Simulation of the Izhikevich model

Izhikevic developed a two-dimensional model that includes a variety of response patterns seen in real neurons. The subthreshold phase is modelled by the following system of non-linear differential equations: \[\begin{aligned}\frac{{dV}}{{dt}} &= 0.04\,{V^2} + 5\,V + 140 - U + {I_{{\rm{stim}}}}\\ \frac{{dU}}{{dt}} &= a \cdot (b \cdot V - U)\end{aligned}\] where \(V\) the membrane potential, \(U\) is the recovery variable, and \(I_\mathrm{stim}\) is the stimulus. The parameter \(a\) describes the time scale of the recovery variable and the parameter \(b\) describes the senitivity of the recovery variable to the subthreshold fluctuations of the membrane potential. When the membrane potential \(V\) reaches a threshold value it is reset to the value \(c\) and the recovery variable \(U\) is increased with the value of \(d\): \[V(t + \delta ) \rightarrow c\quad {\rm{and}}\quad U(t + \delta ) \rightarrow U + d\quad {\rm{when}}\quad \delta \rightarrow 0\]