1. Dynamical Encoding by Networks of Competing Neuron Groups : Winnerless Competition M. Rabinovich 1, A. Volkovskii 1, P. Lecanda 2,3, R. Huerta 1,2, H.D.I. Abarbanel 1,4, and G. Laurent 5 presented by Michael Downes 6 1 Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402 2 GNB, E.T.S. de Ingenieria Informatica, Universidad Autonoma de Madrid, 28049 Madrid, Spain 3 Instituto de Ciencia de Materiales de Madrid, CSIC Cantoblanco, 28049 Madrid, Spain 4 Department of Physics and Marine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 93093-0402 5 California Institute of Technology, Division of Biology, MC 139-74 Pasadena, California 91125 6 Department of Physics, Drexel University, Philadelphia, PA 19104

2. Introduction Competitive or Winnerless Competition Networks Identity or spatiotemporal coding Identity or spatiotemporal coding Deterministic trajectories : heteroclinic orbits Deterministic trajectories : heteroclinic orbits Connect saddle fixed points or saddle limit cycles Connect saddle fixed points or saddle limit cycles Saddle states correspond to neuron activity Saddle states correspond to neuron activity Separatrices correspond to sequential switching Separatrices correspond to sequential switching

3. Introduction (cont.) Features of Neural Encoding – Representation of Input Information: Uses both space and time Sensitively depends on stimulus Deterministic and reproducible Robust against noise Observations Suggest Dissipative dynamical system => “forgetfulness” Information represented as transient trajectories

4. Model and Parameters Neuron Dynamics System of N neurons System of N neurons F[y i (t)] : nonlinear function describing i th neuron dynamics G ij (S) : nonlinear operator describing inhibitory action of j th neuron on i th S(t) :vector-represented stimuli Stimulus acts in 2 ways: Stimulus acts in 2 ways: Excites subset of neurons through S(t) Modifies effective inhibitory connections through G ij (S) Instability in presence of stimulus leads to : Instability in presence of stimulus leads to : Sequence of heteroclinic trajectories Rapid action Robustness against noise Response independent of initial state

5. Model and Parameters (cont.) Numerical Model 9 Fitzhugh-Nagumo Model neurons with constant stimulus 9 Fitzhugh-Nagumo Model neurons with constant stimulus x(t) : membrane potential y(t) : recovery variable z(t) : synaptic current (included inhibition term) f(x) : nonlinear Fitzhugh-Nagumo neuron dynamics G(x) : inhibition function Asserted when membrane potential is greater than zero Asserted when membrane potential is greater than zero “turns on” inhibitory term g ji = 2 for neurons with inhibitory relationships “turns on” inhibitory term g ji = 2 for neurons with inhibitory relationships

6. Models and Parameters (cont.)

7. Results Membrane Potential vs. time for 2 Stimuli S 1 = [0.10,0.15,0.00,0.00,0.15,0.10,0.00,0.00,0.00] S 1 = [0.10,0.15,0.00,0.00,0.15,0.10,0.00,0.00,0.00] S 2 = [0.01,0.03,0.05,0.04,0.06,0.02,0.03,0.05,0.04] S 2 = [0.01,0.03,0.05,0.04,0.06,0.02,0.03,0.05,0.04] Stimulus patterns distinguishable

8. Model and Parameters Information Encoding Input information solely in inhibitory coupling strength between i and j Input information solely in inhibitory coupling strength between i and j Non-symmetric inhibitory connections lead to closed heteroclinic orbits Non-symmetric inhibitory connections lead to closed heteroclinic orbits Global attractors Change in stimulus => new global attractor in orbit vicinity

9. Model and Parameters (cont.) Capacity # of different items the network can encode # of different items the network can encode With N neurons: With N neurons: N-1 cyclically equivalent permutation: (1,2,3,4,5)  (2,3,4,5,1) (N-1)! heteroclinic orbits More heteroclinic orbits associated with N-1, N-2, etc. subspaces For Large N,

10. Conclusion Winnerless Competition model competent to describe data experimental dataWinnerless Competition model competent to describe data experimental data Unique trajectories sensitively dependent on stimulus Unique trajectories sensitively dependent on stimulus Large Encoding Capability Large Encoding Capability

11. Questions?