Transcript What is the language of a single cell?

What is the language of single cells?
What are the elementary symbols of the code?
Most typically, we think about the response as a firing rate, r(t), or a modulated
spiking probability, P(r = spike|s(t)).
Two extremes of description:
A Poisson model, where spikes are generated randomly with rate r(t).
However, most spike trains are not Poisson (refractoriness, internal dynamics).
Fine temporal structure might be meaningful.
Consider spike patterns or “words”, e.g.
• symbols including multiple spikes and the interval between
• retinal ganglion cells: “when” and “how much”
Multiple spike symbols from the fly motion sensitive neuron
Spike Triggered Average
2-Spike Triggered Average
(10 ms separation)
2-Spike Triggered Average
(5 ms)
Spike statistics
Stochastic process that generates a sequence of events: point process
Probability of an event at time t depends only on preceding event: renewal process
All events are statistically independent: Poisson process
Poisson: r(t) = r independent of time, probability to see a spike only depends on
the time you watch.
PT[n] = (rT)n exp(-rT)/n!
Exercise: the mean of this distribution is rT
the variance of this distribution is also rT.
The Fano factor = variance/mean = 1 for Poisson processes.
The Cv = coefficient of variation = STD/mean = 1 for Poisson
Interspike interval distribution P(T) = r exp(-rT)
The Poisson model (homogeneous)
Probability of n spikes in time T
as function of (rate  T)
Poisson approaches Gaussian
for large rT (here = 10)
How good is the Poisson model? Fano Factor
A
B
Area MT
Fano factor
Data fit to:
variance = A  meanB
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
How good is the Poisson Model? CV analysis
Poisson
Coefficients of
Variation for a
set of V1 and MT
Neurons
Poisson with
ref. period
Modeling spike generation
Given a stimulus, when will the system spike?
Decompose the neural computation into a linear stage and a nonlinear stage.
spike-triggering
stimulus feature
stimulus X(t)
f1
x1
P(spike|x1 )
decision function
spike output Y(t)
x1
Simple example: the integrate-and-fire neuron
To what feature in the stimulus is the system sensitive?
Gerstner, spike response model; Aguera y Arcas et al. 2001, 2003; Keat et al., 2001
Predicting the firing rate
Let’s start with a rate response, r(t) and a stimulus, s(t).
The optimal linear estimator is closest to satisfying
Want to solve for K. Multiply by s(t-t’) and integrate over t:
Note that we have produced terms which are simply correlation functions:
Given a convolution, Fourier transform:
Now we have a straightforward algebraic equation for K(w):
Solving for K(t),
Predicting the firing rate
Going back to:
For white noise, the correlation function Css(t) = s2 d(t),
So K(t) is simply Crs(t).
Modeling spike generation
spike-triggering
stimulus feature
stimulus X(t)
f1
x1
P(spike|x1 )
decision function
The decision function is P(spike|x1).
Derive from data using Bayes’ theorem:
spike output Y(t)
x1
P(spike|x1) = P(spike) P(x1 | spike) / P(x1)
P(x1) is the prior : the distribution of all projections onto f1
P(x1 | spike) is the spike-conditional ensemble :
the distribution of all projections onto f1 given there has been a spike
P(spike) is proportional to the mean firing rate
Models of neural function
spike-triggering
stimulus feature
stimulus X(t)
f1
x1
P(spike|x1 )
decision function
spike output Y(t)
x1
Weaknesses
Reverse correlation: a geometric view
Gaussian prior
stimulus distribution
covariance
STA
Spike-conditional
distribution
Dimensionality reduction
The covariance matrix is simply
Cij = < S(t – ti) S(t - tj)> - < STA(t - ti) STA (t - tj)> - < I(t - ti) I(t - tj)>
Stimulus prior
Properties:
•If the computation is low-dimensional, there will be a few eigenvalues
significantly different from zero
•The number of eigenvalues is the relevant dimensionality
•The corresponding eigenvectors span the subspace of the relevant features
Bialek et al., 1997
Functional models of neural function
spike-triggering
stimulus feature
stimulus X(t)
f1
x1
P(spike|x1 )
decision function
spike output Y(t)
x1
Functional models of neural function
spike-triggering stimulus features
f1
x1
multidimensional
decision function
stimulus X(t)
f2
spike output Y(t)
x2
f3
x3
Functional models of neural function
spike-triggering stimulus features
f1
x1
multidimensional
decision function
stimulus X(t)
f2
spike output Y(t)
x2
f3
x3
?
?
?
spike history feedback
Covariance analysis
Let’s develop some intuition for how this works: the Keat model
Keat, Reinagel, Reid and Meister, Predicting every spike. Neuron (2001)
• Spiking is controlled by a single filter
• Spikes happen generally on an upward threshold crossing of
the filtered stimulus
 expect 2 modes, the filter F(t) and its time derivative F’(t)
Covariance analysis
1.0
Eigenvalue
0.5
6
0.0
Projection onto Mode 2
-0.5
-1.0
0
10
20
30
Mode Index
40
50
Spike-Triggered Average
0.6
0.4
4
2
0
-2
0.2
0.0
-2
0
2
4
Projection onto Mode 1
-0.2
-0.4
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
Time before Spike (s)
0.0
6
Covariance analysis
Let’s try some real neurons: rat somatosensory cortex
(Ras Petersen, Mathew Diamond, SISSA: SfN 2003).
Stimulator position ( mm)
Record from single units in barrel cortex
400
200
0
-200
-400
0
200
400
600
Time (ms)
800
1000
Covariance analysis
Spike-triggered average:
1
Normalised velocity
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
180
160
140
120
100
80
60
40
Pre-spike time (ms)
20
0
-20
Covariance analysis
Is the neuron simply not very responsive to a white noise stimulus?
Covariance analysis
Prior
Spiketriggered
Difference
Covariance analysis
Eigenspectrum
Leading modes
4
3
0.3
0.2
2
Velocity
Normalised velocity2
0.4
1
0.1
0
-0.1
-0.2
0
-0.3
-1
0
20
40
60
Feature number
80
100
-0.4
150
100
50
Pre-spike time (ms)
0
Covariance analysis
12
Normalised firing rate
10
Normalised firing rate
Input/output
relations wrt
first two filters,
alone:
10
8
6
4
2
0
-3
-2
-1
0
1
2
8
6
4
2
0
3
Normalised "velocity"
-3
-2
-1
0
1
2
Normalised "acceleration"
10
3
2
"velocity"
and in quadrature:
8
1
0
6
-1
4
-2
2
Normalised firing rate
10
8
6
4
2
-3
-2
0
"acceleration"
2
0
0
0
1
2
2
"vel" +"acc"
3
2
3
Covariance analysis
How about the other modes?
Pair with -ve eigenvalues
Next pair with +ve eigenvalues
Velocity (arbitrary units)
0.5
Velocity (arbitrary units)
0.4
0.3
0.2
0.1
0
0.3
0.2
0.1
0
-0.1
-0.2
-0.1
-0.2
0.4
150
100
50
Pre-spike time (ms)
0
-0.3
150
100
50
Pre-spike time (ms)
0
Covariance analysis
Input/output relations for negative pair
1.5
1.4
1.2
1
1
0
0.8
0.6
-1
0.4
-2
Normalised firing rate
Filter 101
2
1
0.5
0.2
-3
-3
-2
-1
0
Filter 100
1
2
0
0
0.5
1
1.5
"Energy"
Firing rate decreases with increasing projection:
suppressive modes (Simoncelli et al.)
2
2.5
3