FitzHugh–Nagumo and excitability

Biology | Parabolic | Patterns

Here we look at the FitzHugh–Nagumo model, given by

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u +u-u^3-v,\\ \pd{v}{t}&=D\nabla^2v+ \varepsilon_v(u-a_v v-a_z),\end{aligned}\]

where we take $D>1$.

Turing–Hopf bifurcations

We now vary the parameters from the previous simulation so that it supports both pattern formation, but also oscillations. These oscillations come from steady states undergoing Hopf bifurcations. In such regimes, one can often find a range of complex spatial, temporal, and spatiotemporal behaviours, many of which can be simultaneously stable for different initial conditions.

To illustrate this, we consider the initial conditions

\[u(x,y,0) = \cos\left(\frac{m \pi x}{L}\right)\cos\left(\frac{m \pi y}{L}\right), \quad v(x,y,0)=0,\]

for some integer $m$ and domain length $L=280$.

Three-species variant

A three-species variant of the FitzHugh–Nagumo model is

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u +u-u^3-v,\\ \pd{v}{t}&=D_v\nabla^2v+ \varepsilon_v(u-a_v v-a_w w-a_z),\\ \pd{w}{t}&=D_w\nabla^2w+ \varepsilon_w(u-w).\end{aligned}\]