Download - The linear/nonlinear model
![Page 1: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/1.jpg)
The linear/nonlinear model
s*f1
![Page 2: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/2.jpg)
The spike-triggered average
![Page 3: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/3.jpg)
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)
![Page 4: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/4.jpg)
Linear filtering
Linear filtering = convolution = projection
s1
s2
s3
Stimulus feature is a vector in a high-dimensional stimulus space
![Page 5: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/5.jpg)
How to find the components of this model
s*f1
![Page 6: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/6.jpg)
The input/output function is:
which can be derived from data using Bayes’ rule:
Determining the nonlinear input/output function
![Page 7: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/7.jpg)
Tuning curve: P(spike|s) = P(s|spike) P(spike) / P(s)
Nonlinear input/output function
Tuning curve: P(spike|s) = P(s|spike) P(spike) / P(s)
s
P(s|spike) P(s)
s
P(s|spike) P(s)
![Page 8: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/8.jpg)
Models with multiple features
Linear filters & nonlinearity: r(t) = g(f1*s, f2*s, …, fn*s)
![Page 9: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/9.jpg)
Spike-triggeredaverage
Gaussian priorstimulus distribution
Spike-conditional distribution
covariance
Determining linear features from white noise
![Page 10: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/10.jpg)
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
Spike-triggeredaverage
![Page 11: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/11.jpg)
Inner and outer products
u v = S ui vi
u v = projection of u onto vu u = length of u
u x v = M where Mij = ui vj
![Page 12: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/12.jpg)
Eigenvalues and eigenvectors
M u = l u
![Page 13: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/13.jpg)
Singular value decomposition
m
n
=
xx
M
U S V
![Page 14: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/14.jpg)
Principal component analysis
M M* = (U S V)(V*S*U*)= U S (V V*) S*U*
= U S S*U*
C = U L U*
![Page 15: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/15.jpg)
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
Spike-triggeredaverage
![Page 16: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/16.jpg)
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
![Page 17: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/17.jpg)
6
4
2
0
-2P
roje
ctio
n on
to M
ode
26420-2
Projection onto Mode 1
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STAf(t)
f'(t)
6
4
2
0
-2P
roje
ctio
n on
to M
ode
26420-2
Projection onto Mode 1
6
4
2
0
-2P
roje
ctio
n on
to M
ode
26420-2
Projection onto Mode 1
-1.0
-0.5
0.0
0.5
1.0
Eig
enva
lue
50403020100Mode Index
Covariance analysis of a filter-and-fire model
![Page 18: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/18.jpg)
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
ositi
on (
m)
Record from single units in barrel cortex
Some real data
![Page 19: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/19.jpg)
Spike-triggered average:
-20020406080100120140160180-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Pre-spike time (ms)
Norm
alise
d ve
locit
y
White noise analysis in barrel cortex
![Page 20: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/20.jpg)
Is the neuron simply not very responsive to a white noise stimulus?
White noise analysis in barrel cortex
![Page 21: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/21.jpg)
Prior Spike-triggered
Difference
Covariance matrices from barrel cortical neurons
![Page 22: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/22.jpg)
0 20 40 60 80 100-1
0
1
2
3
4
Feature number
Nor
mal
ised
vel
ocity
2
050100150-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Pre-spike time (ms)
Velo
city
Eigenspectrum
Leading modes
Eigenspectrum from barrel cortical neurons
![Page 23: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/23.jpg)
-3 -2 -1 0 1 2 30
2
4
6
8
10
Normalised "acceleration"
Nor
mal
ised
firin
g ra
te
-3 -2 -1 0 1 2 30
2
4
6
8
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
4
6
8
10
0 1 2 30
2
4
6
8
10
"vel"2+"acc"2
Nor
mal
ised
firin
g ra
te
Input/outputrelations wrt
first two filters,alone:
and in quadrature:
Input/output relations from barrel cortical neurons
![Page 24: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/24.jpg)
050100150-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
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
![Page 25: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/25.jpg)
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
0.4
0.6
0.8
1
1.2
1.4
Firing rate decreases with increasing projection: suppressive modes
Input/output relations for negative pair
Negative eigenmode pair
![Page 26: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/26.jpg)
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA Mode 1
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA Mode 1 Mode 2
-1.0
-0.5
0.0
0.5
1.0
Eig
enva
lue
100806040200Mode Index
-6
-4
-2
0
2
4
6
Pro
ject
ion
onto
Mod
e 2
-4 -2 0 2 4
Projection onto Mode 1
Salamander retinal ganglion cells perform a variety of computations
Michael Berry
![Page 27: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/27.jpg)
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA
-6
-4
-2
0
2
4
6
Pro
ject
ion
onto
Mod
e 2
-4 -2 0 2 4
Projection onto Mode 1
-1.0
-0.5
0.0
0.5
Eig
enva
lue
100806040200Mode Index
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2 Mode 3
Almost filter-and-fire-like
![Page 28: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/28.jpg)
1.0
0.5
0.0
-0.5
Eig
enva
lue
100806040200Mode Index
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2 Mode 3
-6
-4
-2
0
2
4
6
Pro
ject
ion
onto
Mod
e 2
-6 -4 -2 0 2 4 6
Projection onto Mode 1
Not a threshold-crossing neuron
![Page 29: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/29.jpg)
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA Mode 1
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA Mode 1 Mode 2
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
STA Mode 1 Mode 2 Mode 3
0.9
0.6
0.3
0.0
-0.3
Eig
enva
lue
100806040200
Mode Index
2.0
1.5
-6
-4
-2
0
2
4
6
Pro
ject
ion
onto
Mod
e 2
-6 -4 -2 0 2 4 6
Projection onto Mode 1
Complex cell like
![Page 30: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/30.jpg)
-0.6
-0.3
0.0
0.3
Eig
enva
lue
100806040200
Mode Index
2.5
2.0
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.6 -0.4 -0.2 0.0Time before Spike (s)
STA
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1
-0.4
-0.2
0.0
0.2
0.4
Spi
ke-T
rigge
red
Aver
age
-0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2 Mode 3
-4
-3
-2
-1
0
1
2
3
4
Pro
ject
ion
onto
Mod
e 2
-6 -4 -2 0 2 4 6
Projection onto Mode 1
Bimodal: two separate features are encoded as either/or
![Page 31: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/31.jpg)
Complex cells in V1
Rust et al., Neuron 2005
![Page 32: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/32.jpg)
• integrators H1, some single cortical neurons• differentiators Retina, simple cells, HH neuron, auditory neurons• frequency-power detectors V1 complex cells, somatosensory, auditory, retina
Basic types of computation
![Page 33: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/33.jpg)
When have you done a good job?
• When the tuning curve over your variable is interesting.
• How to quantify interesting?
![Page 34: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/34.jpg)
Tuning curve: P(spike|s) = P(s|spike) P(spike) / P(s)
Goodness measure: DKL(P(s|spike) | P(s))
When have you done a good job?
Tuning curve: P(spike|s) = P(s|spike) P(spike) / P(s)
s
Boring: spikes unrelated to stimulus
P(s|spike) P(s)
s
Interesting: spikes are selective
P(s|spike) P(s)
![Page 35: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/35.jpg)
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)
![Page 36: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/36.jpg)
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
![Page 37: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/37.jpg)
Less basic coding models
Linear filters & nonlinearity: r(t) = g(f1*s, f2*s, …, fn*s)
…shortcomings?
![Page 38: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/38.jpg)
Less basic coding models
GLM: r(t) = g(f1*s + f2*r)
Pillow et al., Nature 2008; Truccolo, .., Brown, J. Neurophysiol. 2005
![Page 39: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/39.jpg)
Less basic coding models
GLM: r(t) = g(f*s + h*r) …shortcomings?
![Page 40: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/40.jpg)
Less basic coding models
GLM: r(t) = g(f1*s + h1*r1 + h2*r2 +…) …shortcomings?
![Page 41: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/41.jpg)
Poisson spiking
![Page 42: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/42.jpg)
Poisson spiking
Shadlen and Newsome, 1998
![Page 43: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/43.jpg)
Poisson spiking
Properties:
Distribution: PT[k] = (rT)k exp(-rT)/k!Mean: <x> = rT
Variance: Var(x) = rT
Interval distribution: P(T) = r exp(-rT)Fano factor: F = 1
![Page 44: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/44.jpg)
Fano factor Data fit to: variance = A meanB
A
B
Area MT
Poisson spiking in the brain
![Page 45: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/45.jpg)
How good is the Poisson model? ISI analysis
ISI Distribution from an area MT Neuron
ISI distribution generated from a Poisson model with a Gaussian refractory period
Poisson spiking in the brain
![Page 46: The linear/nonlinear model](https://reader035.vdocument.in/reader035/viewer/2022062310/56816504550346895dd77335/html5/thumbnails/46.jpg)
Hodgkin-Huxley neuron driven by noise