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, input_var=True, name=None, mode=None, spike_fun=<function inv_square_grad>)

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 brainpy.math as bm
>>> 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 = bm.array([1.0, 3.5, 2.5, 2.95, 2.8])
>>> Is = bm.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()

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, input_var=True, name=None, mode=None, spike_fun=<function inv_square_grad>)

Methods

__init__(size[, a, b, c, d, r, s, V_rest, ...])
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, y, z, I_ext)
dy(y, t, V)
dz(z, 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