ABSTRACT
Chapter 6 starts with an introduction to neurons and their properties like generation of action potential, signal propagation, neural modeling, and neuronal dynamics. Four biophysical neuron models are examined for evolution of the functional mechanism of the brain (single neuron models and network level). Using conductance-based mathematical models, patterns of spiking activity and qualitative behavior of temporal activity like periodic firing, bursting, chattering, mixed-mode oscillations, and chaotic firing are studied. Model 1 is the Hodgkin–Huxley (HH) model, which describes how action potential in neurons is initiated and propagated. Existence, uniqueness, and regularity of mild solution are proved for a spatial model. Bifurcation, propagation of bursting oscillations, etc., are studied. Model 2 is the 2D FitzHugh–Nagumo (FHN) model, which is a reduced version of the 4D HH model. A review of spatial FHN model as investigated by Zhen and Shen is presented. A modified version of the model which considers external stimulus current with 1D diffusion is also analyzed. Model 3 is the Morris–Lecar (ML) model and describes oscillations in barnacle giant muscle fiber. Multiple-scale analysis is presented near a bifurcation point. Introducing applied current as the third variable, spiking and bursting in a single ML spatial neuron model is investigated. Model 4 is the Hindmarsh–Rose (HR) model. Existence of waves using the reaction–diffusion HR system is studied. A RD system of a modified HR oscillator is considered, which has zero as an equilibrium point and studied the spatial dynamics to explore the wave profiles of traveling wave solutions and spiking and bursting activity.
