# brainpy.dyn.neurons.FractionalFHR#

class brainpy.dyn.neurons.FractionalFHR(size, alpha, num_memory=1000, a=0.7, b=0.8, c=- 0.775, d=1.0, delta=0.08, mu=0.0001, Vth=1.8, V_initializer=OneInit(value=2.5), w_initializer=ZeroInit, y_initializer=ZeroInit, name=None, keep_size=False)[source]#

The fractional-order FH-R model 1.

FitzHugh and Rinzel introduced FH-R model (1976, in an unpublished article), which is the modification of the classical FHN neuron model. The fractional-order FH-R model is described as

$\begin{split}\begin{array}{rcl} \frac{{d}^{\alpha }v}{d{t}^{\alpha }} & = & v-{v}^{3}/3-w+y+I={f}_{1}(v,w,y),\\ \frac{{d}^{\alpha }w}{d{t}^{\alpha }} & = & \delta (a+v-bw)={f}_{2}(v,w,y),\\ \frac{{d}^{\alpha }y}{d{t}^{\alpha }} & = & \mu (c-v-dy)={f}_{3}(v,w,y), \end{array}\end{split}$

where $$v, w$$ and $$y$$ represent the membrane voltage, recovery variable and slow modulation of the current respectively. $$I$$ measures the constant magnitude of external stimulus current, and $$\alpha$$ is the fractional exponent which ranges in the interval $$(0 < \alpha \le 1)$$. $$a, b, c, d, \delta$$ and $$\mu$$ are the system parameters.

The system reduces to the original classical order system when $$\alpha=1$$.

$$\mu$$ indicates a small parameter that determines the pace of the slow system variable $$y$$. The fast subsystem ($$v-w$$) presents a relaxation oscillator in the phase plane where $$\delta$$ is a small parameter. $$v$$ is expressed in mV (millivolt) scale. Time $$t$$ is in ms (millisecond) scale. It exhibits tonic spiking or quiescent state depending on the parameter sets for a fixed value of $$I$$. The parameter $$a$$ in the 2D FHN model corresponds to the parameter $$c$$ of the FH-R neuron model. If we decrease the value of $$a$$, it causes longer intervals between two burstings, however there exists $$a$$ relatively fixed time of bursting duration. With the increasing of $$a$$, the interburst intervals become shorter and periodic bursting changes to tonic spiking.

Examples

Parameters
• size (int, sequence of int) – The size of the neuron group.

• alpha (float, tensor) – The fractional order.

• num_memory (int) – The total number of the short memory.

References

1

Mondal, A., Sharma, S.K., Upadhyay, R.K. et al. Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics. Sci Rep 9, 15721 (2019). https://doi.org/10.1038/s41598-019-52061-4

__init__(size, alpha, num_memory=1000, a=0.7, b=0.8, c=- 0.775, d=1.0, delta=0.08, mu=0.0001, Vth=1.8, V_initializer=OneInit(value=2.5), w_initializer=ZeroInit, y_initializer=ZeroInit, name=None, keep_size=False)[source]#

Methods

 __init__(size, alpha[, num_memory, a, b, c, ...]) clear_input() Function to clear inputs in the neuron group. dV(V, t, w, y) dw(w, t, V) dy(y, 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.