Identifying Neural State Changes due to Gain versus Off-Manifold Displacement.

Memory segmentation is thought to be caused by rapid decorrelation in neural activity, commonly quantified by Euclidean distance and cosine angle. While these metrics are useful for detecting that a transition has occurred, they cannot reveal the nature of that change relative to the repertoire of pre-existing states - a space known as the neural manifold. This distinction matters, since neuromodulators that are thought to drive state transitions also influence neuronal excitability, and there is ongoing controversy over whether learning involves repurposing old representations or creating new ones. Here, I introduce a decomposition that separates changes in neural activity into those that can be understood by gain modulation of a nearby manifold point versus genuine off-manifold displacement. The approach exploits the privileged role of the radial axis in neural population activity by partitioning the normal space of a local region of the neural manifold. A central challenge is identifiability: from a static reference manifold and a single test state, the state from which a perturbation began, its gain magnitude, and, in general, the mechanistic decomposition of the observed displacement cannot be uniquely recovered. I therefore formulate identifiability as a cascade of geometric gates that specify the conditions under which each component can be interpreted. These gates distinguish structural failures, such as the absence of a local manifold chart or incorrect intrinsic dimensionality, from estimation error and systematic bias arising from the manifold reference, gain-axis misalignment, and poor ratio conditioning. I derive exact expressions for how reference error propagates into the reported components and through simulations on idealized manifolds show that several factors-neighborhood size, curvature, sampling density, ambient dimension, and noise-affect the decomposition through a small set of geometric quantities. This framework therefore does not simply assign a neural state change to gain or novelty; it specifies when those assignments are identifiable, how they become biased, and which diagnostics can reveal the relevant failure regime. By quantifying the nature of the neural state change, not just its magnitude, this framework provides a more interpretable description of population dynamics and a principled basis for evaluating candidate mechanisms underlying neural state transitions.