The Brain & Neurons

Your brain: the universe's most amazing computer!

Junior level — plain language, no maths

Inside your skull sits the most complex object in the known universe: your brain. It weighs about 1.4 kg, yet it packs in some 86 billion nerve cells called neurons. Each one can wire up to thousands of others, adding up to around 100 trillion connections - more than there are stars in the Milky Way.

Neurons talk by firing tiny electrical pulses called action potentials. Touch something hot and the neurons in your hand flash the news to your brain in milliseconds; think of a word and millions of them fire together in a precise pattern. And here's the lovely part: when you learn something new - riding a bike, say - the connections between particular neurons physically get stronger. That strengthening, quite literally, is the memory.

Your brain never really switches off. Even as you sleep it replays the day, cements the memories worth keeping, flushes out waste, and readies you for tomorrow - dreams may well be part of that nightly housekeeping. In the simulation below, watch neurons fire and pass signals down a little network, just like the real thing.

Action potentials, synaptic transmission, and neural coding

Student level — the core equations

A neuron at rest holds its inside at about \(-70\) mV, a voltage the Na⁺/K⁺ pump works constantly to keep. Nudge it past a threshold near \(-55\) mV and an action potential fires: voltage-gated sodium channels snap open and the inside rockets to \(+40\) mV, then slam shut as potassium channels open to reset the cell. Crucially it's all-or-none - every spike is the same size - so the brain can't encode anything in a spike's height. It encodes information instead in the rate and the precise timing of firing.

Where one neuron meets the next, at the synapse, the arriving spike opens calcium channels, and that calcium triggers vesicles to dump neurotransmitter - glutamate to excite, GABA to inhibit - across a gap just 20 nanometres wide. On the far side, some receptors are simple ion channels that open on contact, while others launch slower chemical cascades. One stands apart: the NMDA receptor only opens when the cell is both receiving glutamate and already depolarised, making it a built-in coincidence detector.

That coincidence detection is the physical root of learning. When two neurons fire together, calcium pours in through the NMDA receptors and sets off a cascade (through the enzyme CaMKII) that studs the synapse with more receptors, permanently strengthening it - long-term potentiation. Hebb's slogan nails it: "neurons that fire together, wire together." A memory isn't stored inside a cell but in the adjusted strengths of the connections between them.

Computational neuroscience: Hodgkin-Huxley, neural field theory, and the connectome

Scholar level — full mathematical depth

01The equations that captured a spike

In 1952, from painstaking measurements on the giant axon of a squid, Hodgkin and Huxley wrote down a set of coupled nonlinear differential equations that reproduce the action potential in quantitative detail - its shape, its threshold, its refractory pause. The membrane is a capacitor charged and discharged by voltage-dependent ion conductances, \(C_m \dfrac{dV}{dt} = -g_{Na}m^3h(V-E_{Na}) - g_K n^4(V-E_K) - g_L(V-E_L) + I\), with gating variables that open and close as the voltage swings. It won the 1963 Nobel Prize and remains the bedrock of every biophysical neuron model.

02Stripping it down

Four coupled equations per neuron is unwieldy for a network of billions, so theorists built reductions. The FitzHugh–Nagumo model keeps just two variables but preserves the essential phase-plane geometry - the excitable threshold and the limit-cycle firing - while the even simpler integrate-and-fire neuron discards the spike's shape entirely and just tallies input until it crosses threshold. These caricatures sacrifice biophysical detail to make large-scale simulation and mathematical analysis possible.

03From single cells to fields of tissue

Zoom out from individual neurons to a sheet of cortex and you can treat activity as a continuous field. Neural field theory models the mean activity \(u(x,t)\) with an integro-differential equation, \(\tau\,\partial_t u = -u + \int w(x-y)\,F[u(y,t)]\,dy + I\), where the connection kernel \(w\) is typically a "Mexican hat" - nearby neurons excite, distant ones inhibit. It's a remarkably compact way to describe waves, bumps and patterns rolling across the cortical surface.

04The brain makes Turing patterns too

That Mexican-hat coupling has a familiar consequence. Just as in reaction-diffusion chemistry, local excitation with long-range inhibition can spontaneously break uniformity into a periodic pattern - a Turing-like instability, triggered when the kernel favours a particular spatial wavelength. The same mathematics that spots a leopard is invoked to explain cortical columns, the stable "bumps" of activity that hold an item in working memory, and the astonishing hexagonal grid of the entorhinal cortex's navigation cells.

05Mapping every wire: the connectome

Underneath the dynamics lies the wiring diagram - the connectome - and mapping it is brutal. The only complete one belongs to the worm C. elegans: 302 neurons, ~7,000 synapses, finished in 1986. For humans the scale is staggering; slicing and imaging a single cubic millimetre of cortex with electron microscopy generates over 100 terabytes of data (the H01 dataset). Reconstructing an entire human brain remains far out of reach, but the fragments are already reshaping how we think about neural organisation.

06The shape of a healthy network

Where connectomes exist, graph theory reveals a consistent architecture: small-world topology, with dense local clustering and short global paths, plus a "rich club" of heavily interconnected hub regions and a hierarchy of modules. This is efficient wiring under tight metabolic and spatial budgets - and, tellingly, these very network signatures are measurably disrupted in Alzheimer's, schizophrenia and autism, hinting that some brain disorders are, at heart, diseases of connectivity.