How does cue geometry set which decision harmonics an animal experiment can see?

Cue arrangement multiplies the animal's spectrum

The paper starts from a simple reframing. An animal surrounded by landmarks, neighbours or light sources must turn their bearings into one heading, and many experiments probe that mapping by presenting a controlled arrangement of cues and measuring the turning response. Harmonic Theory writes the instantaneous decision landscape over heading as a sum of periodic wells, one per cue, whose Fourier coefficients form the animal's decision spectrum. The first harmonic is a simple pull toward a direction, the second prefers an axis, and higher harmonics encode progressively finer angular structure.

The key step is that the cue contributions combine into a complex cue structure factor, Fn, built from each cue's weight and bearing. If the harmonic kernel is common to the cues present, the landscape collapses to a product of that known complex number and the animal's real coefficients Kn. The experiment enters only through Fn, which the cue arrangement fixes; the animal enters only through Kn. The paper calls this the measurement equation of harmonic spectroscopy and notes that it has the algebraic form of a diffraction amplitude: an intrinsic spectrum multiplied by a structure factor encoding the arrangement of scatterers.

This is why the paper treats cue geometry as an instrument rather than a nuisance to be conditioned on. Because Fn is computable before data collection, a design has a measurable information content at every harmonic. The authors call the expected squared magnitude of Fn the spectral leverage of a design. A single cue has leverage of unity at every harmonic; a pair swept uniformly over a full period has leverage of 2 at every harmonic. Both are spectrally flat, so any decline of measured amplitude with harmonic number is then a property of the animal rather than of the apparatus.

Symmetry deletes the harmonics it forbids

The paper's first proposition turns symmetry into an exact filter. If the weighted cue configuration is invariant under rotation by 2π/p for an integer p of at least 2, then Fn is zero unless p divides n. The proof is short: rotating every bearing by 2π/p multiplies Fn by a phase factor while leaving the configuration, and hence Fn, unchanged, so the only way the equality can hold is for the bracket to vanish, which happens exactly when p divides n.

A p-fold symmetric array therefore acts as a harmonic comb filter. It annihilates every harmonic that is not a multiple of p, including those above p, and passes n = p, 2p, and so on. Three identical cues at 120 degrees delete the first and second harmonics identically, so any orientation structure that survives in such an array is third-harmonic or higher. The paper points out that this inverts the usual reading of a symmetry zero: in an unconstrained regression a predictor that is identically zero is a failure, but here it is the instrument, the same situation as systematic extinctions in crystallography.

A second design gives a tunable notch. For two identical cues at bearings minus and plus half their separation, the structure factor is twice the cosine of n times half the separation. Sweeping the two cues apart draws a notch through each harmonic, and the notch position depends on harmonic number. The paper reports that flies steering between two swept virtual targets reverse their second harmonic near 91 degrees, where that design's structure factor puts its notch; a three-target array moves the predicted notch to about 120 degrees, and the measured reversal moves with it. Two such families measured over the same separations give linearly independent projections of the same spectrum, so cross-geometry transfer is a real test rather than a restatement.

Two channels estimate the same spectrum

The paper offers two ways to read the spectrum from a single trajectory. The dissipative, or drift, channel regresses observed angular increments on the harmonic predictors defined by the cue field, recovering the coefficients directly. The fluctuation, or occupancy, channel uses the conditional stationary distribution of heading at fixed cue configuration, which is a circular exponential family whose natural parameters are the harmonic coefficients in units of the diffusion and whose sufficient statistics are the cue-weighted harmonic moments. It uses every frame rather than every increment and never differentiates an angle.

Because both channels estimate the same physical spectrum, their ratio should equal one for every harmonic whenever the angular dynamics are an overdamped gradient flow of the instantaneous landscape with the stated diffusion. The paper calls this a harmonic-resolved fluctuation-dissipation relation for decisions and a genuine falsification test with no free parameter. It cannot be satisfied by an arbitrary flexible kernel fitted to the same data, because the two channels are computed from different functionals of the trajectory.

The occupancy channel is also more robust to measurement noise. If the observed heading carries frame-to-frame error, the drift regression puts a single draw of that error on both sides of its own equation, with opposite signs, and the fitted coefficient is biased by an amount of order the error variance divided by the sampling interval. That bias grows as sampling is refined, because shortening the interval shrinks the true increment while the camera error stays fixed. The occupancy channel never forms a difference, so the same error cannot enter twice; it only blurs the heading density, and a known blur can be undone inside the likelihood. The paper proves that, under continuous error-free observation in the weak-response limit, the occupancy estimator attains the same asymptotic variance as the continuous-path maximum-likelihood drift estimator.

Detectability makes the first harmonic easiest to see

The paper derives a detectability law that explains a recurring pattern in the literature. Writing the asymptotic signal-to-noise ratio of harmonic n and dividing by the same quantity at the first harmonic gives a ratio with three factors: a sensory term that is Gaussian in the square of n, a neural term set by the bump width, and a design term set by the spectral leverage. The sensory factor dominates.

The numbers make the cost concrete. In a spectrally flat design at a sensory width of 0.7 radians, a third harmonic requires 13 to 50 times more data than a first across bump widths of 60 to 90 degrees, and the requirement diverges as the bump width approaches 120 degrees, where the neural filter itself has a zero at the third harmonic. At a sensory width of 0.9 radians the sensory factor raises the cost to 163 to 652 times.

This is a quantitative prediction of the theory, and the paper uses it to reinterpret a striking regularity: trajectory-based interaction inference rarely resolves angular structure beyond the first harmonic. It is tempting to read that as evidence that angular responses are simple. The detectability law says instead that it is what a broadly tuned sensory front end must produce. The paper is careful about what the fly signal-to-noise ratios do and do not establish: they illustrate this loss of detectability but do not estimate a width, because under Gaussian sensory and neural filters only their combined width enters the spectrum.

The fly compass shows the selection rules directly

The sharpest test is neural, because symmetry can be printed directly onto a visual scene. The paper reads spectra from public recordings of flies, schooling fish, larval zebrafish, mice and a swimming alga, and from the fly's neural compass. In the compass recordings, panoramas with two- and four-fold symmetry suppress the harmonics their symmetry forbids, symmetry-broken twins restore them in the same flies, and every prespecified discriminator shifts in the predicted direction in every animal. The selection rules that organize the behaviour are read out in the neural representation the behaviour is supposed to inherit from.

The same recordings measure the neural filter the theory had only assumed. Measuring the compass form factor favours the theory's smooth bump over its square one. The square bump gives a spectrum that oscillates in sign and decays only as one over n, and the measured profile shows neither signature. A static measurement in the cue frame cannot settle the shape, because on one resolved ratio the two fit identically; a manipulation can. Under a within-fly contrast change, a width fitted to the second harmonic predicts the third: the square bump predicts the wrong sign, and the Gaussian is closer in 15 of 15 flies. Raising cue contrast narrows that bump in every fly.

The paper also reports where the neural evidence is less clean. The compass profile carries a second harmonic of the same order as the behavioural one, though the raw intervals meet only after a correction for placement error. Under the theory's own bridge, even the corrected estimates would require the sensory stage to pass the second harmonic more strongly than the first, which no non-negative kernel that is reflection-symmetric about the cue and nonincreasing with circular distance can do. The uncertainty reaches its own boundary, so the data do not exclude that class at a calibrated level, and a well-resolved ratio above one would still not identify its cause. The paper reports the agreement and the mismatch together rather than combining them.

What does not carry across designs or species

Not everything transfers. Fitted higher harmonics do not carry between the two cue designs, and the two channels agree in shape but not in absolute scale. The paper also declines to offer separability as evidence, since separable kernels already fit fish interaction data. What it claims is not another rule for how animals choose a direction but a way to measure the one they already use, plus a law that says in advance which harmonics an experiment can see.

The alga data provide a mechanistic contrast rather than a failure. Beam separation was varied across five conditions spanning the same two-cue geometrical question, with individual headings measured for 5,300 to 13,100 cells at separations from 25 to 135 degrees. In the independently determined optical reference frame, the second circular moment about the bisector remains positive throughout, declining only from +0.98 to +0.76, with no sign change in 104 file-level bootstrap replicates. The observed distributions remain concentrated near the bisector, with no resolved second lobe at either beam. The paper notes that cue geometry can only filter harmonic landscape terms supplied by the organism's response, so separation alone does not force a fly-like notch within the analysed range; the alga is a single cell with no nervous system and fundamentally different sensorimotor machinery.

The behavioural compromise-to-selection transition is not fly-specific, since it was also reported in larval zebrafish. But the paper could not apply harmonic spectroscopy to that transition, because the public release available to it retains aggregate occupancy grids and position-target trajectories but not direct body orientation or animal and trial identifiers. The transition generalises beyond flies, whereas its harmonic-spectral identification remains established only in the released fly recordings.

A fitted coefficient is not by itself mechanistic

The paper tests whether its structure-factor collapse discriminates among mechanisms. Of eight generative rules passed through a common idealised closed-loop protocol and the same form of occupancy analysis, three reproduce the observed second-harmonic coefficient: winner-take-all, a leaky accumulator and a drift-diffusion process, none of them parameterised as an explicit harmonic landscape. A fitted coefficient alone therefore does not identify a mechanism.

The structure-factor collapse does better, the authors argue, because it tests a location the geometry fixes in advance rather than an amplitude the fit is free to choose. It excludes all six tested alternative parameterisations, and the held-out family comparison extends the test to fitted families, excluding the null, first-harmonic-only, winner-take-all and boxcar families when each is refitted freely and scored on unseen data.

The paper is also explicit about what remains latent. Behaviour identifies only the product of cue-shape, sensory and neural filters, so at least one factor must be varied or measured independently. Cue contrast varied within fly already shows that one factor moves, sharpening the compass bump in 15 of 15 animals, but it leaves open whether the sensory coefficients stay fixed, since they were not measured in the same manipulation. A behavioural spectrum recorded simultaneously would show how the product moves within animals, and an independent cue-size or eccentricity manipulation would separate the two factors.

Design the cue array before trusting a missing harmonic

The practical priority that follows from this paper is a design rule rather than a biological claim. Because a coefficient whose design leverage is zero is unmeasured rather than absent, the paper argues that such a coefficient should be reported as unmeasured, not as absent. A configuration with p-fold symmetry passes only multiples of p, and a swept pair separation is a tunable notch, so the experimenter can choose in advance which harmonics the apparatus is capable of seeing.

The comparison that matters most is between two readings of a flat spectrum. A response that looks first-harmonic-only can mean the animal's angular response is simple, or it can mean the sensory front end suppresses higher harmonics as a Gaussian in the square of n and the design never had the leverage to resolve them. The detectability law makes the second reading quantitative, and the neural recordings show that a measurable third harmonic exists in the compass's own frame even where the cue-referenced frame leaves it unresolved. The paper's own priority is therefore to build a behavioural design with the leverage to read that harmonic, and to vary or measure one filter independently so that the product of cue, sensory and neural terms can be separated.

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