Numerical phase-plane analysis of the Hodgkin-Huxley neuron

hh_phaseplane makes a numerical phase-plane analysis of the Hodgkin-Huxley neuron (hh_psc_alpha). Dynamics is investigated in the V-n space (see remark below). A constant DC can be specified and its influence on the nullclines can be studied.

Remark

To make the two-dimensional analysis possible, the (four-dimensional) Hodgkin-Huxley formalism needs to be artificially reduced to two dimensions, in this case by ‘clamping’ the two other variables, m and h, to constant values (m_eq and h_eq).

import nest
import numpy as np
from matplotlib import pyplot as plt


amplitude = 100.  # Set externally applied current amplitude in pA
dt = 0.1  # simulation step length [ms]

v_min = -100.  # Min membrane potential
v_max = 42.  # Max membrane potential
n_min = 0.1  # Min inactivation variable
n_max = 0.81  # Max inactivation variable
delta_v = 2.  # Membrane potential step length
delta_n = 0.01  # Inactivation variable step length

V_vec = np.arange(v_min, v_max, delta_v)
n_vec = np.arange(n_min, n_max, delta_n)

num_v_steps = len(V_vec)
num_n_steps = len(n_vec)

nest.ResetKernel()
nest.set_verbosity('M_ERROR')

nest.SetKernelStatus({'resolution': dt})
neuron = nest.Create('hh_psc_alpha')

# Numerically obtain equilibrium state
nest.Simulate(1000)

m_eq = neuron.Act_m
h_eq = neuron.Inact_h

neuron.I_e = amplitude  # Apply external current

# Scan state space
print('Scanning phase space')

V_matrix = np.zeros([num_n_steps, num_v_steps])
n_matrix = np.zeros([num_n_steps, num_v_steps])

# pp_data will contain the phase-plane data as a vector field
pp_data = np.zeros([num_n_steps * num_v_steps, 4])

count = 0
for i, V in enumerate(V_vec):
    for j, n in enumerate(n_vec):
        # Set V_m and n
        neuron.set(V_m=V, Act_n=n, Act_m=m_eq, Inact_h=h_eq)
        # Find state
        V_m = neuron.V_m
        Act_n = neuron.Act_n

        # Simulate a short while
        nest.Simulate(dt)

        # Find difference between new state and old state
        V_m_new = neuron.V_m - V
        Act_n_new = neuron.Act_n - n

        # Store in vector for later analysis
        V_matrix[j, i] = abs(V_m_new)
        n_matrix[j, i] = abs(Act_n_new)
        pp_data[count] = np.array([V_m, Act_n, V_m_new, Act_n_new])

        if count % 10 == 0:
            # Write updated state next to old state
            print('')
            print('Vm:  \t', V_m)
            print('new Vm:\t', V_m_new)
            print('Act_n:', Act_n)
            print('new Act_n:', Act_n_new)

        count += 1

# Set state for AP generation
neuron.set(V_m=-34., Act_n=0.2, Act_m=m_eq, Inact_h=h_eq)

print('')
print('AP-trajectory')
# ap will contain the trace of a single action potential as one possible
# numerical solution in the vector field
ap = np.zeros([1000, 2])
for i in range(1000):
    # Find state
    V_m = neuron.V_m
    Act_n = neuron.Act_n

    if i % 10 == 0:
        # Write new state next to old state
        print('Vm: \t', V_m)
        print('Act_n:', Act_n)
    ap[i] = np.array([V_m, Act_n])

    # Simulate again
    neuron.set(Act_m=m_eq, Inact_h=h_eq)
    nest.Simulate(dt)

# Make analysis
print('')
print('Plot analysis')

nullcline_V = []
nullcline_n = []

print('Searching nullclines')
for i in range(0, len(V_vec)):
    index = np.nanargmin(V_matrix[:][i])
    if index != 0 and index != len(n_vec):
        nullcline_V.append([V_vec[i], n_vec[index]])

    index = np.nanargmin(n_matrix[:][i])
    if index != 0 and index != len(n_vec):
        nullcline_n.append([V_vec[i], n_vec[index]])

print('Plotting vector field')
factor = 0.1
for i in range(0, np.shape(pp_data)[0], 3):
    plt.plot([pp_data[i][0], pp_data[i][0] + factor * pp_data[i][2]],
             [pp_data[i][1], pp_data[i][1] + factor * pp_data[i][3]],
             color=[0.6, 0.6, 0.6])

plt.plot(nullcline_V[:][0], nullcline_V[:][1], linewidth=2.0)
plt.plot(nullcline_n[:][0], nullcline_n[:][1], linewidth=2.0)

plt.xlim([V_vec[0], V_vec[-1]])
plt.ylim([n_vec[0], n_vec[-1]])

plt.plot(ap[:][0], ap[:][1], color='black', linewidth=1.0)

plt.xlabel('Membrane potential V [mV]')
plt.ylabel('Inactivation variable n')
plt.title('Phase space of the Hodgkin-Huxley Neuron')

plt.show()