Propagation of the Action Potential

Membrane Excitation and the Generation of Local Currents

  • State of the Excited Membrane Section:

    • During excitation, the polarity of the cell membrane reverses at the affected site.

    • The external surface becomes negatively charged ($-$).

    • The internal surface becomes positively charged ($+$).

    • This is represented as: $- + + + + + -$ (external) and $+ - - - - - +$ (internal) in specific diagrams.

  • Local Currents (Action Currents):

    • These are defined as electric currents that flow on the surfaces of the membrane between the excited and non-excited (resting) sections.

    • Direction of Local Currents:

      • External Surface: Current flows from the unexcited (positive) section towards the excited (negative) section.

      • Internal Surface: Current flows from the excited (positive) section towards the unexcited (negative) section.

    • Loop Formation: The local currents form complete loops across the membrane, resulting in the transport of electrical charge across the lipid bilayer.

Conduction of the Excitation Impulse

  • Mechanism of Propagation:

    • In areas adjacent to the excited section, the membrane potential begins to decrease (it becomes less negative).

    • Once the depolarization reaches the threshold value, a new action potential is generated at that site.

    • At the site of the initial excitation, repolarisation occurs.

    • The interaction of these two processes (excitation at the new site and repolarization at the old site) results in the propagation of the impulse along the membrane.

Modeling the Excitation: The Cable Equation

  • The Electric Cable Model:

    • Nerve impulse conduction is fundamentally driven by electric currents.

    • A nerve fibre can be modeled as an electric cable with specific characteristic properties to study how membrane potential ($\phi$) varies over time ($t$) and distance ($x$) along the fibre.

  • Equivalent Circuit Diagram Parameters:

    • r_i = \text{cytoplasm resistance per unit length (\Omega/m)}

    • r_m = \text{membrane resistance per unit length (\Omega/m)}

    • cm=membrane capacitance per unit length (F/m)c_m = \text{membrane capacitance per unit length (F/m)}

  • The Cable Equation:

    • The variations of the membrane potential are described by the second-order partial differential equation:

    • ϕ=−rmrid2ϕdx2−rmcmdϕdt\phi = - \frac{r_m}{r_i} \frac{d^2\phi}{dx^2} - r_m c_m \frac{d\phi}{dt}

Temporal and Spatial Decay Constants

  • Decay of Membrane Potential Over Time:

    • At any fixed point on the nerve fibre, the membrane potential decays exponentially.

    • The rate of decay is governed by the membrane resistance and capacitance.

    • Formula: ϕ=ϕ0e−t/τ\phi = \phi_0 e^{-t/\tau}

    • The Time Constant ($\tau$): Defined as τ=rmcm\tau = r_m c_m. At time t=τt = \tau, the potential at the point of excitation has decayed by a factor of ee (e≈2.72e \approx 2.72).

  • Distribution of Potential Along the Fibre (Length Constant):

    • The membrane potential decreases exponentially as the distance from the point of excitation increases.

    • The decay rate is determined by the resistances of the membrane and the cytoplasm.

    • Formula: ϕ=ϕ0e−x/λ\phi = \phi_0 e^{-x/\lambda}

    • The Length Constant ($\lambda$): Defined as λ=rmri\lambda = \sqrt{\frac{r_m}{r_i}}. At a distance x=λx = \lambda, the potential is smaller than the potential at the source by a factor of ee.

  • Factors Affecting the Length Constant ($\lambda$):

    • Increases with:

      • Diameter of the nerve fibre (DD): λ∼D\lambda \sim \sqrt{D}

      • Membrane thickness (dd): λ∼d\lambda \sim \sqrt{d}

      • Membrane resistivity (ρm\rho_m)

    • Decreases with:

      • Increasing cytoplasm resistivity (ρi\rho_i)

Action Potentials vs. Graded Potentials

  • Model Validity:

    • The cable model accurately describes the conduction of graded (local) potentials, which include weakly depolarising potentials (below threshold) and hyperpolarising potentials.

  • Nondecremental Conduction of Action Potentials:

    • The cable model is inapplicable to the action potential because nerve impulses (APs) do not decay over time or distance; they are "nondecremental."

    • Local potentials serve to depolarize the membrane to the threshold. Once the threshold is hit, the generated action potential impulse reaches a consistent, maximum amplitude at every point along the fibre.

    • Ion Currents: The amplitude of the impulse is sustained by transient flow of ions across the membrane, specifically Na+Na^+ and K+K^+ currents.

Conduction Speed and Myelination

  • Conduction Speed (vv):

    • The conduction speed is proportional to the length constant: v∼λ∼Dv \sim \lambda \sim \sqrt{D}.

    • Animals such as squid, crayfish, and earthworms utilize giant axons (large diameter fibres) to achieve high conduction speeds.

  • Myelinated Nerve Fibres:

    • Vertebrates use myelination to increase speed without excessively increasing diameter.

    • Myelin Composition: Approximately 80% lipids, 20% proteins, and cholesterol.

    • Electrical Property: Myelin has a resistivity much larger than regular membranes, making it a superior insulator.

  • Anatomy of the Myelin Sheath:

    • The sheath is much thicker than the cell membrane.

    • Nodes of Ranvier: These are regular gaps in the sheath, spaced (1…3) mm(1 \dots 3) \, \text{mm} apart, with a length of several micrometers.

  • Saltatory Conduction:

    • In myelinated fibres, depolarisation and local current crossing occur only at the nodes of Ranvier.

    • The excitation "jumps" from node to node.

    • Advantages:

      • High Speed: Impulses can travel up to 100 m/s100 \, \text{m/s}, which is ten times faster than unmyelinated fibres of identical diameter.

      • Energy Efficiency: Leakage currents are restricted to small node areas rather than the entire membrane surface. This results in smaller shifts in concentration gradients and reduced energy expenditure (ATP) to restore ion balances.

Clinical Blocking and Pathologies of Myelin

  • Nerve Blocking Examples:

    • Single Node Block: If one node is blocked by an anesthetic, the impulse can jump over it to the next node (redundancy for reliability).

    • Double Node Block: If two successive nodes are blocked, propagation stops because the length constant is not large enough to depolarize the third node to the threshold.

  • Demyelination vs. Dismyelination:

    • Demyelination: Loss of the existing myelin sheath.

      • Causes: Immune system disorders (Multiple Sclerosis) or heavy metal poisoning (Mercury).

    • Dismyelination: Defective structure/function of the myelin sheath.

      • Causes: Genetic mutations; an example provided is Schizophrenia.

  • Demyelinating Diseases:

    • CNS Examples: Multiple Sclerosis (MS). The immune system attacks myelin, forming scar tissue (sclerosis). Symptoms include weakness, tingling, numbness, blurred vision, muscle stiffness, and cognitive/urinary problems.

    • PNS Examples: Charcot-Marie-Tooth Disease (CMT). Specifically in CMT1, myelin internodes are lost and replaced by shorter ones or remain demyelinated. This is accompanied by tightly packed neurofilaments and progressive loss of sensory and motor axons.

Propagation in Non-Uniform Fibres and Nerve Trunks

  • Geometry Changes:

    • Expanding Fibres: Speed decreases before the expansion and increases to a new higher value after. If the expansion is too drastic, the impulse may stop.

    • Contracting Fibres: Speed increases before the contraction and decreases after. Blocking does not occur in this scenario.

    • Branching Fibres: The impulse may not enter narrow branches depending on diameter ratios.

      • Summation: Multiple impulses arriving at a junction simultaneously can sum up to excite a wider section.

      • Refractory State: A single delayed impulse might fail to excite a wider section and instead leave it in a refractory (less sensitive) state.

  • Interactions in Nerve Trunks:

    • A nerve trunk is a bundle of independent communication channels (fibres) bound by connective tissue.

    • Ephaptic Interaction: Impulses in one fibre can influence or even trigger impulses in adjacent fibres.

    • Synchronization: Observed if conduction speeds of adjacent fibres differ by less than 10%10\%.

    • Increased Interaction Factors: Interaction strengthens if the resistivity of the surrounding fluid increases or if the diameters of the nerve fibres increase.