Spiking neuron model II

Model description

The general dynamics of the model of spiking neurons is defined by the following ODE.

     \[  \left\{  \begin{array}{rcl} C\cdot \dot{v} & = & k\cdot (v - v_r) \cdot (v - v_t) - u + I \\ \dot{u} & = & a\cdot (b\cdot (v - v_r) - u) \end{array}  \right. \]

wherein the constant parameters are given by  C = 100 ,  v_r = -56 ,  v_t = -42 ,  I = 300 ,  a = 0.03 ,  b = 8 and  k = 1. The values of  v ,  u are reset to  -53 + 0.04\cdot u and  u + 20 respectively when  v \geq 40 - 0.1\cdot u .

Reachability setting

We consider the initial set defined by  v \in [-50.5,-49.5] ,  u \in [-0.5,0.5] .

Result

The following figures show an overapproximation computed by Flow* for the time horizon  [0,200] .

neurons_II_v_u

neurons_II_t_v

References

[1] E. Izhikevich. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press, 2010.