FitzHugh–Nagumo Model

Abstract

Definition The FitzHugh-Nagumo (FHN) model is a mathematical model of neuronal excitability developed by Richard FitzHugh as a reduction of the Hodgkin and Huxley's model of action potential generation in the squid giant axon (FitzHugh 1955). Nagumo et al. subsequently designed, implemented, and analyzed an equivalent electric circuit (Nagumo et al. 1962). In its basic form, the model consists of two coupled, nonlinear ordinary differential equations, one of which describes the fast evolution of the neuronal membrane voltage, the other representing the slower "recovery" action of sodium channel deinactivation and potassium channel deactivation. Phase plane analysis of the FHN model provides qualitative explanations of several aspects of the excitability exhibited by the Hodgkin-Huxley (HH) model, including all-or-none spiking, excitation block, and the apparent absence of a firing threshold. A version of the FHN equations which adds a spatial diffusion term models the propagation of an action potential along an axon as a traveling wave. Due to their relative simplicity and ease of geometric analysis, the FHN model and its variants are commonly used in neuroscience, chemistry, physics, and other disciplines as simple models of excitable dynamics, relaxation oscillations, and reaction-diffusion wave propagation. Detailed Description Mathematical Model FitzHugh constructed the FHN equations by slightly modifying the cubic Bonhoeffer-van der Pol model of relaxation oscillations. The original van der Pol equation is a second-order linear differential equation: € V þ V 2 À 1 À Á _ V þ fV ¼ 0, (1) which may be written as two first-order equations by applying the Lienard transformation: W ¼ À _ V þ V À V 3 =3: _ V ¼ V À V 3 =3 À W (2a) _ W ¼ fV : (2b) For all f > 0, the origin in system Eq. 2 is an unstable fixed point surrounded by a globally stable limit cycle. FitzHugh's modification to Eq. 2 added linear terms which shifted the fixed point and made it stable (FitzHugh 1955): *