Izhikevich

Izhikevich(
    a=0.02,
    b=0.2,
    c=-65.0,
    d=8.0,
    v_thresh=30.0,
    i_offset=0.0,
    noise=0.0,
    tau_refrac=0.0,
    conductance='g_exc - g_inh',
)

Izhikevich quadratic spiking neuron.

Izhikevich, E.M. (2003). Simple Model of Spiking Neurons, IEEE Transaction on Neural Networks, 14:6. http://dx.doi.org/10.1109/TNN.2003.820440

The neural equations are:

\frac{dv}{dt} = 0.04 * v^2 + 5.0 * v + 140.0 - u + I

\frac{du}{dt} = a * (b * v - u)

By default, the conductance is “g_exc - g_inh”, but this can be changed by setting the conductance argument:

neuron = ann.Izhikevich(conductance='g_ampa * (1 + g_nmda) - g_gaba')

The synapses are instantaneous, i.e the corresponding conductance is increased from the synaptic efficiency w at the time step when a spike is received.

The ODEs are solved using the explicit Euler method.

Equivalent code:

Izhikevich = ann.Neuron(
    parameters = dict(
        noise = ann.Parameter(0.0),
        a = ann.Parameter(0.02),
        b = ann.Parameter(0.2),
        c = ann.Parameter(-65.0),
        d = ann.Parameter(8.0),
        v_thresh = ann.Parameter(30.0),
        i_offset = ann.Parameter(0.0),
    ), 
    equations = [
        'I = g_exc - g_inh + noise * Normal(0.0, 1.0) + i_offset',
        ann.Variable('dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I', init = -65.0),
        ann.Variable('du/dt = a * (b*v - u)', init= -13.0),
    ],
    spike = "v > v_thresh",
    reset = "v = c; u += d",
    refractory = 0.0
)

The default parameters are for a regular spiking (RS) neuron. They are defined as local parameters, so their value can be set at the neuron-level. Here is the neuron definition using global parameters:

Izhikevich = ann.Neuron(
    parameters = dict(
        noise = 0.0,
        a = 0.02,
        b = 0.2,
        c = -65.0,
        d = 8.0,
        v_thresh = 30.0,
        i_offset = 0.0,
    ), 
    equations = [
        'I = g_exc - g_inh + noise * Normal(0.0, 1.0) + i_offset',
        ann.Variable('dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I', init = -65.0),
        ann.Variable('du/dt = a * (b*v - u)', init= -13.0),
    ],
    spike = "v > v_thresh",
    reset = "v = c; u += d",
    refractory = 0.0
)

Parameters

Name	Type	Description	Default
a	float	Speed of the recovery variable	0.02
b	float	Scaling of the recovery variable	0.2
c	float	Reset potential.	-65.0
d	float	Increment of the recovery variable after a spike.	8.0
v_thresh	float	Spike threshold (mV).	30.0
i_offset	float	external current (nA).	0.0
noise	float	Amplitude of the normal additive noise.	0.0
tau_refrac	float	Duration of refractory period (ms).	0.0
conductance	str	Conductances used as inputs.	'g_exc - g_inh'