.. currentmodule:: brian2

.. rall:

Example: rall
=============


        .. only:: html

            .. |launchbinder| image:: file:///usr/share/doc/python-brian-doc/docs/badge.svg
            .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/compartmental/rall.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|_

        

A cylinder plus two branches, with diameters according to Rall's formula

::

    from brian2 import *
    
    defaultclock.dt = 0.01*ms
    
    # Passive channels
    gL = 1e-4*siemens/cm**2
    EL = -70*mV
    
    # Morphology
    diameter = 1*um
    length = 300*um
    Cm = 1*uF/cm**2
    Ri = 150*ohm*cm
    N = 500
    rm = 1 / (gL * pi * diameter)  # membrane resistance per unit length
    ra = (4 * Ri)/(pi * diameter**2)  # axial resistance per unit length
    la = sqrt(rm / ra) # space length
    morpho = Cylinder(diameter=diameter, length=length, n=N)
    d1 = 0.5*um
    L1 = 200*um
    rm = 1 / (gL * pi * d1) # membrane resistance per unit length
    ra = (4 * Ri) / (pi * d1**2) # axial resistance per unit length
    l1 = sqrt(rm / ra) # space length
    morpho.L = Cylinder(diameter=d1, length=L1, n=N)
    d2 = (diameter**1.5 - d1**1.5)**(1. / 1.5)
    rm = 1/(gL * pi * d2) # membrane resistance per unit length
    ra = (4 * Ri) / (pi * d2**2) # axial resistance per unit length
    l2 = sqrt(rm / ra) # space length
    L2 = (L1 / l1) * l2
    morpho.R = Cylinder(diameter=d2, length=L2, n=N)
    
    eqs='''
    Im = gL * (EL-v) : amp/meter**2
    I : amp (point current)
    '''
    
    neuron = SpatialNeuron(morphology=morpho, model=eqs, Cm=Cm, Ri=Ri,
                           method='exponential_euler')
    neuron.v = EL
    
    neuron.I[0] = 0.02*nA # injecting at the left end
    run(100*ms, report='text')
    
    plot(neuron.main.distance/um, neuron.main.v/mV, 'k')
    plot(neuron.L.distance/um, neuron.L.v/mV, 'k')
    plot(neuron.R.distance/um, neuron.R.v/mV, 'k')
    # Theory
    x = neuron.main.distance
    ra = la * 4 * Ri/(pi * diameter**2)
    l = length/la + L1/l1
    theory = EL + ra*neuron.I[0]*cosh(l - x/la)/sinh(l)
    plot(x/um, theory/mV, 'r')
    x = neuron.L.distance
    theory = (EL+ra*neuron.I[0]*cosh(l - neuron.main.distance[-1]/la -
                                     (x - neuron.main.distance[-1])/l1)/sinh(l))
    plot(x/um, theory/mV, 'r')
    x = neuron.R.distance
    theory = (EL+ra*neuron.I[0]*cosh(l - neuron.main.distance[-1]/la -
                                     (x - neuron.main.distance[-1])/l2)/sinh(l))
    plot(x/um, theory/mV, 'r')
    xlabel('x (um)')
    ylabel('v (mV)')
    show()
    

