## The FitzHugh-Nagumo model

As mentioned in a previous post the FitzHugh-Nagumo model is a simplified version of the Hodgkin-Huxley model describing the propagation of signals in nerve cells. In this post only the spatially homogeneous case is discussed. It is possible to consider a corresponding system including a diffusion term for one of the unknowns ($x$ in the system below), a subject touched on in a previous post. A different ODE reduction can be obtained by looking at traveling wave solutions. The equations in the homogeneous case form a system of two ODE. It is supposed to capture the essential qualitative behaviour of the system of four ODE defining the Hodgkin-Huxley model while otherwise being as simple as possible. Here the notation of the original paper of FitzHugh (Biophys. J. 1, 445) will be used.The system is

$\dot x=c(y+x-x^3/3+z)$

$\dot y=-(x-a+by)/c$

with constant parameters $a$, $b$ and $c$. The quantity $z$ is in general a prescribed function of $t$. The unknown $x$ corresponds to the voltage in the Hodgkin-Huxley system while $y$ corresponds to the other variables. The parameter $z$ plays the role of the external current in the Hodgkin-Huxley system. The relation to the van der Pol oscillator, as written in the previous post, is easily seen. The parameters $a$ and $b$ and the function $z$ are new. If they are set to zero we get the van der Pol equation in slightly different variables. FitzHugh gives an intuitive description of the dynamics with phase plane pictures. Depending on the values of the parameters there may be a stable stationary solution which is a global attractor, or an unstable stationary solution together with a stable periodic solution. There is more discussion of the qualitative behaviour in Murray’s book ‘Mathematical Biology’.

Theorems about the van der Pol oscillator can be found in many textbooks. Corresponding material on the FitzHugh-Nagumo model seems to be much rarer. There is, nevertheless, a big literature out there. There is a thesis of Matthias Ringqvist online which collects a lot of interesting material on the subject and can serve as a point of entry to the literature for the uninitiated, like myself. He considers the properties of a dynamical system which contains the FitzHugh-Nagumo model as a special case. He motivates this generalization by noting that the more general system includes a number of different systems of interest in different problems in applied mathematics. He discusses the presence of periodic solutions in various parameter regimes and the bifurcations they are involved in. Hopf bifurcations play an important role. Another type of bifurcation which occurs in this context is the Bautin bifurcation. This differs from Hopf case in that one of the non-degeneracy conditions (the one which does not only depend on the linearization at the point of interest) fails. There can be coexistence of more than one periodic solution.

To what extent does the FitzHugh-Nagumo model capture the dynamics present in the Hodgkin-Huxley model? Ringqvist mentions numerical work of Guckenheimer and Oliva from 2002 which suggests that the HH model exhibits chaotic dynamics.The authors do not claim to have proved rigorously that chaos is present but this is a warning that it would be foolish to think that the dynamics of the HH system might be ‘essentially understood’. Chaos is observed in the forced van der Pol oscillator, i.e. the system obtained from the van der Pol oscillator by adding a prescribed function of time. It is more impressive for me to see chaos coming up in a system which is autonomous and which occurs naturally in an application. Of course the Lorenz system, one of the icons of chaos, satisfies the first condition and, at least in a weak sense, the second.