The HH model is non linear and does not possess an explicit solution. As more and more ionic conductances are added, the level of complexity increases dramatically and only numerical methods are possible.

Much of the richness of the HH model comes from the presence of a cubic non-linearity and monotonic recovery function (????). In 1961, Dick Fitzhugh proposed a reduced model that grasp the oscillatory quality of the HH at the expense of many physiological details. The Fitzhugh-Nagumo model is given by the following set of ODEs:

 

dVdt=V−V3/3−w+Idwdt=ϵ(V+a−bw)

 

where V is the membrane voltage and w is a slow recovery variable representing the inactivation of ion channels - namely some form of neural fatigue that prevents the neuron from staying permanently depolarized. The parameters a, ϵ and γ are parameters that are oftentimes fit to data and do not possess a clear biological meaning. The additional term I represents the input - either from an experimental electrode or other neurons.

The Fitzhugh-Nagumo model is very useful to understand the dynamical "machinery" (i.e. the fundamental processes) at play in action potential generation, and is furthermore amenable to analytics (i.e. it is mathematically tractable).

Let us consider the following parameters: a=0.7,b=0.8 and ϵ=0.3.

a)(0.5 point) Find the nullclines of the FHN model above for I=0. (HINT: Instead of solving for V, solve for w)#type the answer as a commentary in the code cell below

b)(0.5 point) Plot the nullclines in the (V,w) plane for V and w ranging from -4 to 4. Dont forget axes labels and title! (HINT: use the commands plt.ylim(-4,4) and plt.xlim(-4,4)).

c)(0.5 point)Using the Euler method, numerically integrate the FHN model for I=0 and plot the solution V(t) as a function of time over the interval 0 to 100ms with steps of dt=0.01. Initial conditions are V(0)=0.1 and w(0)=0.1.

d)(0.5 point) Using the code you have written in c), examine the effect of gradually increasing the input to the neuron I from 0 to 1.2 by steps of 0.1. What do you notice? What is going on?#type the answer as a commentary in the code cell below

e)(1 point) Plot the flow in phase space alongside the nullclines calculated in b) (HINT: As seen in class, all that is needed is to integrate the system for a small amount of time, say 0.1 with steps of dt=0.01. Using two embedded for loops, you can simply change the initial conditions and plot the trajectories in (V,w) space iteratively). What is the effect of changing I on the flow and on the nullclines?