CALLIOPE · BCOM

This work proposes a unifying, mathematically rigorous definition of neural oscillation grounded in algorithmic information theory, topology, and Koopman operator theory. The central claim is that a signal oscillates when a circle-based generative model — rooted in the symmetry group $U(1)$ and the topology of $S^1$ — compresses the data more efficiently than any model lacking periodic structure, formalised as a Kolmogorov complexity criterion. The framework bridges four traditionally disconnected perspectives — classical physics, nonlinear dynamics, neuroscience, and signal processing — by identifying the circle as their common topological substrate. Through normal-form reduction and cohomological analysis, the Stuart–Landau equation is shown to be topologically inevitable near any Hopf bifurcation, with resonant terms arising as obstructions in $H^1(S^1) \cong \mathbb{R}$ that no smooth coordinate change can eliminate. The Koopman eigenfunction is recast as a compression map that reduces nonlinear limit-cycle dynamics to uniform rotation on $S^1$ , linking dynamical-systems theory directly to information-theoretic detection criteria. These results, drawn from Appendix I of the Rosetta Stone of Neural Mass Models (arXiv:2512.10982), carry practical implications for model-independent oscillation detection and explain why excitatory–inhibitory push–pull structure recurs universally across neural mass models.

A signal oscillates when a circle-shaped model compresses it better than anything else — and this paper explains, rigorously, why that has to be true.

Neuroscience, physics, dynamical systems, and signal processing all use the word "oscillation" but mean subtly different things. Physicists want periodicity around equilibrium; mathematicians want stable limit cycles; neuroscientists accept any rhythmic activity; signal processors look for narrow spectral peaks. None of these definitions talks to the others, and none explains why oscillatory structure appears in data in the first place. This paper proposes a single answer that subsumes all four views: a signal oscillates when a generative model built on the circle — the topological object S¹, the group of rotations U(1) — compresses the data more efficiently than any model without periodic structure. That compression gap, formalized as a Kolmogorov complexity criterion, is the definition.

The bridge between nonlinear dynamics and this compression idea runs through the Koopman operator. Even when a physical system is wildly nonlinear, there exists a special coordinate transformation — the Koopman eigenfunction — that maps the full high-dimensional trajectory onto simple uniform rotation on S¹. Finding that transformation is compression: you replace a complicated orbit with an initial phase, a frequency, and a small residual. The paper frames this explicitly as an information-theoretic act, connecting dynamical-systems theory to algorithmic information theory in a way that is usually left implicit.

The topological argument is the sharpest piece. Starting from a generic smooth perturbation of a linear oscillator near a Hopf bifurcation (the point where a stable equilibrium tips into sustained oscillation), the paper shows that systematic coordinate simplification — normal-form reduction — cannot eliminate certain nonlinear terms. Those surviving terms are not accidents; they are topological obstructions, elements of the first cohomology group H¹(S¹) ≅ ℝ, which is one-dimensional precisely because the circle has one "hole." The result is the Stuart–Landau equation, the universal amplitude equation for oscillations near threshold. Crucially, U(1) symmetry was never assumed — it emerges from the topology. You cannot coordinate-change your way out of it.

For neuroscience the payoff is twofold. First, it gives a principled, model-independent detection criterion: ask whether a circle-based model compresses the signal, not whether a spectral peak clears some threshold. This handles noisy, bursty, or intermittent rhythms that fool purely spectral methods. Second, it explains why excitatory–inhibitory push–pull architecture recurs across every neural mass model — from Kuramoto to Wilson–Cowan — without being explicitly designed in: the push–pull motif is just U(1) symmetry expressed in biological coordinates. The circle is not a metaphor; it is the topological reason brain rhythms exist at all.