Can collective behavior emerge from a harmonic theory of directional brain representations?

Rules describe behavior but do not explain it

Traditional models of collective motion treat organisms as self-propelled particles that follow heuristic rules, such as aligning with neighbors or moving toward a local center of mass. Those models show that global coordination can arise without centralized control, but they take the heading vector as the fundamental dynamical variable. The paper's stated problem is that this leaves the organism's decision architecture, and the link between neural representations of space and behavior, outside the explanation.

The authors reframe the question. Instead of asking how collective order emerges from interaction rules, they ask how macroscopic behavior emerges from processing environmental information on a topological map of space. They focus on a ring manifold, a one-dimensional circular representation of heading direction, which they describe as biologically widespread and mathematically tractable.

Coarse-graining fast neurons into a heading landscape

The derivation rests on four assumptions. The neural state is a localized bump on the ring, centered at an angle phi and of width W; the bump's width is read as decision uncertainty, so a narrow bump means a precise, high-confidence decision and a wide bump a diffuse one. Neural dynamics are assumed to relax much faster than behavior, so the bump can be treated as settled while the heading turns. The environment enters through a sensory input with kernel width sigma, and the interaction kernel J_int sets coupling strength and range.

Integrating out the fast neural variables produces an effective Hamiltonian in which the agent's heading couples to the n-th harmonic of the bearing to a target or neighbor. Each harmonic's weight is a product of three terms: the ecological interaction kernel, the Fourier coefficient of the sensory input, which decays as exp(-n^2 sigma^2 / 2), and the n-th moment of the neural bump, which depends on W. Behavior is then modeled as overdamped gradient dynamics on this landscape with rotational noise. The key consequence is that the sensory kernel width sigma controls how strongly higher harmonics contribute: large sigma attenuates them, small sigma lets them shape fine structure.

Sharp choices arise from bifurcations, not switches

In the binary choice problem, the agent faces two targets separated by an angular distance. For large sigma, higher harmonics are attenuated and the landscape is dominated by the first modes. As the agent approaches the targets, the symmetric averaging solution loses stability and gives rise to two symmetric solutions, one per target; the paper identifies this as a supercritical pitchfork bifurcation for the fixed square-bump and Gaussian sensory models. The predicted critical angular separation, the smallest separation above which the symmetric trajectory loses stability, increases with the bump width W. The authors explain this by noting that a choice is expected once the egocentric separation of the targets exceeds the bump width, so the bifurcation angle should scale roughly with W.

This matters because a sudden change of direction does not require a discrete switch between rules. It arises when gradual changes in geometry or harmonic weighting destabilize one branch of the landscape and stabilize another. The paper notes that this frozen-heading classification does not exclude multistability in the full neural or moving-agent dynamics.

Sensory width sets how rich the trajectories become

The same parameter split organizes the individual-level results. With large sigma, the first few resilient harmonics dominate, trajectories are simpler, and the agent often shows back-and-forth motion between targets, which the authors compare to the Buridan's paradigm observed in animals such as Drosophila; the likelihood of direct escape increases as sigma decreases. With smaller sigma, higher harmonics contribute more strongly and trajectories show richer spatial patterns.

Noise behaves differently across this range. Low-order harmonics admit few stable solutions and are therefore resilient to noise, while medium harmonics are more strongly affected. At sufficiently high harmonic order, closely spaced solutions produce wandering, random-walk-like behavior even without external noise. In the binary choice problem, noise at intermediate sigma can help the agent stay near the targets and can speed up choice, which the authors liken to stochastic-resonance effects; at very large sigma the agent already stays deterministically, and at very small sigma high harmonics already induce irregular dynamics.

Accuracy and speed trade off through harmonic order

When the two targets are unequal, the first harmonic alone produces a weighted average of target directions biased toward the stronger option but need not select either target. Adding the second harmonic introduces curvature that favors the stronger target. For large sigma, where dynamics are dominated by the first two modes, the agent strongly prefers the superior target. For smaller sigma, higher harmonics can generate bistability, which the paper reports can lead to faster commitment in some trials but less accurate decisions.

This is a within-paper comparison worth keeping separate from the equal-target case: the same shift in sigma that enriches trajectories also changes the accuracy-speed balance in unequal choice. The paper's account is a model prediction from the harmonic landscape, not a measured behavioral result.

Odd and even harmonics produce different collective modes

The paper classifies collective patterns by harmonic parity. For attractive interactions, minimizing an odd harmonic does not allow consensus on heading; agents must maintain an n-dependent phase difference with respect to their common bearing, which the authors associate with conflict and frustration. For even harmonics, consensus becomes possible: the second harmonic yields alignment and anti-alignment under attraction, and perpendicular alignment and opposed transverse preferences associated with milling under repulsion. Geometric incompatibilities in larger groups can contribute to fission-fusion and frustrated dynamics.

The authors describe harmonics as acting like morphogens in a Turing-like pattern formation, but in the one-dimensional polar space of heading directions rather than in physical space. The first harmonic creates a single well at the average direction and promotes consensus; higher harmonics can alternate between stabilizing and destabilizing as the agent moves. The paper states that this parity-based classification is detailed and numerically confirmed in its supplementary analysis.

Fish data converge on spectral trends, not a parameter mapping

The paper compares the theory with publicly available data from rummy-nose tetra fish. In the simulations, increasing the sensory kernel width attenuates higher angular harmonics and also reduces the temporal spectral complexity and high-frequency content of heading velocity. Across several spectral measures, including spectral entropy, centroid, flatness, bandwidth, rolloff, high-power fraction and peak count, the fish data show analogous trends with increasing illumination. The authors explicitly state that these parallel trends do not establish a mapping between illumination and sigma.

Convergence also appears in how polarization is regulated by the same spectral measures and by speed, and in descriptive similarities in speed and distance-to-wall summaries. The reduced equation prescribes a fixed free-space propulsion speed, so the speed comparison does not establish intrinsic motor-speed regulation. The authors also do not rule out that some convergences could be shown by traditional models, but they argue the breadth across kinematic, spectral and geometric dimensions supports a unifying account in which environmental change tunes the weights of a shared harmonic basis rather than rewriting behavioral rules.

The ring is a starting point, not a full account

The authors state that comprehensive quantitative tests of the theory's predictions are still lacking. The framework makes explicit predictions about how sensory kernel width, interaction range and internal integration should reshape the spatial and temporal spectra of behavior, which they present as targets for future empirical work. The ring case is described as a tractable and biologically widespread starting point for directional behavior, not a claim that all cognition is one-dimensional; the same bottom-up program could be extended to manifolds with other symmetries and dimensions.

Within the paper, the most consequential split is the sensory kernel width sigma. Large sigma leaves a landscape dominated by a few resilient low harmonics, giving simpler trajectories and, in the unequal-target case, accurate preference for the better option. Small sigma admits higher harmonics, producing richer spatial patterns, bistability and faster but less accurate commitment. That trade-off, rather than any single behavior, is the paper's central claim about how directional information is organized.

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