$(5$ pts$)($Spike generation in Fitzhugh$-$Nagumo model$)$ A mathematical model of a bio$-$

logical neuron was first proposed by Hodgkin and Huxley $($they won the Nobel prize for

this work!$).$ Later on$,$ many simplified models were proposed. One popular simplified

neuron model is proposed by Fitzhugh and Nagumo, and the dynamics are written as

$V^{}=V-\frac{V^3}{3}-W+I$

$W^{}=V+a-bW$

where $V$ represents the neuron's membrane voltage, $W$ is an auxiliary 'recovery variable',

$a,b,$ are model parameters, and I is an external current input $($may be regarded as

control$).$ Take $a=1,b=1,=10,$ and initial conditions $(V,W)(0)=(1,0).$

$($a$)(1$ pt$)$ Assuming no external current $)=(0,$ calculate all $($real$)$ equilibrium points

of the system.

$($b$)(2pt)$ Simulate the nonlinear dynamics $(2)$ in Matlab and plot the variables $V(t)$

and $W(t)$ in the time interval $0,100.$ Take all the parameters as listed above and

$I=0.$ Does your solution go to the equilibrium as found in $($a$)?$

$($c$)(2pt)$ Now change the input current to $I=1.$ Then repeat $($b$).$ Does your solution

still go to an equilibrium? If not, what do you observe?