John Parker, Integrated Applied Mathematics, University of New Hampshire
Chaotic Stabilization in Neural Systems
Recent work in dynamical systems theory has shown how chaotic systems are able to be controlled. One control scheme, adapted from Hayes, Grebogi, and Ott, was applied to a chaotic double scroll oscillator and produced stabilized periodic orbits called cupolets (chaotic, unstable, periodic orbit-lets). It was then demonstrated by Morena and Short that interacting cupolets can produce mutual stabilization. Mutual stabilization occurs when two interacting systems maintain a persistent, periodic trajectory that would not exist without the interaction. One interaction function used was based on integrate-and-fire dynamics often seen in neural systems. Here, work is shown that provides evidence of mutual stabilization within interacting neural systems. First, published work is presented that shows how a bidirectional, two-cell FitzHugh-Nagumo neural model can transition from chaotic to periodic behavior through synaptic learning activated by a particular external signal. Then, preliminary work is discussed concerning the existence of cupolets in a Hindmarsh-Rose neuron. Periodic pulses are then generated using these cupolets and coupled into the Hindmarsh-Rose system to recreate the corresponding stabilized periodic orbit.