fitzhugh-nagumo-2d

FitzHugh-Nagumo lattice model in Python and julia

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

Ref.: FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445-466, 1961.

Rendered page: https://frederic-vw.github.io/fitzhugh-nagumo-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)

FitzHugh-Nagumo model

The FHN model is a two-variable system for the abstract representation of action potential as those found in neurons or cardiac myocytes. The voltage-like variable $v$ and the recovery variable $w$ implement depolarization current ($I$) induced spiking and a post-spike refractory behaviour.
Spatial coupling is introduced via diffusion of the voltage-like variable:

Noise is added via Itô-integration:

The main function call running the simulation is: fhn2d(N, T, t0, a, b, c, I, s, D, dt, 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: FHN model parameters, I: 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.

Examples 1, 2 use

• stim = [ [[25,50], [1,N], [3,8]], [[130,150], [n_2-2,n_2+2], [10,25]] ]
• blocks = [ [[2*n_4,3*n_4], [15,20]], [[2*n_4+10,3*n_4+10], [40,45]] ] where n_2 is the (rounded) integer N/2, n_4 is the (rounded) integer N/4.

Example-1

Stimulation near the left border generates a travelling wave which breaks at and merges after the conduction blocks. After-depolarization does not change the qualitative behaviour of a travelling wave. Parameters:
N = 128, T = 1000, t0 = 0, dt = 0.1, s = 0.02, D = 1.0, a = 0.5, b = 0.7, c = 0.3, I = 0.5

Example-2

Increasing the stimulation current while maintaining all other variables constant generates functional re-entry (spiral waves) originating where the 2nd stimulus hits repolarizing tissue. Parameters:
N = 128, T = 1000, t0 = 0, dt = 0.1, s = 0.02, D = 1.0, a = 0.5, b = 0.7, c = 0.3, I = 1.0

Example-3

Without a stimulus current (stim = []), a large background noise can generate re-entry patterns. Parameters on the left:
N = 128, T = 1000, t0 = 0, dt = 0.1, sd = 0.10, D = 1.0, a = 0.5, b = 0.7, c = 0.3, I = 0.5
On the right, with reduced diffusion constant D=0.25, other parameters identical.

Conclusions

The FHN lattice can produce:

• stimulation-induced travelling waves
• stimulation-induced spiral waves
• noise-induced spiral waves

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