Izhikevich

models.Neurons.Izhikevich(
    self,
    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.

Parameters:

• a = 0.02 : Speed of the recovery variable
• b = 0.2: Scaling of the recovery variable
• c = -65.0 : Reset potential.
• d = 8.0 : Increment of the recovery variable after a spike.
• v_thresh = 30.0 : Spike threshold (mV).
• i_offset = 0.0 : external current (nA).
• noise = 0.0 : Amplitude of the normal additive noise.
• tau_refrac = 0.0 : Duration of refractory period (ms).

Variables:

• I : input current (user-defined conductance/current + external current + normal noise):

  I = conductance + i_offset + noise * Normal(0.0, 1.0)

• v : membrane potential in mV (init = c):

  dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I

• u : recovery variable (init= b * c):

  du/dt = a * (b * v - u)

Spike emission:

v > v_thresh

Reset:

v = c
u += d 

The ODEs are solved using the explicit Euler method.

Equivalent code:

    Izhikevich = ann.Neuron(
        parameters = """
            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
            dv/dt = 0.04 * v^2 + 5.0 * v + 140.0 - u + I : init = -65.0
            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 derived from the above mentioned article.

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'