.. currentmodule:: brian2

.. Touboul_Brette_2008:

Example: Touboul_Brette_2008
============================


        .. only:: html

            .. |launchbinder| image:: http://mybinder.org/badge.svg
            .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/frompapers/Touboul_Brette_2008.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|_

        

Chaos in the AdEx model
-----------------------
Fig. 8B from:
Touboul, J. and Brette, R. (2008). Dynamics and bifurcations of the adaptive
exponential integrate-and-fire model. Biological Cybernetics 99(4-5):319-34.

This shows the bifurcation structure when the reset value is varied
(vertical axis shows the values of w at spike times for a given a reset value
Vr).

::

    from brian2 import *
    
    defaultclock.dt = 0.01*ms
    
    C = 281*pF
    gL = 30*nS
    EL = -70.6*mV
    VT = -50.4*mV
    DeltaT = 2*mV
    tauw = 40*ms
    a = 4*nS
    b = 0.08*nA
    I = .8*nA
    Vcut = VT + 5 * DeltaT  # practical threshold condition
    N = 200
    
    eqs = """
    dvm/dt=(gL*(EL-vm)+gL*DeltaT*exp((vm-VT)/DeltaT)+I-w)/C : volt
    dw/dt=(a*(vm-EL)-w)/tauw : amp
    Vr:volt
    """
    
    neuron = NeuronGroup(N, model=eqs, threshold='vm > Vcut',
                         reset="vm = Vr; w += b", method='euler')
    neuron.vm = EL
    neuron.w = a * (neuron.vm - EL)
    neuron.Vr = linspace(-48.3 * mV, -47.7 * mV, N)  # bifurcation parameter
    
    init_time = 3*second
    run(init_time, report='text')  # we discard the first spikes
    
    states = StateMonitor(neuron, "w", record=True, when='start')
    spikes = SpikeMonitor(neuron)
    run(1 * second, report='text')
    
    # Get the values of Vr and w for each spike
    Vr = neuron.Vr[spikes.i]
    w = states.w[spikes.i, int_((spikes.t-init_time)/defaultclock.dt)]
    
    figure()
    plot(Vr / mV, w / nA, '.k')
    xlabel('Vr (mV)')
    ylabel('w (nA)')
    show()
    

.. image:: ../resources/examples_images/frompapers.Touboul_Brette_2008.1.png