izhikevich

izhikevich - Izhikevich neuron model

Description

Implementation of the simple spiking neuron model introduced by Izhikevich 1. The dynamics are given by:

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

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

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

1. the new value of \(u\) is calculated based on the new value of \(v\), rather than the previous value

2. the variable \(v\) is updated using a time step half the size of that used to update variable \(u\).

This model will instead be simulated using the numerical solver that is recommended by ODE-toolbox during code generation.

References

1(1,2)

Izhikevich, Simple Model of Spiking Neurons, IEEE Transactions on Neural Networks (2003) 14:1569-1572

Parameters

Name	Physical unit	Default value	Description
a	real	0.02	describes time scale of recovery variable
b	real	0.2	sensitivity of recovery variable
c	mV	-65mV	after-spike reset value of V_m
d	real	8.0	after-spike reset value of U_m
V_m_init	mV	-65mV	initial membrane potential
V_min	mV	-inf * mV	Absolute lower value for the membrane potential.
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_m	mV	V_m_init	Membrane potential
U_m	real	b * V_m_init	Membrane potential recovery variable

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\frac{ 0.04 \cdot V_{m} \cdot V_{m} } { \mathrm{mV} } + 5.0 \cdot V_{m} + (140 - U_{m}) \cdot \mathrm{mV} + ((I_{e} + I_{stim}) \cdot \mathrm{GOhm})) } \right)\]

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

Source code

neuron izhikevich:
  state:
    V_m mV = V_m_init # Membrane potential
    U_m real = b * V_m_init # Membrane potential recovery variable
  end
  equations:
    V_m'=(0.04 * V_m * V_m / mV + 5.0 * V_m + (140 - U_m) * mV + ((I_e + I_stim) * GOhm)) / ms
    U_m'=a * (b * V_m - U_m * mV) / (mV * ms)
  end

  parameters:
    a real = 0.02 # describes time scale of recovery variable
    b real = 0.2 # sensitivity of recovery variable
    c mV = -65mV # after-spike reset value of V_m
    d real = 8.0 # after-spike reset value of U_m
    V_m_init mV = -65mV # initial membrane potential
    V_min mV = -inf * mV # Absolute lower value for the membrane potential.
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  input:
    spikes mV <-spike
    I_stim pA <-current
  end

  output: spike

  update:
    integrate_odes()
    # Add synaptic current

    # Add synaptic current
    V_m += spikes
    # lower bound of membrane potential
    V_m = (V_m < V_min)?V_min:V_m
    # threshold crossing
    if V_m >= 30mV:
      V_m = c
      U_m += d
      emit_spike()
    end
  end

end

Characterisation

Synaptic response

f-I curve