# brainpy.dyn.neurons.FHN#

class brainpy.dyn.neurons.FHN(size, a=0.7, b=0.8, tau=12.5, Vth=1.8, V_initializer=ZeroInit, w_initializer=ZeroInit, noise=None, method='exp_auto', keep_size=False, name=None, mode=NormalMode, spike_fun=<jax._src.custom_derivatives.custom_vjp object>)[source]#

FitzHugh-Nagumo neuron model.

Model Descriptions

The FitzHugh–Nagumo model (FHN), named after Richard FitzHugh (1922–2007) who suggested the system in 1961 1 and J. Nagumo et al. who created the equivalent circuit the following year, describes a prototype of an excitable system (e.g., a neuron).

The motivation for the FitzHugh-Nagumo model was to isolate conceptually the essentially mathematical properties of excitation and propagation from the electrochemical properties of sodium and potassium ion flow. The model consists of

• a voltage-like variable having cubic nonlinearity that allows regenerative self-excitation via a positive feedback, and

• a recovery variable having a linear dynamics that provides a slower negative feedback.

\begin{split}\begin{aligned} {\dot {v}} &=v-{\frac {v^{3}}{3}}-w+RI_{\rm {ext}}, \\ \tau {\dot {w}}&=v+a-bw. \end{aligned}\end{split}

The FHN Model is an example of a relaxation oscillator because, if the external stimulus $$I_{\text{ext}}$$ exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables $$v$$ and $$w$$ relax back to their rest values. This behaviour is typical for spike generations (a short, nonlinear elevation of membrane voltage $$v$$, diminished over time by a slower, linear recovery variable $$w$$) in a neuron after stimulation by an external input current.

Model Examples

>>> import brainpy as bp
>>> fhn = bp.dyn.FHN(1)
>>> runner = bp.dyn.DSRunner(fhn, inputs=('input', 1.), monitors=['V', 'w'])
>>> runner.run(100.)
>>> bp.visualize.line_plot(runner.mon.ts, runner.mon.w, legend='w')
>>> bp.visualize.line_plot(runner.mon.ts, runner.mon.V, legend='V', show=True)


Model Parameters

 Parameter Init Value Unit Explanation a 1 Positive constant b 1 Positive constant tau 10 ms Membrane time constant. V_th 1.8 mV Threshold potential of spike.

Model Variables

 Variables name Initial Value Explanation V 0 Membrane potential. w 0 A recovery variable which represents the combined effects of sodium channel de-inactivation and potassium channel deactivation. input 0 External and synaptic input current. spike False Flag to mark whether the neuron is spiking. t_last_spike -1e7 Last spike time stamp.

References

1

FitzHugh, Richard. “Impulses and physiological states in theoretical models of nerve membrane.” Biophysical journal 1.6 (1961): 445-466.

2

https://en.wikipedia.org/wiki/FitzHugh%E2%80%93Nagumo_model

3

http://www.scholarpedia.org/article/FitzHugh-Nagumo_model

__init__(size, a=0.7, b=0.8, tau=12.5, Vth=1.8, V_initializer=ZeroInit, w_initializer=ZeroInit, noise=None, method='exp_auto', keep_size=False, name=None, mode=NormalMode, spike_fun=<jax._src.custom_derivatives.custom_vjp object>)[source]#

Methods

 __init__(size[, a, b, tau, Vth, ...]) clear_input() Function to clear inputs in the neuron group. dV(V, t, w, I_ext) dw(w, 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.