Nonlinear Dynamics of Electrophysiological Neural Models

Arizona State University, Scottsdale AZ

Investigators

Abstract

Hoppensteadt 0109001 The investigator and his colleague study the relation between bifurcations in neural dynamics and electrophysiological features of neurons. There is growing understanding of the role of bifurcations in neuron dynamics. Indeed, knowledge of all bifurcations in a neuron model can explain some features of neuron activity, such as the existence of oscillatory potentials, synchronization, bursting, excitability, etc., without knowing electrophysiological details. However, there is little understanding of the relationship between electrophysiology and bifurcations in neural dynamics. For example, why do certain ionic currents or combinations of them result in an Andronov-Hopf bifurcation but not a saddle-node on limit cycle bifurcation? (These two types of bifurcations describe Hodgkin's classification of excitable neurons.) Over 100 different kinds of recognizable bifurcations result in bursting patterns of activity. A key question is whether or not one can determine for a given type of bursting activity what combinations of ionic currents must (or must not) be involved. The investigator and his colleague introduce here a notion of minimal electrophysiological models classified by voltage gated, second messenger gated, and variable Nernst potentials. They map this classification onto the existing classification of bifurcation mechanisms in neuron dynamics. This yields a deeper understanding of nonlinear dynamics of electrophysiological systems. How does a brain encode and process information? Mathematics has played an important role in studies of this problem to date by creating methods for analyzing results obtained by life science and medical researchers. There is a well-developed electrophysiological theory of how neural tissue can sustain and propagate electrical activity using ionic currents. At the same time there is an emerging mathematical theory based on canonical models of bifurcations, which are observable changes in a biological preparation in response to stimulation. This project brings these two lines of inquiry together by classifying dynamical systems (mathematical structures) in terms that are consistent with electrophysiology. The result is a deeper understanding of nonlinear phenomena in electrophysiological systems.