Hy guys,
I am trying to examine the effects of the parameter $a$ in the Izhikevich neuron model that is represented by 2 first order differential equations solved via numerical integration.
Model: \begin{align*} v'&= 0.04v^2 + 5v + 140 - u + I \\ u'&= a(bv - u) \end{align*}
$v,u$ are variables representing membrane voltage and recovery variable of the membrane voltage; $a,b$ are constants defining the dynamics of the neuron model. Initial values are usually $u=-10$, $v=-65$. $I$ represents incoming current, that in a network is again a function of $v$ (and the connections among the neurons).
The equations are usually solved with Euler integration or alternatively Runge-Kutta.
Izhikevich claims on the above site that $a$ determines the decay of $u$, and therefore a larger value results in faster spiking. However clearly the effects of $a$ depend on the sign of $v$ and $u$.
I did some simulations in MATLAB plotting $a$ vs. $v$ and saw all sorts of relationships (cubic mostly), but never linear. The results were also highly dependent on the Euler step size.
I would be super grateful for any hits to solve this problem analytically.
Cheers.