# Example: Brunel_Wang_2001

> 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):

## Sample-specific persistent activity

Five subpopulations with three selective and one reset stimuli example. Analog to figure 6b in the paper.

BRUNEL, Nicolas et WANG, Xiao-Jing. Effects of neuromodulation in a cortical network model of object working memory dominated by recurrent inhibition. Journal of computational neuroscience, 2001, vol. 11, no 1, p. 63-85.

    from brian2 import *

    # populations
    N = 1000
    N_E = int(N * 0.8)  # pyramidal neurons
    N_I = int(N * 0.2)  # interneurons

    # voltage
    V_L = -70. * mV
    V_thr = -50. * mV
    V_reset = -55. * mV
    V_E = 0. * mV
    V_I = -70. * mV

    # membrane capacitance
    C_m_E = 0.5 * nF
    C_m_I = 0.2 * nF

    # membrane leak
    g_m_E = 25. * nS
    g_m_I = 20. * nS

    # refractory period
    tau_rp_E = 2. * ms
    tau_rp_I = 1. * ms

    # external stimuli
    rate = 3 * Hz
    C_ext = 800

    # synapses
    C_E = N_E
    C_I = N_I

    # AMPA (excitatory)
    g_AMPA_ext_E = 2.08 * nS
    g_AMPA_rec_E = 0.104 * nS * 800. / N_E
    g_AMPA_ext_I = 1.62 * nS
    g_AMPA_rec_I = 0.081 * nS * 800. / N_E
    tau_AMPA = 2. * ms

    # NMDA (excitatory)
    g_NMDA_E = 0.327 * nS * 800. / N_E
    g_NMDA_I = 0.258 * nS * 800. / N_E
    tau_NMDA_rise = 2. * ms
    tau_NMDA_decay = 100. * ms
    alpha = 0.5 / ms
    Mg2 = 1.

    # GABAergic (inhibitory)
    g_GABA_E = 1.25 * nS * 200. / N_I
    g_GABA_I = 0.973 * nS * 200. / N_I
    tau_GABA = 10. * ms

    # subpopulations
    f = 0.1
    p = 5
    N_sub = int(N_E * f)
    N_non = int(N_E * (1. - f * p))
    w_plus = 2.1
    w_minus = 1. - f * (w_plus - 1.) / (1. - f)

    # modeling
    eqs_E = '''
    dv / dt = (- g_m_E * (v - V_L) - I_syn) / C_m_E : volt (unless refractory)

    I_syn = I_AMPA_ext + I_AMPA_rec + I_NMDA_rec + I_GABA_rec : amp

    I_AMPA_ext = g_AMPA_ext_E * (v - V_E) * s_AMPA_ext : amp
    I_AMPA_rec = g_AMPA_rec_E * (v - V_E) * 1 * s_AMPA : amp
    ds_AMPA_ext / dt = - s_AMPA_ext / tau_AMPA : 1
    ds_AMPA / dt = - s_AMPA / tau_AMPA : 1

    I_NMDA_rec = g_NMDA_E * (v - V_E) / (1 + Mg2 * exp(-0.062 * v / mV) / 3.57) * s_NMDA_tot : amp
    s_NMDA_tot : 1

    I_GABA_rec = g_GABA_E * (v - V_I) * s_GABA : amp
    ds_GABA / dt = - s_GABA / tau_GABA : 1
    '''

    eqs_I = '''
    dv / dt = (- g_m_I * (v - V_L) - I_syn) / C_m_I : volt (unless refractory)

    I_syn = I_AMPA_ext + I_AMPA_rec + I_NMDA_rec + I_GABA_rec : amp

    I_AMPA_ext = g_AMPA_ext_I * (v - V_E) * s_AMPA_ext : amp
    I_AMPA_rec = g_AMPA_rec_I * (v - V_E) * 1 * s_AMPA : amp
    ds_AMPA_ext / dt = - s_AMPA_ext / tau_AMPA : 1
    ds_AMPA / dt = - s_AMPA / tau_AMPA : 1

    I_NMDA_rec = g_NMDA_I * (v - V_E) / (1 + Mg2 * exp(-0.062 * v / mV) / 3.57) * s_NMDA_tot : amp
    s_NMDA_tot : 1

    I_GABA_rec = g_GABA_I * (v - V_I) * s_GABA : amp
    ds_GABA / dt = - s_GABA / tau_GABA : 1
    '''

    P_E = NeuronGroup(N_E, eqs_E, threshold='v > V_thr', reset='v = V_reset', refractory=tau_rp_E, method='euler')
    P_E.v = V_L
    P_I = NeuronGroup(N_I, eqs_I, threshold='v > V_thr', reset='v = V_reset', refractory=tau_rp_I, method='euler')
    P_I.v = V_L

    eqs_glut = '''
    s_NMDA_tot_post = w * s_NMDA : 1 (summed)
    ds_NMDA / dt = - s_NMDA / tau_NMDA_decay + alpha * x * (1 - s_NMDA) : 1 (clock-driven)
    dx / dt = - x / tau_NMDA_rise : 1 (clock-driven)
    w : 1
    '''

    eqs_pre_glut = '''
    s_AMPA += w
    x += 1
    '''

    eqs_pre_gaba = '''
    s_GABA += 1
    '''

    # E to E
    C_E_E = Synapses(P_E, P_E, model=eqs_glut, on_pre=eqs_pre_glut, method='euler')
    C_E_E.connect('i != j')
    C_E_E.w[:] = 1

    for pi in range(N_non, N_non + p * N_sub, N_sub):

        # internal other subpopulation to current nonselective
        C_E_E.w[C_E_E.indices[:, pi:pi + N_sub]] = w_minus

        # internal current subpopulation to current subpopulation
        C_E_E.w[C_E_E.indices[pi:pi + N_sub, pi:pi + N_sub]] = w_plus

    # E to I
    C_E_I = Synapses(P_E, P_I, model=eqs_glut, on_pre=eqs_pre_glut, method='euler')
    C_E_I.connect()
    C_E_I.w[:] = 1

    # I to I
    C_I_I = Synapses(P_I, P_I, on_pre=eqs_pre_gaba, method='euler')
    C_I_I.connect('i != j')

    # I to E
    C_I_E = Synapses(P_I, P_E, on_pre=eqs_pre_gaba, method='euler')
    C_I_E.connect()

    # external noise
    C_P_E = PoissonInput(P_E, 's_AMPA_ext', C_ext, rate, '1')
    C_P_I = PoissonInput(P_I, 's_AMPA_ext', C_ext, rate, '1')

    # at 1s, select population 1
    C_selection = int(f * C_ext)
    rate_selection = 25 * Hz
    stimuli1 = TimedArray(np.r_[np.zeros(40), np.ones(2), np.zeros(100)], dt=25 * ms)
    input1 = PoissonInput(P_E[N_non:N_non + N_sub], 's_AMPA_ext', C_selection, rate_selection, 'stimuli1(t)')

    # at 2s, select population 2
    stimuli2 = TimedArray(np.r_[np.zeros(80), np.ones(2), np.zeros(100)], dt=25 * ms)
    input2 = PoissonInput(P_E[N_non + N_sub:N_non + 2 * N_sub], 's_AMPA_ext', C_selection, rate_selection, 'stimuli2(t)')

    # at 4s, reset selection
    stimuli_reset = TimedArray(np.r_[np.zeros(120), np.ones(2), np.zeros(100)], dt=25 * ms)
    input_reset_I = PoissonInput(P_E, 's_AMPA_ext', C_ext, rate_selection, 'stimuli_reset(t)')
    input_reset_E = PoissonInput(P_I, 's_AMPA_ext', C_ext, rate_selection, 'stimuli_reset(t)')

    # monitors
    N_activity_plot = 15
    sp_E_sels = [SpikeMonitor(P_E[pi:pi + N_activity_plot]) for pi in range(N_non, N_non + p * N_sub, N_sub)]
    sp_E = SpikeMonitor(P_E[:N_activity_plot])
    sp_I = SpikeMonitor(P_I[:N_activity_plot])

    r_E_sels = [PopulationRateMonitor(P_E[pi:pi + N_sub]) for pi in range(N_non, N_non + p * N_sub, N_sub)]
    r_E = PopulationRateMonitor(P_E[:N_non])
    r_I = PopulationRateMonitor(P_I)

    # simulate, can be long >120s
    net = Network(collect())
    net.add(sp_E_sels)
    net.add(r_E_sels)
    net.run(4 * second, report='stdout')

    # plotting
    title('Population rates')
    xlabel('ms')
    ylabel('Hz')

    plot(r_E.t / ms, r_E.smooth_rate(width=25 * ms) / Hz, label='nonselective')
    plot(r_I.t / ms, r_I.smooth_rate(width=25 * ms) / Hz, label='inhibitory')

    for i, r_E_sel in enumerate(r_E_sels[::-1]):
        plot(r_E_sel.t / ms, r_E_sel.smooth_rate(width=25 * ms) / Hz,
             label=f"selective {p - i}")

    legend()
    figure()

    title(f"Population activities ({N_activity_plot} neurons/pop)")
    xlabel('ms')
    yticks([])

    plot(sp_E.t / ms, sp_E.i + (p + 1) * N_activity_plot, '.', markersize=2,
         label="nonselective")
    plot(sp_I.t / ms, sp_I.i + p * N_activity_plot, '.', markersize=2, label="inhibitory")

    for i, sp_E_sel in enumerate(sp_E_sels[::-1]):
        plot(sp_E_sel.t / ms, sp_E_sel.i + (p - i - 1) * N_activity_plot, '.', markersize=2,
             label=f"selective {p - i}")

    legend()
    show()