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Computational Issues when Modeling Neural Coding Schemes Albert E. Parker Center for Computational Biology Department of Mathematical Sciences Montana State University Collaborators: Alexander Dimitrov John P. Miller Zane Aldworth Thomas Gedeon Brendan Mumey

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Computational Issues when

Modeling Neural Coding Schemes

Albert E. Parker

Center for Computational Biology

Department of Mathematical Sciences

Montana State University

Collaborators:

Alexander Dimitrov

John P. Miller

Zane Aldworth Thomas Gedeon

Brendan Mumey

neural coding and decoding
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
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:
    • Define a coding scheme as a relation between stimulus and neural response classes.
    • Construct a coding scheme that is stochastic on the finer scale yet almost deterministic on the classes.
  • Deal with the major problem:
    • Use whatever quantity of data is available to construct coarse but optimally informative approximations of the coding scheme.
    • Refine the coding scheme as more data becomes available.
  • Investigate the cricket cercal sensory system.
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Information Theoretic Quantities

A 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 Yfgiven Y

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stimulus sequences

X

Y

stimulus/response

sequence pairs

response sequences

distinguishable classes of stimulus/response pairs

Stimulus and Response Classes

the model
The Model

Problem: 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.

equivalent optimization problems
Equivalent Optimization Problems
  • Maximum entropy: maximizeF(q(yf|y)) = H(Yf|Y)constrainedbyI(X;Yf )  Io Iodetermines the informativeness of the reproduction.
  • Deterministic annealing (Rose, ’98): maximizeF(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:
slide8

?

?

slide9

Modeling the cricket cercal sensory system as a communication channel

Nervous system

Signal

Communicationchannel

slide10

Wind Stimulus and Neural Response in the cricket cercal systemNeural Responses (over a 30 minute recording) caused by white noise wind stimulus.

X

Y

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

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

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

T, ms

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probabilistic

refined

Y

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

Yf

Y

high performance computing
High Performance Computing

Tools:

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

Algorithms:

  • Model Building
  • Optimization
  • Bootstrapping
conclusions
Conclusions

We

  • 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.