izhikevich_psc_alpha

izhikevich_psc_alpha - Detailed Izhikevich neuron model with alpha-kernel post-synaptic current

Description

Implementation of the simple spiking neuron model introduced by Izhikevich [1], with membrane potential in (milli)volt and current-based synapses.

The dynamics are given by:

\[ \begin{align}\begin{aligned}C_m \frac{dV_m}{dt} = k (V - V_t)(V - V_t) - u + I + I_{syn,ex} + I_{syn,in} \frac{dU_m}{dt} = a(b(V_m - E_L) - U_m)\\\begin{split}&\text{if}\;\;\; V_m \geq V_{th}:\\ &\;\;\;\; V_m \text{ is set to } c &\;\;\;\; U_m \text{ is incremented by } d\end{split}\end{aligned}\end{align} \]

On each spike arrival, the membrane potential is subject to an alpha-kernel current of the form:

\[I_syn = I_0 \cdot t \cdot \exp\left(-t/\tau_{syn}\right) / \tau_{syn}\]

References

Parameters

Name	Physical unit	Default value	Description
C_m	pF	200pF	Membrane capacitance
k	pF / (ms mV)	8pF / mV / ms	Spiking slope
V_r	mV	-65mV	Resting potential
V_t	mV	-45mV	Threshold potential
a	1 / ms	0.01 / ms	Time scale of recovery variable
b	nS	9nS	Sensitivity of recovery variable
c	mV	-65mV	After-spike reset value of V_m
d	pA	60pA	After-spike reset value of U_m
V_peak	mV	0mV	Spike detection threshold (reset condition)
tau_syn_exc	ms	0.2ms	Synaptic time constant of excitatory synapse
tau_syn_inh	ms	2ms	Synaptic time constant of inhibitory synapse
refr_T	ms	2ms	Duration of refractory period
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_m	mV	-65mV	Membrane potential
U_m	pA	0pA	Membrane potential recovery variable
refr_t	ms	0ms	Refractory period timer
is_refractory	boolean	false

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (k \cdot (V_{m} - V_{r}) \cdot (V_{m} - V_{t}) - U_{m} + I_{e} + I_{stim} + I_{syn,exc} - I_{syn,inh}) } \right)\]

\[\frac{ dU_{m} } { dt }= a \cdot (b \cdot (V_{m} - V_{r}) - U_{m})\]

Source code

The model source code can be found in the NESTML models repository here: izhikevich_psc_alpha.