izhikevich_cond_beta – Conductance-based Izhikevich neuron model

Description

Implementation of the simple spiking neuron model introduced by Izhikevich [1] with synaptic conductance modeled by a beta function, as described in [2]. The dynamics are given by:

\[\begin{split}\frac{dV_m}{dt} &= 0.04 V_m^2 + 5 V_m + 140 - u + I \\ \frac{du}{dt} &= a (b V_m - u))\end{split}\]

\[\begin{split}&\text{if}\;\;\; V_m \geq V_{th}:\\ &\;\;\;\; V_m \text{ is set to } c\\ &\;\;\;\; u \text{ is incremented by } d\\ & \, \\ &v \text{ jumps on each spike arrival by the weight of the spike}\end{split}\]

This implementation uses the standard technique for forward Euler integration. This model is multisynapse, so it allows an arbitrary number of synaptic rise time and decay time constants. The number of receptor ports must be specified at neuron creation (default value is 1) and the receptor index starts from 0 (and not from 1 as in NEST multisynapse models). The time constants are supplied by by two arrays, tau_rise and tau_decay for the synaptic rise time and decay time, respectively. The synaptic reversal potentials are supplied by the array E_rev. Port numbers are automatically assigned in the range from 0 to n_receptors-1. During connection, the ports are selected with the synapse property receptor.

Parameters

The following parameters can be set in the status dictionary.

V_m	mV	Membrane potential
u	mV	Membrane potential recovery variable
V_th	mV	Spike threshold
a	real	Describes time scale of recovery variable
b	real	Sensitivity of recovery variable
c	mV	After-spike reset value of V_m
d	mV	After-spike reset value of u
I_e	pA	Constant input current
t_ref	ms	Refractory time
den_delay	ms	Dendritic delay
E_rev	mV	Leak reversal potential
tau_rise	ms	Rise time constant of synaptic conductance
tau_decay	ms	Decay time constant of synaptic conductance
h_min_rel	real	Starting step in ODE integration relative to time resolution
h0_rel	real	Minimum step in ODE integration relative to time resolution

References

[1]

Izhikevich EM (2003). Simple model of spiking neurons. IEEE Transactions on Neural Networks, 14:1569-1572. DOI:

https://doi.org/10.1109/TNN.2003.820440

[2]

A. Roth and M. C. W. van Rossum, Computational Modeling Methods for Neuroscientists, MIT Press 2013, Chapter 6. DOI: https://doi.org/10.7551/mitpress/9780262013277.003.0007

See also

Neuron, Integrate-And-Fire