izhikevich-2d

Izhikevich lattice model in Python and julia

This repo contains implementations of the Izhikevich model of cellular excitability on a 2D lattice.
The code is available in Python and Julia.

References:
Izhikevich, E., Simple model of spiking neurons. IEEE Transactions on Neural Networks 14(6):1569-1572, 2003.
Izhikevich, E., Which model to use for cortical spiking neurons? IEEE Transactions on Neural Networks 15(5):1063-1070, 2004.

Rendered page: https://frederic-vw.github.io/izhikevich-2d/

Python Requirements:

1. python installation, latest version tested 3.6.9
2. python packages (pip install package-name)
   • NumPy
   • Matplotlib
   • opencv-python (save data as movie)

Julia Requirements:

1. julia installation, latest version tested 1.6.1
2. julia packages (julia > using Pkg; Pkg.add("Package Name"))
   • NPZ
   • PyCall (load Matplotlib animation functionality)
   • PyPlot
   • Statistics
   • VideoIO (save data as movie)

Izhikevich model

The Izhikevich model uses two variables to model membrane potential dynamics in response to current injections. The variable v represents voltage, u is a recovery variable, modelling K⁺ and other currents. The free parameters (a, b, c, d, v0, vpeak) can produce a variety of spiking, bursting and other interesting firing patterns.
Spatial coupling is introduced via diffusion of the voltage-like variable:

When the voltage threshold is exceeded, $v \ge v_{peak}$, the following reset/update is performed:

Noise is added via stochastic integration of the variable $v$:

The main function call running the simulation is: izh2d(N, T, t0, dt, s, D, a, b, c, d, v0, vpeak, I, stim, blocks):

• N: lattice size (N,N)
• T: number of simulation time steps
• t0: number of ‘warm-up’ iterations
• dt: integration time step
• s: noise intensity (σ)
• D: diffusion constant
• a,b,c,d,vpeak: Izhikevich model parameters, I: stimulation current amplitude
• v0: initial membrane voltage
• stim: stimulation current parameters, array of time-, x-, and y-interval borders
• blocks: conduction blocks, array of x- and y-interval borders

Outputs: (T,N,N) array as NumPy .npy format and as .mp4 movie.

In example 1, use

• stim = [ [[1,550], [1,5], [1,10]], [[2400,2800], [45,50], [1,30]] ]
• blocks = [ [[1,20], [10,15]] ] (Julia parameters, in Python change 1 to 0)

Example-1

Stimulation in the lower left corner generates a travelling wave along a narrow channel formed by the left boundary and the conduction block defined by blocks, an early after-depolarization within the partially refractory region generates a spiral wave. Parameters:
N = 128, T = 15000, t0 = 500, dt = 0.05, s = 1.0, D = 0.075, a = 0.02, b = 0.2, c = -50, d = 2.0, v0 = -70, vpeak = 30, I = 15

Conclusions

The Izhikevich lattice model using bursting neurons can produce spiral waves and other travelling waves.

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