Excitable membranes and the action potential

Animals sense, decide and act quickly, and almost all of that speed depends on electrical signals carried by excitable cells: neurons, muscle fibres and some endocrine and sensory cells. An escaping squid jets away within a fraction of a second of seeing a predator; a cockroach turns from a puff of air in about 20 milliseconds. This lesson explains how membranes generate and propagate the action potential, the all-or-none electrical impulse that carries information over long distances. BIOL-214 introduced ion pumps, channels and the resting membrane potential, including the Nernst and Goldman equations. Here we build on that foundation to explain the action potential quantitatively, using the model that Alan Hodgkin and Andrew Huxley developed from their experiments on the squid giant axon, and to ask how animals have made conduction faster.

The resting potential revisited

Every animal cell has a voltage across its plasma membrane, inside negative relative to outside, typically −60 to −90 millivolts in neurons and muscle. Two ingredients produce it. First, the sodium–potassium pump keeps the concentrations of potassium high inside and sodium high outside. Second, the resting membrane is much more permeable to potassium than to sodium, because many potassium leak channels are open. Potassium ions diffuse out down their concentration gradient, leaving behind unbalanced negative charge, until the electrical force pulling them back balances the chemical force pushing them out. The voltage at which the two balance for a single ion is its equilibrium potential, given by the Nernst equation.

At 20 °C, RT/F is about 25.3 mV, so for a monovalent cation the Nernst potential is about 58 mV times the base-10 logarithm of the concentration ratio. When a membrane is permeable to several ions, the resting potential lies between their equilibrium potentials, weighted by the relative permeabilities, as described by the Goldman–Hodgkin–Katz equation. Because the resting membrane is mostly permeable to potassium, the resting potential lies close to the potassium equilibrium potential, but a small sodium permeability pulls it a few millivolts toward the sodium equilibrium potential.

import math

R, F = 8.314, 96485.0
def nernst(z, out, inn, T=293.15):
    return 1000 * R * T / (z * F) * math.log(out / inn)

def ghk(pk, pna, pcl, ko, ki, nao, nai, clo, cli, T=293.15):
    num = pk * ko + pna * nao + pcl * cli
    den = pk * ki + pna * nai + pcl * clo
    return 1000 * R * T / F * math.log(num / den)

# Squid giant axon, approximate concentrations (mM) in sea water
ko, ki, nao, nai, clo, cli = 20, 400, 440, 50, 560, 52
print(f"E_K  = {nernst(1, ko, ki):6.1f} mV")
print(f"E_Na = {nernst(1, nao, nai):6.1f} mV")
print(f"E_Cl = {nernst(-1, clo, cli):6.1f} mV")
for label_, pna in (("rest   (PK:PNa:PCl = 1:0.04:0.45)", 0.04), ("peak   (PK:PNa:PCl = 1:20:0.45)", 20.0)):
    print(f"{label_}: Vm = {ghk(1.0, pna, 0.45, ko, ki, nao, nai, clo, cli):6.1f} mV")

E_K  =  -75.7 mV
E_Na =   54.9 mV
E_Cl =  -60.0 mV
rest   (PK:PNa:PCl = 1:0.04:0.45): Vm =  -59.9 mV
peak   (PK:PNa:PCl = 1:20:0.45): Vm =   42.4 mV

With the classic permeability ratios at rest, the Goldman equation gives a resting potential of about −60 mV, close to the measured value for squid axons. If sodium permeability rises about 500-fold, as it does at the peak of an action potential, the membrane potential swings to about +40 mV, approaching the sodium equilibrium potential. The action potential is therefore, at heart, a brief, dramatic change in relative permeability, and the ions that cross the membrane to produce it are a tiny fraction of those present. A rough calculation shows that only about one potassium ion in a million inside a squid axon needs to cross the membrane to charge the membrane capacitance during one impulse. The concentration gradients barely change, and the pump restores them slowly in the background.

Hodgkin and Huxley and the squid giant axon

The squid giant axon, up to a millimetre in diameter, controls the mantle muscles used in jet propulsion. Its size let Hodgkin and Huxley, working at Plymouth in England in the late 1940s and early 1950s, thread a wire electrode inside it and use the voltage clamp, a feedback circuit developed by Kenneth Cole and George Marmont, to hold the membrane potential at chosen values while measuring the current that flowed. By replacing external sodium with an impermeant cation, they separated the current into a sodium component and a potassium component. Their analysis, published in 1952, earned them the 1963 Nobel Prize in Physiology or Medicine, which they shared with John Eccles.

The voltage clamp revealed three features of the membrane. A depolarizing step produced an early inward current carried by sodium, which rose within a millisecond and then declined even though the voltage was held constant, and a delayed outward current carried by potassium, which rose more slowly and was sustained. The sodium conductance therefore had two processes: activation, which opened it quickly on depolarization, and inactivation, which closed it more slowly. The potassium conductance had only delayed activation. Hodgkin and Huxley described each conductance with hypothetical gating particles, m and h for sodium and n for potassium, whose probabilities of being in the permissive position changed with voltage and time.

Each gating variable relaxes toward a voltage-dependent steady state with a voltage-dependent time constant. Decades later, cloning and structural biology showed that the gating particles correspond to real molecular parts. Voltage-gated sodium and potassium channels have four voltage-sensing domains, each containing a positively charged S4 helix that moves outward when the membrane depolarizes, pulling open the channel's gate. Fast inactivation of sodium channels is produced by a short intracellular loop, the inactivation gate, that swings in to block the pore. The fourth power in the potassium term matches the four subunits of the potassium channel, a satisfying confirmation of a model built from electrical measurements alone.

Simulating an action potential

The Hodgkin–Huxley equations can be integrated numerically with a simple Euler method. The code below uses the standard parameters for the squid axon at 6.3 °C, written in the modern convention with a resting potential near −65 mV. It applies a brief current pulse and reports the main features of the resulting action potential, and then tests a weaker pulse to show the threshold.

import math

# Hodgkin-Huxley model (squid axon, 6.3 degC), modern sign convention
gNa, gK, gL = 120.0, 36.0, 0.3          # mS/cm^2
ENa, EK, EL = 50.0, -77.0, -54.4         # mV
Cm = 1.0                                 # uF/cm^2

def rates(V):
    am = 0.1 * (V + 40) / (1 - math.exp(-(V + 40) / 10))
    bm = 4.0 * math.exp(-(V + 65) / 18)
    ah = 0.07 * math.exp(-(V + 65) / 20)
    bh = 1.0 / (1 + math.exp(-(V + 35) / 10))
    an = 0.01 * (V + 55) / (1 - math.exp(-(V + 55) / 10))
    bn = 0.125 * math.exp(-(V + 65) / 80)
    return am, bm, ah, bh, an, bn

def run(amp, dur=0.5, T=20.0, dt=0.005):
    V = -65.0
    am, bm, ah, bh, an, bn = rates(V)
    m, h, n = am / (am + bm), ah / (ah + bh), an / (an + bn)
    peak, t_peak, vmin, t = V, 0.0, V, 0.0
    while t < T:
        I = amp if 1.0 <= t < 1.0 + dur else 0.0
        am, bm, ah, bh, an, bn = rates(V)
        m += dt * (am * (1 - m) - bm * m)
        h += dt * (ah * (1 - h) - bh * h)
        n += dt * (an * (1 - n) - bn * n)
        Iion = gNa * m**3 * h * (V - ENa) + gK * n**4 * (V - EK) + gL * (V - EL)
        V += dt * (I - Iion) / Cm
        if V > peak: peak, t_peak = V, t
        if t > t_peak and V < vmin: vmin = V
        t += dt
    return peak, t_peak, vmin

for amp in (5.0, 20.0, 80.0):
    peak, tp, vmin = run(amp)
    spike = "spike" if peak > 0 else "no spike"
    print(f"pulse {amp:4.1f} uA/cm2 x 0.5 ms: peak {peak:6.1f} mV at {tp:4.2f} ms, "
          f"after-hyperpolarization {vmin:6.1f} mV  ({spike})")

pulse  5.0 uA/cm2 x 0.5 ms: peak  -62.8 mV at 1.50 ms, after-hyperpolarization  -65.6 mV  (no spike)
pulse 20.0 uA/cm2 x 0.5 ms: peak   39.6 mV at 3.06 ms, after-hyperpolarization  -76.2 mV  (spike)
pulse 80.0 uA/cm2 x 0.5 ms: peak   41.8 mV at 1.82 ms, after-hyperpolarization  -76.2 mV  (spike)

The weakest pulse produces only a small, passive depolarization that decays; the stronger pulses produce a full action potential peaking near +40 mV, followed by an undershoot below the resting potential. A pulse four times stronger produces a spike of essentially the same size; it simply arrives sooner, because threshold is reached faster. This is the all-or-none property: once threshold is crossed, the positive feedback between depolarization and sodium channel opening runs to completion, and the shape of the impulse is set by the membrane, not by the stimulus. Information is therefore carried not by the size of action potentials but by their frequency and timing.

Threshold, refractoriness and firing patterns

Threshold is the voltage at which the inward sodium current produced by a depolarization just exceeds the outward potassium and leak currents, so that the depolarization becomes self-sustaining. After an action potential, sodium channels remain inactivated for a millisecond or so, and potassium channels remain open, so a second impulse cannot be triggered at all during the absolute refractory period and needs a stronger stimulus during the relative refractory period. Refractoriness limits the maximum firing frequency, to several hundred impulses per second in fast neurons, and it makes action potentials travel in one direction along an axon, because the membrane behind an advancing impulse is refractory.

Real neurons contain many more kinds of channels than the squid axon. Calcium channels, calcium-activated potassium channels, hyperpolarization-activated cation channels and slowly inactivating potassium channels give different neurons different firing patterns: regular firing, adaptation in which the rate slows during a sustained stimulus, bursting, and rhythmic pacemaking. The heart's pacemaker cells and the neurons that drive breathing and walking use these channels to generate rhythms without any outside input. Comparative studies show that the basic voltage-gated sodium and potassium channels are ancient, with homologues in single-celled organisms, while the diversity of channel types has expanded greatly in animals with complex nervous systems.

Toxins that target channels

Many animals and plants defend themselves with toxins that block or modify ion channels, and these toxins have been invaluable tools. Tetrodotoxin, made by bacteria living in pufferfish, rough-skinned newts of the Pacific Northwest and blue-ringed octopuses, blocks voltage-gated sodium channels from the outside. Saxitoxin, made by dinoflagellates and concentrated by shellfish during harmful algal blooms, does the same and causes paralytic shellfish poisoning, which is why shellfish harvesting on both coasts is closed when monitoring detects it. Tetraethylammonium blocks potassium channels. Garter snakes that feed on rough-skinned newts have evolved sodium channels resistant to tetrodotoxin, and the newts in turn have evolved higher toxin levels, a famous example of a coevolutionary arms race studied by Edmund Brodie and colleagues.

Passive spread and the cable properties of axons

An action potential spreads along an axon because the current that enters at an active patch of membrane flows along the axon's interior and depolarizes the membrane ahead to threshold. How far this local current spreads depends on the axon's cable properties. Current flowing along the inside of the axon meets the axial resistance of the cytoplasm, while some of it leaks out across the membrane through the membrane resistance. The length constant, λ, is the distance over which a steady voltage decays to about 37 per cent of its starting value. It increases with the ratio of membrane resistance to internal resistance, and because the internal resistance per unit length falls with the square of the diameter while the membrane resistance per unit length falls only with the diameter, λ increases with the square root of axon diameter.

Membrane capacitance adds a time element. Before the membrane voltage can change, its capacitance must be charged, and the time constant τ, the product of membrane resistance and capacitance, describes how quickly this happens. Conduction velocity in an unmyelinated axon depends on both: a larger λ means current reaches farther ahead, and a smaller τ means the membrane there charges faster. Theory and measurements agree that conduction velocity in unmyelinated axons is roughly proportional to the square root of diameter. That explains the giant axons of squid, earthworms and cockroaches: doubling conduction velocity requires a fourfold increase in diameter, which is affordable only for a few axons dedicated to escape.

Myelin and saltatory conduction

Vertebrates found a better solution. In the peripheral nervous system, Schwann cells, and in the central nervous system, oligodendrocytes, wrap axons in many layers of membrane called myelin. Myelin increases the membrane resistance and decreases the capacitance of the wrapped segments, so current spreads much farther with little loss and charges the membrane much less. Voltage-gated sodium channels are concentrated at gaps in the myelin, the nodes of Ranvier, about 1 to 2 µm long and spaced 0.2 to 2 mm apart. The action potential appears to jump from node to node, saltatory conduction, and conduction velocity becomes roughly proportional to diameter rather than to its square root. A myelinated axon 10 µm across conducts at about 50 to 60 metres per second, about the same speed as a squid giant axon 50 to 100 times wider.

Myelin evolved in jawed vertebrates and, independently, in some invertebrates, notably certain shrimp and copepods, whose myelin-like sheaths support very fast conduction in small axons. Myelination has a cost: myelin must be made and maintained, and in demyelinating diseases such as multiple sclerosis, loss of myelin slows or blocks conduction. But it allows vertebrates to pack many fast axons into a small nerve, which is essential for complex, rapid behaviour in large bodies. A human sciatic nerve contains tens of thousands of axons; if they all had to be unmyelinated and as fast as they are, the nerve would be many centimetres thick.

import math

# Conduction velocity scaling (rules of thumb):
#   unmyelinated: v ~ k1 * sqrt(d)      myelinated: v ~ k2 * d  (d in um)
k1 = 22.0 / math.sqrt(500.0)   # calibrated: squid axon, 500 um -> 22 m/s
k2 = 5.5                       # about 5.5 m/s per um (Hursh-type rule for mammals)

print(" target v (m/s)   unmyelinated d needed   myelinated d needed")
for v in (2, 10, 25, 60, 100):
    d_un = (v / k1) ** 2
    d_my = v / k2
    print(f"   {v:6.0f}         {d_un:10.0f} um            {d_my:6.1f} um")

# time for a signal from a moose's brain to a hind foot about 2.5 m away
for v in (1.0, 60.0):
    print(f"2.5 m at {v:5.1f} m/s takes {2.5 / v * 1000:7.0f} ms")

 target v (m/s)   unmyelinated d needed   myelinated d needed
        2                  4 um               0.4 um
       10                103 um               1.8 um
       25                646 um               4.5 um
       60               3719 um              10.9 um
      100              10331 um              18.2 um
2.5 m at   1.0 m/s takes    2500 ms
2.5 m at  60.0 m/s takes      42 ms

To reach 60 metres per second without myelin, an axon would need to be almost 4 mm across; with myelin, about 11 µm suffices. For a large animal, conduction delays matter. A signal from a moose's brain to its hind foot would take more than two seconds at the speed of a slow unmyelinated fibre, but about 40 milliseconds in a fast myelinated motor axon. Even so, large animals live with longer neural delays than small ones, and studies of reaction times across mammals show that the largest species respond more slowly to disturbances in proportion to their size.