Recovering Hidden Neural Oscillatory Modes from Partial Observations through Delay Embedding and Perturbation

Characterising the oscillatory behaviour generated by neural interactions is central to systems neuroscience because these dynamical properties are linked to brain function. However, inferring latent neural dynamics is challenging because recordings sample only part of the system. Time-delay embedding allows dynamical modes to be estimated from partial recordings, using methods such as Hankel dynamic mode decomposition (Hankel DMD). Despite its promise, two questions remain: what dynamical information is recoverable through time-delay embedding, and how can recovery be improved? We address both questions using control theory. Our framework centres on factorising the Hankel matrix of delay-embedded observations into an observability matrix and a trajectory matrix with the form of a controllability matrix. We show that their ranks impose complementary constraints on what any delay-embedding method can recover: delay embedding can increase observability rank, while perturbation can increase trajectory rank by exciting otherwise inactive modes. We validate this framework using Hankel DMD in simulations and human electrocorticography (ECoG). In both, perturbation increased the number of fast-decaying mode estimates, whereas greater delay depth increased the number of reliably estimated modes and, in ECoG, extended their frequency range. These results clarify what partial recordings can reveal about neural dynamics and how to reveal more.