Spiking neurons | Theory of Hybrid Systems

Classification

# of variables	# of modes	# of jumps
2	1	1

Type	Continuous dynamics	Guards & Invariants	Resets
hybrid	linear polynomial	linear polynomial	linear polynomial

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Flow* model I	neuron_I.model
Flow* model II	neuron_II.model

Description of model I

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 $C = 100$ , $v_r = -60$ , $v_t = -40$ , $I = 70$ , $a = 0.03$ and $b = -2$ . The value of $k$ is $0.7$ when $v \leq v_t$ , otherwise it is $7$ . Whenever the value of $v$ reaches $35$ , its value is reset to $-50$ and meanwhile $u$ is updated to $u + 100$ .

Reachability settings for model I

We consider the initial set defined by $v \in [-61,-59]$ , $u \in [-1,1]$ .

Results for model I

The following figures show an overapproximation computed by Flow* for the time horizon $[0,1000]$ .

Description of model II

As the second example, 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 $v \geq 40 - 0.1\cdot u$ .