hindmarsh-rose-2d

Hindmarsh-Rose lattice model in Python and julia

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This repo contains implementations of the Hindmarsh-Rose model of neuronal excitability on a 2D lattice.
The code is available in Python and Julia.

Ref.: Hindmarsh, J.L., Rose, R.M., A model of neuronal bursting using three coupled first order differental equations. Proc R Soc Lond B 221:87-102, 1984.

Rendered page: https://frederic-vw.github.io/hindmarsh-rose-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)

Hindmarsh-Rose model

The Hindmarsh-Rose model uses three variables to model membrane potential dynamics in response to current injections. The model is an extension of the FitzHugh-Nagumo model. The third variable of the Hindmarsh-Rose model implements slow inactivation dynamics of the membrane potential and allows to model neuronal bursting. Spatial coupling is introduced via diffusion of the voltage variable ($D \Delta V$):

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

The main function call running the simulation is: hr2d(N, T, t0, dt, sd, D, a, b, c, d, s, r, x_1, I0, stim, blocks):

• N: lattice size (N,N)
• T: number of simulation time steps
• t0: number of ‘warm-up’ iterations
• dt: integration time step
• sd: noise intensity (σ)
• D: diffusion constant
• a,b,c,d,s,r,x_1: Hindmarsh-Rose model parameters, I0: stimulation current amplitude
• 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 = [ [[0,15000], [5,N-5], [5,N-5]] ]
• blocks = [] (works in Python, in Julia change 0 to 1)

Example-1

Stimulation of a large block of the simulation area generates a depolarization in that area in which different wave patterns develop, including spiral waves. Parameters:
N = 64, T = 15000, t0 = 2500, dt = 0.05, sd = 0.05, D = 2.0, a = 1.0, b = 3.0, c = 1.0, d = 5.0, s = 4.0, r = 0.001, x_1 = -1.6, I0 = 3.5

Conclusions

The Hindmarsh-Rose lattice model can produce multiple wave patterns.

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