Understanding Nernst and Goldman Equations | Membrane Potential

Voltage-gated ion channels are membrane proteins that open and close in response to changes in membrane potential; they exist in three states—closed (paddles oriented downward, blocking the pore), open (paddles oriented upward, allowing ion flow), and inactivated (a positively charged ball-and-chain structure blocks the pore shortly after opening)—enabling precise regulation of ion movement across cell membranes.

Sodium ions move through voltage-gated sodium channels with two gates: an outer activation gate and an inner inactivation gate. During resting potential, the activation gate is closed while the inactivation gate is open, preventing sodium entry. When a stimulus is applied, the activation gate opens, allowing sodium to enter. This opening is called 'activation' because it activates the channel's function. The inactivation gate closes when sodium enters, blocking further entry. Potassium channels have only one gate and open slowly in response to membrane depolarization. These channel characteristics determine the timing and shape of the action potential.

Voltage-gated sodium channels have two gates: an outer gate and an inner gate. In the resting state, the outer gate is closed (because sodium concentration is higher outside the cell) while the inner gate is open. When a stimulus causes membrane depolarization, the outer gate (activation gate) opens, allowing sodium ions to enter the cell and causing rapid depolarization. The inner gate (inactivation gate) then closes to prevent further sodium entry. Voltage-gated potassium channels have only one gate (the inner gate) located on the intracellular side. Unlike sodium channels, these channels open slowly in response to depolarization. They are responsible for repolarization by allowing potassium ions to exit the cell. The slow opening of these channels is crucial for the repolarization phase of the action potential.

Voltage-gated ion channels are transmembrane proteins that open in response to changes in membrane electrical potential, allowing specific ions to pass through. Voltage-gated sodium channels possess four transmembrane domains with six segments each, featuring a voltage sensor (S4 segment), a selectivity filter containing negatively charged amino acids that attract sodium ions, and two gates: an activation gate that opens when membrane potential shifts from -90mV to +35mV, and an inactivation gate that closes when the potential returns to rest. Voltage-gated potassium channels have a similar four-domain structure with a single inactivation gate that opens when the membrane potential becomes positive, enabling potassium ion passage through their selectivity filter.

Voltage-gated ion channels are membrane proteins that open in response to changes in membrane voltage. They contain voltage-sensitive helices with charged amino acids that move when membrane potential changes, causing conformational changes that open the channel pore. These channels have selectivity filters (poor loops) that allow only specific ions to pass. The voltage-gated sodium channel uniquely possesses a ball and chain structure that creates three states: closed, open, and inactivated. The inactivated state occurs when the 'ball' plugs into the open channel, blocking ion flow. This structural feature is essential for the refractory period that prevents immediate re-firing of action potentials.

The Hodgkin-Huxley model treats neuronal membranes as electrical circuits: membrane capacitance stores charge, while ionic channels act as conductances allowing specific ions to pass. The fundamental equation governing membrane potential is C_m * dV/dt = I_total, where C_m is membrane capacitance, V is membrane potential, and I_total is total current. Ionic currents follow Ohm's law: I = g * (V - E), where g is conductance, V is membrane potential, and E is reversal potential. This circuit analogy enabled mathematical description of complex neuronal behavior using principles from electrical engineering.

The Hodgkin Huxley model represents neuronal membrane dynamics using an electrical equivalent circuit consisting of a membrane capacitance (representing charge storage in the lipid bilayer), leak conductance (representing passive ion channels), and voltage-gated ion channels (sodium and potassium) modeled as variable conductances controlled by membrane voltage; this framework enabled Hodgkin and Huxley to mathematically describe and separate the different currents contributing to the action potential by using voltage clamp techniques and chemical substitutions to isolate individual ionic contributions.

The Hodgkin-Huxley model explains how neurons generate action potentials using an equivalent electrical circuit. Two key ion channels are modeled: sodium conductance with a +50 mV battery and potassium conductance with a -75 mV battery. These conductances cooperate to produce action potentials—sodium conductance depolarizes the membrane to +55 mV, while potassium conductance repolarizes it to -75 mV. The total membrane current consists of three components: sodium current (voltage/time-dependent), potassium current (voltage/time-dependent), and leak current (fixed at -50 mV). The membrane potential evolves according to the differential equation dV/dt = -I_membrane/C, where I_membrane is the sum of all currents.

The Hodgkin-Huxley model describes neurons as equivalent electrical circuits with three conductances (sodium, potassium, leak) and membrane capacitance. Sodium conductance connects to a positive sodium battery, while potassium conductance connects to a negative potassium battery. Together, these voltage- and time-dependent conductances generate action potentials through coordinated opening and closing. The model uses differential equations to describe how membrane potential changes in response to ionic currents, providing a quantitative framework for understanding neuronal excitability.

The Hodgkin-Huxley model (1952) provides a comprehensive mathematical framework for understanding electrical signaling in excitable cells. The model treats the cell membrane as an electrical circuit with a capacitor representing the lipid bilayer's insulating properties, and variable resistors representing sodium, potassium, and leak channels. The governing equation balances capacitive current with ionic currents, where sodium and potassium channels exhibit voltage-dependent gating kinetics described by first-order differential equations with rate constants derived from experimental data. Leak channels follow Ohm's law as constant resistors. This system of coupled ordinary differential equations enables prediction of membrane potential dynamics and ionic currents, serving as a foundational model for studying cellular electrophysiology in neurons and cardiac cells.

Electrolyte imbalances affect excitability: hypokalemia increases K+ gradient, moves equilibrium potential away from zero, requires stronger stimuli, causes muscle weakness; hyperkalemia decreases K+ gradient, moves equilibrium potential toward zero, requires weaker stimuli, causes hyperexcitability. Hyponatremia decreases Na+ gradient, bringing equilibrium potential closer to zero; hypernatremia increases Na+ gradient, moving it away from zero.

Electrolyte imbalances significantly affect membrane excitability. Hyperkalemia (high potassium outside the cell) reduces the concentration gradient for potassium, making the resting membrane potential less negative (depolarized). This makes the cell closer to threshold and more excitable initially, but severe hyperkalemia can cause membrane inexcitability because sodium channels become inactivated. Hypercalcemia (high calcium outside the cell) reduces excitability by stabilizing sodium channels, making them less likely to open. Hypocalcemia (low calcium outside the cell) increases excitability by making sodium channels more likely to open. These imbalances can cause muscle weakness, cramps, tetany, and cardiac arrhythmias.

Electrolyte imbalances significantly affect cellular excitability. Potassium (normal 3.5-5.5 mEq/L) is the primary determinant of resting membrane potential. Hyperkalemia depolarizes the membrane, increasing excitability initially but causing sodium channel inactivation at severe levels. Hypokalemia hyperpolarizes the membrane, decreasing excitability. Calcium (normal total 8.5-10.5 mg/dL) affects threshold: hypercalcemia raises threshold (less excitable), hypocalcemia lowers threshold (more excitable). Calcium acts as a membrane stabilizer. These imbalances affect all excitable tissues including neurons, muscle, and cardiac cells.

Electrolyte concentrations significantly affect membrane excitability. Calcium acts as a protective barrier for sodium channels: high calcium (hypercalcemia) reduces excitability by blocking sodium channels, while low calcium (hypocalcemia) increases excitability. Potassium affects excitability through its concentration gradient: high extracellular potassium (hyperkalemia) reduces the potassium gradient, making the resting potential less negative (closer to threshold) and increasing excitability, while low extracellular potassium (hypokalemia) increases the gradient, making the resting potential more negative and reducing excitability.

Hyperkalemia (elevated blood potassium) reduces the potassium concentration gradient, decreasing potassium efflux and making cells less negative—increasing excitability in mild cases. However, severe hyperkalemia causes persistent depolarization that inactivates sodium channels, paradoxically decreasing excitability. Calcium acts as a membrane stabilizer: hypercalcemia decreases sodium permeability and excitability, while hypocalcemia increases both. Sodium does not affect resting membrane potential because sodium channels are impermeable at rest. These principles explain how electrolyte disturbances clinically impact neuronal and cardiac function.