Tuning curves

What is a neuron?

Every neuron in the brain is a cell with a clear job: take in voltage changes from upstream cells, integrate them, and — when the integrated signal is large enough — emit its own electrical pulse downstream. The parts of the cell are specialised for that job. Dendrites branch off the cell body to collect input from many sources. The soma (cell body) does the integration. A single long axon carries the output.

The output is binary — an action potential, or “spike”. The cell sits at a resting voltage of about $-70$ mV; when input pushes the membrane above a threshold around $-55$ mV, voltage-gated sodium channels open and the membrane rapidly depolarises to about $+30$ mV, then just as rapidly repolarises and dips slightly below resting before returning. The whole event takes about a millisecond, and the resulting pulse travels down the axon at meters per second toward the cell’s downstream targets.

STIMULATE PAUSE

The all-or-nothing nature of spikes is what makes this physiology so useful. Neurons don’t broadcast a continuously varying voltage — they emit a stream of yes/no events at certain times. Information lives in the rate and pattern of those spikes, not in their amplitude. A neuron firing 30 spikes per second is communicating something different from the same neuron firing 5 per second; the meaning of “different” is set by what the downstream cells do with that input.

What is a network?

When an action potential reaches an axon terminal, it doesn’t just stop there. It triggers the release of chemical neurotransmitters into the small gap between cells called the synapse. The neurotransmitter diffuses across the gap and either excites the downstream neuron (making it more likely to fire) or inhibits it (making it less likely). Whether a synapse is excitatory or inhibitory is determined by the neurotransmitter and the receptors it binds — glutamate at AMPA/NMDA receptors is excitatory; GABA at GABA-A receptors is inhibitory.

A single cortical neuron typically receives input from thousands of upstream cells through thousands of synapses. Whether it fires depends on the integrated effect of all of them. The figure below sketches a much smaller network — just six neurons connected by a mix of excitatory (arrowheads) and inhibitory (flat ends) synapses. Click any neuron to inject a current, or use the buttons to stimulate the input cells. Watch how an excitatory cascade can be cut short by an inhibitory connection, and how the output neuron’s firing depends on which combination of upstream cells is active.

STIMULATE A STIMULATE B RESET

This is the smallest interesting unit of “network activity” — small enough to follow each event, large enough to show the basic pattern: a spike at the input doesn’t deterministically produce a spike at the output, because inhibitory connections and the timing of competing inputs matter. Real cortical networks scale this up by six or seven orders of magnitude — the human cortex has roughly $10^{10}$ neurons and $10^{14}$ synapses — but the qualitative behaviour is the same.

The animated raster on this site’s landing page is the next step up: a population of one thousand neurons, where each spot of brightness is one yes/no spike from one cell. The rest of this piece is about how that population, despite being made of these noisy yes/no events, manages to encode a continuous value — the angle of the cursor.

A single neuron

The animated raster on the landing page is built out of one thousand copies of the same idea: a neuron whose firing rate is highest when the stimulus matches a specific preferred direction and falls off gracefully as the stimulus angle moves away from it. The standard mathematical shape for that is a Gaussian tuning curve:

$$f(\theta) = \exp\!\left(-\frac{(\theta - \phi)^2}{2\sigma^2}\right)$$

Here $\theta$ is the stimulus angle, $\phi$ is the neuron’s preferred direction, and $\sigma$ is the tuning width — how broadly the neuron responds to nearby angles. Drag the sliders below: $\phi$ slides the peak left and right; $\sigma$ controls how sharp or broad the response is.

φ 0.79 σ 0.40

In the landing-page raster, each neuron is created with a random $\phi$ drawn uniformly from $[0, 2\pi)$ and a random $\sigma$ drawn uniformly from $[0.2, 0.7]$. So a “narrow” neuron in this population has $\sigma \approx 0.2$ rad and only fires for a narrow band of stimulus directions; a “broad” neuron has $\sigma \approx 0.7$ and is excited across most of the angular range.

The biological lineage of this curve is older than the math. Hubel and Wiesel won the Nobel Prize for showing, in 1959, that neurons in cat primary visual cortex have preferred orientations — a single cell fires hard for a bar tilted at one angle and barely at all for a bar tilted ninety degrees away. The same selectivity has since been found in motion-sensitive area MT (tuned to direction of movement), in parietal cortex (tuned to attended location), and in primary motor cortex (tuned to direction of arm movement). The Gaussian above is the canonical first-pass model for any of these systems.

A neuron in time

A tuning curve, on its own, is a theoretical object. What you measure in a real recording is a spike train: a series of action potentials emitted at unpredictable times whose rate is governed by the tuning curve. To go from rate to spikes, the standard assumption is a Poisson process — the probability of emitting a spike in a small interval $dt$ is $f(\theta) \cdot dt$, scaled to a peak firing rate that’s biologically plausible (in the figure below, ~30 spikes per second at the peak of the curve).

That’s why the raster on the landing page never looks the same twice. Even a perfectly stationary stimulus and a perfectly tuned neuron will produce a different sequence of spikes every time — same average, never the same instance.

σ 0.40 SPEED 0.60 PAUSE

The exact same machinery — angular distance from the preferred direction, Gaussian envelope, Poisson sampling — runs once per neuron per frame in the raster simulation. With 1,000 neurons and rates between 0 and ~30 Hz, the raster you see is a thousand independent random processes, all coupled only through their shared dependence on the cursor angle.

A population, not a single cell

A single direction-tuned neuron is, frankly, a bad measurement device. Its peak firing rate has noise on the order of $\sqrt{rate}$ — pure Poisson statistics — so reading off the precise stimulus angle from one neuron’s output requires a long integration window and is still wrong by a wide margin. The brain doesn’t try.

Instead, the cortex uses a population code. Many neurons, each with its own preferred angle, respond simultaneously; the stimulus is inferred from their joint pattern of activity. The figure below shows this directly: each dot is one neuron, sitting at its preferred angle on the unit circle, with its radial distance from the center proportional to its current firing rate. As the stimulus rotates, the polygon connecting the neurons morphs through the population’s collective response shape. No single dot “knows” the stimulus angle. The shape, taken as a whole, does.

σ 0.50 SPEED 0.45 PAUSE

This is more than a visualisation trick. The Hubel-and-Wiesel-era idea that a single “grandmother cell” represents a single concept has been largely displaced, in the orientation/direction-tuning literature, by distributed coding. Strong evidence comes from Apostolos Georgopoulos’s monkey reaching experiments in the 1980s, where simultaneously-recorded motor cortex neurons each responded broadly to many movement directions, but their combined output predicted the direction the monkey was about to reach in with striking accuracy. We’ll get to that decoding step in the last section.

Three neuron types

So far each neuron in the simulation has been a Poisson generator modulated by a Gaussian tuning curve. Real cortex isn’t that homogeneous, and neither is the landing-page raster. Three sub-types coexist there and in the figure below:

• Standard. A Poisson cell. Spikes arrive independently with a rate set by the tuning curve. Most cortical neurons behave like this most of the time — this is the “Poisson cortex” approximation that underlies a lot of computational neuroscience.
• Bursty. Around 15% of the simulation’s neurons. When activated, they don’t emit a single spike — they emit a short train of 3–8 spikes packed close together, mimicking intrinsic-bursting (IB) cells found in cortical layer 5. Bursts carry more information than isolated spikes; downstream cells treat them as a stronger signal.
• Persistent. Around 7% of the simulation’s neurons. These cells carry a residual of recent input that decays slowly after the stimulus is gone. Real cortical neurons of this type are heavily studied in working-memory contexts: they keep a stimulus “in mind” by sustaining their firing well past the offset of the thing that drove them. Romo and colleagues showed this directly in monkey prefrontal cortex with parametric working-memory tasks.

The figure below sends the same drive impulse to all three types and shows how the output differs. Standard fires a sparse Poisson scatter. Bursty fires in clusters tightly time-locked to the drive. Persistent keeps firing well after the drive has ended.

DRIVE NOW PAUSE

In the population on the landing page, all three types fire alongside each other every frame. The visual difference is subtle when you’re not looking for it, but it’s why the raster has texture — not all spikes are statistically equivalent.

Reading the population back

If the cortex represents a stimulus as a pattern across many neurons, then the natural inverse problem is: given the pattern, recover the stimulus. The simplest decoding rule that works is the population vector, formalised by Georgopoulos in 1986:

$$\mathbf{v}(\text{stim}) = \sum_{i=1}^{N} r_i \cdot (\cos \phi_i, \sin \phi_i)$$

Each neuron contributes a vector pointing in its own preferred direction, weighted by how strongly it’s currently firing. Sum over the population and the resulting vector points (roughly) at the stimulus angle. With a small number of neurons the estimate is jittery — Poisson noise wins. With more, the estimate sharpens, and the angular error shrinks roughly as $1/\sqrt{N}$.

N 12 σ 0.50 PAUSE

The reason the population vector matters beyond a textbook example: it was the first method robust enough to support a working brain–computer interface. Neural prosthetics that decode intended movement from motor-cortex recordings are direct descendants of Georgopoulos’s arithmetic. The mathematical content here — Gaussian tuning, population coding, vector decoding — is the same content, in the same order, that underwrites the field.

The animation on the landing page is a 1,000-neuron population doing exactly this. The cursor is the stimulus; the spikes are the response. Whether it’s “doing” anything useful is up to the reader. But it is, at least, doing the same thing the cortex does.