# Example: Clopath_et_al_2010_homeostasis

> 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):

This code contains an adapted version of the voltage-dependent triplet STDP rule from: Clopath et al., Connectivity reflects coding: a model of voltage-based STDP with homeostasis, Nature Neuroscience, 2010 (http://dx.doi.org/10.1038/nn.2479)

The plasticity rule is adapted for a leaky integrate & fire model in Brian2. More specifically, the filters `v_lowpass1` and `v_lowpass2` are incremented by a constant at every post-synaptic spike time, to compensate for the lack of an actual spike in the integrate & fire model.

As an illustration of the rule, we simulate the competition between inputs projecting on a downstream neuron. We would like to note that the parameters have been chosen arbitrarily to qualitatively reproduce the behavior of the original work, but need additional fitting.

We kindly ask to cite the article when using the model presented below.

This code was written by Jacopo Bono, 12/2015

    from brian2 import *

    ################################################################################
    # PLASTICITY MODEL
    ################################################################################

    #### Plasticity Parameters

    V_rest = -70.*mV        # resting potential
    V_thresh = -55.*mV      # spiking threshold
    Theta_low = V_rest      # depolarization threshold for plasticity
    x_reset = 1.            # spike trace reset value
    taux = 15.*ms           # spike trace time constant
    A_LTD = 1.5e-4          # depression amplitude
    A_LTP = 1.5e-2          # potentiation amplitude
    tau_lowpass1 = 40*ms    # timeconstant for low-pass filtered voltage
    tau_lowpass2 = 30*ms    # timeconstant for low-pass filtered voltage
    tau_homeo = 1000*ms     # homeostatic timeconstant
    v_target = 12*mV**2     # target depolarisation

    #### Plasticity Equations

    # equations executed at every timestepC
    Syn_model =   ('''
                w_ampa:1                # synaptic weight (ampa synapse)
                ''')

    # equations executed only when a presynaptic spike occurs
    Pre_eq = ('''
                g_ampa_post += w_ampa*ampa_max_cond                                                               # increment synaptic conductance
                A_LTD_u = A_LTD*(v_homeo**2/v_target)                                                             # metaplasticity
                w_minus = A_LTD_u*(v_lowpass1_post/mV - Theta_low/mV)*int(v_lowpass1_post/mV - Theta_low/mV > 0)  # synaptic depression
                w_ampa = clip(w_ampa-w_minus, 0, w_max)                                                           # hard bounds
                ''' )

    # equations executed only when a postsynaptic spike occurs
    Post_eq = ('''
                v_lowpass1 += 10*mV                                                                                        # mimics the depolarisation effect due to a spike
                v_lowpass2 += 10*mV                                                                                        # mimics the depolarisation effect due to a spike
                v_homeo += 0.1*mV                                                                                          # mimics the depolarisation effect due to a spike
                w_plus = A_LTP*x_trace_pre*(v_lowpass2_post/mV - Theta_low/mV)*int(v_lowpass2_post/mV - Theta_low/mV > 0)  # synaptic potentiation
                w_ampa = clip(w_ampa+w_plus, 0, w_max)                                                                     # hard bounds
                ''' )

    ################################################################################
    # I&F Parameters and equations
    ################################################################################

    #### Neuron parameters

    gleak = 30.*nS                  # leak conductance
    C = 300.*pF                     # membrane capacitance
    tau_AMPA = 2.*ms                # AMPA synaptic timeconstant
    E_AMPA = 0.*mV                  # reversal potential AMPA

    ampa_max_cond = 5.e-8*siemens  # Ampa maximal conductance
    w_max = 1.                      # maximal ampa weight

    #### Neuron Equations

    # We connect 10 presynaptic neurons to 1 downstream neuron

    # downstream neuron
    eqs_neurons = '''
    dv/dt = (gleak*(V_rest-v) + I_ext + I_syn)/C: volt      # voltage
    dv_lowpass1/dt = (v-v_lowpass1)/tau_lowpass1 : volt     # low-pass filter of the voltage
    dv_lowpass2/dt = (v-v_lowpass2)/tau_lowpass2 : volt     # low-pass filter of the voltage
    dv_homeo/dt = (v-V_rest-v_homeo)/tau_homeo : volt       # low-pass filter of the voltage
    I_ext : amp                                             # external current
    I_syn = g_ampa*(E_AMPA-v): amp                          # synaptic current
    dg_ampa/dt = -g_ampa/tau_AMPA : siemens                 # synaptic conductance
    dx_trace/dt = -x_trace/taux :1                          # spike trace
    '''

    # input neurons
    eqs_inputs = '''
    dv/dt = gleak*(V_rest-v)/C: volt                        # voltage
    dx_trace/dt = -x_trace/taux :1                          # spike trace
    rates : Hz                                              # input rates
    selected_index : integer (shared)                       # active neuron
    '''

    ################################################################################
    # Simulation
    ################################################################################

    #### Parameters

    defaultclock.dt = 500.*us                        # timestep
    Nr_neurons = 1                                   # Number of downstream neurons
    Nr_inputs = 5                                    # Number of input neurons
    input_rate = 35*Hz                               # Rates
    init_weight = 0.5                                # initial synaptic weight
    final_t = 20.*second                             # end of simulation
    input_time = 100.*ms                             # duration of an input

    #### Create neuron objects

    Nrn_downstream = NeuronGroup(Nr_neurons, eqs_neurons, threshold='v>V_thresh',
                                 reset='v=V_rest;x_trace+=x_reset/(taux/ms)',
                                 method='euler')
    Nrns_input = NeuronGroup(Nr_inputs, eqs_inputs, threshold='rand()<rates*dt',
                             reset='v=V_rest;x_trace+=x_reset/(taux/ms)',
                             method='exact')

    #### create Synapses

    Syn = Synapses(Nrns_input, Nrn_downstream,
                   model=Syn_model,
                   on_pre=Pre_eq,
                   on_post=Post_eq
                   )

    Syn.connect(i=numpy.arange(Nr_inputs), j=0)

    #### Monitors and storage
    W_evolution = StateMonitor(Syn, 'w_ampa', record=True)

    #### Run

    # Initial values
    Nrn_downstream.v = V_rest
    Nrn_downstream.v_lowpass1 = V_rest
    Nrn_downstream.v_lowpass2 = V_rest
    Nrn_downstream.v_homeo = 0
    Nrn_downstream.I_ext = 0.*amp
    Nrn_downstream.x_trace = 0.
    Nrns_input.v = V_rest
    Nrns_input.x_trace = 0.
    Syn.w_ampa = init_weight

    # Switch on a different input every 100ms
    Nrns_input.run_regularly('''
                             selected_index = int(floor(rand()*Nr_inputs))
                             rates = input_rate * int(selected_index == i)  # All rates are zero except for the selected neuron
                             ''', dt=input_time)
    run(final_t, report='text')

    ################################################################################
    # Plots
    ################################################################################
    stitle = 'Synaptic Competition'

    fig = figure(figsize=(8, 5))
    for kk in range(Nr_inputs):
        plt.plot(W_evolution.t, W_evolution.w_ampa[kk], '-', linewidth=2)
    xlabel('Time [ms]', fontsize=22)
    ylabel('Weight [a.u.]', fontsize=22)
    plt.subplots_adjust(bottom=0.2, left=0.15, right=0.95, top=0.85)
    title(stitle, fontsize=22)
    plt.show()