.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_gallery/pde_coupled.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_gallery_pde_coupled.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_gallery_pde_coupled.py:


Custom Class for coupled PDEs
=============================

This example shows how to solve a set of coupled PDEs, the
spatially coupled `FitzHugh–Nagumo model 
<https://en.wikipedia.org/wiki/FitzHugh–Nagumo_model>`_, which is a simple model
for the excitable dynamics of coupled Neurons:

.. math::

    \partial_t u &= \nabla^2 u + u (u - \alpha) (1 - u) + w \\
    \partial_t w &= \epsilon u

Here, :math:`\alpha` denotes the external stimulus and :math:`\epsilon` defines
the recovery time scale. We implement this as a custom PDE class below.

.. GENERATED FROM PYTHON SOURCE LINES 18-48



.. image-sg:: /examples_gallery/images/sphx_glr_pde_coupled_001.png
   :alt: pde coupled
   :srcset: /examples_gallery/images/sphx_glr_pde_coupled_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

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.. code-block:: Python


    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()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 8.296 seconds)


.. _sphx_glr_download_examples_gallery_pde_coupled.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: pde_coupled.ipynb <pde_coupled.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: pde_coupled.py <pde_coupled.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: pde_coupled.zip <pde_coupled.zip>`