Date of Award

Spring 2021

Project Type

Dissertation

Program or Major

Applied Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Kevin M Short

Second Advisor

John Gibson

Third Advisor

Mark Lyon

Abstract

Recent work has demonstrated that interacting chaotic systems can establish persistent, periodic behavior, called mutual stabilization, when certain information is passed through interaction functions. In particular, this was first shown with two interacting cupolets (Chaotic Unstable Periodic Orbit-lets) of the double scroll oscillator. Cupolets are highly accurate approximations of unstable periodic orbits of a chaotic attractor that can be generated through a control scheme that repeatedly applies perturbations along Poincaré sections. The decision to perturb or not to perturb the trajectory is determined by a bit in a binary control sequence. One interaction function used in the original cupolet research was based on integrate-and-fire dynamics that are often seen in neural and laser systems and was used to demonstrate mutual stabilization between two double scroll oscillators. This result provided the motivation for this thesis where the stabilization of chaos in mathematical models of communicating neurons is investigated.

This thesis begins by introducing mathematical models of neurons and discusses the biological realism of the models. Then, we consider the two-dimensional FitzHugh-Nagumo (FHN) neural model and we show how two FHN neurons can exhibit chaotic behavior when communication is mediated by a coupling constant, g, representative of the synaptic strength between the neurons. Through a bifurcation analysis, where the synaptic strength is the bifurcation parameter, we analyze the space of possible long-term behaviors of this model. After identifying regions of periodic and chaotic behavior, we show how a synaptic sigmoidal learning rule transitions the chaotic dynamics of the system to periodic dynamics in the presence of an external signal. After the signal passes through the synapse, synaptic learning alters the synaptic strength and the two neurons remain in a persistent, mutually stabilized periodic state even after the signal is removed. This result provides a proof-of-concept for chaotic stabilization in communicating neurons.

Next, we focus on the 3-dimensional Hindmarsh-Rose (HR) neural model that is known to exhibit chaotic behavior and bursting neural firing. Using this model, we create a control scheme using two Poincaré sections in a manner similar to the control scheme for the double scroll system. Using the control scheme we establish that it is possible to generate cupolets in the HR model. We use the HR model to create neural networks where the communication between neurons is mediated by an integrate-and-fire interaction function. With this interaction, we show how a signal can propagate down a unidirectional chain of chaotic neurons. We further show how mutual stabilization can occur if two neurons communicate through this interaction function. Lastly, we expand the investigation to more complicated networks including a feedback network and a chain of neurons that ends in a feedback loop between the two terminal neurons. Mutual stabilization is found to exist in all cases. At each stage, we comment on the potential biological implications and extensions of these results.

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