Introduction

1. Dynamical Encoding by Networks of Competing Neuron Groups : Winnerless Competition M. Rabinovich1, A. Volkovskii1, P. Lecanda2,3, R. Huerta1,2, H.D.I. Abarbanel1,4, and G. Laurent5presented byMichael Downes6 1Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402 2GNB, E.T.S. de Ingenieria Informatica, Universidad Autonoma de Madrid, 28049 Madrid, Spain 3Instituto de Ciencia de Materiales de Madrid, CSIC Cantoblanco, 28049 Madrid, Spain 4Department of Physics and Marine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 93093-0402 5California Institute of Technology, Division of Biology, MC 139-74 Pasadena, California 91125 6Department of Physics, Drexel University, Philadelphia, PA 19104

2. Introduction • Competitive or Winnerless Competition Networks • Identity or spatiotemporal coding • Deterministic trajectories : heteroclinic orbits • Connect saddle fixed points or saddle limit cycles • Saddle states correspond to neuron activity • 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 • F[yi(t)] : nonlinear function describing ith neuron dynamics • Gij(S) : nonlinear operator describing inhibitory action of jth neuron on ith • S(t) : vector-represented stimuli • Stimulus acts in 2 ways: • Excites subset of neurons through S(t) • Modifies effective inhibitory connections through Gij(S) • 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 • 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 • “turns on” inhibitory term gji = 2 for neurons with inhibitory relationships

6. Models and Parameters (cont.)

7. Results • Membrane Potential vs. time for 2 Stimuli • S1 = [0.10,0.15,0.00,0.00,0.15,0.10,0.00,0.00,0.00] • S2 = [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 • 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 • 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 data • Unique trajectories sensitively dependent on stimulus • Large Encoding Capability

11. Questions?