordon Conference on Sensory
Coding and the Natural Environment, Bates College, Me.
This work became Kastner et al. (accepted).
3. Kastner DB, Baccus SA (2011, talk). The adaptive field and predictive object
representation in the retina. Nature Preceding. Conference Abstract: Computa-
tional and Systems Neuroscience (COSYNE)
Part of this work is contained in Chapter 5.
4. Kastner DB, Baccus SA, Sharpee TO (2012, poster). Optimal placement of
dynamic range by coordinated populations of retinal ganglion cells. Conference
Abstract: Computational and Systems Neuroscience (COSYNE), Salt Lake City,
UT.
Part of this work is contained in Chapter 4.
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