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”

Spike Triggered Average 2-Spike Triggered Average (10 ms separation)

2-Spike Triggered Average(5 ms)

Multiple spike symbols from the fly motion sensitive neuron

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 onthe 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

Fano factor Data fit to: variance = A meanB

A

BArea MT

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

Coefficients of

Variation for a

set of V1 and

MT Neurons

Poisson

Poisson with ref. period

spike-triggering stimulus feature

stimulus X(t)

decision function

spike output Y(t)x1

f1

P(s

pike

|x1

)x1

Decompose the neural computation into a linear stage and a nonlinear stage.

Modeling spike generation

Given a stimulus, when will the system spike?

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

Simple example: the integrate-and-fire neuron

Let’s start with a rate response, r(t) and a stimulus, s(t).

The optimal linear estimator is closest to satisfying

Predicting the firing rate

Want to solve for K. Multiply by s(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

For white noise, the correlation function Css() = So K() is simply Crs().

Going back to:

spike-triggering stimulus feature

stimulus X(t)

decision function

spike output Y(t)x1

f1

P(s

pike

|x1

)

x1

The decision function is P(spike|x1). Derive from data using Bayes’ theorem:

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

Modeling spike generation

Models of neural function

spike-triggering stimulus feature

stimulus X(t)

decision function

spike output Y(t)x1

f1

P(s

pike

|x1

)

x1

Weaknesses

STA

Gaussian priorstimulus distribution

Spike-conditional distribution

covariance

Reverse correlation: a geometric view

Dimensionality reduction

Cij = < S(t – ti) S(t - tj)> - < STA(t - ti) STA (t - tj)> - < I(t - ti) I(t - tj)>

The covariance matrix is simply

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

Stimulus prior

Bialek et al., 1997

Functional models of neural function

spike-triggering stimulus feature

stimulus X(t)

decision function

spike output Y(t)x1

f1

P(s

pike

|x1

)

x1

spike-triggering stimulus features

stimulus X(t)

multidimensionaldecision function

spike output Y(t)

x1

x2

x3

f1

f2

f3

Functional models of neural function

?

spike-triggering stimulus features

?

stimulus X(t)

multidimensionaldecision function

spike history feedback

spike output Y(t)

x1

x2

x3

f1

f2

f3

?

Functional models of neural function

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)

-1.0

-0.5

0.0

0.5

1.0

Eig

en

valu

e

50403020100Mode Index

-1.0

-0.5

0.0

0.5

1.0

Eig

en

valu

e

50403020100Mode Index

-1.0

-0.5

0.0

0.5

1.0

Eig

en

valu

e

50403020100Mode Index

6

4

2

0

-2P

roje

ctio

n o

nto

Mo

de

26420-2

Projection onto Mode 1

6

4

2

0

-2P

roje

ctio

n o

nto

Mo

de

2

6420-2

Projection onto Mode 1

6

4

2

0

-2P

roje

ctio

n o

nto

Mo

de

2

6420-2

Projection onto Mode 1

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Sp

ike

-Tri

gg

ere

d A

vera

ge

-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)

Covariance analysis

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Sp

ike

-Tri

gg

ere

d A

vera

ge

-0.5 -0.4 -0.3 -0.2 -0.1 0.0Time before Spike (s)

Covariance analysis

Let’s try some real neurons: rat somatosensory cortex (Ras Petersen, Mathew Diamond, SISSA: SfN 2003).

0 200 400 600 800 1000

-400

-200

0

200

400

Time (ms)

Stim

ulat

or

posi

tion

(m

)

Record from single units in barrel cortex

Covariance analysis

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)

No

rma

lise

d ve

loci

ty

Covariance analysis

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

Covariance analysis

Prior Spike-triggered

Difference

Covariance analysis

0 20 40 60 80 100-1

0

1

2

3

4

Feature number

No

rmal

ised

vel

oci

ty2

050100150-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Pre-spike time (ms)

Vel

oci

ty

Eigenspectrum

Leading modes

-3 -2 -1 0 1 2 30

2

4

6

8

10

Normalised "acceleration"

No

rmal

ised

firi

ng r

ate

-3 -2 -1 0 1 2 30

2

4

6

8

10

12

Normalised "velocity"

No

rmal

ised

firi

ng r

ate

-2 0 2

-3

-2

-1

0

1

2

3

"acceleration"

"vel

oci

ty"

0

2

4

6

8

10

0 1 2 30

2

4

6

8

10

"vel"2+"acc"2

No

rmal

ised

firi

ng r

ate

Input/outputrelations wrtfirst two filters,alone:

and in quadrature:

Covariance analysis

050100150-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Pre-spike time (ms)

Velo

city (

arb

itra

ry u

nits)

050100150-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Pre-spike time (ms)V

elo

city

(a

rbitr

ary

un

its)

How about the other modes?

Next pair with +ve eigenvalues Pair with -ve eigenvalues

Covariance analysis

Covariance analysis

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

"Energy"N

orm

alis

ed f

iring

rat

e

-3 -2 -1 0 1 2-3

-2

-1

0

1

2

Filter 100

Filt

er 1

01

0.2

0.4

0.6

0.8

1

1.2

1.4

Firing rate decreases with increasing projection: suppressive modes (Simoncelli et al.)

Input/output relations for negative pair