A compass can hide two destinations

Picture a circle of tiny lanterns. Each lantern prefers a direction: north, northeast, east, and so on. When an animal faces east, lanterns near east glow brightly. As it turns, the bright patch travels around the circle.

This is a useful picture of a population code, a representation carried by the activity pattern of many neurons together. A downstream circuit might summarize that pattern with a compass arrow. But what happens when two equally bright patches face opposite ways?

The arrow vanishes. It also vanishes when every lantern glows equally. Two strong alternatives and diffuse activity have reached the same summary. Let’s keep the full pattern on the page and investigate what the arrow preserves.

Give each neuron a little direction

Label a neuron’s preferred direction by an angle ${\varphi }_i$, measured in radians. Choose a nonnegative activity weight $w_i$, such as a firing rate divided by a stated rate unit. Its unit direction is the two-component vector

We can now collect total activity $A$ and a weighted directional sum $\mathbf{P}$:

Assume . The vector $\mathbf{z}$ is the normalized first directional moment, an average arrow. Its length $R$, called the resultant length, lies between zero and one. If $\mathbf{P}\ne 0$, its angle defines the decoded mean direction. At $\mathbf{P}=0$, the mean direction is undefined.

These are chosen activity coordinates. Interpreting them as a posterior probability distribution would require a separate encoding model. Nevertheless, their positive geometry is already exact.

A common increase in activity, $w_i\mapsto c{w}_i$ with , scales both $A$ and $\mathbf{P}$ by $c$. It leaves $\mathbf{z}$ unchanged. The decoded direction and normalized resultant can stay identical while the circuit’s overall activity changes substantially.

Open the compass and inspect its gears

Make a prediction: which preset will distinguish two opposite bumps from diffuse activity: the first moment, the second moment, or the total activity?

Turn the population in the playground.

The second directional moment averages doubled angles:

Doubling the angle sends opposite directions to the same place because $2(\varphi +\pi )$ and $2\varphi$ differ by a full turn. Two opposite bumps therefore reinforce each other in this summary. Uniform activity around the circle cancels in both moments.

An idealized pair of equal point-like bumps has $R=0$ and $R_2=1$. Uniform activity has $R=R_2=0$. The demo uses smooth bumps, so its opposite-pair value of $R_2$ falls below one. Higher moments provide further views of a pattern; any finite list can still discard structure.

This is also a decoder question. If a task asks for one heading, an average arrow may be convenient. If it asks whether either of two opposite routes is available, the same compression discards the distinction the task needs. The Information Theory readings follows related questions about which differences a representation preserves.

Positive activity lands inside a cone

The triangle inequality gives $|\mathbf{P}|\le A$. Consequently the quadratic quantity

is nonnegative. Expand the two squares and the cancellation becomes especially revealing:

The notation means every distinct pair is included once. Each pair contributes according to its two weights and their angular disagreement. Identical directions contribute zero; opposite directions contribute the largest factor, two, inside the brackets.

This has the same algebraic form as the time-minus-space quadratic in the Epistemic Physics cone. Here $M$ is defined from a chosen population encoding. Its units follow the activity weights; the identity alone supplies no physical mass or circuit dynamics.

The demo’s two opposite bumps and diffuse activity share $A$, zero first moment, and therefore $M=A$. Keeping amount alongside direction recovers more than direction alone, but still leaves those patterns indistinguishable. A full directional ledger is richer than its cone summary.

If we later place a symmetric Dirichlet prior on $K$ direction bins, its parameters can be ${\alpha }_i=\tau /K$ for a fixed total concentration $\tau$. Refining the bins then preserves the total prior amount. Assigning a whole new prior to each newly introduced bin would instead change the model as its resolution changes. The demo uses activity weights directly and adds no such prior.

How a ring remembers which way it faces

Our lantern circle is a static snapshot. To hold a heading through time, the circuit needs a rule that updates its activity.

A recurrent network sends activity back through connections among its neurons. A ring attractor is a recurrent system with a family of persistent activity patterns indexed by a circular variable such as heading. A bump can be stable against some disturbances while still moving around the ring when turning input arrives.

Kim, Rouault, Druckmann, and Jayaraman combined imaging and optogenetic perturbation in flies to investigate such a heading representation. An artificially imposed population pattern could persist with naturalistic dynamics. Their findings support a circuit architecture involving local excitation and global inhibition.

This gives our static patterns a new question: which can the dynamics sustain? An arbitrary two-bump configuration allowed by positive-weight geometry might be unstable in a circuit that maintains a unique bump. Geometry describes the available encoding; recurrent dynamics select trajectories and persistent states within it.

There is a tempting experiment here. Keep the same initial activity pattern but vary recurrent coupling, then compare the time needed to recover a bump after a perturbation. The cone quantities provide interpretable summaries to track. A proposed relationship between them and recovery time would need the actual dynamical equations and measurements to earn its place.

Shared fluctuations have a direction too

Neurons vary across repeated presentations of the same stimulus. Noise correlation describes how those trial-to-trial deviations co-vary. If two neurons both fluctuate upward on the same trial, their deviations have positive covariance.

The pattern of those correlations matters. Imagine moving the whole activity bump a little clockwise. A downstream decoder sees something very similar to an actual clockwise change in heading. By comparison, multiplying the whole pattern by a common positive factor leaves our normalized arrow unchanged.

Moreno-Bote and colleagues formalized how certain correlation patterns limit information. In particular, variability aligned with changes in the mean response as the stimulus changes can mimic the signal itself. This motivates examining the direction of shared variability, rather than treating every nonzero correlation as equally harmful.

For a local mathematical picture, let $\mathbf{f}(\theta )$ be the vector of mean neuronal responses at heading $\theta$. A small random heading-like perturbation $\delta$ gives

The prime means derivative with respect to heading. If $\delta$ has variance ${\sigma }_{\delta }^2$, this shared contribution has approximate covariance matrix

The superscript $\mathsf{T}$ turns a column into a row, so the product records every pair of response changes. This is the relevant direction in population space: fluctuations can look like a change in the variable the animal is trying to estimate. The formula is a local small-perturbation construction, not a claim that every observed covariance has this form.

Let the population keep its clock

A population snapshot also leaves out temporal organization. If a neural state persists for several time bins, activity at neighboring times can share that state. Averaging more bins then differs from collecting the same number of freshly redrawn states.

Explore the activity clock in the playground.

This companion experiment concerns shared state across time. The correlation argument above concerns relationships across neurons. Both invite the same practical habit: specify what is being repeated and what remains shared before treating a count as a collection of separate trials.

A neural compass becomes much more interesting once it carries its whole traveling light show with it: the activity pattern, the amount, the first and higher moments, the shared fluctuations, and the dynamics that keep it moving. Each answers a different question about what the animal can represent and remember. See the disclaimer for the wider scope of these exploratory connections.