FitzHugh–Nagumo and excitability

    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 $L=280$.

    Action potentials and travelling waves

    This model is often considered in the context of action potentials, which are nerve impulses that travel along neurons. We look at this model with an alternative parameterisation

    \[\begin{aligned}\pd{v}{t}&=D\nabla^2 v +v(v-a)(1-v) - w,\\ \pd{w}{t}&=\varepsilon(\gamma v - w),\end{aligned}\]

    where we’ve used $v$ and $w$ to follow the convention that $v$ represents the electrical voltage across the neuron. Here, $D$ and $\varepsilon$ are typically small positive constants.

    1. Load the interactive simulation.

    2. Try clicking briefly. Small disturbances simply decay back to equilibrium.

    3. Now try clicking for longer to see large-amplitude travelling waves emerge, which model nerve impulses down neurons. This is an example of a threshold phenomenon, which occurs generically in so-called excitable systems.

    4. Try exploring this model in 2D by changing the domain via → Domain → Dimension to see waves spread out radially.

    Three-species variant

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

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