.. currentmodule:: brian2

.. continuous_interaction:

Example: continuous_interaction
===============================


        .. only:: html

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

        

Synaptic model with continuous interaction
------------------------------------------

This example implements a conductance base synapse that is continuously linking
two neurons, i.e. the synaptic gating variable updates at each time step.
Two Reduced Traub-Miles Model (RTM) neurons are connected to each other
through a directed synapse from neuron 1 to 2.

Here, the complexity stems from the fact that the synaptic conductance is a
continuous function of the membrane potential, instead of being triggered by
individual spikes. This can be useful in particular when analyzing models
mathematically but it is not recommended in most cases because they tend to be
less efficient. Also note that this model only works with (pre-synaptic) neuron
models that model the action potential in detail, i.e. not with
integrate-and-fire type models.

There are two broad approaches (``s`` as part of the pre-synaptic neuron or
``s`` as part of the Synapses object), all depends on whether the time
constants are the same across all synapses or whether they can vary between
synapses. In this example, the time constant is assumed to be the same and
``s`` is therefore part of the pre-synaptic neuron model.

References:

-  Introduction to modeling neural dynamics, Börgers, chapter 20
- `Discussion in Brian forum <https://brian.discourse.group/t/how-to-implement-a-conductance-base-synapse/77/2>`_

::

    from brian2 import *
    
    I_e = 1.5*uA
    simulation_time = 100*ms
    # neuron RTM parameters
    El = -67 * mV
    EK = -100 * mV
    ENa = 50 * mV
    ESyn = 0 * mV
    gl = 0.1 * msiemens
    gK = 80 * msiemens
    gNa = 100 * msiemens
    
    C = 1 * ufarad
    
    weight = 0.25
    gSyn = 1.0 * msiemens
    tau_d = 2 * ms
    tau_r = 0.2 * ms
    
    # forming RTM model with differential equations
    eqs = """
    alphah = 0.128 * exp(-(vm + 50.0*mV) / (18.0*mV))/ms :Hz
    alpham = 0.32/mV * (vm + 54*mV) / (1.0 - exp(-(vm + 54.0*mV) / (4.0*mV)))/ms:Hz
    alphan = 0.032/mV * (vm + 52*mV) / (1.0 - exp(-(vm + 52.0*mV) / (5.0*mV)))/ms:Hz
    
    betah  = 4.0 / (1.0 + exp(-(vm + 27.0*mV) / (5.0*mV)))/ms:Hz
    betam  = 0.28/mV * (vm + 27.0*mV) / (exp((vm + 27.0*mV) / (5.0*mV)) - 1.0)/ms:Hz
    betan  = 0.5 * exp(-(vm + 57.0*mV) / (40.0*mV))/ms:Hz
    
    membrane_Im = I_ext + gNa*m**3*h*(ENa-vm) +
                  gl*(El-vm) + gK*n**4*(EK-vm) + gSyn*s_in*(-vm): amp
    I_ext : amp
    s_in  : 1
    
    dm/dt = alpham*(1-m)-betam*m : 1
    dn/dt = alphan*(1-n)-betan*n : 1
    dh/dt = alphah*(1-h)-betah*h : 1
    
    ds/dt = 0.5 * (1 + tanh(0.1*vm/mV)) * (1-s)/tau_r - s/tau_d : 1
    
    dvm/dt = membrane_Im/C : volt
    """
    
    neuron = NeuronGroup(2, eqs, method="exponential_euler")
    
    # initialize variables
    neuron.vm = [-70.0, -65.0]*mV
    neuron.m = "alpham / (alpham + betam)"
    neuron.h = "alphah / (alphah + betah)"
    neuron.n = "alphan / (alphan + betan)"
    neuron.I_ext = [I_e, 0.0*uA]
    
    S = Synapses(neuron,
                 neuron,
                 's_in_post = weight*s_pre:1 (summed)')
    S.connect(i=0, j=1)
    
    # tracking variables
    st_mon = StateMonitor(neuron, ["vm", "s", "s_in"], record=[0, 1])
    
    # running the simulation
    run(simulation_time)
    
    # plot the results
    fig, ax = plt.subplots(2, figsize=(10, 6), sharex=True,
                           gridspec_kw={'height_ratios': (3, 1)})
    
    ax[0].plot(st_mon.t/ms, st_mon.vm[0]/mV,
               lw=2, c="r", alpha=0.5, label="neuron 0")
    ax[0].plot(st_mon.t/ms, st_mon.vm[1]/mV,
               lw=2, c="b", alpha=0.5, label='neuron 1')
    ax[1].plot(st_mon.t/ms, st_mon.s[0],
               lw=2, c="r", alpha=0.5, label='s, neuron 0')
    ax[1].plot(st_mon.t/ms, st_mon.s_in[1],
               lw=2, c="b", alpha=0.5, label='s_in, neuron 1')
    ax[0].set(ylabel='v [mV]', xlim=(0, np.max(st_mon.t / ms)),
              ylim=(-100, 50))
    ax[1].set(xlabel="t [ms]", ylabel="s", ylim=(0, 1))
    
    ax[0].legend()
    ax[1].legend()
    
    plt.show()
    

.. image:: ../resources/examples_images/synapses.continuous_interaction.1.png