Complexity-Indexed Observability of Latent Dynamics in Fast–Slow Systems

This manuscript develops a theoretical framework for understanding how the observable dimensionality of a dynamical system depends on the temporal complexity of the protocol used to probe it. Many systems with separated timescales—such as neurons, ecological systems, and regulatory networks—contain multiple interacting slow variables that remain effectively hidden under low-complexity stimulation. In these regimes, reduced models with fewer dimensions can accurately reproduce the system’s behaviour because the slow subsystem remains confined within regions of state space where different internal configurations produce indistinguishable observable outputs. The framework introduces the concept of complexity-indexed observability, in which stimulus or protocol complexity determines whether latent dynamical dimensions become accessible to observation. Systems with fast–slow structure naturally contain tolerance regions in their slow state space where internal variation does not affect observable trajectories when stimulation is temporally simple. When protocol complexity remains below the system’s slow activation frequency, these tolerance regions persist and the system behaves as if it were low-dimensional. As protocol complexity increases and approaches the characteristic frequency associated with the slowest dynamical process, the tolerance region collapses and the slow subsystem becomes dynamically active. This transition exposes latent structure that cannot be captured by reduced models. The manuscript formalises this behaviour through a theorem linking protocol complexity to the activation of hidden dimensions and describes a sharp transition surface in protocol space where reduced and full models diverge. Empirical signatures of this transition are demonstrated through the geometry of spike-threshold drift and related slope and curvature statistics. These observables provide a practical method for detecting the activation of slow variables and identifying when reduced dynamical models cease to accurately represent the system. Although motivated by neuronal dynamics and comparisons between conductance-based models and reduced neuron models, the framework applies broadly to complex dynamical systems with separated timescales. By linking stimulus complexity to the emergence of latent dimensions, the theory provides a principled method for designing experimental protocols that reveal hidden structure, estimate slow timescales, and evaluate the validity of reduced models.