The Hodgkin-Huxley (H-H) model is a mathematical framework developed in the early 1950s that provides a quantitative description of how excitable cells, such as neurons, generate and transmit electrical signals. This model uses a set of nonlinear differential equations to link the movement of ions across the cell membrane to the macroscopic electrical events of the nerve impulse. It remains a cornerstone of computational neuroscience, providing the blueprint for understanding electrical signaling in the nervous system.
The Biological Problem: Action Potentials
The fundamental biological event the Hodgkin-Huxley model describes is the action potential, which is the brief, rapid change in voltage across a neuron’s membrane. At rest, a neuron maintains a negative internal charge, known as the resting potential, typically around -70 millivolts (mV). This state is maintained by a difference in ion concentrations, primarily high sodium (\(\text{Na}^{+}\)) outside the cell and high potassium (\(\text{K}^{+}\)) inside the cell.
An action potential begins when a stimulus causes the membrane voltage to rise to a specific threshold. Once reached, a rapid depolarization occurs as voltage-gated sodium channels quickly open, allowing a massive influx of positively charged \(\text{Na}^{+}\) ions. This sudden inward current causes the membrane potential to momentarily reverse polarity, becoming positive inside the cell.
The peak is quickly followed by repolarization, which restores the negative potential. This is achieved as the fast \(\text{Na}^{+}\) channels inactivate, while voltage-gated potassium channels open with a slight delay. The resulting efflux of positively charged \(\text{K}^{+}\) ions rapidly drives the membrane potential back toward its negative resting level. The \(\text{K}^{+}\) channels often remain open slightly longer, causing a brief hyperpolarization, a temporary dip below the resting potential, before the membrane fully returns to its stable resting state.
Representing the Neuron: The Circuit Analogy
Hodgkin and Huxley translated the neuron’s cell membrane into an equivalent electrical circuit to provide a physical basis for their equations. The lipid bilayer acts as a capacitor (\(\text{C}_m\)) because it stores electrical charge across its thin, insulating structure.
The different types of ion channels embedded within the membrane are represented as parallel electrical pathways. Voltage-gated ion channels, which control the flow of \(\text{Na}^{+}\) and \(\text{K}^{+}\) during an action potential, are modeled as variable resistors or conductances.
The electrochemical forces that drive the ions across the membrane are represented by batteries, or voltage sources, in series with each channel pathway. Each battery’s voltage, known as the Nernst potential or reversal potential, is determined by the specific concentration gradient for that ion. A third pathway, the leak channel, is represented by a fixed resistor and battery to account for the steady, non-specific flow of other ions, like chloride (\(\text{Cl}^{-}\)), that influence the resting potential.
The Role of Ion Gating Variables
The dynamic behavior of the Hodgkin-Huxley model is captured by its gating variables. These variables are mathematical constructs that describe the instantaneous, voltage-dependent probability that an ion channel is in an open or closed state.
The voltage-gated \(\text{Na}^{+}\) current is governed by two distinct gating variables: activation (\(m\)) and inactivation (\(h\)). The activation variable (\(m\)) represents three hypothetical, independent gates that must all be open for the \(\text{Na}^{+}\) channel to conduct current, modeled as \(m^3\). This \(m\)-gate responds very quickly to depolarization, causing the rapid upstroke of the action potential.
Conversely, the inactivation variable (\(h\)) represents a single, slower gate that closes to block the \(\text{Na}^{+}\) flow, even if the \(m\)-gates are open. The slow closure of the \(h\)-gate limits the duration of the depolarization phase and initiates the repolarization.
The voltage-gated \(\text{K}^{+}\) current is controlled by a single activation variable, \(n\), represented as \(n^4\). This variable corresponds to four independent gates that respond to depolarization much more slowly than the \(\text{Na}^{+}\) \(m\)-gate. The delayed opening of the \(n\)-gates allows the \(\text{Na}^{+}\) influx to dominate first, and their subsequent opening facilitates the rapid \(\text{K}^{+}\) efflux that drives the membrane potential back down.
Enduring Influence on Neuroscience
The Hodgkin-Huxley model, published in 1952, was based on experimental data gathered from the giant axon of the squid, an organism chosen for its unusually large nerve cell diameter. The authors, Alan Hodgkin and Andrew Huxley, were jointly awarded the Nobel Prize in Physiology or Medicine in 1963 for their discoveries concerning the ionic mechanisms of nerve cell membranes.
The formalism established by the model remains the foundation of modern computational neuroscience, providing the language to describe ion channel kinetics. It serves as the basis for models used to simulate complex neural networks, study the effects of pharmacological agents on channel activity, and understand how neurons encode information through the timing of action potentials. Subsequent research has largely validated the model’s conceptual predictions about activation and inactivation gates. The Hodgkin-Huxley equations are still the standard building blocks for simulations of neuronal dynamics.