Implementing Izhikevich's spiking neuron model

Question

I am trying to implement Izhikevich's simple spiking neuron model which is given by

V'(t) = 0.04 V^2+5V+140-u+I
    u'(t) = a(bv-u)
    if V=30 mV then V=c, u=u+d

I am using a=0.02, b=0.2, c=-65, d=8

I can solve the coupled equations using NDSolve with the initial conditions of V and u, which gives me the interpolating functions for V(t) and u(t) and then I can plot them within a time window. But my problem is implementing the "If" condition within the boundary condition of NDSolve. Could someone please help me to figure out how to implement that If condition within NDSolve? Another problem is, how to then apply the resulting spiking neuron signals to a system? For example, in Simulink we can simply connect a signal source to a system. Let's say here I find the transfer function of my system where I want to apply the spiking neuron as the input. Typically I we use Outputresponse in mathematica to get output of the system against a standard signal input. But in this case, is there a way to use a similar method to get the system's response with the derived spiking neuron train as input? Thanks

Answer 1

We can use input data from the page. Tonic Spiking model can be implemented as follows

{a, b, c, d} = {0.02, 0.40, -65.0, 2.0}; 
i[t_] := 10 UnitStep[t - 50];
sol = NDSolve[{v'[t] == .04 v[t]^2 + 5 v[t] + 140 - u[t] + i[t], 
   u'[t] == a (b v[t] - u[t]), u[0] == 0, v[0] == 18, 
   WhenEvent[v[t] == 30, {v[t] -> c, u[t] -> u[t] + d}]}, {u, v}, {t, 
   0, 200}]

Visualization

Plot[{v[t] /. sol[[1]], i[t]}, {t, 0, 200}, PlotRange -> All, 
 Frame -> True, FrameLabel -> Automatic, PlotPoints -> 200, 
 PlotStyle -> {Blue, Red}, PlotLegends -> {"v(t)", "I(t)"}]