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:

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

See also

izhikevich, iaf_psc_alpha

References

Parameters

Name

Physical unit

Default value

Description

C_m

pF

200pF

Membrane capacitance

k

pF / (mV ms)

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

t_ref

ms

2ms

Refractory period

I_e

pA

0pA

constant external input current

State variables

Name

Physical unit

Default value

Description

r

integer

0

Number of steps in the current refractory phase

V_m

mV

-65mV

Membrane potential

U_m

pA

0pA

Membrane potential recovery variable

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.

Characterisation