.. currentmodule:: brian

.. index::
   pair: example usage; NeuronGroup
   pair: example usage; StateSpikeMonitor
   pair: example usage; run

.. _example-frompapers_Touboul_Brette_2008:

Example: Touboul_Brette_2008 (frompapers)
=========================================

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 brian 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=500
    
    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=Vcut,reset="vm=Vr;w+=b")
    neuron.vm=EL
    neuron.w=a*(neuron.vm-EL)
    neuron.Vr=linspace(-48.3*mV,-47.7*mV,N) # bifurcation parameter
    
    run(3*second,report='text') # we discard the first spikes
    
    M=StateSpikeMonitor(neuron,("Vr","w")) # record Vr and w at spike times
    run(2*second,report='text')
    
    Vr,w=M.values("Vr"),M.values("w")
    
    figure()
    plot(Vr/mV,w/nA,'.k')
    xlabel('Vr (mV)')
    ylabel('w (nA)')
    show()