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What is the neural code?. What is the neural code?. Alan Litke, UCSD. What is the neural code?. What is the neural code?. Encoding : how does a stimulus cause a pattern of responses? what are the responses and what are their characteristics? neural models:

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slide2

What is the neural code?

Alan Litke, UCSD

what is the neural code2

What is the neural code?

  • 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?
what is the neural code3

What is the neural code?

Single cells:

spike rate

spike times

spike intervals

what is the neural code4

What is the neural code?

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?

what is the neural code5

What is the neural code?

Populations of cells:

population coding

correlations between responses

synergy and redundancy

receptive fields and tuning curves

Receptive fields and tuning curves

Tuning curve: r = f(s)

Gaussian tuning curve of a cortical (V1) neuron

receptive fields and tuning curves1

Receptive fields and tuning curves

Tuning curve: r = f(s)

Hand reaching direction

Cosine tuning curve of a motor cortical neuron

receptive fields and tuning curves2

Receptive fields and tuning curves

Retinal disparity for a “near”

object

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

slide11

Higher brain areas represent increasingly complex features

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

slide15

More generally, we are interested in determining the relationship:

encoding

P(response | stimulus)

P(stimulus | response)

decoding

Due to noise, this is a stochastic description.

Problem of dimensionality, both in response and in stimulus

reverse correlation

Reverse correlation

Fast modulation of firing by

dynamic stimuli

Feature extraction

Use reverse correlation to decide what each of these spiking events

stands for, and so to either:

-- predict the time-varying firing rate

-- reconstruct the stimulus from the spikes

reverse correlation1

Basic idea: throw random stimuli at the system and collect the ones

that cause a response

Reverse correlation

Typically, use Gaussian, white noise stimulus: an unbiased stimulus

which samples all directions equally

S(t)

r(t)

reverse correlation2

Reverse correlation

Spike-conditional

ensemble

Goal: simplify!

slide19

Example: a neuron in the ELL of a fish

stimulus = fluctuating potential

(generates electric field)

Spike-triggered Average

modeling spike encoding

spike-triggering

stimulus feature

decision function

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

stimulus X(t)

f1

spike output Y(t)

x1

P(spike|x1 )

x1

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

Modeling spike encoding

Given a stimulus, when will the system spike?

Simple example: the integrate-and-fire neuron

modeling spike encoding1

spike-triggering

stimulus feature

decision function

stimulus X(t)

f1

spike output Y(t)

x1

P(spike|x1 )

x1

Modeling spike encoding

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

models of neural function

Models of neural function

spike-triggering

stimulus feature

decision function

stimulus X(t)

f1

spike output Y(t)

x1

P(spike|x1 )

x1

Weaknesses

reverse correlation a geometric view

covariance

Reverse correlation: a geometric view

Gaussian prior

stimulus distribution

STA

Spike-conditional

distribution

dimensionality reduction

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

Dimensionality reduction

The covariance matrix is given by

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

Functional models of neural function

spike-triggering

stimulus feature

decision function

stimulus X(t)

f1

spike output Y(t)

x1

P(spike|x1 )

x1

slide27

Functional models of neural function

spike-triggering stimulus features

f1

multidimensional

decision function

x1

stimulus X(t)

f2

spike output Y(t)

x2

f3

x3

slide28

Functional models of neural function

?

?

?

spike-triggering stimulus features

f1

x1

decision function

stimulus X(t)

f2

spike output Y(t)

x2

f3

x3

spike history feedback

covariance analysis

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 analysis1

Covariance analysis

Let’s try a real neuron: rat somatosensory cortex

(Ras Petersen, Mathew Diamond, SISSA)

Record from single units in barrel cortex

covariance analysis2

Normalisedvelocity

Pre-spike time (ms)

Covariance analysis

Spike-triggered average:

covariance analysis3

Covariance analysis

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

covariance analysis4

Covariance analysis

Prior

Spike-

triggered

Difference

covariance analysis5

0.4

0.3

0.2

0.1

Velocity

0

-0.1

-0.2

-0.3

-0.4

150

100

50

0

Pre-spike time (ms)

Covariance analysis

Eigenspectrum

Leading modes

slide36

Covariance analysis

Input/output

relations wrt

first two filters,

alone:

and in quadrature:

slide37

0.4

0.3

0.2

Velocity (arbitrary units)

0.1

0

-0.1

-0.2

-0.3

150

100

50

0

Pre-spike time (ms)

Covariance analysis

How about the other modes?

Pair with -ve eigenvalues

Next pair with +ve eigenvalues

covariance analysis6

Covariance analysis

Input/output relations for negative pair

Firing rate decreases with increasing projection:

suppressive modes

slide39

Beyond covariance analysis

  • Single, best filter determined by the first moment
  • A family of filters derived using the second moment
  • Use the entire distribution: information theoretic methods

 Find the dimensions that maximize the mutual information

between stimulus and spike

Removes requirement for Gaussian stimuli

slide40

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!