Membrane Potentials: Passive Properties and Types

Passive Properties and Types of Membrane Potentials

Membrane as an Electrical Circuit

• The neuron's plasma membrane behaves as an electrical circuit, incorporating properties of both a resistor and a capacitor.

• Capacitor (C): Stores charge, specifically lipid bilayers that separate charge. The membrane stores electrical charge.

• Resistor (R): Ion channels continuously conduct charge across the membrane.

Time Constant ($\tau$)

• Definition: The time required for the membrane voltage ($V<em>m$) to change from a starting potential to $63\%$ of its final potential ($V</em>{final}$).

• Effect: Determines how quickly a membrane voltage can depolarize or repolarize in response to a current.

• Formula: $\tau = R<em>m C</em>m$

  • $R_m$ = membrane resistance (resistance across the membrane)

  • $C_m$ = membrane capacitance (capacity to store charge across the membrane)

Charging the Membrane (Circuit Analogy)

• When current is applied (e.g., from an electrode):

  1. Positive charges predominantly flow onto the capacitor first, due to less initial resistance to current flow compared to ion channels (resistors).

  2. As the capacitor progresses towards its maximum charge, more charges begin to flow through the parallel resistor (ion channels).

  3. Once the capacitor is fully charged (e.g., at $-55 \text{mV}$), all subsequent charges flow through the resistor.

• Mathematical Expression for Charging: $V<em>{m}(t) = V</em>{final}(1 - e^{-t/(R<em>m C</em>m)}) = V_{final}(1 - e^{-t/\tau})$

• Mathematical Expression for Discharging (Decay): $V<em>{m}(t) = V</em>{0}e^{-t/(R<em>m C</em>m)} = V_{0}e^{-t/\tau}$

• Analogy: The membrane's voltage change is not instantaneous like a simple on/off switch; it is gradual, similar to a dimmer light switch, because of the time required to charge the membrane capacitor.

Significance of the Time Constant

• The time constant affects how quickly $V_m$ can change in response to the opening of voltage-gated or ligand-gated ion channels.

• Impact on Neuronal Excitability:

  • Increased $\tau$: Caused by an increase in $R<em>m$, $C</em>m$, or both. This makes the cell hypoexcitable because it takes longer for the membrane potential to change and reach an action potential threshold.

  • Decreased $\tau$: Caused by a decrease in $R<em>m$, $C</em>m$, or both. This makes the cell hyperexcitable because it can reach the action potential threshold faster.

• Capacitance Analogy: A larger capacitor takes longer to fill up with charge to reach a maximal voltage compared to a smaller capacitor. Therefore, higher capacitance leads to a larger time constant and slower voltage changes.

Length Constant ($\lambda$)

• Definition: The distance over which the change in membrane voltage ($V_m$) declines to $37\%$ (or decays by $63\%$) of its initial peak value from the point of origin.

• Effect: Determines how far a signal can passively travel along the membrane before its voltage decays below a critical level.

• Formula: $\lambda = \sqrt{\frac{R<em>m}{R</em>i + R_o}}$

  • $R_m$ = membrane resistance

  • $R_i$ = internal cytoplasm resistance (longitudinal resistance of the axoplasm)

  • $R_o$ = outside resistance (longitudinal resistance of the extracellular fluid, often considered constant or negligible in some contexts)

Signal Spread and Decay

• When a membrane potential is changed (e.g., by an electrode, an action potential, or a graded potential), the passive electrical properties of the membrane cause the change in potential to diminish with distance from the source.

• Decremental Signal Decay: This decay is primarily caused by the resistive properties of the neuron's membrane, specifically:

  • Leak Channels: Allow current to