In 1952, two British physiologists stuck a glass electrode into a giant squid’s axon and recorded something nobody had seen before: a precise, reproducible electrical pulse — an action potential — traveling down the nerve fiber like a burning fuse.
They measured it. They modeled it. And what they found would eventually earn them a Nobel Prize and lay the mathematical foundation for everything we now call computational neuroscience.
The model is called the Hodgkin-Huxley model, and it explains, in four coupled differential equations, exactly how a neuron decides to fire.
You can explore it interactively at ElysiaTools.
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The Problem Nobody Could Solve
Before Hodgkin and Huxley, physiologists knew neurons communicated electrically. They could measure the voltage across a neuron’s membrane — the thin boundary separating the inside of the cell from the outside fluid. They knew that at rest, a neuron sat at about -65 millivolts (inside negative relative to outside). And they knew that during a nerve impulse, this voltage spiked sharply positive before returning to baseline.
What nobody could explain was why. What membrane mechanism generated this pulse? Why did it always have the same shape? Why did it propagate without dying out?
The breakthrough came not just from better measurements, but from a better mathematical description of what was being measured.
The Core Equation: A Circuit Made of Salt Water
The neuron’s membrane is only about 7 nanometers thick — thinner than any human-made membrane. Yet it acts like an electrical circuit with three parallel conductance pathways (for sodium ions Na⁺, potassium ions K⁺, and a leak current), each driven by its own voltage gradient:
“`
I_ext = C_m × dV/dt + g_Na × m³h × (V − E_Na)
- g_K × n⁴ × (V − E_K)
- g_L × (V − E_L)
“`
Don’t let the notation scare you off. The intuition is simple: the total injected current (I_ext) either charges the membrane capacitance (C_m × dV/dt) or flows out through the three ion channels. The m³h and n⁴ terms are not exponents for convenience — they reflect the gating structure of the actual channels. Sodium channels need three activating particles (m) to open AND one inactivating particle (h) to close. Potassium channels need four activating particles (n) to open.
This was the key insight: ion channels aren’t simple holes. They have gates that open and close depending on voltage. And you can write down the math.
Threshold: Why Neurons Don’t Fire Randomly
One of the most striking predictions of the model — now confirmed experimentally — is that neurons have a firing threshold.
Below about 6.3 microamps per square centimeter of injected current, the neuron stays silent. The membrane voltage drifts slightly but returns to rest, like a slightly pushed swing returning to equilibrium.
Cross that threshold, however, and something qualitatively different happens: the neuron fires — and keeps firing regularly as long as the current persists. This is called regular spiking.
This threshold behavior emerges entirely from the nonlinear interaction between the sodium and potassium conductances. It’s not a hardcoded rule in the model — it self-organizes from the equations.
The Four Equations, Explained
The full model consists of four coupled ODEs:
- Membrane equation: Governs how membrane voltage changes based on total ionic current flow
- Sodium activation (m): Controls the opening of sodium channels — fast activation
- Sodium inactivation (h): Controls the closing of sodium channels — slow inactivation
- Potassium activation (n): Controls the opening of potassium channels — slower than sodium
Each gating variable follows first-order kinetics — its rate of change depends only on its current value and a voltage-dependent rate constant:
“ dm/dt = α_m(V) × (1 − m) − β_m(V) × m “
The functions α(V) and β(V) are empirically measured voltage-clamp curves. Hodgkin and Huxley determined them by holding the membrane voltage at fixed values and measuring how quickly the gating variables moved.
Why the Nobel Prize Was Deserved
In 1963, Hodgkin and Huxley received the Nobel Prize in Physiology or Medicine “for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheral and central portions of the nerve cell membrane.”
The key word is discoveries — plural. They didn’t just describe the phenomenon. They discovered the mechanism: voltage-gated sodium and potassium channels, described quantitatively, explain the action potential completely.
The model has since been extended, simplified (the FitzHugh-Nagumo model reduces it to two equations), and applied to cardiac cells, pancreatic beta cells, and any electrically-excitable cell. But the core architecture — the four equations — remains the foundation.
What the Interactive Visualization Shows
The Hodgkin-Huxley visualization at ElysiaTools lets you play with the model directly:
– Membrane potential plot: Watch the characteristic spike waveform in real time as you adjust parameters
– Gating variables plot: See how m, h, and n evolve during each spike — sodium activation (m) shoots up fast, potassium activation (n) follows more slowly
– Ion conductance plot: Observe how g_Na and g_K trade dominance during the spike — sodium conductance peaks first (depolarization), then potassium conductance dominates (repolarization)
– Phase portrait: Plot dV/dt against V to reveal the underlying limit cycle — the closed orbit the system traces during repetitive firing
The presets let you explore different firing regimes:
– Resting: No current, membrane at rest (-65 mV)
– Threshold: Current just at the edge of firing — watch subthreshold oscillations
– Regular Spiking: The canonical neuron behavior
– Fast Spiking: High-frequency firing with reduced spike adaptation
– Strong Drive: Rapid, intense spiking with prominent afterhyperpolarization
Why This Still Matters
Modern artificial neural networks abstract away the biological neuron to a simple threshold or sigmoid function. This was a deliberate simplification — computing with spiking neurons is much harder than computing with real numbers.
But as researchers push toward neuromorphic computing, brain-inspired chips, and more biologically-plausible learning rules, the Hodgkin-Huxley model is making a quiet comeback. Understanding how real neurons spike, adapt, and synchronize gives us a richer vocabulary for building machines that behave less like brute-force matrix multipliers and more like the three-pound biological computers we actually carry in our skulls.
The equations that Hodgkin and Huxley wrote down in a cramped lab in Plymouth, measuring impulses in squid neurons with hand-crafted glass electrodes, are now running in real-time simulations on supercomputers, in silicon chips designed to mimic cortical circuits, and — thanks to ElysiaTools — in your browser.
Try it: Explore the Hodgkin-Huxley Neuron Model — drag the injected current slider and watch the spike train emerge from pure mathematics.
