# brainmodels.neurons.HindmarshRose

class brainmodels.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, method='euler', **kwargs)[source]

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 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, monitors=['V'])
>>> group = bp.math.jit(group)
>>> group.run(1e3, inputs=['input', Is], report=0.1)
>>>
>>> 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(group.mon.ts, group.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, method='euler', **kwargs)[source]

Methods

 __init__(size[, a, b, c, d, r, s, V_rest, ...]) build_inputs([inputs, show_code]) build_monitors([show_code]) cpu() cuda() derivative(V, y, z, t, Iext) ints([method]) Collect all integrators in this node and the children nodes. load_states(filename[, verbose, check]) Load the model states. nodes([method, _paths]) Collect all children nodes. register_constant_delay(key, size, delay[, ...]) Register a constant delay. run(duration[, dt, report, inputs, extra_func]) The running function. save_states(filename[, all_vars]) Save the model states. step(t_and_dt, **kwargs) to(devices) tpu() 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

 implicit_nodes Used to wrap the implicit children nodes which cannot be accessed by self.xxx implicit_vars Used to wrap the implicit variables which cannot be accessed by self.xxx target_backend Used to specify the target backend which the model to run.