Center for Computational BiologyDepartment of Mathematical Sciences

Montana State University

Collaborators:Alexander Dimitrov

John P. MillerZane Aldworth

Thomas Gedeon Brendan Mumey

Computational Issues whenModeling Neural Coding Schemes

Albert E. Parker

Neural Coding and Decoding.

Goal: Determine a coding scheme: How does neural ensemble activity represent information about sensory stimuli?

Demands: • An animal needs to recognize the same object on repeated

exposures. Coding has to be deterministic at this level.• The code must deal with uncertainties introduced by the environment

and neural architecture. Coding is by necessity stochastic at this finer scale.

Major Problem: The search for a coding scheme requires large amounts of data

How to determine a coding scheme?

Idea: Model a part of a neural system as a communication channel using Information Theory. This model enables us to:

• Meet the demands of a coding scheme:o Define a coding scheme as a relation between stimulus and neural

response classes.

o Construct a coding scheme that is stochastic on the finer scale yet almost deterministic on the classes.

• Deal with the major problem:o Use whatever quantity of data is available to construct coarse but

optimally informative approximations of the coding scheme.

o Refine the coding scheme as more data becomes available.

• Investigate the cricket cercal sensory system.

Information Theoretic QuantitiesA quantizer or encoder, Q, relates the environmental stimulus, X, to the neural response, Y, through a process called quantization. In general, Q is a stochastic map

The Reproduction space Y is a quantization of X. This can be repeated: Let Yf be a reproduction of Y. So there is a quantizer

Use Mutual information to measure the degree of dependence between X and Yf.

Use Conditional Entropy to measure the self-information of Yf given Y

q y y Y Yf f( | ):

I X Y q y y p x y

q y y p x y

p x p y q y yf f

fy

fy

y y f

( , ) ( , ) ( , ) log

( , ) ( , )

( ) ( ) ( , ),

H Y Y p y q y y q y yf fy y

ff

( | ) ( ) ( | ) log( ( | )),

Q y x X Y( | ):

Y

X stimulus sequences

resp

onse

seq

uenc

es stimulus/response sequence pairs

distinguishable classes of stimulus/response pairs

Stimulus and Response Classes

The ModelProblem: To determine a coding scheme between X and Y requires large amounts of

data

Idea: Determine the coding scheme between X and Yf, a squashing (reproduction) of Y, such that: Yf preserves as much information (mutual information) with X as possible and the self-information (entropy) of Yf |Y is maximized. That is, we are searching for an optimal mapping (quantizer):

that satisfies these conditions.

Justification: Jayne's maximum entropy principle, which states that of all the quantizers that satisfy a given set of constraints, choose the one that maximizes the entropy.

q y y Y Yf f*( | ):

f

Iq

Iq

y

yp

qD

yy

yp

qD

f ee

yyq

,|

Equivalent Optimization Problems

Maximum entropy:

maximize F(q(yf|y)) = H(Yf|Y) constrained by

I(X;Yf ) Io Io determines the informativeness of the reproduction.

Deterministic annealing (Rose, ’98):

maximize F(q(yf|y)) = H(Yf|Y) - DI(Y,Yf ).Small favor maximum entropy, large - minimum DI.

Simplex Algorithm:

maximize I(X,Yf ) over vertices of constraint space

Implicit solution:

?

?

Signal

Nervous system

Communicationchannel

Modeling the cricket cercal sensory system as a communication channel

Wind Stimulus and Neural Response in the cricket cercal system

Neural Responses (over a 30 minute recording) caused by white noise wind stimulus.

T, ms

Neural Responses (these are all doublets) for a 12 ms window

Some of the air current stimuli preceding one of the neural responses

Time in ms. A t T=0, the first spike occurs

X

Y

YfY

Quantization:A quantizer is any map f: Y -> Yf from Y to a reproduction space Yf with finitely many elements. Quantizers can be

deterministic or

refined

yy f

Y

probabilistic

yyq f |

Applying the algorithm to cricket sensory data.

Y

1

2

3

1

2

Yf

Yf

High Performance Computing

Tools: Bigdog: an SGI Origin 2000 MATLAB 5.3 Parallel Toolbox

Algorithms: Model Building Optimization Bootstrapping

ConclusionsWe• model a part of the neural system as a communication

channel.

• define a coding scheme through relations between classes of stimulus/response pairs.

- Coding is probabilistic on the individual elements of X and Y.- Coding is almost deterministic on the stimulus/response

classes.

To recover such a coding scheme, we• propose a new method to quantify neural spike trains.

- Quantize the response patterns to a small finite space (Yf).

- Use information theoretic measures to determine optimal quantizer for a fixed reproduction size.

- Refine the coding scheme by increasing the reproduction size.

• present preliminary results with cricket sensory data.