Center for Computational Biology Department of Mathematical Sciences Montana State University

1. 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

2. 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

3. 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.

4. 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

5. stimulus sequences X Y stimulus/response sequence pairs response sequences distinguishable classes of stimulus/response pairs Stimulus and Response Classes

6. 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.

7. 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:

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9. Modeling the cricket cercal sensory system as a communication channel Nervous system Signal Communicationchannel

10. 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

11. 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

12. Applying the algorithm to cricket sensory data. Yf 1 2 1 2 3 Yf Y

13. High Performance Computing Tools: • Bigdog: an SGI Origin 2000 • MATLAB 5.3 • Parallel Toolbox Algorithms: • Model Building • Optimization • Bootstrapping

14. 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.