A unified theoretical framework for the geometry of cognitive dynamics

How the brain flexibly switches between different cognitive contexts and deploys internal representational models of vastly different geometries is a core puzzle in neuroscience. Large-scale neural recordings have recently revealed a contradictory phenomenon: during different tasks, the low-dimensional manifold of neural activity sometimes manifests as zerodimensional points (fixed points), sometimes as one-dimensional lines, and at other times as two-dimensional or even higher-dimensional continuous spaces [1–3]. However, a unified theoretical framework capable of explaining why and how these attractors of disparate dimensions can coexist within the same biological system has been missing. This paper proposes a dynamical paradigm based on first principles that provides a concise and powerful solution to this problem. This paradigm reveals a two-layer, decoupled topological design principle: a top-level "camp" competition network determines the number of discrete, stable contexts (multistability) [4, 5], while a bottom-level "citadel" topology precisely endows each context with its local geometric form, such as points, lines, or planes [6, 7]. Through a series of numerical experiments, we not only demonstrate that this system can stably emerge coexisting attractors of different dimensions but also, by simulating targeted interventions, causally prove the switching mechanisms of multistability and the construction rules for attractor dimensions. This theoretical framework unifies a seemingly chaotic experimental landscape into a simple, programmable topological rule, offering a fundamental theoretical insight into the neural basis of cognitive flexibility and context-dependent computation.