The FitzHugh-Nagumo (FHN) model is a simplified mathematical model that describes the behavior of excitable systems, such as neurons. It is a two-dimensional reduction of the more complex Hodgkin-Huxley model, which details the electrical activity of nerve cells. The FHN model captures the essential features of neuronal excitation, including the generation of action potentials (spikes) in response to a stimulus.

First, recall the FitzHugh-Nagumo model (in a standard nondimensional form):

$$ \begin{aligned} \dot{v} & =v-\frac{v^3}{3}-w+I \\ \dot{w} & =\epsilon(v+a-b w) \end{aligned} $$

where:

• $v=$ voltage-like variable,
• $w=$ recovery-like variable,
• $\epsilon \ll 1$ = small parameter (slow recovery),
• $a, b$ are constants,
• $I=$ external input current (the parameter you are varying).

Step 1: Find the fixed point

Fixed points are where $\dot{v}=0$ and $\dot{w}=0$. Set:

$$ \begin{gathered} v-\frac{v^3}{3}-w+I=0 \quad \text { (1) } \\ \epsilon(v+a-b w)=0 \Rightarrow v+a-b w=0 \quad \Rightarrow \quad w=\frac{v+a}{b} \end{gathered} $$

Substitute (2) into (1):

$$ v-\frac{v^3}{3}-\frac{v+a}{b}+I=0 $$

Rearranged:

$$ v-\frac{v^3}{3}-\frac{v}{b}-\frac{a}{b}+I=0 $$

Or:

$$ -\frac{v^3}{3}+v\left(1-\frac{1}{b}\right)+\left(I-\frac{a}{b}\right)=0 $$

Thus, finding $v$ amounts to solving this cubic equation depending on $I$.

Step 2: Linearize around the fixed point