FitzHugh- Nagumo Model conversion into state space

Question

Is it possible to convert the Fitzhugh Nagumo model into State-space? Is it done using linearization using Jacobian?

Answer 1

You cite \begin{align} \varepsilon \frac{dv}{dt}&=F(v)-w+I\\ \frac{dw}{dt}&=v-\gamma w \end{align} with $F(v)=v(1-v)(v+a)$, $ε\ll1$ and all other variables constants ($I$ might also be a time-variable external input like in https://stackoverflow.com/questions/49256309/matlab-euler-explicit).

This is already an explicit first-order system with state space $(v,w)\in\Bbb R^2$. While linearizations at equilibrium points serve to determine the nature of these, in general there is no linear formulation of this system, the non-linear term $F(v)$ is essential.

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The expected behavior is that the first equation forces $v$ in time-scales $\sim ε$ to "jump" to one of the roots of $F(v)=w-I$ with a negative $F$ derivative. The second equation moves $w$ relatively slowly towards equality with $v/γ$. This includes situations where the current solution branch of $F(v)=w-I$ ends in a fold bifurcation, so that $v$ has to jump to the other solution branch (which always exist as $F$ is cubic with negative leading coefficient).

As one can see in these plots with $ε=0.01$, solutions jump quickly to the cubic curve and then move in direction of the line of the second equation. If there is an intersection, it is a stable equilibrium. If there is no intersection before the minimum or maximum, the solution jumps to the other value of this level. If the line is arranged to pass below the minimum and above the maximum like in the third panel, a limit cycle gets established.