what is the neural code?

What is the neural code?

Alan Litke, UCSD

Reading out the neural code


Kwan, HFSP 2010


M. Berry


Single neurons

Populations

Encoding: how does a stimulus cause a pattern of responses?

• what are the responses and what are their characteristics?• neural models:

-- from stimulus to response-- descriptive mechanistic models

Decoding: what do these responses tell us about the stimulus?

• Implies some kind of decoding algorithm• How to evaluate how good our algorithm is?

Encoding and decoding

Basic coding model: linear filtering

Spatial filter: r = f(x,y) I(x0-x,y0-y) dx dy

Basic coding model: temporal filtering

Linear filter: r(t) = f(t-t) s(t) dt

…shortcomings?

Nonlinearity: tuning curves

Gaussian tuning curve of a cortical (V1) neuron

Nonlinear function: r = g(s)


Cosine tuning curve of a motor cortical neuron

Nonlinear function: r = g(s) Hand reaching direction


Sigmoidal tuning curve of a V1 cortical neuron

Nonlinear function: r = g(s) Hand reaching direction


Quian Quiroga, Reddy, Kreiman, Koch and Fried, Nature (2005)

What is s?


Quian Quiroga, Reddy, Kreiman, Koch and Fried, Nature (2005)

What is s?

Coding models

P(response | stimulus)

P(stimulus | response)

encoding

decoding

• What is response?• What is stimulus?• What is the function P?

Basic coding model: temporal filtering

Linear filter: r(t) = f(t-t) s(t) dt

Next most basic coding model

Linear filter & nonlinearity: r(t) = g( f(t-t) s(t) dt)

s*f1

How to find the components of this model

s*f1

Reverse correlation: the spike-triggered average

Spike-conditionalensemble

The spike-triggered average

More generally, one can conceive of the action of the neuron or neural system as selecting a low dimensional subset of its inputs.

Start with a very high dimensional description (eg. an image or a time-varying waveform)

and pick out a small set of relevant dimensions.

s(t)

s1s2 s3

s1

s2

s3

s4 s5 s.. s.. s.. sn S(t) = (S1,S2,S3,…,Sn)

Dimensionality reduction

P(response | stimulus) P(response | s1, s2, .., sn)

Linear filtering

Linear filtering = convolution = projection

s1

s2

s3

Stimulus feature is a vector in a high-dimensional stimulus space

Spike-triggeredaverage

Gaussian priorstimulus distribution

Spike-conditional distribution

Determining linear features from white noise

How to find the components of this model

s*f1

The input/output function is:

which can be derived from data using Bayes’ rule:

Determining the nonlinear input/output function

Tuning curve: P(spike|s) = P(s|spike) P(spike) / P(s)

Nonlinear input/output function


s

P(s|spike) P(s)

s

P(s|spike) P(s)

Next most basic coding model

Linear filter & nonlinearity: r(t) = g( f(t-t) s(t) dt)

s*f1

…shortcomings?

Less basic coding models

Linear filters & nonlinearity: r(t) = g(f1*s, f2*s, …, fn*s)


Gaussian priorstimulus distribution

Spike-conditional distribution

covariance

Determining linear features from white noise

The covariance matrix is

Properties:

• The number of eigenvalues significantly different from zero is the number of relevant stimulus features

• The corresponding eigenvectors are the relevant features (or span the relevant subspace)

Stimulus prior

Bialek et al., 1988; Brenner et al., 2000; Bialek and de Ruyter, 2005

Identifying multiple features

Spike-triggered stimulus correlation


Let’s develop some intuition for how this works: a filter-and-fire threshold-crossing model with AHP

Keat, Reinagel, Reid and Meister, Predicting every spike. Neuron (2001)

• Spiking is controlled by a single filter, f(t)• Spikes happen generally on an upward threshold crossing of

the filtered stimulus expect 2 relevant features, the filter f(t) and its time derivative f’(t)

A toy example: a filter-and-fire model

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Projection onto Mode 1

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STAf(t)

f'(t)

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50403020100Mode Index

Covariance analysis of a filter-and-fire model

Example: rat somatosensory (barrel) cortex Ras Petersen and Mathew Diamond, SISSA

0 200 400 600 800 1000

-400

-200

0

200

400

Time (ms)

Stim

ulat

or p

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on (

m)

Record from single units in barrel cortex

Let’s try it

Spike-triggered average:

-20020406080100120140160180-1

-0.8

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0

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1

Pre-spike time (ms)

Norm

alise

d ve

locit

y

White noise analysis in barrel cortex

Is the neuron simply not very responsive to a white noise stimulus?

White noise analysis in barrel cortex

Prior Spike-triggered

Difference

Covariance matrices from barrel cortical neurons

0 20 40 60 80 100-1

0

1

2

3

4

Feature number

Nor

mal

ised

vel

ocity

2

050100150-0.4

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0

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0.4

Pre-spike time (ms)

Velo

city

Eigenspectrum

Leading modes

Eigenspectrum from barrel cortical neurons

-3 -2 -1 0 1 2 30

2

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10

Normalised "acceleration"

Nor

mal

ised

firin

g ra

te

-3 -2 -1 0 1 2 30

2

4

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10

12

Normalised "velocity"

Nor

mal

ised

firin

g ra

te

-2 0 2

-3

-2

-1

0

1

2

3

"acceleration"

"vel

ocity

"

0

2

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10

0 1 2 30

2

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"vel"2+"acc"2

Nor

mal

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firin

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te

Input/outputrelations wrt

first two filters,alone:

and in quadrature:

Input/output relations from barrel cortical neurons

050100150-0.2

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0

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Pre-spike time (ms)

Vel

ocity

(arb

itrar

y un

its)

050100150-0.3-0.2-0.1

00.10.20.30.4

Pre-spike time (ms)Ve

locit

y (a

rbitr

ary

units

)

How about the other modes?

Next pair with +ve eigenvaluesPair with -ve eigenvalues

Less significant eigenmodes from barrel cortical neurons

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

"Energy"N

orm

alis

ed fi

ring

rate

-3 -2 -1 0 1 2-3

-2

-1

0

1

2

Filter 100

Filte

r 101

0.2

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1.2

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Firing rate decreases with increasing projection: suppressive modes

Input/output relations for negative pair

Negative eigenmode pair

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Salamander retinal ganglion cells perform a variety of computations

Michael Berry

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STA Mode 1 Mode 2 Mode 3

Almost filter-and-fire-like

1.0

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Not a threshold-crossing neuron

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0.9

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100806040200

Mode Index

2.0

1.5

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Complex cell like

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100806040200

Mode Index

2.5

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-0.6 -0.4 -0.2 0.0Time before Spike (s)

STA

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Bimodal: two separate features are encoded as either/or

When have you done a good job?

• When the tuning curve over your variable is interesting.

• How to quantify interesting?


Goodness measure: DKL(P(s|spike) | P(s))

When have you done a good job?


s

Boring: spikes unrelated to stimulus

P(s|spike) P(s)

s

Interesting: spikes are selective

P(s|spike) P(s)

Maximally informative dimensions

Choose filter in order to maximize DKL between spike-conditional and prior distributions

Equivalent to maximizing mutual information between stimulus and spike

Sharpee, Rust and Bialek, Neural Computation, 2004

Does not depend on white noise inputs

Likely to be most appropriatefor deriving models fromnatural stimuli

s = I*f

P(s|spike) P(s)

1. Single, best filter determined by the first moment

2. A family of filters derived using the second moment

3. Use the entire distribution: information theoretic methodsRemoves requirement for Gaussian stimuli

Finding relevant features


Linear filters & nonlinearity: r(t) = g(f1*s, f2*s, …, fn*s)

…shortcomings?


GLM: r(t) = g(f1*s + f2*r)

Pillow et al., Nature 2008; Truccolo, .., Brown, J. Neurophysiol. 2005


GLM: r(t) = g(f*s + h*r) …shortcomings?


GLM: r(t) = g(f1*s + h1*r1 + h2*r2 +…) …shortcomings?

what is the neural code?

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