Coherence Resonance in the FitzHugh-Nagumo system

The simplest models used to describe physical systems are often linear. However, several natural phenomena can only be accurately modeled by relatively complex nonlinear systems of equations. There is a wealth of interesting dynamics displayed by systems consisting of nonlinear elements, for example systems in which the dynamics depend delicately on the initial conditions, viz, chaos, and systems which generate fractal structures. These types of dynamics are especially noticeable in models of biological phenomena such as cardiac arrythmia, fluctuations in predator-prey populations, or neuronal cascades in the human brain. Of special interest to us are the class of nonlinear systems which are excitable by external stimuli. Excitability is characterized by a response that is highly nonlinear; for small perturbations, the system remains steady, but once a threshold is reached, the system “spikes”, leaving the resting state entirely and going on an excursion in phase space. This type of behavior is characteristic of the action potential train through a neuron, described by the Hodgkin-Huxley (HH) model [1]. In the early 1950s, Hodgkin and Huxley performed action potential measurements on the squid giant axon and proposed a model of action potential transmission, typically represented as an electrical circuit as shown in Figure 1. The classical model of potential transmission through a neuron was of the form