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.

  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%|          | 0.24/100.0 [00:00<01:31,  1.08it/s]
  1%|          | 0.74/100.0 [00:00<00:35,  2.83it/s]
  3%|3         | 3.22/100.0 [00:00<00:13,  7.19it/s]
  9%|8         | 8.99/100.0 [00:00<00:08, 10.18it/s]
 18%|#7        | 17.72/100.0 [00:01<00:07, 10.71it/s]
 28%|##7       | 27.66/100.0 [00:02<00:06, 11.20it/s]
 39%|###8      | 38.66/100.0 [00:03<00:05, 11.67it/s]
 51%|#####     | 50.66/100.0 [00:04<00:04, 11.61it/s]
 62%|######2   | 62.36/100.0 [00:05<00:03, 11.85it/s]
 75%|#######4  | 74.69/100.0 [00:06<00:02, 11.88it/s]
 87%|########6 | 86.87/100.0 [00:07<00:01, 11.92it/s]
 99%|#########9| 99.07/100.0 [00:08<00:00, 12.01it/s]
 99%|#########9| 99.07/100.0 [00:08<00:00, 11.89it/s]
100%|##########| 100.0/100.0 [00:08<00:00, 12.01it/s]
100%|##########| 100.0/100.0 [00:08<00:00, 12.01it/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"):
        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()