Question

Write a program that simulates a cable of cells (1D array, 100 cells) that all individually have standard cubic FHN dynamic with k=4,ε=0.01,α=0.2. Couple the cells by diffusion of the activator u with diffusion constant D (like we discussed in class).

a) Stimulate one end of the cable (set the first 5 cells to u=0.8 ). What is the minimum value Dc of D that you need to see propagation?

Answer 1

The Python program simulates a 1D array of coupled cells using standard cubic FHN dynamics with diffusion. The minimum diffusion constant, Dc, required for propagation is determined and marked on the plot.

Creating a simulation for this scenario involves solving the FitzHugh-Nagumo (FHN) equations for each cell in the 1D array and incorporating diffusion for the activator u. Below is a Python program using the Euler method for numerical integration:

import numpy as np

import matplotlib.pyplot as plt

# Parameters

k = 4

epsilon = 0.01

alpha = 0.2

D_values = np.linspace(0.01, 0.5, 100) # Diffusion constants to test

num_cells = 100

num_steps = 1000

dt = 0.01

# Initial conditions

u = np.zeros((num_cells,))

u[:5] = 0.8 # Stimulate the first 5 cells

# Function to update FHN equations with diffusion

def update_fhn(u, D):

du = np.zeros_like(u)

dv = np.zeros_like(u)

for i in range(1, num_cells - 1):

du[i] = epsilon * (u[i] - u[i] ** 3 / 3 - u[i] ** 2 * u[i + 1] / 6 + u[i] * u[i + 1] ** 2 / 6 + u[i] ** 2 * u[i - 1] / 6 - u[i - 1] ** 2 * u[i] / 6)

du[i] += D * (u[i + 1] - 2 * u[i] + u[i - 1])

u += du * dt

return u

# Simulation loop for different diffusion constants

for D in D_values:

u = np.zeros((num_cells,))

u[:5] = 0.8 # Stimulate the first 5 cells

for _ in range(num_steps):

u = update_fhn(u, D)

# Check for propagation (finding the minimum Dc)

if np.max(u) > 0.8:

Dc = D

break

# Plotting

plt.plot(D_values, np.zeros_like(D_values), label='Stable')

plt.scatter(Dc, 0, color='red', label=f'Dc = {Dc:.3f}', marker='x')

plt.xlabel('Diffusion Constant (D)')

plt.title('Propagation Threshold')

plt.legend()

plt.show()

This program initializes a cable of cells, stimulates the first 5 cells, and iteratively updates the FHN equations with diffusion. The minimum value `Dc` for which propagation is observed is identified and marked on the plot.

Note: This is a simplified simulation, and you may need to adjust parameters and the update function for a more accurate representation based on your specific requirements.