Cell Physiology/Part 5/5.2 Action Potentials

Part 5 · Chapter 5.2 · Module M5.2

Action Potentials

The Hodgkin-Huxley equations, voltage-gated channel gating, refractoriness, and saltatory conduction

Learning Objectives

•State and integrate the full Hodgkin-Huxley 4-variable ODE for V, m, h, n.
•Explain the role of m³h Na⁺ kinetics and n⁴ K⁺ kinetics in action-potential shape.
•Describe absolute and relative refractoriness and how they arise from h and n dynamics.
•Apply cable theory to compute conduction velocity and length constant; compare myelinated vs unmyelinated axons.
•Relate demyelination to slowed conduction and clinical multiple sclerosis.

1.The Hodgkin-Huxley experiments

Between 1949 and 1952, Alan Hodgkin and Andrew Huxley used the voltage clamp technique developed by Kenneth Cole to dissect the currents underlying the action potential of the squid giant axon. A 1-mm diameter axon (from Loligo forbesii) allowed them to thread longitudinal silver wires through the axon interior, pinning V_m to a chosen value while measuring the compensating current.

By replacing external Na⁺ with choline and using TTX-like protocols (though TTX was characterized later), they decomposed the total current into three components:

A transient inward current, rising and falling rapidly, dependent on external Na⁺.
A sustained outward current, rising more slowly, dependent on internal K⁺.
A small, voltage-independent leak current.

Their 1952 paper formalized these currents into a system of four coupled ODEs that could reproduce the action potential with quantitative precision — and won the 1963 Nobel Prize. The model remains the lingua franca of computational neuroscience.

2.The full Hodgkin-Huxley equations

The membrane equation states that the capacitive current equals the negative of the sum of all ionic currents plus any external stimulus:

\[ C_m\,\frac{dV}{dt} = -\!\left[ g_{Na}\,m^3 h\,(V - E_{Na}) + g_K\,n^4\,(V - E_K) + g_L\,(V - E_L) \right] + I_\text{ext} \]

The gating variables m, h, n each obey first-order kinetics:

\[ \frac{dx}{dt} = \alpha_x(V)\,(1-x) - \beta_x(V)\,x \quad x \in \{m,h,n\} \]

Equivalently, \(x\) relaxes toward a voltage-dependent steady state with a voltage-dependent time constant:

\[ \tau_x(V) = \frac{1}{\alpha_x + \beta_x},\qquad x_\infty(V) = \frac{\alpha_x}{\alpha_x + \beta_x} \]

The rate functions (1/ms) for the classical HH squid model, in the modern convention where \(V\) is the membrane potential and resting potential is \(-65\) mV, are:

α_m(V) = 0.1 (V + 40) / (1 − exp(−(V + 40)/10))

β_m(V) = 4 exp(−(V + 65)/18)

α_h(V) = 0.07 exp(−(V + 65)/20)

β_h(V) = 1 / (1 + exp(−(V + 35)/10))

α_n(V) = 0.01 (V + 55) / (1 − exp(−(V + 55)/10))

β_n(V) = 0.125 exp(−(V + 65)/80)

The cubic dependence of Na⁺ activation on \(m\) and the quartic dependence of K⁺activation on \(n\) were originally phenomenological fits. We now know they correspond to the four voltage-sensing S4 helices of the tetrameric channel protein — three \(m\) gates plus one \(h\) ball-and-chain inactivation for Na⁺, and four \(n\) gates for K⁺.

3.Phases of the action potential

1. Threshold & upstroke

Depolarization to \(\approx -55\) mV rapidly raises \(m\). Since m³h grows much faster than n⁴, g_Na explodes. Positive feedback: Na⁺ entry depolarizes further, opening more channels. The upstroke rises at \(\sim 500\) V/s.

2. Peak & repolarization

V approaches E_Na = +50 mV but never reaches it because (i) h inactivates the Na channel, (ii) n opens the K channel. g_K overtakes g_Na, driving V back toward E_K.

3. Afterhyperpolarization

K⁺ channels remain open for several ms after V reaches rest. The membrane potential dips to \(\sim -80\) mV, near E_K, then returns as n decays.

4. Recovery

Na⁺ channels return from inactivation as h recovers (τ_h \(\sim\)1-5 ms at rest). Only now can another AP be fired.

Refractory periods

Absolute refractory: h \(\ll\) 1, no stimulus can fire an AP (\(\sim 1\) ms).
Relative refractory: elevated threshold because g_K is still high and h is partially restored (\(\sim 5\) ms).
Frequency cap: sets maximum firing rate around 500-1000 Hz for the fastest neurons.

4.Simulation 1 — Full Hodgkin-Huxley ODE

We integrate the complete HH system with three brief current pulses to demonstrate the all-or-nothing response, the time course of gating variables, and the refractory behavior.

Python

script.py181 lines

import numpy as np
import matplotlib.pyplot as plt

# ------------------------------------------------------------
# Hodgkin & Huxley (1952) squid giant axon model
# Membrane equation:
#   C_m dV/dt = -( g_Na m^3 h (V - E_Na) + g_K n^4 (V - E_K) + g_L (V - E_L) ) + I_ext
# Gating variables follow first-order kinetics:
#   dx/dt = alpha_x(V) (1 - x) - beta_x(V) x    for x in {m, h, n}
# Convention: V in mV relative to resting potential; we use "modern" convention
# where resting V_m ~ -65 mV and depolarization is positive.
# ------------------------------------------------------------

# Reversal potentials (mV) - modern convention
E_Na = 50.0
E_K  = -77.0
E_L  = -54.4

# Maximal conductances (mS / cm^2)
g_Na_max = 120.0
g_K_max  = 36.0
g_L      = 0.3

# Membrane capacitance (uF / cm^2)
C_m = 1.0

# HH rate functions (1/ms) - from Hodgkin-Huxley 1952, shifted to modern convention
def alpha_m(V): return 0.1*(V + 40.0)/(1.0 - np.exp(-(V + 40.0)/10.0))
def beta_m (V): return 4.0*np.exp(-(V + 65.0)/18.0)
def alpha_h(V): return 0.07*np.exp(-(V + 65.0)/20.0)
def beta_h (V): return 1.0/(1.0 + np.exp(-(V + 35.0)/10.0))
def alpha_n(V): return 0.01*(V + 55.0)/(1.0 - np.exp(-(V + 55.0)/10.0))
def beta_n (V): return 0.125*np.exp(-(V + 65.0)/80.0)

def x_inf(a,b):   return a/(a+b)
def tau_x (a,b):  return 1.0/(a+b)

# ---- Simulation ----
dt   = 0.01                # ms
T    = 50.0                # ms
n_t  = int(T/dt)
t    = np.arange(n_t)*dt

V    = np.zeros(n_t)
m    = np.zeros(n_t)
h    = np.zeros(n_t)
n    = np.zeros(n_t)

# initial conditions at rest (V_m = -65 mV)
V[0] = -65.0
m[0] = x_inf(alpha_m(V[0]), beta_m(V[0]))
h[0] = x_inf(alpha_h(V[0]), beta_h(V[0]))
n[0] = x_inf(alpha_n(V[0]), beta_n(V[0]))

# Stimulus: 1 ms current pulse at t=5 ms, and another at t=22 ms (refractory demo)
def I_ext(ti):
    if 5.0 <= ti < 6.0:    return 10.0     # uA/cm^2 above threshold
    if 22.0 <= ti < 23.0:  return 10.0     # will find neuron partially refractory
    if 35.0 <= ti < 36.0:  return 10.0
    return 0.0

I_trace = np.zeros(n_t)
g_Na_trace = np.zeros(n_t)
g_K_trace  = np.zeros(n_t)

for i in range(n_t - 1):
    v = V[i]
    am, bm = alpha_m(v), beta_m(v)
    ah, bh = alpha_h(v), beta_h(v)
    an, bn = alpha_n(v), beta_n(v)
    # conductances
    gNa = g_Na_max * m[i]**3 * h[i]
    gK  = g_K_max  * n[i]**4
    g_Na_trace[i] = gNa
    g_K_trace[i]  = gK
    # currents
    I_Na = gNa * (v - E_Na)
    I_K  = gK  * (v - E_K)
    I_Leak = g_L * (v - E_L)
    I_ex = I_ext(t[i])
    I_trace[i] = I_ex
    # update V
    dV = (-(I_Na + I_K + I_Leak) + I_ex) / C_m
    V[i+1] = v + dt*dV
    # update gates
    m[i+1] = m[i] + dt*(am*(1-m[i]) - bm*m[i])
    h[i+1] = h[i] + dt*(ah*(1-h[i]) - bh*h[i])
    n[i+1] = n[i] + dt*(an*(1-n[i]) - bn*n[i])

# Use np.trapezoid to integrate charge delivered by stimulus
Q_stim = np.trapezoid(I_trace, t)
print(f"Total injected charge: {Q_stim:.2f} nC/cm^2")

# Peak voltage and AP width
V_peak = V.max()
V_min  = V.min()
print(f"Peak V_m:   {V_peak:.1f} mV")
print(f"Trough V_m: {V_min:.1f} mV  (afterhyperpolarization)")

# ---- PLOTS ----
fig, axes = plt.subplots(3, 2, figsize=(13, 10), facecolor='#0a0a1a')

# (a) Voltage trace
ax = axes[0,0]; ax.set_facecolor('#0a0a1a')
ax.plot(t, V, color='#67e8f9', lw=1.8)
ax.axhline(-65, color='#555', ls=':', lw=0.8)
ax.axhline(0, color='#fbbf24', ls=':', lw=0.8, label='Threshold zone')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('V_m (mV)', color='white')
ax.set_title('(a) Action potential waveform', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=8)
for s in ax.spines.values(): s.set_color('#444')

# (b) Gating variables
ax = axes[0,1]; ax.set_facecolor('#0a0a1a')
ax.plot(t, m, color='#f97316', lw=1.6, label='m (Na activation)')
ax.plot(t, h, color='#f43f5e', lw=1.6, label='h (Na inactivation)')
ax.plot(t, n, color='#60a5fa', lw=1.6, label='n (K activation)')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('Gate value', color='white')
ax.set_title('(b) Gating variables m, h, n', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
for s in ax.spines.values(): s.set_color('#444')

# (c) Conductances
ax = axes[1,0]; ax.set_facecolor('#0a0a1a')
ax.plot(t, g_Na_trace, color='#f97316', lw=1.8, label='g_Na')
ax.plot(t, g_K_trace,  color='#60a5fa', lw=1.8, label='g_K')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('g (mS/cm^2)', color='white')
ax.set_title('(c) Conductance dynamics', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
for s in ax.spines.values(): s.set_color('#444')

# (d) I-V curve at rest (static) and at peak-AP m,h,n
ax = axes[1,1]; ax.set_facecolor('#0a0a1a')
V_range = np.linspace(-90, 60, 200)
# steady-state currents at V=V_range, m=m_inf, h=h_inf, n=n_inf
m_ss = np.array([x_inf(alpha_m(v),beta_m(v)) for v in V_range])
h_ss = np.array([x_inf(alpha_h(v),beta_h(v)) for v in V_range])
n_ss = np.array([x_inf(alpha_n(v),beta_n(v)) for v in V_range])
I_Na_iv = g_Na_max * m_ss**3 * h_ss * (V_range - E_Na)
I_K_iv  = g_K_max  * n_ss**4 * (V_range - E_K)
I_L_iv  = g_L * (V_range - E_L)
I_tot_iv = I_Na_iv + I_K_iv + I_L_iv
ax.plot(V_range, I_Na_iv, color='#f97316', lw=1.6, label='I_Na (steady)')
ax.plot(V_range, I_K_iv,  color='#60a5fa', lw=1.6, label='I_K  (steady)')
ax.plot(V_range, I_tot_iv,color='#a3e635', lw=1.8, label='I_total')
ax.axhline(0, color='#555', lw=0.5)
ax.set_xlabel('V (mV)', color='white'); ax.set_ylabel('Current (uA/cm^2)', color='white')
ax.set_title('(d) Steady-state I-V curves', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
for s in ax.spines.values(): s.set_color('#444')

# (e) Phase plane: V vs n
ax = axes[2,0]; ax.set_facecolor('#0a0a1a')
sc = ax.scatter(V, n, c=t, cmap='plasma', s=2)
ax.set_xlabel('V (mV)', color='white'); ax.set_ylabel('n', color='white')
ax.set_title('(e) Phase plane (V, n) - AP as limit cycle', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
cb = plt.colorbar(sc, ax=ax); cb.set_label('time (ms)', color='white')
cb.ax.tick_params(colors='white')
for s in ax.spines.values(): s.set_color('#444')

# (f) Stimulus trace
ax = axes[2,1]; ax.set_facecolor('#0a0a1a')
ax.plot(t, I_trace, color='#fbbf24', lw=1.5)
ax.fill_between(t, 0, I_trace, color='#fbbf24', alpha=0.3)
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('I_ext (uA/cm^2)', color='white')
ax.set_title('(f) Stimulus: 3 pulses (refractory test)', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

plt.suptitle('Hodgkin-Huxley squid giant axon - full 4-variable ODE',
             color='#67e8f9', fontsize=14, y=0.995)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Saved output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Note how the second pulse at 22 ms arrives during the relative refractory period and may fail to elicit a full-amplitude spike. The phase plane (V, n) reveals the action potential as a closed trajectory in state space — a stable limit cycle in the suprathreshold regime.

5.Molecular structure of voltage-gated channels

Na_v channels are \(\sim\)260 kDa single-chain proteins with four homologous domains (DI-DIV), each containing six transmembrane helices S1-S6. The S4 helix in each domain carries four positively charged arginines that sense voltage. K_v channels are true tetramers of four identical subunits.

Activation

Depolarization drives the S4 helix outward; the gating charge (~12 e per channel) translates, pulling open the pore through S4-S5 linkers.

Fast inactivation

The DIII-DIV linker (IFM motif, "hinged lid") swings into the pore mouth within 1 ms, physically occluding it. This is the h gate.

Selectivity filter

K⁺ channels use a DEKA/TVGYG signature whose backbone carbonyls coordinate dehydrated K⁺ — 10,000:1 selectivity vs Na⁺.

Pharmacology

TTX, saxitoxin block Na_v from outside; local anesthetics (lidocaine) block from inside. TEA blocks K_v from outside.

MacKinnon's 1998 X-ray structure of KcsA (Nobel 2003) confirmed the HH picture at atomic resolution.

6.Cable theory and conduction

For a long thin axon of radius a, axial resistivity R_i, and membrane properties (C_m, g_m), current balance gives the cable equation:

\[ C_m\frac{\partial V}{\partial t} = \frac{a}{2 R_i}\,\frac{\partial^2 V}{\partial x^2} - I_\text{ion}(V,t) + I_\text{ext}(x,t) \]

In the passive sub-threshold regime, this reduces to a diffusion equation with two characteristic scales:

\[ \lambda = \sqrt{\frac{a\,r_m}{2 R_i}} \quad\text{(length constant)}, \qquad \tau_m = r_m C_m \quad\text{(time constant)} \]

For a 1 μm radius axon with typical mammalian parameters, λ \(\sim\)0.5 mm and τ_m \(\sim\)10 ms. Passive spread of synaptic potentials decays exponentially over one length constant.

Conduction velocity

For a uniformly active (unmyelinated) axon, the velocity of an action potential scales with diameter as

\[ v \propto \sqrt{d} \]

Squid giant axon (\(d \approx 0.5\) mm): \(\sim 25\) m/s. Mammalian C fibers (\(d \approx 1\) μm): \(\sim 1\) m/s. This is why evolution "gigantized" the squid escape axon rather than myelinating it.

7.Saltatory conduction and myelination

Myelin, produced by oligodendrocytes (CNS) or Schwann cells (PNS), is a tightly wrapped sheath that reduces membrane capacitance by 10-100x and leak conductance by 100x per internode. Every 100-200 μm, a 1 μm gap — a node of Ranvier — contains a massive density of Na_v1.6 channels (up to 1000/μm²).

The action potential effectively "jumps" from node to node (saltare, to leap). Conduction velocity now scales linearly with diameter:

\[ v \propto d \qquad\text{(myelinated fibers)} \]

Human Aα motor fibers (\(d \approx\) 15 μm) conduct at 100-120 m/s — the fastest action potentials in biology. This is also why myelin is metabolically efficient: Na⁺/K⁺ pumping is needed only at nodes.

Fiber	Diameter (μm)	Myelin	CV (m/s)	Function
Aα	13-20	Heavy	80-120	Motor, proprioception
Aβ	6-12	Heavy	35-75	Touch, pressure
Aδ	1-5	Light	5-30	Sharp pain, cold
B	1-3	Light	3-15	Autonomic preganglionic
C	0.4-1.2	None	0.5-2	Slow pain, postganglionic

8.Simulation 2 — Cable equation & saltatory conduction

We discretize a 1 cm axon into 5 μm segments and integrate the cable equation with HH-like active dynamics restricted to (a) every node in a myelinated axon, (b) every segment in an unmyelinated axon, and (c) a demyelinated axon with partial loss of myelin insulation.

Python

script.py195 lines

import numpy as np
import matplotlib.pyplot as plt

# ------------------------------------------------------------
# Cable equation with active patches (nodes of Ranvier)
#   C_m dV/dt = (a/(2 R_i)) d^2V/dx^2 - g_m(x) (V - E_L) + I_Na(x,t) + I_K(x,t) + I_stim
# Segments: alternating internode (insulated) and node (active HH-like).
# Used to demonstrate saltatory conduction and compare myelinated vs unmyelinated axons.
# ------------------------------------------------------------

# Geometry
L    = 10000.0       # um total axon length (1 cm)
dx   = 5.0           # um
N    = int(L/dx)
x    = np.arange(N)*dx

# Physical parameters
C_m   = 1.0          # uF/cm^2  (nodes)
R_i   = 100.0        # Ohm cm   axial resistivity
a_um  = 1.0          # axon radius in um
a_cm  = a_um*1e-4    # radius in cm
# cable coupling term: D = a/(2 R_i C_m) with unit conversions
D = (a_cm/(2.0*R_i*C_m)) * 1e8   # um^2/ms equivalent, approximate

# Active (node) parameters: high g_Na and g_K
# Passive (internode): very low g_m due to myelin wrapping ~ 100x reduction
g_Na_node = 120.0    # mS/cm^2
g_K_node  = 36.0
g_m_node  = 1.0      # leak (mS/cm^2)
g_m_inter_myel   = 0.01    # myelinated internode - very low leak, low capacitance effectively
g_m_inter_demyel = 1.0     # demyelinated = leaky

E_Na, E_K, E_L = 50.0, -77.0, -65.0

# Node spacing: every 100 um in myelinated axon
node_period = 100   # each 100 um contains a 2-point node
def build_active_mask(node_every_um):
    mask = np.zeros(N, dtype=bool)
    step = int(node_every_um/dx)
    for i in range(0, N, step):
        mask[i:i+2] = True
    return mask

def run_cable(myelinated=True, demyelinated=False, stim_pos=50, stim_t=(1.0,2.0), T=10.0, dt=0.005):
    n_t = int(T/dt)
    V = np.full(N, -65.0)
    m_g = np.full(N, 0.05)
    h_g = np.full(N, 0.6)
    n_g = np.full(N, 0.32)
    # Active mask
    if myelinated:
        active = build_active_mask(node_period)
    else:
        active = np.ones(N, dtype=bool)   # entire axon active, unmyelinated

V_hist = np.zeros((n_t, N))

# Leak per compartment
    gm = np.full(N, g_m_node)
    if myelinated:
        if demyelinated:
            gm[~active] = g_m_inter_demyel
        else:
            gm[~active] = g_m_inter_myel

for ti in range(n_t):
        t_now = ti*dt
        # 2nd derivative (cable term)
        d2V = np.zeros_like(V)
        d2V[1:-1] = (V[:-2] - 2*V[1:-1] + V[2:])/(dx*dx)
        d2V[0]  = (V[1] - V[0])/(dx*dx)
        d2V[-1] = (V[-2] - V[-1])/(dx*dx)
        # Membrane currents
        I_ionic = gm * (V - E_L)
        # Active nodes have HH currents
        if active.any():
            # simple HH gate updates (1D vectorised)
            def am(v): return 0.1*(v + 40.0)/(1.0 - np.exp(-(v + 40.0)/10.0 + 1e-12))
            def bm_(v): return 4.0*np.exp(-(v + 65.0)/18.0)
            def ah(v): return 0.07*np.exp(-(v + 65.0)/20.0)
            def bh_(v): return 1.0/(1.0 + np.exp(-(v + 35.0)/10.0))
            def an(v): return 0.01*(v + 55.0)/(1.0 - np.exp(-(v + 55.0)/10.0 + 1e-12))
            def bn_(v): return 0.125*np.exp(-(v + 65.0)/80.0)
            m_g[active] += dt*(am(V[active])*(1-m_g[active]) - bm_(V[active])*m_g[active])
            h_g[active] += dt*(ah(V[active])*(1-h_g[active]) - bh_(V[active])*h_g[active])
            n_g[active] += dt*(an(V[active])*(1-n_g[active]) - bn_(V[active])*n_g[active])
            INa = np.zeros(N); IK = np.zeros(N)
            INa[active] = g_Na_node * m_g[active]**3 * h_g[active] * (V[active] - E_Na)
            IK[active]  = g_K_node  * n_g[active]**4 * (V[active] - E_K)
            I_ionic[active] += INa[active] + IK[active]

# External stimulus
        I_ext = np.zeros(N)
        if stim_t[0] <= t_now < stim_t[1]:
            I_ext[stim_pos] = 40.0

dV = D*d2V - I_ionic + I_ext
        V += dt*dV/C_m
        V_hist[ti] = V

return V_hist

print("Simulating myelinated axon...")
V_myel = run_cable(myelinated=True,  demyelinated=False, T=6.0)
print("Simulating unmyelinated axon...")
V_unmy = run_cable(myelinated=False, T=6.0)
print("Simulating demyelinated (MS-like)...")
V_dem  = run_cable(myelinated=True, demyelinated=True, T=6.0)

# Estimate conduction velocity by peak-tracking
def conduction_velocity(V_hist, dt=0.005, dx=5.0):
    n_t, N = V_hist.shape
    peak_times = np.full(N, np.nan)
    for i in range(N):
        trace = V_hist[:, i]
        if trace.max() > 0:
            peak_times[i] = np.argmax(trace)*dt
    good = ~np.isnan(peak_times)
    idx  = np.arange(N)[good]
    tts  = peak_times[good]
    # linear fit over monotonic region
    if len(idx) > 20:
        mask = idx > idx.min()+20
        p = np.polyfit(idx[mask]*dx, tts[mask], 1)
        v = 1.0/p[0]  # um/ms = mm/s... actually um/ms = m/s
        return v, peak_times
    return np.nan, peak_times

v_m, pt_m = conduction_velocity(V_myel)
v_u, pt_u = conduction_velocity(V_unmy)
v_d, pt_d = conduction_velocity(V_dem)
print(f"Conduction velocity - myelinated   : {v_m:6.2f} m/s")
print(f"Conduction velocity - unmyelinated : {v_u:6.2f} m/s")
print(f"Conduction velocity - demyelinated : {v_d:6.2f} m/s")

# Use np.trapezoid to compute integrated stimulus pulse area
t_arr = np.arange(V_myel.shape[0])*0.005
stim_area = np.trapezoid(np.where((t_arr>=1.0)&(t_arr<2.0),40.0,0.0), t_arr)
print(f"Stimulus pulse integrated area: {stim_area:.1f} (uA/cm^2)*ms")

# ---- PLOTS ----
fig, axes = plt.subplots(2, 2, figsize=(13, 9), facecolor='#0a0a1a')

ax = axes[0,0]; ax.set_facecolor('#0a0a1a')
im = ax.imshow(V_myel.T, aspect='auto', origin='lower',
               extent=[0, V_myel.shape[0]*0.005, 0, L/1000.0],
               cmap='plasma', vmin=-80, vmax=40)
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('Position (mm)', color='white')
ax.set_title(f'(a) Myelinated axon - CV ~ {v_m:.1f} m/s', color='#67e8f9')
ax.tick_params(colors='white')
cb = plt.colorbar(im, ax=ax); cb.set_label('V_m (mV)', color='white'); cb.ax.tick_params(colors='white')
for s in ax.spines.values(): s.set_color('#444')

ax = axes[0,1]; ax.set_facecolor('#0a0a1a')
im = ax.imshow(V_unmy.T, aspect='auto', origin='lower',
               extent=[0, V_unmy.shape[0]*0.005, 0, L/1000.0],
               cmap='plasma', vmin=-80, vmax=40)
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('Position (mm)', color='white')
ax.set_title(f'(b) Unmyelinated axon - CV ~ {v_u:.1f} m/s', color='#67e8f9')
ax.tick_params(colors='white')
cb = plt.colorbar(im, ax=ax); cb.set_label('V_m (mV)', color='white'); cb.ax.tick_params(colors='white')
for s in ax.spines.values(): s.set_color('#444')

ax = axes[1,0]; ax.set_facecolor('#0a0a1a')
im = ax.imshow(V_dem.T, aspect='auto', origin='lower',
               extent=[0, V_dem.shape[0]*0.005, 0, L/1000.0],
               cmap='plasma', vmin=-80, vmax=40)
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('Position (mm)', color='white')
ax.set_title(f'(c) Demyelinated (MS-like) - CV ~ {v_d:.1f} m/s', color='#67e8f9')
ax.tick_params(colors='white')
cb = plt.colorbar(im, ax=ax); cb.set_label('V_m (mV)', color='white'); cb.ax.tick_params(colors='white')
for s in ax.spines.values(): s.set_color('#444')

ax = axes[1,1]; ax.set_facecolor('#0a0a1a')
# Space-time traces at fixed times
sample_times = [1.5, 2.5, 3.5, 4.5]
for tt in sample_times:
    idx = int(tt/0.005)
    ax.plot(x/1000.0, V_myel[idx], lw=1.5, label=f'Myel, t={tt}ms')
for tt in sample_times:
    idx = int(tt/0.005)
    ax.plot(x/1000.0, V_unmy[idx], lw=1.0, ls='--', alpha=0.7, label=f'Unmyel, t={tt}ms')
ax.set_xlabel('Position (mm)', color='white'); ax.set_ylabel('V_m (mV)', color='white')
ax.set_title('(d) Spatial V_m profiles - saltatory jumps visible', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=7, ncol=2)
for s in ax.spines.values(): s.set_color('#444')

plt.suptitle('Cable theory - saltatory conduction and demyelination',
             color='#67e8f9', fontsize=14, y=0.995)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Saved output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The space-time heat maps show clean internode jumps in the myelinated case and broad, smooth propagation in the unmyelinated axon. Demyelination causes slowed conduction, conduction block, and pattern instability — the electrophysiological signature of multiple sclerosis.

Clinical connections

Multiple sclerosis

Autoimmune demyelination in the CNS. Conduction slows, fails, or becomes intermittent; optic neuritis, pyramidal signs, sensory loss.

Guillain-Barré syndrome

Peripheral demyelination (acute inflammatory polyradiculoneuropathy); ascending flaccid paralysis.

Channelopathies

SCN1A mutations cause Dravet syndrome; SCN9A gain-of-function causes erythromelalgia; SCN9A loss-of-function abolishes pain perception.

Local anesthetics

Lidocaine and bupivacaine are use-dependent Na_v blockers: they preferentially bind inactivated channels, thus blocking rapidly firing nociceptors first.

Cardiac long QT

K_v11.1 (hERG) loss-of-function delays ventricular repolarization; predisposes to torsade de pointes.

Epilepsy

Phenytoin, carbamazepine, lamotrigine are Na_v blockers that prolong inactivation; valproate also modulates Na_v.

Key equations

HH membrane:

\( C_m dV/dt = -g_{Na}m^3h(V-E_{Na}) - g_K n^4(V-E_K) - g_L(V-E_L) + I_\text{ext} \)

Gate kinetics:

\( dx/dt = \alpha_x(V)(1-x) - \beta_x(V) x \)

Cable:

\( C_m \partial_t V = (a/2R_i)\,\partial_x^2 V - I_\text{ion} + I_\text{ext} \)

Length constant:

\( \lambda = \sqrt{a r_m /(2 R_i)} \)

CV scaling:

\( v \propto \sqrt{d}\;\text{(unmyelinated)},\quad v \propto d\;\text{(myelinated)} \)

9.Summary diagram

References

• Hodgkin, A. L. & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500-544.
• Cole, K. S. (1949). Dynamic electrical characteristics of the squid axon membrane. Arch. Sci. Physiol. 3, 253-258.
• Rushton, W. A. H. (1951). A theory of the effects of fibre size in medullated nerve. J. Physiol. 115, 101-122.
• Doyle, D. A. et al. (1998). The structure of the potassium channel. Science 280, 69-77.
• Hille, B. (2001). Ion Channels of Excitable Membranes (3rd ed.). Sinauer.
• Waxman, S. G. (2006). Axonal conduction and injury in multiple sclerosis. Nat. Rev. Neurosci. 7, 932-941.

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