The ODEs of the FitzHugh-Nagumo model, \(\dot{\vec x} = f(\vec x)\), are given by: \[ \begin{align} \dot{v}(t) & = v(t)\cdot \left( a-v(t)\right)\cdot\left(v(t)-1\right)-w(t)+I_{\mathrm{app}}, \\ \dot{w}(t) & = \epsilon\cdot\left(v(t)-\gamma\cdot w(t)\right) \end{align} \] where \(\mathbb{R} \ni t \mapsto v(t),w(t) \in \mathbb{R}\) and the parameters \((a, \epsilon, \gamma) \in \mathbb{R}_{+}^3\) are positive.
Write a function FN <- function(time, ini, pars) that can be passed as func argument to ode() of the package deSolve. Add two functions out2config <- function(out, parameter) and out2phase <- function(out, parameter) that arrange the output of ode() in a config or phase space data.frame.
ode() to integrate the system for \(a = 0.25\), \(\epsilon=0.002\), \(\gamma = 1.1\) and \(I_{\mathrm{app}}=0\) for different initial values \((v(0), w(0))\) varying \(v(0)\) around \(a = 0.25\) and keeping \(w(0) = 0\).
Write a function FNnc <- function(v, pars) returning the nullclines of the system \(w(v)\) for \(\dot{v}(t) = 0\) and \(w(v)\) for \(\dot{w}(t) = 0\). Add them to your phase-space plot.
Now test \(I_{app}>0\). Plot solutions in config space for Iapp = c(0, 0.02, 0.1, 0.5, 0.7) and use the nullclines to understand the behaviour in phase space. The initial value \((v(0), w(0))\) can be fixed to \((0, 0)\).
The southern portal dates from the Renaissance. What is the connection between the Renaissance portal, the reformation, the counter reformation and Erasmus from Rotterdam?