Hodgkin-Huxley model

In this section you will interact with the famous Hodgkin-Huxley (HH) Neuron Model. This was a mathematical model of a neuron’s membrane potential based on careful experiments that studied the electrical properties of a squid’s giant axon, which is visible to the naked eye.

Image credit: National Institute of Health, USA

The HH Model is more realistic than many commonly used models of neurons. and captures the rich dynamics of a neuron’s membrane as a function of its transmembrane conductances.

!pip install interact --quiet
!pip install interactive --quiet

import matplotlib.pyplot as plt
import numpy as np
import ipywidgets as widgets
from ipywidgets import interact, interactive, fixed, interact_manual
from IPython.display import display

def HH_model(C,Gna,Gk,Gleak,Iin):
  dt      = 0.5; # all time units in ms
  t_end = 1000;
  num_timepoints = t_end/dt;
  time    = int(num_timepoints);

#  C       = 1; # (muF/cm^2)
#  Gna     = 2.0; # (2mS/cm^2)
#  Gk      = 1.44; # (1.44mS/cm^2)
#  Gleak   = 0.2; # (0.2mS/cm^2)


  Ena     = 50.0; # 50(mV)
  Ek      = -80.0; # -77(mV)
  Eleak   = -65.0; # -54(mV)

  kn      = 15.0; # (mV)
  km      = 7.0; # (mV);
  tau     = 10.0; # (ms)
  Vn      = -45.0; # (mV)

  n       = np.zeros((time,1));      # this is our gating parameter
  V       = Eleak*np.ones((time,1)); # this will store our membrane potential
#  Iin     = 2.0;                     # and this is an input current (muA/cm^2)

  x = np.arange(1, time, 1, dtype=int);


  for t in x:
      minf    = 1/(1+np.exp((-40.0-V[t-1])/km));
      ninf    = 1/(1+np.exp((Vn-V[t-1])/kn));
      h       = 0.89 - 1.1*n[t-1];

      dvdt = (1/C)*(-Gna*minf**3*h*(V[t-1]-Ena) - Gk*n[t-1]**4*(V[t-1]-Ek) - Gleak*(V[t-1]-Eleak) + Iin);
      dndt = (1/tau)*(ninf - n[t-1]);

      V[t] = V[t-1] + dvdt*dt;
      n[t] = n[t-1] + dndt*dt;

  t = np.arange(0, t_end, 0.5, dtype=float);


  plt.plot(t,V)
  plt.ylabel('Voltage (mV)');
  plt.xlabel('Time (ms)');
  plt.ylim(-100,100);
  plt.show


interact_manual(HH_model, C = widgets.FloatSlider(
    value=1.0,
    min=0.1,
    max=2.0,
    step=0.1,
    continuous_update=False,
    style = {'description_width': 'initial'},
    description='C membrane capacitance'),
    Gna = widgets.FloatSlider(
    value=2.0,
    min=0.0,
    max=5.0,
    step=0.1,
    continuous_update=False,
    style = {'description_width': 'initial'},
    description='gNa conductance'),
    Gk = widgets.FloatSlider(
    value=1.44,
    min=0.0,
    max=5.0,
    step=0.1,
    continuous_update=False,
    style = {'description_width': 'initial'},
    description='gK conductance'),
    Gleak = widgets.FloatSlider(
    value=0.2,
    min=0.0,
    max=1.0,
    step=0.1,
    continuous_update=False,
    style = {'description_width': 'initial'},
    description='gLeak conductance'),
    Iin = widgets.FloatSlider(
    value=2.0,
    min=0.0,
    max=10.0,
    step=0.1,
    continuous_update=False,
    description='Input current')
)

<function __main__.HH_model(C, Gna, Gk, Gleak, Iin)>