morris-lecar-2d

Morris-Lecar lattice model in Python and julia

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

Ref.: Morris, C., Lecar, H., Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193-213, 1981.

Rendered page: https://frederic-vw.github.io/morris-lecar-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)

Morris-Lecar model

The Morris-Lecar model uses two variables to model membrane potential dynamics in response to current injections, as modelled form experiments in the barnacle giant muscle fiber. The two variables represent voltage ($V$), and the fraction of open K⁺ channels. Spatial coupling is introduced via diffusion of the voltage variable ($D \Delta V$):

and the steady-state functions

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

The main function call running the simulation is: ml2d(N, T, t0, dt, s, D, gL, VL, gCa, VCa, gK, VK, C, I, V0, V1, V2, V3, V4, phi, 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
• gL,VL,gCa,VCa,gK,VK,C,V1,V2,V3,V4,phi: Morris-Lecar 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 = [ [[50,250], [0,5], [0,5]], [[6200,6700], [25,30], [0,15]] ]
• blocks = [ [[0,15], [5,10]] ] (works in Python, in Julia change 0 to 1)

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 = 10000, t0 = 1000, dt = 0.01, s = 2.0, D = 0.8, gL = 2.0, VL = -60.0, gCa = 5.0, VCa = 120.0, gK = 6.0, VK = -84, C = 5.0, V0 = -60.0, V1 = -1.2, V2 = 18.0, V3 = 12.0, V4 = 17.4, phi=1/15, I=90.0

Conclusions

The Morris-Lecar lattice model can produce spiral waves and other travelling waves.

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