Dynamical Systems in Biology

Exercise 1:

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.

• Use 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 $w(0) = 0$.
  • Plot solutions in configuration space.
  • Plot solutions in phase space.

• 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 $(0, 0)$.

Cathedral exercise

The southern portal dates from the Renaissance. What is the connection between the Renaissance portal, the reformation, the counter reformation and Erasmus from Rotterdam?