1. A Bifurcation Theoretical Approach to the Solving the Neural Coding Problem June 28 Albert E. Parker Complex Biological Systems Department of Mathematical Sciences Center for Computational Biology Montana State University Collaborators: Tomas Gedeon, Alex Dimitrov, John Miller, and Zane Aldworth

2.  The Neural Coding Problem  A Clustering Problem  The Role of Bifurcation Theory  A new algorithm to solve the Neural Coding Problem Outline

3. The Neural Coding Problem GOAL: To understand the neural code. EASIER GOAL: We seek an answer to the question, How does neural activity represent information about environmental stimuli? “The little fly sitting in the fly’s brain trying to fly the fly”

4. stimulus X response Y Looking for the dictionary to the neural code … decoding encoding

5. … but the dictionary is not deterministic! Given a stimulus, an experimenter observes many different neural responses: X Y i | X i = 1, 2, 3, 4

6. … but the dictionary is not deterministic! Given a stimulus, an experimenter observes many different neural responses: Neural coding is stochastic!! X Y i | X i = 1, 2, 3, 4

7. Similarly, neural decoding is stochastic: Y X i |Y i = 1, 2, …, 9

8. Probability Framework X Y environmental stimuli neural responses decoder: P(X|Y) encoder: P(Y|X)

9. The Neural Coding Problem: How to determine the encoder P(Y|X) or the decoder P(X|Y)? Common Approaches: parametric estimations, linear methods Difficulty: There is never enough data. As we attempt search for an answer to the neural coding problem, we proceed in the spirit of John Tukey: It is better to be approximately right than exactly wrong.

10. One Approach: Cluster the responses X Y StimuliResponses YNYN q(Y N |Y) Clusters K objects {y i } N objects {y Ni }L objects {x i } p(X,Y)

11. One Approach: Cluster the responses To address the insufficient data problem, one clusters the outputs Y into clusters Y N so that the information that one can learn about X by observing Y N, I(X;Y N ), is as close as possible to the mutual information I(X;Y) X Y StimuliResponses YNYN q(Y N |Y) Clusters K objects {y i } N objects {y Ni }L objects {x i } p(X,Y)

12. Information Bottleneck Method (Tishby, Pereira, Bialek 1999) min I(Y,Y N ) constrained by I(X;Y N )  I 0 max –I(Y,Y N ) +  I(X;Y N ) Information Distortion Method (Dimitrov and Miller 2001) max H(Y N |Y) constrained by I(X;Y N )  I 0 max H(Y N |Y) +  I(X;Y N ) Two optimization problems which use this approach

13. An annealing algorithm to solve max q  (G(q)+  D(q)) Let q 0 be the maximizer of max q G(q), and let  0 =0. For k  0, let (q k,  k ) be a solution to max q G(q) +  D(q ). Iterate the following steps until  K =  max for some K. 1.Perform  -step: Let  k+1 =  k + d k where d k >0 2.The initial guess for q k+1 at  k+1 is q k+1 (0) = q k +  for some small perturbation . 3.Optimization: solve max q (G(q) +  k+1 D(q)) to get the maximizer q k+1, using initial guess q k+1 (0).

14. Application of the annealing method to the Information Distortion problem max q  (H(Y N |Y) +  I(X;Y N )) when p(X,Y) is defined by four gaussian blobs Stimuli Responses X Y 52 responses 52 stimuli p(X,Y) YYNYN q(Y N |Y) 52 responses4 clusters

15. Evolution of the optimal clustering: Observed Bifurcations for the Four Blob problem: We just saw the optimal clusterings q * at some  * =  max. What do the clusterings look like for  <  max ??

17. Application to cricket sensory data E(X|Y N ): stimulus means conditioned on each of the classes typical spike patterns optimal quantizer

18. We have used bifurcation theory in the presence of symmetries to totally describe how the the optimal clusterings of the responses must evolve…

19. Symmetries?? Y YNYN q(Y N |Y) : a clustering K objects {y i } N objects {y Ni } class 1 class 3

20. Y YNYN q(Y N |Y) : a clustering K objects {y i } N objects {y Ni } class 3 class 1 Symmetries??

21. Observed Bifurcation Structure

22. Observed Bifurcation Structure Group Structure

23.  q* Observed Bifurcation Structure

24. Continuation techniques provide numerical confirmation of the theory

26. Additional structure!!

28. Conclusions … We have a complete theoretical picture of how the clusterings of the responses evolve for any problem of the form max q  (G(q)+  D(q)) oWhen clustering to N classes, there are N-1 bifurcations. oIn general, there are only pitchfork and saddle-node bifurcations. oWe can determine whether pitchfork bifurcations are either subcritical or supercritical (1 st or 2 nd order phase transitions) oWe know the explicit bifurcating directions SO WHAT?? This yields a new and improved algorithm for solving the neural coding problem …

29. A numerical algorithm to solve max(G(q)+  D(q)) Let q 0 be the maximizer of max q G(q),  0 =1 and  s > 0. For k  0, let (q k,  k ) be a solution to max q G(q) +  D(q ). Iterate the following steps until  K =  max for some K. 1.Perform  -step: solve for and select  k+1 =  k + d k where d k = (  s sgn(cos  )) /(||   q k || 2 + ||   k || 2 +1) 1/2. 2.The initial guess for (q k+1, k+1 ) at  k+1 is (q k+1 (0), k+1 (0) ) = (q k, k ) + d k (   q k,   k ). 3.Optimization: solve max q (G(q) +  k+1 D(q)) using pseudoarclength continuation to get the maximizer q k+1, and the vector of Lagrange multipliers k+1 using initial guess (q k+1 (0), k+1 (0) ). 4.Check for bifurcation: compare the sign of the determinant of an identical block of each of  q [G(q k ) +  k D(q k )] and  q [G(q k+1 ) +  k+1 D(q k+1 )]. If a bifurcation is detected, then set q k+1 (0) = q k + d_k u where u is bifurcating direction and repeat step 3.