brainpy.neurons.HindmarshRose

class brainpy.neurons.HindmarshRose(size, a=1.0, b=3.0, c=1.0, d=5.0, r=0.01, s=4.0, V_rest=-1.6, V_th=1.0, V_initializer=ZeroInit, y_initializer=OneInit(value=-10.0), z_initializer=ZeroInit, noise=None, method='exp_auto', keep_size=False, name=None, mode=None, spike_fun=<brainpy._src.math.surrogate._utils.VJPCustom object>)

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 \(x(t)\), \(y(t)\), and \(z(t)\). They read:

\[\begin{split}\begin{aligned} \frac{d V}{d t} &= y - a V^3 + b V^2 - z + I \\ \frac{d y}{d t} &= c - d V^2 - y \\ \frac{d z}{d t} &= r (s (V - V_{rest}) - z) \end{aligned}\end{split}\]

where \(a, b, c, d\) model the working of the fast ion channels, \(I\) models the slow ion channels.

Model Examples

>>> import jax.numpy as jnp
>>> import brainpy as bp
>>> import matplotlib.pyplot as plt
>>>
>>> bp.math.set_dt(dt=0.01)
>>> bp.ode.set_default_odeint('rk4')
>>>
>>> types = ['quiescence', 'spiking', 'bursting', 'irregular_spiking', 'irregular_bursting']
>>> bs = jnp.array([1.0, 3.5, 2.5, 2.95, 2.8])
>>> Is = jnp.array([2.0, 5.0, 3.0, 3.3, 3.7])
>>>
>>> # define neuron type
>>> group = bp.neurons.HindmarshRose(len(types), b=bs)
>>> runner = bp.DSRunner(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()

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

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.

__init__(size, a=1.0, b=3.0, c=1.0, d=5.0, r=0.01, s=4.0, V_rest=-1.6, V_th=1.0, V_initializer=ZeroInit, y_initializer=OneInit(value=-10.0), z_initializer=ZeroInit, noise=None, method='exp_auto', keep_size=False, name=None, mode=None, spike_fun=<brainpy._src.math.surrogate._utils.VJPCustom object>)

Methods

__init__(size[, a, b, c, d, r, s, V_rest, ...])
clear_input()	Function to clear inputs in the neuron group.
cpu()	Move all variable into the CPU device.
cuda()	Move all variables into the GPU device.
dV(V, t, y, z, I_ext)
dy(y, t, V)
dz(z, t, V)
get_batch_shape([batch_size])
get_delay_data(identifier, delay_step, *indices)	Get delay data according to the provided delay steps.
load_state_dict(state_dict[, warn])	Copy parameters and buffers from state_dict into this module and its descendants.
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[, node_cls])
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.
state_dict()	Returns a dictionary containing a whole state of the module.
to(device)	Moves all variables into the given device.
tpu()	Move all variables into the TPU device.
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)	New in version 2.3.1.
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	Global delay data, which stores the delay variables and corresponding delay targets.
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.