.. currentmodule:: brian2

.. Hindmarsh_Rose_1984:

Example: Hindmarsh_Rose_1984
============================


        .. only:: html

            .. |launchbinder| image:: http://mybinder.org/badge.svg
            .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/frompapers/Hindmarsh_Rose_1984.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|_

        

Burst generation in the Hinsmarsh-Rose model. Reproduces Figure 6 of:

Hindmarsh, J. L., and R. M. Rose.
“A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations.”
Proceedings of the Royal Society of London. Series B, Biological Sciences 221, no. 1222 (1984): 87–102.

::

    from brian2 import *
    
    # In the original model, time is measured in arbitrary time units
    time_unit = 1*ms
    defaultclock.dt = time_unit/10
    
    x_1 = -1.6  # leftmost equilibrium point of the model without adaptation
    a = 1; b = 3; c = 1; d = 5
    r = 0.001; s = 4
    eqs = '''
    dx/dt = (y - a*x**3 + b*x**2 + I - z)/time_unit : 1
    dy/dt = (c - d*x**2 - y)/time_unit : 1
    dz/dt = r*(s*(x - x_1) - z)/time_unit : 1
    I : 1 (constant)
    '''
    
    # We run the model with three different currents
    neuron = NeuronGroup(3, eqs, method='rk4')
    
    # Set all variables to their equilibrium point
    neuron.x = x_1
    neuron.y = 'c - d*x**2'
    neuron.z = 'r*(s*(x - x_1))'
    
    # Set the constant current input
    neuron.I = [0.4, 2, 4]
    
    # Record the "membrane potential"
    mon = StateMonitor(neuron, 'x', record=True)
    
    run(2100*time_unit)
    
    ax_top = plt.subplot2grid((2, 3), (0, 0), colspan=3)
    ax_bottom_l = plt.subplot2grid((2, 3), (1, 0), colspan=2)
    ax_bottom_r = plt.subplot2grid((2, 3), (1, 2))
    for ax in [ax_top, ax_bottom_l, ax_bottom_r]:
        ax.spines['top'].set_visible(False)
        ax.spines['right'].set_visible(False)
        ax.set(ylim=(-2, 2), yticks=[-2, 0, 2])
    
    ax_top.plot(mon.t/time_unit, mon.x[0])
    
    ax_bottom_l.plot(mon.t/time_unit, mon.x[1])
    ax_bottom_l.set_xlim(700, 2100)
    
    ax_bottom_r.plot(mon.t/time_unit, mon.x[2])
    ax_bottom_r.set_xlim(1400, 2100)
    ax_bottom_r.set_yticks([])
    
    plt.show()
    

.. image:: ../resources/examples_images/frompapers.Hindmarsh_Rose_1984.1.png