Note
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2.20. Custom Class for coupled PDEs
This example shows how to solve a set of coupled PDEs, the spatially coupled FitzHugh–Nagumo model, which is a simple model for the excitable dynamics of coupled Neurons:
\[\begin{split}\partial_t u &= \nabla^2 u + u (u - \alpha) (1 - u) + w \\
\partial_t w &= \epsilon u\end{split}\]
Here, \(\alpha\) denotes the external stimulus and \(\epsilon\) defines the recovery time scale. We implement this as a custom PDE class below.

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from pde import FieldCollection, PDEBase, UnitGrid
class FitzhughNagumoPDE(PDEBase):
"""FitzHugh–Nagumo model with diffusive coupling"""
def __init__(self, stimulus=0.5, τ=10, a=0, b=0, bc="auto_periodic_neumann"):
super().__init__()
self.bc = bc
self.stimulus = stimulus
self.τ = τ
self.a = a
self.b = b
def evolution_rate(self, state, t=0):
v, w = state # membrane potential and recovery variable
v_t = v.laplace(bc=self.bc) + v - v**3 / 3 - w + self.stimulus
w_t = (v + self.a - self.b * w) / self.τ
return FieldCollection([v_t, w_t])
grid = UnitGrid([32, 32])
state = FieldCollection.scalar_random_uniform(2, grid)
eq = FitzhughNagumoPDE()
result = eq.solve(state, t_range=100, dt=0.01)
result.plot()
Total running time of the script: (0 minutes 10.745 seconds)