Source code for brainmodels.neurons.HindmarshRose

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

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

__all__ = [
  'HindmarshRose'
]


[docs]class HindmarshRose(Neuron): r"""Hindmarsh-Rose neuron model. **Model Descriptions** The Hindmarsh–Rose model [1]_ [2]_ of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments made with a single neuron. The model has the mathematical form of a system of three nonlinear ordinary differential equations on the dimensionless dynamical variables :math:`x(t)`, :math:`y(t)`, and :math:`z(t)`. They read: .. math:: \begin{aligned} \frac{d V}{d t} &= y - a V^3 + b V^2 - z + I \\ \frac{d y}{d t} &= c - d V^2 - y \\ \frac{d z}{d t} &= r (s (V - V_{rest}) - z) \end{aligned} where :math:`a, b, c, d` model the working of the fast ion channels, :math:`I` models the slow ion channels. **Model Examples** .. plot:: :include-source: True >>> import brainpy as bp >>> import brainmodels >>> import matplotlib.pyplot as plt >>> >>> bp.math.set_dt(dt=0.01) >>> bp.set_default_odeint('rk4') >>> >>> types = ['quiescence', 'spiking', 'bursting', 'irregular_spiking', 'irregular_bursting'] >>> bs = bp.math.array([1.0, 3.5, 2.5, 2.95, 2.8]) >>> Is = bp.math.array([2.0, 5.0, 3.0, 3.3, 3.7]) >>> >>> # define neuron type >>> group = brainmodels.neurons.HindmarshRose(len(types), b=bs) >>> runner = bp.StructRunner(group, monitors=['V'], inputs=['input', Is],) >>> runner.run(1e3) >>> >>> fig, gs = bp.visualize.get_figure(row_num=3, col_num=2, row_len=3, col_len=5) >>> for i, mode in enumerate(types): >>> fig.add_subplot(gs[i // 2, i % 2]) >>> plt.plot(runner.mon.ts, runner.mon.V[:, i]) >>> plt.title(mode) >>> plt.xlabel('Time [ms]') >>> plt.show() **Model Parameters** ============= ============== ========= ============================================================ **Parameter** **Init Value** **Unit** **Explanation** ------------- -------------- --------- ------------------------------------------------------------ a 1 \ Model parameter. Fixed to a value best fit neuron activity. b 3 \ Model parameter. Allows the model to switch between bursting and spiking, controls the spiking frequency. c 1 \ Model parameter. Fixed to a value best fit neuron activity. d 5 \ Model parameter. Fixed to a value best fit neuron activity. r 0.01 \ Model parameter. Controls slow variable z's variation speed. Governs spiking frequency when spiking, and affects the number of spikes per burst when bursting. s 4 \ Model parameter. Governs adaption. ============= ============== ========= ============================================================ **Model Variables** =============== ================= ===================================== **Member name** **Initial Value** **Explanation** --------------- ----------------- ------------------------------------- V -1.6 Membrane potential. y -10 Gating variable. z 0 Gating variable. spike False Whether generate the spikes. input 0 External and synaptic input current. t_last_spike -1e7 Last spike time stamp. =============== ================= ===================================== **References** .. [1] Hindmarsh, James L., and R. M. Rose. "A model of neuronal bursting using three coupled first order differential equations." Proceedings of the Royal society of London. Series B. Biological sciences 221.1222 (1984): 87-102. .. [2] Storace, Marco, Daniele Linaro, and Enno de Lange. "The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations." Chaos: An Interdisciplinary Journal of Nonlinear Science 18.3 (2008): 033128. """
[docs] def __init__(self, size, a=1., b=3., c=1., d=5., r=0.01, s=4., V_rest=-1.6, V_th=1.0, method='exp_auto', name=None): # initialization super(HindmarshRose, self).__init__(size=size, method=method, name=name) # parameters self.a = a self.b = b self.c = c self.d = d self.r = r self.s = s self.V_th = V_th self.V_rest = V_rest # variables self.z = bm.Variable(bm.zeros(self.num)) self.y = bm.Variable(bm.ones(self.num) * -10.)
def dV(self, V, t, y, z, Iext): return y - self.a * V * V * V + self.b * V * V - z + Iext def dy(self, y, t, V): return self.c - self.d * V * V - y def dz(self, z, t, V): return self.r * (self.s * (V - self.V_rest) - z) @property def derivative(self): return bp.JointEq([self.dV, self.dy, self.dz])
[docs] def update(self, _t, _dt): V, y, z = self.integral(self.V, self.y, self.z, _t, self.input, dt=_dt) self.spike.value = bm.logical_and(V >= self.V_th, self.V < self.V_th) self.t_last_spike.value = bm.where(self.spike, _t, self.t_last_spike) self.V.value = V self.y.value = y self.z.value = z self.input[:] = 0.