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

.. only:: html

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

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

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

.. _sphx_glr_auto_models_lif.py:


Leaky Integrate-and-Fire Spiking Neuron Model
=============================================

The `leaky integrate-and-fire <https://en.wikipedia.org/wiki/Biological_neuron_model#Leaky_integrate-and-fire>`_ (LIF)
neuron model is the most commonly used spiking neuron model.
It has a single state-variable :math:`v_i`, the dynamics of which are governed by

.. math::

    \dot v_i = -\frac{v_i}{\tau} + I_i(t) + k s_i^{in}.

The two constants governing the dynamics of :math:`v_i` are the global decay time constant :math:`\tau` and the global
coupling constant :math:`k`. Any time :math:`v_i \geq v_{peak}`, a spike is counted and the reset condition
:math:`v_i \rightarrow v_{reset}` is applied. This introduces a discontinuity around the spike event to the dynamics
of :math:`v_i`, and makes the state variable similar to the membrane potential of a neuron close to the soma.
Spikes or spike-driven synapses should be used as input to other neurons in an RNN.
Here, we use a spike-driven synaptic activation :math:`s_i` with the following dynamics:

.. math::

    \tau_s \dot s_i = -\frac{s_i}{\tau_s} + \delta(v_i - v_{peak}),

where :math:`\tau_s` is the synaptic time constant and :math:`\delta` is the `Dirac delta <https://en.wikipedia.org/wiki/Dirac_delta_function>`_ function,
which is one whenever a spike is elicited by the :math:`i^{th}` neuron and zero otherwise.
The variable :math:`s_i^{in}` serves as target variable for such synaptic activation variables.

In this example, we will examine an RNN of LIF neurons with spike-coupling.

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Step 1: Network initialization
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We will start out by creating a random coupling matrix and implementing an RNN model of rate-coupled LIF neurons.

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.. code-block:: Python


    import numpy as np

    from rectipy import Network

    # define network parameters
    node = "neuron_model_templates.spiking_neurons.lif.lif"
    N = 5
    J = np.random.randn(N, N)*20.0

    # initialize network
    net = Network(dt=1e-3)

    # add LIF population to network
    net.add_diffeq_node("lif", node, weights=J, source_var="s", target_var="s_in", input_var="I_ext",
                        output_var="s", spike_var="spike", reset_var="v", op="lif_op")


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The above code implements a `rectipy.Network` with :math:`N = 5` LIF neurons with random coupling weights drawn from
a standard Gaussian distribution with mean :math:`\mu = 0` and standard deviation :math:`\sigma = 10`. This particular
choice of source and target variables for the coupling implements :math:`s_i^{in} = \sum_{j=1}^N J_{ij} s_j`.

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Step 2: Simulation of the network dynamics
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Now, lets examine the network dynamics by integrating the evolution equations over 10000 steps, using the integration
step-size of :code:`dt = 1e-3` as defined above.

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.. code-block:: Python


    # define network input
    steps = 10000
    inp = np.zeros((steps, N)) + 200.0

    # perform numerical simulation
    obs = net.run(inputs=inp, sampling_steps=10)


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We created a timeseries of constant input and fed that input to the input variable of the RNN at each
integration step. Let's have a look at the resulting network dynamics.

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.. code-block:: Python


    from matplotlib.pyplot import show, legend

    obs.plot("out")
    legend([f"s_{i}" for i in range(N)])
    show()


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As can be seen, the different LIF neurons spiked at different times, even though they received the same
extrinsic input, due to the randomly sampled coupling weights and the RNN dynamics emerging from that.


.. _sphx_glr_download_auto_models_lif.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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