The linear/nonlinear model s*f 1. The spike-triggered average.
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Transcript of The linear/nonlinear model s*f 1. 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
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
Tuning curve: P(spike|s) = P(s|spike) P(spike) / P(s)
s
P(s|spike) P(s)
s
P(s|spike) P(s)
Spike-triggeredaverage
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
Spike-triggeredaverage
Inner and outer products
u v = S ui vi
u v = projection of u onto v
u u = length of u
u x v = M where Mij = ui vj
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
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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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
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Time (ms)
Stim
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Record from single units in barrel cortex
Some real data
Spike-triggered average:
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Pre-spike time (ms)
Norm
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White noise analysis in barrel cortex
Is the neuron simply not very responsive to a white noise stimulus?
White noise analysis in barrel cortex
0 20 40 60 80 100-1
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Feature number
No
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ty2
050100150-0.4
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0
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Pre-spike time (ms)
Ve
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Eigenspectrum
Leading modes
Eigenspectrum from barrel cortical neurons
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2
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Normalised "acceleration"
No
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firi
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ate
-3 -2 -1 0 1 2 30
2
4
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12
Normalised "velocity"
No
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ised
firi
ng r
ate
-2 0 2
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-1
0
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"acceleration"
"vel
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0 1 2 30
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"vel"2+"acc"2
No
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ng r
ate
Input/outputrelations wrt
first two filters,alone:
and in quadrature:
Input/output relations from barrel cortical neurons
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Pre-spike time (ms)
Velo
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arb
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nits)
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Pre-spike time (ms)Velo
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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
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"Energy"N
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Filter 100
Filt
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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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-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)
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STA Mode 1
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STA Mode 1 Mode 2
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100806040200Mode Index
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Salamander retinal ganglion cells perform a variety of computations
Michael Berry
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STA Mode 1 Mode 2
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STA Mode 1 Mode 2 Mode 3
Almost filter-and-fire-like
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STA Mode 1
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-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2
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-0.8 -0.6 -0.4 -0.2 0.0Time before Spike (s)
STA Mode 1 Mode 2 Mode 3
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ject
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-6 -4 -2 0 2 4 6
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Not a threshold-crossing neuron
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STA Mode 1 Mode 2
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STA Mode 1 Mode 2 Mode 3
0.9
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Eig
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Mode Index
2.0
1.5
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ject
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-6 -4 -2 0 2 4 6
Projection onto Mode 1
Complex cell like
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Mode Index
2.5
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-0.6 -0.4 -0.2 0.0Time before Spike (s)
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STA Mode 1
-0.4
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STA Mode 1 Mode 2 Mode 3
-4
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Pro
ject
ion
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to M
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e 2
-6 -4 -2 0 2 4 6
Projection onto Mode 1
Bimodal: two separate features are encoded as either/or
• 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
When have you done a good job?
• When the tuning curve over your variable is interesting.
• How to quantify interesting?
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)
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 methods
Removes requirement for Gaussian stimuli
Finding relevant features
Less basic coding models
Linear filters & nonlinearity: r(t) = g(f1*s, f2*s, …, fn*s)
…shortcomings?
Less basic coding models
GLM: r(t) = g(f1*s + f2*r)
Pillow et al., Nature 2008; Truccolo, .., Brown, J. Neurophysiol. 2005
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
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