HHLTC

class brainpy.dyn.HHLTC(size, sharding=None, keep_size=False, mode=None, name=None, method='exp_auto', init_var=True, ENa=50.0, gNa=120.0, EK=-77.0, gK=36.0, EL=-54.387, gL=0.03, V_th=20.0, C=1.0, V_initializer=Uniform(min_val=-70, max_val=-60.0, rng=[ 479025946 4206239744]), m_initializer=None, h_initializer=None, n_initializer=None)

Hodgkin–Huxley neuron model with liquid time constant.

Model Descriptions

The Hodgkin-Huxley (HH; Hodgkin & Huxley, 1952) model [1] for the generation of the nerve action potential is one of the most successful mathematical models of a complex biological process that has ever been formulated. The basic concepts expressed in the model have proved a valid approach to the study of bio-electrical activity from the most primitive single-celled organisms such as Paramecium, right through to the neurons within our own brains.

Mathematically, the model is given by,

\[ \begin{align}\begin{aligned}C \frac {dV} {dt} = -(\bar{g}_{Na} m^3 h (V &-E_{Na}) + \bar{g}_K n^4 (V-E_K) + g_{leak} (V - E_{leak})) + I(t)\\\frac {dx} {dt} &= \alpha_x (1-x) - \beta_x, \quad x\in {\rm{\{m, h, n\}}}\\&\alpha_m(V) = \frac {0.1(V+40)}{1-\exp(\frac{-(V + 40)} {10})}\\&\beta_m(V) = 4.0 \exp(\frac{-(V + 65)} {18})\\&\alpha_h(V) = 0.07 \exp(\frac{-(V+65)}{20})\\&\beta_h(V) = \frac 1 {1 + \exp(\frac{-(V + 35)} {10})}\\&\alpha_n(V) = \frac {0.01(V+55)}{1-\exp(-(V+55)/10)}\\&\beta_n(V) = 0.125 \exp(\frac{-(V + 65)} {80})\end{aligned}\end{align} \]

The illustrated example of HH neuron model please see this notebook.

The Hodgkin–Huxley model can be thought of as a differential equation system with four state variables, \(V_{m}(t),n(t),m(t)\), and \(h(t)\), that change with respect to time \(t\). The system is difficult to study because it is a nonlinear system and cannot be solved analytically. However, there are many numeric methods available to analyze the system. Certain properties and general behaviors, such as limit cycles, can be proven to exist.

References

[1]

Hodgkin, Alan L., and Andrew F. Huxley. “A quantitative description of membrane current and its application to conduction and excitation in nerve.” The Journal of physiology 117.4 (1952): 500.

[2]

https://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model

[3]

Ashwin, Peter, Stephen Coombes, and Rachel Nicks. “Mathematical frameworks for oscillatory network dynamics in neuroscience.” The Journal of Mathematical Neuroscience 6, no. 1 (2016): 1-92.

Examples

Here is a simple usage example:

import brainpy as bp

neu = bp.dyn.HHLTC(1)

# raise input current from 4 mA to 40 mA
inputs = bp.inputs.ramp_input(4, 40, 700, 100, 600,)

runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)

bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)

Parameters:

• size (sequence of int, int) – The size of the neuron group.

• ENa (float, ArrayType, Initializer, callable) – The reversal potential of sodium. Default is 50 mV.

• gNa (float, ArrayType, Initializer, callable) – The maximum conductance of sodium channel. Default is 120 msiemens.

• EK (float, ArrayType, Initializer, callable) – The reversal potential of potassium. Default is -77 mV.

• gK (float, ArrayType, Initializer, callable) – The maximum conductance of potassium channel. Default is 36 msiemens.

• EL (float, ArrayType, Initializer, callable) – The reversal potential of learky channel. Default is -54.387 mV.

• gL (float, ArrayType, Initializer, callable) – The conductance of learky channel. Default is 0.03 msiemens.

• V_th (float, ArrayType, Initializer, callable) – The threshold of the membrane spike. Default is 20 mV.

• C (float, ArrayType, Initializer, callable) – The membrane capacitance. Default is 1 ufarad.

• V_initializer (ArrayType, Initializer, callable) – The initializer of membrane potential.

• m_initializer (ArrayType, Initializer, callable) – The initializer of m channel.

• h_initializer (ArrayType, Initializer, callable) – The initializer of h channel.

• n_initializer (ArrayType, Initializer, callable) – The initializer of n channel.

• method (str) – The numerical integration method.

• name (str) – The group name.

reset_state(batch_size=None, **kwargs)

Reset function which resets local states in this model.

Simply speaking, this function should implement the logic of resetting of local variables in this node.

See https://brainpy.readthedocs.io/en/latest/tutorial_toolbox/state_resetting.html for details.

update(x=None)

The function to specify the updating rule.