Action Potential Propagation, Cable Theory, and Myelination

Action Potential Initiation and Core Properties

• Determinants of Action Potential Threshold:

  • Voltage-gated sodium ($Na_V$) channel expression and biophysical gating kinetics.
  • Density of $Na_V$ channels localized at the axon initial segment (AIS).
  • Resting membrane resistance ($r_m$).
  • Temperature (modulates ion channel gating kinetics and conductance rates).

• Fundamental Properties of Action Potentials:

  • All-or-None Phenomenon: Within any given individual neuron, action potentials display a uniform, invariant amplitude. The spike either fires completely if threshold is reached or does not fire at all.
  • Frequency Coding of Depolarization: The magnitude or strength of depolarization arriving at the AIS is encoded exclusively by the frequency (number) of action potentials generated over time, never by changes in individual spike amplitude.
  • Ionic Dependence of Amplitude and Waveform: Alterations in transmembrane electrochemical gradients or extracellular/intracellular ion concentrations directly modify action potential peak amplitude, overshoot, and repolarization trajectory.

• Morphological Organization and Regional Excitability:

  • Dendrites and Soma (Cell Body): Receive synaptic inputs that produce graded postsynaptic potentials (PSPs). Synaptic potentials spread electrotonically toward the cell body and axon hillock.
  • Axon Initial Segment (AIS): Located immediately adjacent to the axon hillock and unmyelinated. Exhibits the lowest threshold for action potential firing across the entire neuron due to an extraordinarily high concentration and density of $Na_V$ channels.
  • Threshold Comparison: Membrane regions along the dendrites and soma maintain a high excitation threshold, whereas the threshold plunges dramatically at the AIS.
  • All-or-None Threshold Firing Rule: When the arriving synaptic depolarization exceeds threshold at the AIS, an action potential is generated. When the voltage remains below threshold, no action potential is initiated.

Synaptic Potential Summation and Passive Electrotonic Spread

• Depolarization and Summation Requirements:

  • Individual postsynaptic potentials are typically subthreshold; generating sufficient depolarization to reach threshold at the AIS requires summation of multiple synaptic events.
  • Both temporal and spatial summation rely on the passive, electrotonic propagation of synaptic potentials from dendritic input sites across the soma to the AIS.

• Summation Modalities:

  • No Summation: An isolated presynaptic action potential triggers a single, low-amplitude dendritic postsynaptic potential (PSP). Because it decays passively over space and time, the membrane potential ($V_m$) at the AIS never crosses threshold, resulting in zero action potentials fired.
  • Spatial Summation: Action potentials arriving simultaneously from separate presynaptic terminals activate distinct synapses on the same dendritic tree. The individual EPSPs integrate additively into a larger depolarizing wave. When two inputs summate above threshold, two action potentials are triggered at the AIS.
  • Enhanced Spatial Summation: Concurrent activation across three distinct synaptic sites produces a substantial, extended compound PSP above threshold, resulting in a rapid burst of six action potentials.
  • Temporal Summation: A single presynaptic axon delivers repetitive action potentials in rapid sequential succession before the previous PSP has decayed. The consecutive depolarizations build on one another in a stepped pattern (each step forming a "hump"). By the third hump, the integrated PSP crosses threshold, initiating action potentials.

• Voltage Recording Comparisons Between Soma and Axon:

  • Axon Hillock / Soma Traces: Exhibit the underlying graded synaptic potentials upon which action potential spikes are superimposed.
  • Axonal Internode / Presynaptic Terminal Traces: Exhibit isolated, uniform action potential spikes lacking any graded PSP component. The firing frequency and temporal timing of spikes along the axon match those initiated at the soma.

Action Potential Refractory Periods and Channel Kinetics

• Paired-Pulse Refractory Period Experiments (Squid Giant Axon):
  • Reconstructed voltage-clamp and paired current pulse experiments evaluate the time course of neuronal excitability following an initial action potential:
  • Interval at $4.5\,\text{ms}$: Delivery of a second suprathreshold stimulus step fails to evoke any secondary spike. The axon is totally unexcitable.
  • Interval at $6\,\text{ms}$: Delivery of the second stimulus initiates an action potential, but with significantly diminished amplitude ($\approx 0.9$ of control amplitude).
  • Interval at $9\,\text{ms}$: Delivery of the second stimulus produces a fully restored action potential with standard control amplitude ($\approx 1.0$).
  • Plotting normalized action potential amplitude against inter-stimulus intervals produces an exponential recovery curve that characterizes axonal refractory phases.

• Absolute Refractory Period:

  • Definition: A definitive interval during which no secondary action potential can be triggered, regardless of the strength or duration of injected depolarizing current.
  • Biophysical Mechanism: Voltage-gated $Na^+$ channels remain in a non-conducting, inactivated state. The intracellular inactivation gate (ball-and-chain / hinged-lid motif) obstructs the open channel pore following depolarization.
  • Temporal Extent: Encompasses the entire rapid depolarizing upstroke and the initial repolarization phase, extending from $0\,\text{ms}$ to approximately $1.2\,\text{ms}$.

• Relative Refractory Period:

  • Definition: A subsequent interval during which a secondary action potential can be triggered, but requires a significantly larger depolarizing current stimulus to cross threshold.
  • Biophysical Mechanism: Caused by two overlapping factors:
  • Delayed-rectifier voltage-gated potassium ($K_V$) channels remain open, producing an elevated outward $K^+$ conductance that hyperpolarizes the membrane (afterhyperpolarization undershoot below $-70\,\text{mV}$) and shunts incoming depolarizing current.
  • A portion of $Na_V$ channels remains inactivated; only a fraction has recovered to the closed, resting state.
  • Temporal Extent: Encompasses the afterhyperpolarization phase, lasting from approximately $1.2\,\text{ms}$ to $4.0\,\text{ms}$.

• Conformational States of $Na_V$ and $K_V$ Channels During Depolarization:
  • Voltage Step: Membrane potential is clamped from $-100\,\text{mV}$ to a depolarized step of $+50\,\text{mV}$.
  • $Na_V$ Channel Transitions:
  • Resting State (Pre-stimulus, $-100\,\text{mV}$): Closed; activation gates shut, inactivation gates open.
  • Activation (Time $0\,\text{ms}$): Depolarization triggers rapid outward movement of voltage sensors, opening the pore to allow rapid inward $Na^+$ influx.
  • Inactivating / Inactivated ($5\text{--}10\,\text{ms}$): Inactivation particle swings into the intracellular mouth of the pore, blocking ion permeation even while the membrane remains depolarized.
  • De-inactivation (Time $15\,\text{ms}$): Upon repolarization back to $-100\,\text{mV}$, the activation gates shut and the inactivation plug disengages, resetting the channel to the closed, excitable state.
  • $K_V$ Channel Transitions:
  • Resting State: Closed; non-conducting at negative membrane potentials.
  • Delayed Activation ($5\text{--}10\,\text{ms}$): Gating occurs with a slower kinetic delay relative to $Na_V$ channels. Channels open to allow outward $K^+$ efflux, mediating membrane repolarization.
  • Deactivation (Time $15\,\text{ms}$): Return to resting membrane potential drives slow closing of the activation gates.

Passive Membrane Properties and Cable Theory

• Passive Electrotonic Propagation:

  • Defined as the physical movement of electrical charges through the intracellular cytoplasm and across the lipid bilayer in the absence of voltage-gated ion channel activation.
  • Intracellular charges move longitudinally down their electrochemical gradient inside an axon or dendrite cylinder while simultaneously leaking outward through passive membrane leak pathways.

• The Length (Space) Constant ($\lambda$):

  • Mathematical Formula for Voltage Decay:$\Delta V(x) = \Delta V_0 e^{-x/\lambda}$
  • $\Delta V(x)$: Change in membrane potential at distance $x$ from the injection point.
  • $\Delta V_0$: Steady-state voltage change at the point of current injection ($x = 0$).
  • $x$: Distance along the neurite axis.
  • $\lambda$: Axon length constant.
  • Definition of Length Constant ($\lambda$): The precise longitudinal distance from the current source at which the membrane potential decay falls to $1/e$ ($\approx 37\%$) of its initial peak value ($\Delta V_0$ at $100\%$).
  • Formula Relating Cable Resistances:$\lambda = \sqrt{\frac{r_m}{r_a}}$
  • $r_m$: Transmembrane resistance of a unit length of cylindrical membrane ($\Omega\cdot\text{cm}$), representing leak resistance.
  • $r_a$: Axial resistance (internal cytoplasmic resistance) per unit length ($\Omega/\text{cm}$), opposing longitudinal charge flow.

• Equivalent Electrical Cable Model:
  • Neurites are modeled as a ladder network of interconnected electrical components:
  • Longitudinal cytoplasmic flow is governed by internal resistors in series ($r_a$).
  • Transmembrane current pathways consist of parallel circuits containing a membrane resistor ($r_m$) and a membrane capacitor ($c_m$).
  • The external environment is represented by a low-resistance extracellular fluid pathway.

• Influence of Axon Caliber on Passive Spread:
  • Internal Resistance ($r_a$): Inversely proportional to axon cross-sectional area ($\pi r^2$). Large-diameter axons have low internal resistance; small-diameter axons have high internal resistance.
  • Membrane Resistance ($r_m$): Inversely proportional to axon circumference ($2\pi r$). Specific membrane resistance across equivalent surface areas remains similar between large and small unmyelinated axons.
  • Length Constant Comparison: Because $r_a$ drops much faster with increasing radius than $r_m$, the ratio $r_m / r_a$ increases. Consequently, large unmyelinated axons exhibit a larger length constant ($\lambda$) and superior passive spread compared to small unmyelinated axons.
  • Membrane Capacitance ($c_m$) and Charge Immobilization: The internal resistance $r_a$ determines how rapidly current can access and charge or discharge the membrane capacitance ($c_m$).

Active and Saltatory Action Potential Propagation

• Active Propagation in Continuous Unmyelinated Axons:
  • Action potential transmission requires an alternating cycle of internal passive electrotonic current flow and active membrane regeneration.
  • Temporal and Spatial Sequence:
  • Time $t = 1$ (Point A): Depolarizing stimulus drives $Na^+$ influx through open $Na_V$ channels at Point A. Intracellular positive charges passively flow downstream toward Point B, forming local current loops that leak outward through the membrane.
  • Time $t = 2$ (Point B): Passive downstream charge spread depolarizes Point B above threshold, opening local $Na_V$ channels and generating a regenerated action potential. Simultaneously, Point A repolarizes via $K^+$ efflux through open $K_V$ channels and enters the refractory state.
  • Time $t = 3$ (Point C): Inward current at Point B flows passively downstream to depolarize Point C to threshold, initiating an action potential at Point C. Point B repolarizes and enters refractoriness, while Point A recovers toward rest.
  • Unidirectionality: The action potential cannot travel backward into recently activated membrane because upstream $Na_V$ channels are inactivated and $K_V$ channels remain open (refractory period). This enforces strictly forward conduction from the AIS toward the axon terminus.

• Saltatory Conduction in Myelinated Axons:
  • Structural Organization: Oligodendrocytes (in the central nervous system) or Schwann cells (in the peripheral nervous system) wrap multiple layers of compact lipid membrane around internodal segments, interrupted periodically by uninsulated gaps known as nodes of Ranvier.
  • Channel Localization: Voltage-gated $Na^+$ and $K^+$ channels are clustered at high density exclusively at the nodes of Ranvier. Internodal stretches beneath the myelin sheath are largely devoid of channels.
  • Insulative Function: Myelin prevents trans-membrane current loss across internodes, forcing intracellular current to travel rapidly and passively through the cytoplasm to the subsequent node of Ranvier.
  • Saltatory Mechanism: The passive longitudinal current depolarizes the downstream node to threshold, triggering an active spike. The action potential appears to jump discontinuous distances from node to node, markedly accelerating conduction velocity.

• Comparative Cable Parameters (Unmyelinated vs. Myelinated Axons):
  • Membrane Resistance ($r_m$):$UA\,r_m \ll MA\,r_m$
  • Myelin adds multiple lipid bilayers in series, drastically increasing transverse membrane resistance ($MA\,r_m$) and preventing current leak.
  • Membrane Capacitance ($c_m$):$UA\,c_m \gg MA\,c_m$
  • Capacitance is inversely proportional to the distance separating charges ($C = \frac{\varepsilon A}{d}$). Myelin increases membrane thickness ($d$), dramatically lowering membrane capacitance ($MA\,c_m$). Reduced capacitance decreases the number of charges immobilized on the membrane surfaces, allowing charge to flow rapidly downstream rather than being consumed charging the capacitor.
  • Effective Axial/Internal Resistance ($r_a$):$UA\,r_a \gg MA\,r_a$
  • By drastically decreasing charge immobilization ($c_m$) and trans-membrane leakage ($r_m$), myelination promotes longitudinal electrotonic charge propagation.

• Conduction Velocity and Spatial Progress Comparison:
  • In an unmyelinated axon, the wave of depolarization must regenerate continuously at every micrometer of the axolemma, introducing cumulative gating delays.
  • In a myelinated axon, rapid electrotonic spread along internodes bypasses continuous channel activation. Over identical elapsed intervals ($t = 1, 2, 3$), the action potential traverses a dramatically greater distance in myelinated axons than in unmyelinated axons.

Axon Conduction Velocity Metrics and Clinical Assessment

• Axon Diameter versus Conduction Velocity Trade-Off:

  • Action potential propagation speed is determined by both active channel gating and passive electrotonic spread.
  • Increasing physical axon diameter lowers internal resistance ($r_a$), accelerating passive current transfer to the next regenerative patch.
  • Invertebrates achieve high conduction speeds using massive, unmyelinated fibers (such as the squid giant axon), which occupy significant volume.
  • Vertebrates utilize myelination, enabling ultra-fast conduction velocities in fibers that maintain small physical diameters.

• Quantitative Conduction Velocities Across Fiber Classes:

  • Squid Giant Axon:
  • Conduction Velocity: $25\,\text{m/s}$
  • Fiber Diameter: $500\,\mu\text{m}$
  • Myelinated: No
  • Human Motor Axons:
  • $A\alpha$ type: Conduction Velocity: $80\text{--}120\,\text{m/s}$; Diameter: $13\text{--}20\,\mu\text{m}$; Myelinated: Yes
  • $A\gamma$ type: Conduction Velocity: $4\text{--}24\,\text{m/s}$; Diameter: $5\text{--}8\,\mu\text{m}$; Myelinated: Yes
  • Human Sensory Axons:
  • $A\alpha$ type: Conduction Velocity: $80\text{--}120\,\text{m/s}$; Diameter: $13\text{--}20\,\mu\text{m}$; Myelinated: Yes
  • $A\beta$ type: Conduction Velocity: $35\text{--}75\,\text{m/s}$; Diameter: $6\text{--}12\,\mu\text{m}$; Myelinated: Yes
  • $A\delta$ type: Conduction Velocity: $3\text{--}35\,\text{m/s}$; Diameter: $1\text{--}5\,\mu\text{m}$; Myelinated: Thin
  • $C$ type: Conduction Velocity: $0.5\text{--}2.0\,\text{m/s}$; Diameter: $0.2\text{--}1.5\,\mu\text{m}$; Myelinated: No
  • Human Autonomic Axons:
  • Preganglionic $B$ type: Conduction Velocity: $3\text{--}15\,\text{m/s}$; Diameter: $1\text{--}5\,\mu\text{m}$; Myelinated: Yes
  • Postganglionic $C$ type: Conduction Velocity: $0.5\text{--}2.0\,\text{m/s}$; Diameter: $0.2\text{--}1.5\,\mu\text{m}$; Myelinated: No

• Nerve Conduction Velocity (NCV) Diagnostic Testing:
  • Methodology: Non-invasive electrodiagnostic procedure used to evaluate peripheral nerve function in human patients.
  • Electrode Configuration: A pair of surface electrodes is secured along the anatomical course of a peripheral nerve (such as the tibial or peroneal nerve in the leg):
  • A stimulating electrode delivers a calibrated electrical current pulse to evoke compound action potentials.
  • A recording electrode placed at a precise distance downstream detects the arriving electrical wave.
  • Data Acquisition: Both electrodes interface with an electrodiagnostic monitor that records the response latency, duration, and peak voltage amplitude.
  • Clinical Utility:
  • Conduction velocity is calculated as the known distance divided by signal latency ($\text{Velocity} = \frac{\Delta x}{\Delta t}$).
  • Pathological slowing indicates demyelinating neuropathies (damage to the myelin sheath), whereas reductions in signal amplitude indicate axonal loss.

Self-Check Review Questions and Answers

• Question 1: Action potential amplitude is proportional to the stimulus strength.

  • Options: A) True, B) False
  • Answer: B) False. Action potentials follow an all-or-none property; stimulus strength is encoded by action potential frequency, not amplitude.

• Question 2: A large current influx would lead to multiple action potentials because (choose the best answer):

  • Options:
  • A) The depolarization would trigger $Na_V$ at several regions in the axon.
  • B) The large amount of charge at the AIS would stay long enough to initiate several cycles of $Na_V$/$K_V$ opening.
  • C) The large current influx would override $Na_V$ inactivation.
  • D) The large influx would prevent $K_V$ closing and would allow $Na_V$ to reopen immediately.
  • Answer: B) The large amount of charge at the AIS would stay long enough to initiate several cycles of $Na_V$/$K_V$ opening. Persistent sustained depolarization re-excites the AIS as soon as channels recover from their refractory state.

• Question 3: Action potential threshold does not depend on (choose the best answer):

  • Options:
  • A) $E_{Na}$
  • B) $Na_V$ density at the AIS
  • C) $K^+$ leak
  • D) All of the above
  • E) B & C
  • Answer: A) $E_{Na}$. The equilibrium potential of sodium ($E_{Na}$) dictates action potential peak overshoot and driving force, whereas threshold is determined by channel density, leak conductances, and channel gating kinetics.

• Question 4: Passive propagation of current is faster in larger axons and dendrites because:

  • Options:
  • A) Internal resistance is lower
  • B) Membrane resistance is lower
  • C) The Membrane Capacitance is lower
  • D) A & C
  • Answer: D) A & C. Lower internal resistance ($r_a$) combined with reduced charging delay of membrane capacitance ($c_m$) accelerates passive electrotonic current flow.

• Question 5: Passive propagation is slower than action potential propagation in unmyelinated axons.

  • Options: A) True, B) False
  • Answer: B) False. Passive electrotonic spread occurs nearly instantaneously, whereas active action potential propagation is constrained by the slower kinetics of channel gating and protein conformational changes.

• Question 6: Myelin speeds up action potential propagation because:

  • Options:
  • A) Insulation helps passive propagation of current
  • B) Fewer regeneration of AP is faster
  • C) Decreases $K^+$ leak
  • D) All of the above
  • Answer: D) All of the above. Myelination insulates the axon to minimize leak, reduces capacitive load, and eliminates the time-consuming necessity of regenerating action potentials across continuous membrane spans.

Summary of Core Principles

• Passive Current Spread in Cylinders: Charge propagation inside a cylindrical neurite lacking voltage-gated channels occurs via passive electrotonic spread. It is strictly limited by outward current leakage through the membrane ($r_m$) and by internal cytoplasmic resistance ($r_a$). Depolarization decays exponentially with distance, and decays more rapidly in smaller-caliber neurites.
• Regenerative Spikes: Action potentials are regenerative events that reproduce with identical amplitude along the length of an unmyelinated axon, preserving signal fidelity without attenuation.
• Coupling of Passive and Active Conduction: Action potential propagation fundamentally requires passive longitudinal charge transfer to transport depolarization forward to the next physical region containing $Na_V$ and $K_V$ channels, which then regenerates the full spike.
• Caliber Dependency: Action potential propagation velocity is directly dependent on axon diameter because larger diameters reduce internal axial resistance ($r_a$).
• Saltatory Conduction via Myelin: Myelinated internodes combined with high channel density at nodes of Ranvier enable fast saltatory propagation, allowing thin vertebrate axons to conduct action potentials at high velocities.
• Dendritic Computation: Complex dendritic arborizations utilize passive electrotonic properties to integrate, summate, and compute synaptic inputs before delivering charge to the axon initial segment.