The Electric Brain

Your brain runs on tiny sparks!

Junior level — plain language, no maths

Inside your skull, right now, some 86 billion neurons are flashing like tiny lightning bolts, flinging messages along at up to 120 metres a second. Those sparks are how you're reading this sentence, how you can picture a friend's face, and how your hand yanks back from something hot before you've even decided to move.

A neuron is a specialised cell shaped a bit like a tree. Its branching dendrites are the roots, gathering incoming signals from neighbours, and its long tail, the axon, is the trunk that carries the signal onward. When enough incoming signals pile up, the neuron "fires", launching a brief electric pulse racing down the axon toward the next cell in line.

Between two neurons lies a narrow gap, the synapse, and here the signal changes form. Rather than leaping across as electricity, the arriving pulse releases little chemical messengers - neurotransmitters like dopamine and serotonin - that drift over the gap and tell the next neuron whether to fire or hold still. Every thought, feeling, memory and movement you've ever had was carried by chains of these sparks hopping across billions of synapses.

The Hodgkin-Huxley Action Potential

Student level — the core equations

A neuron at rest holds a membrane potential of about \(-70\) mV - its interior kept more negative than the outside by the Na⁺/K⁺ pump, which tirelessly ejects three sodium ions for every two potassium it hauls in. Depolarise the membrane past a threshold near \(-55\) mV and voltage-gated sodium channels burst open, letting sodium flood in and driving the voltage all the way to \(+40\) mV in under a millisecond.

That lightning reversal is the action potential, and it is strictly all-or-nothing: once it trips, it always reaches roughly the same peak no matter how hard it was pushed. So a neuron signals intensity by how often it fires, not how big each spike is. Within a millisecond the sodium channels inactivate and slower potassium channels swing the voltage back down, briefly overshooting into a refractory lull in which no fresh spike can fire - a hard ceiling on how fast a neuron can talk.

Then the spike travels: each patch of depolarised membrane tips its neighbour past threshold, and the pulse marches down the axon without fading. Nature bolted on a turbocharger - fatty myelin insulation forces the signal to leap between bare gaps called nodes of Ranvier (saltatory conduction), pushing speed toward 120 m/s while slashing the energy bill. Hodgkin and Huxley mapped this whole cascade in 1952, from ionic currents to spike, and took the 1963 Nobel Prize for it.

Hodgkin-Huxley Equations, Conductance Models, and Neural Coding

Scholar level — full mathematical depth

01The spike as a circuit

Hodgkin and Huxley's genius was to treat the membrane as an electrical circuit: a capacitor in parallel with voltage-controlled resistors for each ion. Conservation of current gives \(C_m \dfrac{dV}{dt} = -g_{Na}m^3h(V-E_{Na}) - g_K n^4(V-E_K) - g_L(V-E_L) + I\). Each term is one ionic pathway, each battery \(E\) the ion's Nernst potential. Solve it and a full action potential appears, unbidden - proof that the spike is not a special biological trick but a consequence of a handful of voltage-dependent channels obeying Ohm's law.

02Channels that open and close in time

The nonlinearity lives in the gating variables \(m, h, n\), each the fraction of channels in an open configuration and each relaxing toward a voltage-dependent target, \(\dfrac{dx}{dt} = \alpha_x(V)(1-x) - \beta_x(V)x\). Sodium activates fast (\(m\)) but then inactivates (\(h\)); potassium activates slowly (\(n\)). That staggered timing - quick in, delayed out - is what makes the voltage overshoot and recover rather than simply relax, turning a threshold crossing into a stereotyped pulse.

03Boiling four equations down to two

Four coupled variables are hard to picture, so FitzHugh and Nagumo bundled the fast pair and the slow pair into just two, \(\dot v = v - v^3/3 - w + I\) and \(\dot w = \varepsilon(v + a - bw)\). Now the whole story fits on a 2-D phase plane, where the action potential is simply a large loop the state makes when a stimulus shoves it across a threshold curve - below the curve it slides quietly home, above it takes the grand excursion.

04Two flavours of excitability

That geometric view classifies real neurons. If firing switches on through a saddle-node bifurcation, the cell is Type I - it can fire arbitrarily slowly, ramping its rate smoothly with input. If it switches on through a Hopf bifurcation, it's Type II - firing appears abruptly at a finite frequency. This isn't idle taxonomy: whether a neuron integrates inputs or resonates with them shapes how a whole circuit computes.

05How signals spread: cable theory

A dendrite is a leaky electrical cable, and how far a passive signal reaches before dwindling is set by the space constant \(\lambda = \sqrt{r_m/r_a}\) - the balance between membrane leak and axial resistance. This is why distant synapses are heard only faintly at the soma, why myelin (raising \(r_m\)) lets signals travel farther between nodes, and why the branching architecture of a neuron is itself a form of computation, not mere plumbing.

06What the spikes actually mean

Finally, what do these pulses encode? The default answer is rate coding - information in the firing frequency - but timing clearly matters too: the auditory system pins sound direction from microsecond spike differences, and hippocampal place cells signal location through the exact phase at which they fire. The cortex leans on sparse coding, with only ~1% of neurons active at once, squeezing maximum representation from minimum metabolic spend - the same efficiency logic now driving neuromorphic "spiking" chips built to compute like a brain.