Stability Analysis of FitzHugh-Nagumo with Smooth Periodic Forcing

Abstract

Alan Lloyd Hodgkin and Andrew Huxley received the 1963 Nobel Prize in Physiology for their work describing the propagation of action potentials in the squid giant axon.  Major analysis of their system of differential equations was performed by Richard FitzHugh, and later by Jin-Ichi Nagumo who created a tunnel diode circuit based upon FitzHugh’s work.  The resulting differential model, known as the FitzHugh-Nagumo (FH-N) oscillator, represents a simplification of the Hodgkin-Huxley (H-H) model, but still replicates the original neuronal dynamics (Izhikevich, 2010).  We begin by providing a thorough grounding in the physiology behind the equations, then continue by introducing some of the results established by Kostova et al. for FH-N without forcing (Kostova et al., 2004).  Finally, this sets up our own exploration into stimulating the system with smooth periodic forcing.  Subsequent quantification of the chaotic phase portraits using a Lyapunov exponent are discussed, as well as the relevance of these results to electrocardiography.