.. currentmodule:: brian2

.. Ornstein_Uhlenbeck:

Example: Ornstein_Uhlenbeck
===========================


        .. only:: html

            .. |launchbinder| image:: http://mybinder.org/badge.svg
            .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/advanced/Ornstein_Uhlenbeck.ipynb

            .. note::
               You can launch an interactive, editable version of this
               example without installing any local files
               using the Binder service (although note that at some times this
               may be slow or fail to open): |launchbinder|_

        

Ornstein-Uhlenbeck process

Figure 2: Two realizations of the Ornstein-Uhlenbeck process for
parameters τ=1.0 and σ=0.1 (black curve), and for τ=0.1 and σ=0.31622
(red curve). In both cases the noise intensity is σ^2*τ=0.01 . The red
curve represents a noise that more closely mimics Gaussian white
noise. Both realizations begin here at x(0)=1.0 , after which the mean
decays exponentially to zero with time constant τ.

Andre Longtin (2010) Stochastic dynamical systems. Scholarpedia, 5(4):1619.

Sebastian Schmitt, 2022

::

    
    import matplotlib.pyplot as plt
    import numpy as np
    
    from brian2 import run
    from brian2 import NeuronGroup, StateMonitor
    from brian2 import second, ms
    
    N = NeuronGroup(
        2,
        """
        tau : second
        sigma : 1
        dy/dt = -y/tau + sqrt(2*sigma**2/tau)*xi : 1
        """,
        method="euler"
    )
    
    N.tau = np.array([1, 0.1]) * second
    N.sigma = np.array([0.1, 0.31622])
    N.y = 1
    
    M = StateMonitor(N, "y", record=True)
    
    run(10 * second)
    
    plt.plot(M.t / second, M.y[1], color="red", label=r"$\tau$=0.1 s, $\sigma$=0.31622")
    plt.plot(M.t / second, M.y[0], color="k", label=r"$\tau$=1 s, $\sigma$=0.1")
    
    plt.xlim(0, 10)
    plt.ylim(-1.1, 1.1)
    
    plt.xlabel("time (sec)")
    plt.ylabel("Ornstein-Uhlenbeck process")
    
    plt.legend()
    
    plt.show()
    

.. image:: ../resources/examples_images/advanced.Ornstein_Uhlenbeck.1.png