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

Memory segmentation is thought to arise from rapid decorrelation in neural activity, often quantified by Euclidean distance or cosine angle. Although these metrics detect a transition, they do not reveal how the new state relates to the repertoire represented by the neural manifold. This matters because neuromodulators that drive state transitions also alter excitability, and learning may repurpose existing representations or create new ones. Here, I introduce a geometric decomposition that separates changes attributable to gain modulation of a nearby manifold state from movement within the manifold and genuine off-manifold displacement. The approach uses the radial axis of neural population activity to partition the normal space of a local manifold region. A central challenge is identifiability: given only a static reference manifold and a single test state, neither the state from which a perturbation began nor its gain magnitude and mechanistic decomposition can generally be recovered uniquely. I therefore formulate identifiability as a cascade of geometric gates specifying when each component can be interpreted. The gates distinguish structural failures, including the absence of a local chart or incorrect intrinsic dimensionality, from estimation error and systematic bias caused by reference sampling, tangent-frame error, gain-axis misalignment, anchor displacement, and poor ratio conditioning. Simulations show that neighborhood size, curvature, sampling density, ambient dimension, and noise act through a small set of geometric quantities. The framework specifies when assignments to gain or novelty are identifiable, how they become biased, and which diagnostics reveal the relevant failure regime. By quantifying the nature rather than only the magnitude of neural state change, it provides a clear, readily interpretable framework for evaluating mechanisms of neural state transitions.