What is the neural code?

What is the neural code?

Alan Litke, UCSD




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

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

what takes us from stimulus to response;descriptive and mechanistic models, and the relation between them.

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

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


Single cells:spike ratespike timesspike intervals


Single cells:spike rate: what does the firing rate correspond to?spike times: what in the stimulus triggers a spike?spike intervals: can patterns of spikes convey extra information?


Populations of cells:population codingcorrelations between responsessynergy and redundancy

Receptive fields and tuning curves

Tuning curve: r = f(s)

Gaussian tuning curve of a cortical (V1) neuron


Tuning curve: r = f(s)

Cosine tuning curve of a motor cortical neuron

Hand reaching direction


Sigmoid/logistic tuning curve of a “stereo” V1 neuron

Retinal disparity for a “near”object

Higher brain areas represent increasingly complex features

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

More generally, we are interested in determining the relationship:

P(response | stimulus)

Due to noise, this is a stochastic description.

Problem of dimensionality, both in response and in stimulus

P(stimulus | response)

encoding

decoding

Reverse correlation

Fast modulation of firing by dynamic stimuli

Feature extraction

Use reverse correlation to decide what each of these spiking eventsstands for, and so to either:

-- predict the time-varying firing rate-- reconstruct the stimulus from the spikes

Reverse correlation

Typically, use Gaussian, white noise stimulus: an unbiased stimulus which samples all directions equally

S(t)

r(t)

Basic idea: throw random stimuli at the system and collect the onesthat cause a response

Reverse correlation

Spike-conditionalensemble

Goal: simplify!

stimulus = fluctuating potential

(generates electric field)

Example: a neuron in the ELL of a fish

Spike-triggered Average

This can be done with other dimensions of stimulus as well

Spatio-temporal receptive field

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 encoding

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


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 encoding

Models of neural function


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 given by

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


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


?

spike-triggering stimulus features

?

stimulus X(t)

decision function

spike history feedback

spike output Y(t)

x1

x2

x3

f1

f2

f3

?


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


-1.0

-0.5

0.0

0.5

1.0

Eig

en

valu

e


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


6

4

2

0

-2P

roje

ctio

n o

nto

Mo

de

2

6420-2


-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 a real neuron: rat somatosensory cortex (Ras Petersen, Mathew Diamond, SISSA)

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

Input/output relations for negative pair

Beyond covariance analysis

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

Find the dimensions that maximize the mutual informationbetween stimulus and spike

Removes requirement for Gaussian stimuli

Limitations

Not a completely “blind” procedure: have to have some idea of the appropriate stimulus space

Very complex stimuli: does a geometrical picture work or make sense?

Adaptation:stimulus representations change with experience!

Documents

What is the neural code?