# Numerical phase-plane analysis of the Hodgkin-Huxley neuron¶

Run this example as a Jupyter notebook:

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 = 100.0  # Set externally applied current amplitude in pA
dt = 0.1  # simulation step length [ms]

v_min = -100.0  # Min membrane potential
v_max = 42.0  # Max membrane potential
n_min = 0.1  # Min inactivation variable
n_max = 0.81  # Max inactivation variable
delta_v = 2.0  # 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.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=-34.0, 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()
```

Gallery generated by Sphinx-Gallery