brainpy.neurons.FHN

class brainpy.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, input_var=True, name=None, mode=None)

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.neurons.FHN(1)
>>> runner = bp.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, input_var=True, name=None, mode=None)

Methods

__init__(size[, a, b, tau, Vth, ...])
add_aft_update(key, fun)	Add the after update into this node
add_bef_update(key, fun)	Add the before update into this node
add_inp_fun(key, fun)	Add an input function.
clear_input()	Empty function of clearing inputs.
cpu()	Move all variable into the CPU device.
cuda()	Move all variables into the GPU device.
dV(V, t, w, I_ext)
dw(w, t, V)
get_aft_update(key)	Get the after update of this node by the given key.
get_batch_shape([batch_size])
get_bef_update(key)	Get the before update of this node by the given key.
get_delay_data(identifier, delay_pos, *indices)	Get delay data according to the provided delay steps.
get_delay_var(name)
get_inp_fun(key)	Get the input function.
get_local_delay(var_name, delay_name)	Get the delay at the given identifier (name).
has_aft_update(key)	Whether this node has the after update of the given key.
has_bef_update(key)	Whether this node has the before update of the given key.
init_param(param[, shape, sharding])	Initialize parameters.
init_variable(var_data, batch_or_mode[, ...])	Initialize variables.
jit_step_run(i, *args, **kwargs)	The jitted step function for running.
load_state(state_dict, **kwargs)	Load states from a dictionary.
load_state_dict(state_dict[, warn, compatible])	Copy parameters and buffers from state_dict into this module and its descendants.
nodes([method, level, include_self])	Collect all children nodes.
register_delay(identifier, delay_step, ...)	Register delay variable.
register_implicit_nodes(*nodes[, node_cls])
register_implicit_vars(*variables[, var_cls])
register_local_delay(var_name, delay_name[, ...])	Register local relay at the given delay time.
reset(*args, **kwargs)	Reset function which reset the whole variables in the model (including its children models).
reset_local_delays([nodes])	Reset local delay variables.
reset_state([batch_size])	Reset function which resets local states in this model.
return_info()	rtype: Union[Variable, ReturnInfo]
save_state(**kwargs)	Save states as a dictionary.
setattr(key, value)	rtype: None
state_dict(**kwargs)	Returns a dictionary containing a whole state of the module.
step_run(i, *args, **kwargs)	The step run function.
sum_inputs(*args[, init, label])	Summarize all inputs by the defined input functions .cur_inputs.
to(device)	Moves all variables into the given device.
tpu()	Move all variables into the TPU device.
tracing_variable(name, init, shape[, ...])	Initialize the variable which can be traced during computations and transformations.
train_vars([method, level, include_self])	The shortcut for retrieving all trainable variables.
tree_flatten()	Flattens the object as a PyTree.
tree_unflatten(aux, dynamic_values)	Unflatten the data to construct an object of this class.
unique_name([name, type_])	Get the unique name for this object.
update([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
mode	Mode of the model, which is useful to control the multiple behaviors of the model.
name	Name of the model.
supported_modes	Supported computing modes.
varshape	The shape of variables in the neuron group.
cur_inputs