English: ```python
from matplotlib.widgets import AxesWidget
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
from scipy import optimize
for eps in [0.2]:
# Define the parameter values
a = 0.7
b = 2.0 # If b < 1.5, then there are stable loops. Else there are no loops.
tau = 12.5
R = 0.1
I_ext = ((a-1)/b + 2/3)/R + eps
# Define the system of ODEs
def system(t, y):
v, w = y
dv = v - (v ** 3) / 3 - w + R * I_ext
dw = (1 / tau) * (v + a - b * w)
return [dv, dw]
def system_reversed(t, y):
v, w = y
dv = v - (v ** 3) / 3 - w + R * I_ext
dw = (1 / tau) * (v + a - b * w)
return [-dv, -dw]
vmin, vmax, wmin, wmax = -2, 2, -2+R*I_ext, 2+R*I_ext
t_span = [0, 100]
trajectory_resolution = 10
def fun(x):
v = x[0]
return v-v**3/3 + R * I_ext - (v+a)/b
sol = optimize.root(fun, [0], method='hybr')
x_root = sol.x[0]
y_root = (x_root+a)/b
# vmin, vmax, wmin, wmax = -1.5, -0.5, -1.1 +1/3 + R * I_ext, -0.8 +1/3 + R * I_ext
# initial_conditions = [(-1.0, y) for y in np.linspace(-0.16, -0.03, 30)]
initial_conditions = [(x, y) for x in np.linspace(vmin, vmax, trajectory_resolution) for y in np.linspace(wmin, wmax, trajectory_resolution)]
epsilon = 0.005
initial_conditions += [(x, y) for x in np.linspace(x_root - epsilon, x_root+epsilon, trajectory_resolution) for y in np.linspace(y_root - epsilon, y_root+epsilon, trajectory_resolution)]
sols = {}
for ic in initial_conditions:
sols[ic] = solve_ivp(system, t_span, ic, dense_output=True, max_step=0.1)
sols_reversed = {}
for ic in initial_conditions:
sols_reversed[ic] = solve_ivp(system_reversed, t_span, ic, dense_output=True, max_step=0.1)
vs = np.linspace(vmin, vmax, 200)
v_axis = np.linspace(vmin, vmax, 20)
w_axis = np.linspace(wmin, wmax, 20)
v_values, w_values = np.meshgrid(v_axis, w_axis)
dv = v_values - (v_values ** 3) / 3 - w_values + R * I_ext
dw = (1 / tau) * (v_values + a - b * w_values)
fig, ax = plt.subplots(figsize=(16,16))
# integral curves
for ic in initial_conditions:
sol = sols[ic]
ax.plot(sol.y[0], sol.y[1], color='k', alpha=0.4, linewidth=0.5)
sol = sols_reversed[ic]
ax.plot(sol.y[0], sol.y[1], color='k', alpha=0.4, linewidth=0.5)
# vector fields
arrow_lengths = np.sqrt(dv**2 + dw**2)
alpha_values = 1 - (arrow_lengths / np.max(arrow_lengths))**0.4
ax.quiver(v_values, w_values, dv, dw, color='blue', linewidth=0.5, scale=25, alpha=alpha_values)
# nullclines
ax.plot(vs, vs - vs**3/3 + R * I_ext, color="green", alpha=0.4, label="v nullcline")
ax.plot(vs, (vs + a) / b, color="red", alpha=0.4, label="w nullcline")
# ax.set_xlabel('v')
# ax.set_ylabel('w')
ax.set_title(f'FitzHugh-Nagumo Model\n$b={b:.2f}$\t\t$I_
= {I_ext:.2f
})