izhikevich – Izhikevich neuron model

Description

izhikevich implements the simple spiking neuron model introduced by Izhikevich [1]. This model reproduces spiking and bursting behavior of known types of cortical neurons.

Membrane potential evolution, spike emission, and refractoriness

The model is defined by the following differential equations:

\[\frac{dV_{\text{m}}}{dt} = 0.04 V_{\text{m}}^2 + 5 V_{\text{m}} + 140 - U_{\text{m}} + I_{\text{e}}\]

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

where \(V_{\text{m}}\) is the membrane potential, \(U_{\text{m}}\) is the recovery variable, and \(I_{\text{e}}\) is the input current.

A spike is emitted when \(V_{\text{m}}\) reaches a threshold \(V_{\text{th}}\). At this point, the membrane potential and recovery variable are updated according to:

\[\begin{split}&\text{if}\;\ V_m \geq V_{th}:\\ &\;\ V_m \text{ is set to } c\\ &\;\ U_m \text{ is incremented by } d\\\end{split}\]

In addition, each incoming spike increases \(V_{\text{m}}\) by the synaptic weight associated with the spike.

As published in [1], the numerics differs from the standard forward Euler technique in two ways:

• the recovery variable \(U_{\text{m}}\) is updated based on the new value of \(V_{\text{m}}\), rather than the previous one.

• the membrane potential \(V_{\text{m}}\) is updated with a time step half the size of that used for \(U_{\text{m}}\).

This model offers both forms of integration, they can be selected using the boolean parameter consistent_integration:

• consistent_integration = false: use the published form of the dynamics (for replicating published results).

• consistent_integration = true (default): use the standard Euler method (recommended for general use).

Note

For a detailed analysis of the numerical differences between these integration schemes and their impact on simulation results, see [2].

Parameters

The following parameters can be set in the status dictionary.

Parameter	Default	Math equivalent	Description
V_m	-65 mV	\(V_{\text{m}}\)	Membrane potential
U_m	-13 mV	\(U_{\text{m}}\)	Membrane potential recovery variable
V_th	30 mV	\(V_{\text{th}}\)	Spike threshold
I_e	0 pA	\(I_{\text{e}}\)	Constant input current (R=1)
V_min	-1.79 mV	\(V_{\text{min}}\)	Absolute lower value for the membrane potential
a	0.02 real	\(a\)	Describes time scale of recovery variable
b	0.2 real	\(b\)	Sensitivity of recovery variable
c	-65 mV	\(c\)	After-spike reset value of V_m
d	8 mV	\(d\)	After-spike reset value of U_m
consistent_integration	true	None	Use standard integration technique

References

[1] (1,2)

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]

Pauli R, Weidel P, Kunkel S, Morrison A (2018). Reproducing polychronization: A guide to maximizing the reproducibility of spiking network models. Frontiers in Neuroinformatics, 12. DOI: https://www.frontiersin.org/article/10.3389/fninf.2018.00046

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

See also

iaf_psc_delta, mat2_psc_exp

Examples using this model

None