Effective Hamiltonians Across Scales in Octopus Camouflage Patterns

Octopus camouflage is a dynamic visual computation: the body changes texture and color within sub-second timescales to match its surroundings. These transformations arise from millions of tiny pigment organs (chromatophores) that expand and contract under neural control, making the skin an active, adaptive surface. Octopus skin thus provides a unique window into neural computation that is both distributed and directly observable. Yet we still lack a quantitative framework that captures how local neural control produces the coherent, large-scale visual textures observed. Because coordination between local control and global texture suggests collective organization, it is natural to frame the problem in statistical-mechanics terms, where measured correlations define effective interactions. Using high-resolution recordings during active sleep, a phase when the skin spontaneously replays patterns, we quantify spatial and temporal chromatophore correlations. The correlations decay over a finite range, motivating microscopic variables for individual chromatophores and coarse-grained variables for correlated patches. From these statistics we infer effective Hamiltonians whose couplings reproduce the observed one- and two-point functions at each level. The resulting energy landscape captures how local interactions generate macroscopic texture and predicts the dominant modes of pattern fluctuation, providing a concrete, testable account of pattern formation under neural control.