2.19. 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.

Out:

  0%|          | 0/100.0 [00:00<?, ?it/s]
Initializing:   0%|          | 0/100.0 [00:00<?, ?it/s]
  0%|          | 0/100.0 [00:00<?, ?it/s]
  0%|          | 0.23/100.0 [00:03<21:49, 13.12s/it]
  0%|          | 0.36/100.0 [00:03<13:57,  8.41s/it]
  2%|1         | 1.87/100.0 [00:03<02:43,  1.66s/it]
  7%|7         | 7.15/100.0 [00:03<00:43,  2.11it/s]
 17%|#7        | 17.21/100.0 [00:03<00:18,  4.40it/s]
 31%|###       | 30.99/100.0 [00:04<00:10,  6.68it/s]
 47%|####7     | 47.19/100.0 [00:05<00:06,  8.60it/s]
 65%|######4   | 64.75/100.0 [00:06<00:03, 10.06it/s]
 83%|########2 | 82.84/100.0 [00:07<00:01, 11.22it/s]
 83%|########2 | 82.84/100.0 [00:08<00:01, 10.01it/s]
100%|##########| 100.0/100.0 [00:08<00:00, 12.09it/s]
100%|##########| 100.0/100.0 [00:08<00:00, 12.09it/s]

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"):
        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()