Source code for brainmodels.neurons.FHN

# -*- coding: utf-8 -*-

import brainpy as bp
import brainpy.math as bm
from .base import Neuron

__all__ = [
'FHN'
]

[docs]class FHN(Neuron): r"""FitzHugh-Nagumo neuron model. **Model Descriptions** The FitzHugh–Nagumo model (FHN), named after Richard FitzHugh (1922–2007) who suggested the system in 1961 _ 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. .. math:: \begin{aligned} {\dot {v}} &=v-{\frac {v^{3}}{3}}-w+RI_{\rm {ext}}, \\ \tau {\dot {w}}&=v+a-bw. \end{aligned} The FHN Model is an example of a relaxation oscillator because, if the external stimulus :math:I_{\text{ext}} exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables :math:v and :math:w relax back to their rest values. This behaviour is typical for spike generations (a short, nonlinear elevation of membrane voltage :math:v, diminished over time by a slower, linear recovery variable :math:w) in a neuron after stimulation by an external input current. **Model Examples** .. plot:: :include-source: True >>> 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) **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** ..  FitzHugh, Richard. "Impulses and physiological states in theoretical models of nerve membrane." Biophysical journal 1.6 (1961): 445-466. ..  https://en.wikipedia.org/wiki/FitzHugh%E2%80%93Nagumo_model ..  http://www.scholarpedia.org/article/FitzHugh-Nagumo_model """
[docs] def __init__(self, size, a=0.7, b=0.8, tau=12.5, Vth=1.8, method='exp_auto', name=None): # initialization super(FHN, self).__init__(size=size, method=method, name=name) # parameters self.a = a self.b = b self.tau = tau self.Vth = Vth # variables self.w = bm.Variable(bm.zeros(self.num))
def dV(self, V, t, w, Iext): return V - V * V * V / 3 - w + Iext def dw(self, w, t, V): return (V + self.a - self.b * w) / self.tau @property def derivative(self): return bp.JointEq([self.dV, self.dw])
[docs] def update(self, _t, _dt): V, w = self.integral(self.V, self.w, _t, self.input, dt=_dt) self.spike.value = bm.logical_and(V >= self.Vth, self.V < self.Vth) self.t_last_spike.value = bm.where(self.spike, _t, self.t_last_spike) self.V.value = V self.w.value = w self.input[:] = 0.