brainmodels.neurons.FHN

class brainmodels.neurons.FHN(size, a=0.7, b=0.8, tau=12.5, Vth=1.8, method='exp_auto', name=None)[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
>>> import brainmodels
>>>
>>> # simulation
>>> fnh = brainmodels.neurons.FHN(1)
>>> runner = bp.StructRunner(fnh, 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)

(Source code, png, hires.png, pdf)

../../_images/brainmodels-neurons-FHN-1.png

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, method='exp_auto', name=None)[source]

Methods

__init__(size[, a, b, tau, Vth, method, name])

child_ds([method, include_self])

Return the children instance of dynamical systems.

dV(V, t, w, Iext)

dw(w, t, V)

ints([method])

Collect all integrators in this node and the children nodes.

load_states(filename[, verbose, check_missing])

Load the model states.

nodes([method, _paths])

Collect all children nodes.

register_constant_delay(key, size, delay[, ...])

Register a constant delay, whose update method will be appended into the self.steps in this host class.

register_implicit_nodes(nodes)

register_implicit_vars(variables)

save_states(filename[, all_vars])

Save the model states.

train_vars([method])

The shortcut for retrieving all trainable variables.

unique_name([name, type_])

Get the unique name for this object.

update(_t, _dt)

The function to specify the updating rule.

vars([method])

Collect all variables in this node and the children nodes.

Attributes

derivative