An introduction to dynamical systems and bifurcations applied to the Hodgkin-Huxley neuronal model

Abstract

The purpose of this project is to study the fundamental concepts and techniques from continuous dynamical systems and, furthermore, their applications to neuroscience. We propose to investigate the relation between electrophysiology, nonlinear dynamics and neurocomputational properties, using primarily the Hodgkin-Huxley's differential equations. A neuron is a dynamical system with parameters, that is, a system whose behavior is described by differential equations over time. The transition from the resting state to the spiking state is a bifurcation, and the distinct types of bifurcations that can occur can be investigated mathematically, such as excitability, oscillations and thresholds, for instance. Therefore, not only the electrophysiological features determine whether a neuron shoots off, but also its internal dynamics. Such aspects are important in the computational neuroscience research field. Hence, studying those models from a formal mathematical approach is a key to understand the intricate behavior of neural activity. (AU)