.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/excitable/plot_01_fitzhugh_nagumo.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_ode_dynamics_excitable_plot_01_fitzhugh_nagumo.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_excitable_plot_01_fitzhugh_nagumo.py:


FitzHugh-Nagumo: excitability and spiking
===============================================

With no injected current the model neuron rests at a stable fixed point.
Between the two Hopf currents that fixed point is unstable, and the
neuron fires repetitively on a relaxation limit cycle that runs along
the outer branches of the cubic :math:`v`-nullcline.

.. GENERATED FROM PYTHON SOURCE LINES 12-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.excitable import FitzHughNagumo, fitzhugh_nagumo_hopf_currents

    i_low, i_high = fitzhugh_nagumo_hopf_currents()
    print(f"rest state unstable for {i_low:.4f} < I < {i_high:.4f}")





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

 .. code-block:: none

    rest state unstable for 0.3313 < I < 1.4187




.. GENERATED FROM PYTHON SOURCE LINES 21-23

Phase plane and voltage traces
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-45

.. code-block:: Python


    v = np.linspace(-2.5, 2.5, 300)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    for current, color in ((0.0, "tab:blue"), (0.5, "tab:red")):
        system = FitzHughNagumo([-1.0, 1.0], current=current)
        traj = system.integrate((0.0, 200.0), dt=1e-2, method="rk4")
        axes[0].plot(traj.y[:, 0], traj.y[:, 1], color=color, lw=1, label=f"I={current}")
        axes[0].plot(v, v - v**3 / 3 + current, color=color, ls=":", lw=1)
        axes[1].plot(traj.t, traj.y[:, 0], color=color, label=f"I={current}")
    axes[0].plot(v, (v + system.a) / system.b, "k--", lw=1, label="w-nullcline")
    axes[0].set_ylim(-1, 2.5)
    axes[0].set_xlabel("v")
    axes[0].set_ylabel("w")
    axes[0].set_title("Nullclines (dotted: v-nullcline) and orbits")
    axes[0].legend()
    axes[1].set_xlabel("t")
    axes[1].set_ylabel("v")
    axes[1].set_title("Rest (I=0) vs. periodic spiking (I=0.5)")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/excitable/images/sphx_glr_plot_01_fitzhugh_nagumo_001.png
   :alt: Nullclines (dotted: v-nullcline) and orbits, Rest (I=0) vs. periodic spiking (I=0.5)
   :srcset: /api/gallery/ode_dynamics/excitable/images/sphx_glr_plot_01_fitzhugh_nagumo_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.591 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_excitable_plot_01_fitzhugh_nagumo.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_fitzhugh_nagumo.ipynb <plot_01_fitzhugh_nagumo.ipynb>`

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

      :download:`Download Python source code: plot_01_fitzhugh_nagumo.py <plot_01_fitzhugh_nagumo.py>`

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

      :download:`Download zipped: plot_01_fitzhugh_nagumo.zip <plot_01_fitzhugh_nagumo.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_