# 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()

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