# brainpy.dyn.neurons.FractionalIzhikevich#

class brainpy.dyn.neurons.FractionalIzhikevich(size, alpha, num_memory, a=0.02, b=0.2, c=- 65.0, d=8.0, f=0.04, g=5.0, h=140.0, R=1.0, tau=1.0, V_th=30.0, V_initializer=OneInit(value=- 65.0), u_initializer=OneInit(value=- 13.0), keep_size=False, name=None)[source]#

Fractional-order Izhikevich model 10.

The fractional-order Izhikevich model is given by

\begin{split}\begin{aligned} &\tau \frac{d^{\alpha} v}{d t^{\alpha}}=\mathrm{f} v^{2}+g v+h-u+R I \\ &\tau \frac{d^{\alpha} u}{d t^{\alpha}}=a(b v-u) \end{aligned}\end{split}

where $$\alpha$$ is the fractional order (exponent) such that $$0<\alpha\le1$$. It is a commensurate system that reduces to classical Izhikevich model at $$\alpha=1$$.

The time $$t$$ is in ms; and the system variable $$v$$ expressed in mV corresponds to membrane voltage. Moreover, $$u$$ expressed in mV is the recovery variable that corresponds to the activation of K+ ionic current and inactivation of Na+ ionic current.

The parameters $$f, g, h$$ are fixed constants (should not be changed) such that $$f=0.04$$ (mV)−1, $$g=5, h=140$$ mV; and $$a$$ and $$b$$ are dimensionless parameters. The time constant $$\tau=1$$ ms; the resistance $$R=1$$ Ω; and $$I$$ expressed in mA measures the injected (applied) dc stimulus current to the system.

When the membrane voltage reaches the spike peak $$v_{peak}$$, the two variables are rest as follow:

$\begin{split}\text { if } v \geq v_{\text {peak }} \text { then }\left\{\begin{array}{l} v \leftarrow c \\ u \leftarrow u+d \end{array}\right.\end{split}$

we used $$v_{peak}=30$$ mV, and $$c$$ and $$d$$ are parameters expressed in mV. When the spike reaches its peak value, the membrane voltage $$v$$ and the recovery variable $$u$$ are reset according to the above condition.

Examples

References

10

Teka, Wondimu W., Ranjit Kumar Upadhyay, and Argha Mondal. “Spiking and bursting patterns of fractional-order Izhikevich model.” Communications in Nonlinear Science and Numerical Simulation 56 (2018): 161-176.

__init__(size, alpha, num_memory, a=0.02, b=0.2, c=- 65.0, d=8.0, f=0.04, g=5.0, h=140.0, R=1.0, tau=1.0, V_th=30.0, V_initializer=OneInit(value=- 65.0), u_initializer=OneInit(value=- 13.0), keep_size=False, name=None)[source]#

Methods

 __init__(size, alpha, num_memory[, a, b, c, ...]) clear_input() Function to clear inputs in the neuron group. dV(V, t, u, I_ext) du(u, t, V) get_batch_shape([batch_size]) get_delay_data(identifier, delay_step, *indices) Get delay data according to the provided delay steps. load_states(filename[, verbose]) Load the model states. nodes([method, level, include_self]) Collect all children nodes. offline_fit(target, fit_record) offline_init() online_fit(target, fit_record) online_init() register_delay(identifier, delay_step, ...) Register delay variable. register_implicit_nodes(*nodes, **named_nodes) register_implicit_vars(*variables, ...) reset([batch_size]) Reset function which reset the whole variables in the model. reset_local_delays([nodes]) Reset local delay variables. reset_state([batch_size]) Reset function which reset the states in the model. save_states(filename[, variables]) Save the model states. train_vars([method, level, include_self]) The shortcut for retrieving all trainable variables. unique_name([name, type_]) Get the unique name for this object. update(tdi[, x]) The function to specify the updating rule. update_local_delays([nodes]) Update local delay variables. vars([method, level, include_self]) Collect all variables in this node and the children nodes.

Attributes

 derivative global_delay_data mode Mode of the model, which is useful to control the multiple behaviors of the model. name Name of the model. varshape The shape of variables in the neuron group.