FitzHugh-Nagumo

The forthcoming R Journal has an interesting article about phaseR: An R Package for Phase Plane Analysis of Autonomous ODE Systems by Michael J. Grayling. The package has some nice functions to analysis one and two dimensional dynamical systems. As an example I use here the FitzHugh-Nagumo system introduced earlier: \[ \begin{aligned} \dot{v}=&2 (w + v - \frac{1}{3}v^3) + I_0 \\\\\\ \dot{w}=&\frac{1}{2}(1 - v - w)\\\\\\ \end{aligned} \] The FitzHugh-Nagumo system is a simplification of the Hodgkin-Huxley model of spike generation in squid giant axon.

I discussed earlier how the action potential of a neuron can be modelled via the Hodgkin-Huxely equations. Here I will present a simple model that describes how action potentials can be generated and propagated across neurons. The tricky bit here is that I use delay differential equations (DDE) to take into account the propagation time of the signal across the network. My model is based on the paper: Epileptiform activity in a neocortical network: a mathematical model by F.