# Example: Izhikevich_2003

> 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):

Fig. 3 of

Simple Model of Spiking Neurons IEEE Transactions on Neural Networks ( Volume: 14, Issue: 6, Nov. 2003) Eugene M. Izhikevich

based on net.m by Eugene M. Izhikevich (http://izhikevich.org/publications/spikes.htm)

Akif Erdem Sağtekin and Sebastian Schmitt, 2022

    import matplotlib.pyplot as plt
    import numpy as np

    from brian2 import NeuronGroup, Synapses, SpikeMonitor, StateMonitor
    from brian2 import ms, mV
    from brian2 import defaultclock, run

    tfinal = 1000 * ms
    Ne = 800
    Ni = 200

    re = np.random.uniform(size=Ne)
    ri = np.random.uniform(size=Ni)
    weights = np.hstack(
        [
            0.5 * np.random.uniform(size=(Ne + Ni, Ne)),
            -np.random.uniform(size=(Ne + Ni, Ni)),
        ]
    ).T

    defaultclock.dt = 1 * ms

    eqs = """dv/dt = (0.04*v**2 + 5*v + 140 - u + I + I_noise )/ms : 1
             du/dt = (a*(b*v - u))/ms  : 1
             I : 1
             I_noise : 1
             a : 1
             b : 1
             c : 1
             d : 1
           """

    N = NeuronGroup(Ne + Ni, eqs, threshold="v>=30", reset="v=c; u+=d", method="euler")
    N.v = -65

    N_exc = N[:Ne]
    N_inh = N[Ne:]

    spikemon = SpikeMonitor(N)
    statemon = StateMonitor(N, 'v', record=0, when='after_thresholds')
    N_exc.a = 0.02
    N_exc.b = 0.2
    N_exc.c = -65 + 15 * re**2
    N_exc.d = 8 - 6 * re**2

    N_inh.a = 0.02 + 0.08 * ri
    N_inh.b = 0.25 - 0.05 * ri
    N_inh.c = -65
    N_inh.d = 2

    N_exc.u = "b*v"
    N_inh.u = "b*v"

    S = Synapses(
        N,
        N,
        "w : 1",
        on_pre={"up": "I += w", "down": "I -= w"},
        delay={"up": 0 * ms, "down": 1 * ms},
    )
    S.connect()
    S.w[:] = weights.flatten()

    N_exc.run_regularly("I_noise = 5*randn()", dt=1 * ms)
    N_inh.run_regularly("I_noise = 2*randn()", dt=1 * ms)

    run(tfinal)

    fig, (ax, ax_voltage) = plt.subplots(2, 1, sharex=True,
                                         gridspec_kw={'height_ratios': (3, 1)})

    ax.scatter(spikemon.t / ms, spikemon.i[:], marker="_", color="k", s=10)
    ax.set_xlim(0, tfinal / ms)
    ax.set_ylim(0, len(N))
    ax.set_ylabel("neuron number")
    ax.set_yticks(np.arange(0, len(N), 100))
    ax.spines['right'].set_visible(False)
    ax.spines['top'].set_visible(False)
    ax.axhline(Ne, color="k")
    ax.text(500, 900, 'inhibitory', backgroundcolor='w', color='k', ha='center')
    ax.text(500, 400, 'excitatory', backgroundcolor='w', color='k', ha='center')

    ax_voltage.plot(statemon.t / ms, np.clip(statemon.v[0], -np.inf, 30),
                   color='k')
    ax_voltage.text(25, 0, 'v₁(t)')
    ax_voltage.set_xticks(np.arange(0, tfinal / ms, 100))
    ax_voltage.spines['right'].set_visible(False)
    ax_voltage.spines['top'].set_visible(False)
    ax_voltage.set_xlabel("time, ms")

    plt.show()