Action Potential Propagation and Cable Properties

An axon is, electrically, a cylinder of conductive cytoplasm wrapped in a resistive, capacitive membrane and bathed in conductive extracellular fluid. When current is injected locally — say, by synaptic input at a dendrite or by the opening of channels during a spike — that current has two possible fates at every point along the membrane: it can leak outward across the membrane resistance, or it can continue to flow axially down the cytoplasm to the next patch of membrane. Because some current always leaks out, the amount of current available to depolarize distant membrane patches shrinks with distance. This decremental, non-regenerative spread is called passive or electrotonic conduction, and it obeys the mathematics of an electrical cable — hence "cable theory," originally borrowed from the theory of transatlantic telegraph cables. Passive spread by itself cannot carry a signal across a whole neuron with any fidelity, because it decays exponentially. What rescues long-distance signaling is that the axon membrane is not electrically inert: it is studded with voltage-gated Na⁺ and K⁺ channels identical to those that generate the action potential at the axon hillock. Wherever passive spread manages to depolarize the membrane to threshold, these channels open and regenerate a full-amplitude spike locally. That fresh spike then spreads passively to the next patch of membrane, depolarizes it to threshold, and the cycle repeats — a chain reaction of local regeneration riding on top of passive spread. The action potential does not "travel" in the sense of a wave moving through a fixed medium; it is re-created anew, at full amplitude, at every point along the axon. This is why the impulse arrives at the terminal exactly as large as it left the soma, regardless of axon length, provided the axon is healthy. Two further ideas complete the picture. First, conduction has a direction, and it is enforced not by channel geometry but by time: the region of membrane just traversed by the spike is left in a refractory period, during which its Na⁺ channels are inactivated and cannot reopen no matter how much depolarizing current arrives. This makes the "trailing edge" of the wave un-excitable, so the only direction the wave can regenerate into is forward, into fresh, unfired membrane. Second, the speed of this traveling regeneration — the conduction velocity — is set almost entirely by passive cable properties: how far and how fast passive current spreads determines how quickly downstream membrane reaches threshold. Anything that widens the effective "reach" of passive spread (larger axon diameter, myelin insulation) speeds up conduction without changing the active channel kinetics at all.

Step 1 — The axon as an electrical cable. Treat a short cylindrical segment of axon as a parallel combination of a membrane resistance (Rm, arising from resting ion channels) and a membrane capacitance (Cm, arising from the lipid bilayer separating charge), in series with the axial resistance of the cytoplasm (Ri) that current must overcome to travel down the inside of the axon. Current injected at one point can either charge/discharge the local Cm and leak across Rm, or travel further down Ri to the next segment — mathematically, this is described by the cable equation, a partial differential equation relating membrane potential V to distance x and time t: λ² (∂²V/∂x²) = τ (∂V/∂t) + V

Step 2 — Two constants govern everything. Solving this equation for a steady, subthreshold current injection gives an exponential decay of voltage with distance: V(x) = V₀ e^(−x/λ) where the length constant (space constant) λ is λ = √(Rm / Ri) Typical values: λ ≈ 0.1–1.0 mm in a small unmyelinated axon (bigger axons and myelinated axons have much larger λ, into the centimeter range). λ is the distance over which a steady voltage falls to 1/e (≈37%) of its original value. A larger λ means passive current spreads farther before it decays away, so distant membrane can be brought to threshold faster and more easily. For time-varying signals (like the rising phase of a synaptic potential or a spike), the membrane time constant τ governs how fast voltage changes in response to injected current: τ = Rm × Cm Typical values: τ ≈ 1–20 ms, depending on channel density and membrane area. τ determines how much temporal summation is possible and how quickly the membrane can be charged to threshold.

Step 3 — Passive spread ahead of the active spike. During a propagating action potential, the depolarized, Na⁺-flooded patch of membrane is at roughly +40 mV while the membrane immediately ahead is still at rest (≈ −65 mV). This steep local voltage gradient drives local circuit current: positive charge flows intracellularly from the depolarized zone into the adjacent resting cytoplasm, and correspondingly flows extracellularly in the opposite direction to complete the circuit. This local current is exactly the passive, decaying signal described by the cable equation — but it does not need to travel far, only far enough (a few hundred micrometers to a couple of millimeters) to depolarize the immediately adjacent patch to threshold (~ −55 mV).

Step 4 — Regeneration ("boosting") at threshold. Once the adjacent patch reaches threshold, its own voltage-gated Na⁺ channels open explosively (because Na⁺ channel activation is itself steeply voltage-dependent and cooperative in effect, even though each channel is an independent stochastic switch), producing a full new action potential at that location with the same peak amplitude as the one that triggered it. This new spike then generates its own local circuit currents, which depolarize the next patch of membrane in turn. The process is self-similar at every point along the axon: passive spread only has to bridge a short local gap, and active regeneration restores full amplitude after every gap. This is why propagation is often described as non-decremental: unlike passive spread, an actively propagated action potential does not shrink with distance.

Step 5 — The refractory wake enforces direction. Immediately behind the advancing spike, the membrane has just undergone Na⁺ channel inactivation (a distinct conformational state from closed, requiring repolarization to remove) and delayed-rectifier K⁺ channels are still open, hyperpolarizing the membrane. During this absolute refractory period (roughly 1–2 ms), essentially no amount of depolarizing current can trigger another spike there, because inactivated Na⁺ channels cannot open. During the following relative refractory period, spikes can be triggered but only with a stronger-than-normal stimulus, because some Na⁺ channels have recovered but K⁺ conductance is still elevated, raising threshold. Local circuit current does flow backward into this just-fired membrane as well as forward — the biophysics is symmetric — but because it is refractory, only the forward-spreading current onto fresh, resting membrane can trigger a new spike. This is the mechanistic basis of unidirectional, "orthodromic" conduction under normal conditions (the same logic explains why an action potential initiated experimentally in the middle of an axon propagates in both directions, since both flanks are unfired resting membrane).

Step 6 — What sets conduction velocity. Velocity depends on how quickly local circuit current can depolarize the next patch of membrane to threshold, which depends on λ (how far current reaches) and τ (how fast the membrane charges). Approximately, conduction velocity scales with λ/τ, or equivalently √(diameter). Larger-diameter axons have lower internal axial resistance (Ri falls as cross-sectional area rises), increasing λ; this is why invertebrates evolved giant axons (like the squid giant axon, up to ~1 mm in diameter) for escape reflexes requiring speed, and why vertebrate axons instead evolved myelin, which raises effective Rm and lowers effective Cm along internodes, achieving even greater speed increases without requiring enormous diameter — the strategy detailed in Myelination and the Insulation of Axons and Saltatory Conduction.

The central claims here — that axons obey cable equations, that Na⁺ channels regenerate the spike locally, and that refractoriness blocks backward propagation — were established through a sequence of biophysical experiments, most famously in the squid giant axon, chosen by J.Z. Young and exploited by Alan Hodgkin, Andrew Huxley, and Bernard Katz precisely because its huge diameter (up to ~1 mm) permitted insertion of internal electrodes impossible in ordinary vertebrate axons.

Measuring passive cable constants (Hodgkin and Rushton, and Hodgkin & Huxley, 1930s–1940s). To test cable theory directly, Hodgkin and colleagues injected small, subthreshold steady currents into the squid axon through a fine electrode and recorded the resulting voltage at increasing distances along the same axon with a second electrode. As cable theory predicted, the steady-state voltage fell off exponentially with distance rather than linearly or not at all, and plotting log(voltage) against distance gave a straight line, exactly as expected from V(x) = V₀e^(−x/λ). The slope of that line yielded a direct experimental measurement of λ. Independently varying the external saline resistance and comparing observed λ against the cable equation's prediction (λ = √(Rm/Ri)) confirmed that the underlying assumption — that the axon behaves as a leaky resistor-capacitor cable with fixed Rm and Ri — was quantitatively correct, not just qualitatively plausible.

Space-clamping to isolate active from passive currents (Cole and Marmont, 1949; Hodgkin, Huxley & Katz, 1952). A separate, decisive methodological advance was the voltage clamp, in which a long wire electrode was threaded down the inside of the squid axon to hold membrane potential uniform along a stretch of membrane (eliminating passive, cable-like spread of current within that stretch) while a feedback circuit measured exactly the current needed to hold voltage at any commanded level. This let Hodgkin and Huxley separate the active membrane currents (voltage- and time-dependent Na⁺ and K⁺ currents) from the passive cable properties, and to show that the same Na⁺-channel population responsible for local depolarization at the axon hillock is present and behaves identically at every point along the axon — the physical basis for regeneration at each successive patch. This work (published in a landmark 1952 series in the Journal of Physiology) earned Hodgkin and Huxley the 1963 Nobel Prize in Physiology or Medicine, shared with John Eccles.

Demonstrating unidirectional conduction and the refractory block directly. Independent classical experiments stimulated an isolated axon or nerve trunk simultaneously at two points, or stimulated in the middle and recorded on both sides, confirming that a spike triggered mid-axon travels in both directions away from the stimulus site (since fresh, resting membrane exists on both flanks), while a second stimulus delivered to the same site during the absolute refractory period (within roughly 1 ms of the first, in the squid axon) fails entirely to produce a second impulse — direct behavioral proof that inactivated Na⁺ channels, not any anatomical valve or directional gate, are what block backward re-firing. Later patch-clamp work (Neher and Sakmann, 1970s–80s, earning the 1991 Nobel Prize) resolved single Na⁺ channel currents and confirmed directly, at the single-molecule level, that channels enter a distinct non-conducting inactivated state after opening, providing the molecular mechanism underlying the refractory period that Hodgkin and Huxley had inferred purely from macroscopic current recordings decades earlier.

Conduction velocity measurements confirming the diameter/λ relationship. By stimulating an axon at one end and recording the arrival time of the compound action potential at a known distance downstream (a technique still used clinically as nerve conduction velocity testing), physiologists showed conduction velocity increases systematically with axon diameter across many invertebrate and vertebrate axon populations, consistent with λ (and hence velocity) scaling with the square root of diameter — a relationship later shown to be dramatically outcompeted, per unit diameter, by myelination.

Example 1 — Comparing two axons quantitatively. Suppose axon A is unmyelinated with diameter 1 μm, and axon B is unmyelinated with diameter 4 μm (both from the same neuron type, same Rm and cytoplasmic resistivity). Axial resistance Ri per unit length scales inversely with cross-sectional area, i.e., inversely with the square of diameter (d²). Since λ = √(Rm/Ri), and Ri ∝ 1/d², we get λ ∝ √(d²) = d. So a 4-fold increase in diameter should produce roughly a 4-fold increase in λ (in reality the relationship is not perfectly linear because Rm per unit length also depends on d, but the qualitative scaling — bigger axons have proportionally larger λ — holds, and conduction velocity for unmyelinated axons is known empirically to scale roughly with √diameter). If axon A has λ = 0.25 mm, we predict axon B has λ ≈ 1 mm. Since conduction velocity scales with λ/τ (τ unchanged, since specific membrane properties per unit area are the same), axon B should conduct roughly 4× faster than axon A. This is exactly the logic that favored evolution of giant axons in squid (up to ~1 mm diameter, conduction velocities around 20 m/s) for the fast mantle-contraction escape response, where speed is a matter of survival and myelin is unavailable to invertebrates.

Example 2 — Refractory period and maximum firing rate. A myelinated sensory axon has an absolute refractory period of 1.0 ms and a relative refractory period lasting an additional 3.0 ms during which threshold is elevated but firing is possible with strong enough stimulation. (a) What is the theoretical maximum firing rate limited purely by the absolute refractory period? If each cycle requires at minimum 1.0 ms before the next spike can fire at all, the maximum possible rate is 1/(1.0 ms) = 1000 spikes/second. (b) In practice, sustained natural firing rarely approaches this because during the relative refractory period, more current is needed to reach the (raised) threshold, so achievable sustained rates under strong but physiological stimulation are typically far lower — commonly in the range of tens to a few hundred Hz for most neurons, with the true theoretical ceiling only approached transiently, e.g., in brief high-frequency bursts. This distinction — an absolute physical ceiling set by channel recovery kinetics, versus a lower practically-observed maximum shaped by the relative refractory period and the driving stimulus strength — is why textbooks quote "several hundred Hz maximum" for most neurons even though the bare absolute refractory period alone would allow up to ~1000 Hz.

Alan Lloyd Hodgkin trained as a zoologist and physiologist at Cambridge and, together with the slightly younger Andrew Huxley, spent the years around 1939 and then 1947–1952 dissecting the electrical behavior of the squid giant axon at the Marine Biological Laboratory in Plymouth, England. Their choice of the squid axon was itself a methodological breakthrough credited initially to the zoologist J.Z. Young, who had recognized that this unusually large invertebrate axon (evolved for fast escape jetting) was wide enough to allow a fine wire electrode to be threaded down its interior — something impossible in any mammalian axon of the day. Hodgkin and Huxley combined this preparation with the newly devised voltage clamp (built with Bernard Katz) to dissect, current by current, the separate contributions of Na⁺ and K⁺ conductances to the action potential, and used simple intracellular current injection and recording along the axon's length to establish that passive spread obeyed classical cable theory. Their 1952 series of five papers in the Journal of Physiology combined these passive and active measurements into a single quantitative, predictive mathematical model — arguably the most successful quantitative model in all of cellular physiology, still taught and used essentially unmodified today.

Cable and propagation theory is not a historical curiosity; it underlies core diagnostic and therapeutic tools in modern medicine and neuroscience. Nerve conduction studies, a routine clinical test, stimulate a peripheral nerve at one point and record the compound action potential at a known distance to calculate conduction velocity directly; abnormally slow velocities are a hallmark of demyelinating diseases such as Guillain-Barré syndrome and multiple sclerosis, where loss of myelin lowers Rm and raises effective Cm, shrinking λ and sometimes causing outright conduction block when passive spread can no longer reach the next node of Ranvier at threshold. Local anesthetics such as lidocaine work by blocking voltage-gated Na⁺ channels, preventing the active regeneration step described above; without regenerable Na⁺ current, only weak, rapidly decaying passive spread remains, and the signal simply dies out before reaching the spinal cord, producing loss of sensation. Understanding cable properties also guides the design of cochlear implants and other neural prostheses, where engineers must predict how far injected current will spread passively through tissue and axon membranes to selectively activate target fibers without cross-stimulating neighbors. In axon injury and regeneration research, restoring adequate λ (via remyelination or by encouraging larger regenerated axon caliber) is a direct therapeutic target for recovering functional conduction velocity after nerve damage. Finally, cable theory is the mathematical backbone of essentially all realistic computational neuron models (from single-compartment integrate-and-fire models to full multicompartment simulations in software like NEURON), which discretize a neuron's dendrites and axon into cable segments governed by exactly the equations described here.

Ask the tutor: How exactly does myelin's effect on Rm and Cm translate into the discrete, node-to-node jumping described as saltatory conduction?