MV
M. Valencia Torres
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Hippocampal oscillations during non-rapid eye movement sleep play a central role in memory consolidation, yet modelling their dynamics from partial observations remains an open problem. Data-driven methods based on the Koopman operator can represent nonlinear dynamics through a linear propagator acting on a lifted state space, but existing approaches mostly require full-state measurements or rely on fixed dictionaries that are sensitive to noise and difficult to build. This thesis first analyses the Hankel Alternative View of Koopman framework, the most established delay-based Koopman method for scalar time series, and shows that its forcing term loses its dynamical interpretation under measurement noise and non-periodic inputs, collapsing to a signal indistinguishable from the residual of a standard autoregressive model. These limitations motivate the Delay-Embedded Deep Koopman pipeline developed in this work, which combines Takens delay embedding with a neural-network encoder and a finite-dimensional Koopman matrix trained end-to-end on a multi-step prediction loss. The pipeline operates on a single scalar channel and includes a linear input extension for forced systems, validated on a nonlinear system with known ground truth. Applied to hippocampal local field potential recordings during non-rapid eye movement sleep from two independent datasets recorded in different laboratories and species, the model consistently outperformed a linear autoregressive baseline across all animals, with the advantage growing at longer prediction horizons. The eigenvalue decomposition of the learned Koopman matrix revealed three oscillatory modes at approximately 6.3, 9.5, and 13\,Hz within the sharp-wave passband, stable across independently trained models and both datasets. This three-mode structure was not recoverable by variance-based decomposition methods applied to the same data, indicating that the multi-step training loss selects for dynamical persistence rather than spectral energy. A modal amplitude analysis further showed that the three modes are not uniformly engaged during sharp-wave ripple events, suggesting distinct dynamical roles rather than a redundant frequency decomposition. Future directions are also discussed, including extensions to forced ripple dynamics and applications to closed-loop neuromodulation.
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Hippocampal oscillations during non-rapid eye movement sleep play a central role in memory consolidation, yet modelling their dynamics from partial observations remains an open problem. Data-driven methods based on the Koopman operator can represent nonlinear dynamics through a linear propagator acting on a lifted state space, but existing approaches mostly require full-state measurements or rely on fixed dictionaries that are sensitive to noise and difficult to build. This thesis first analyses the Hankel Alternative View of Koopman framework, the most established delay-based Koopman method for scalar time series, and shows that its forcing term loses its dynamical interpretation under measurement noise and non-periodic inputs, collapsing to a signal indistinguishable from the residual of a standard autoregressive model. These limitations motivate the Delay-Embedded Deep Koopman pipeline developed in this work, which combines Takens delay embedding with a neural-network encoder and a finite-dimensional Koopman matrix trained end-to-end on a multi-step prediction loss. The pipeline operates on a single scalar channel and includes a linear input extension for forced systems, validated on a nonlinear system with known ground truth. Applied to hippocampal local field potential recordings during non-rapid eye movement sleep from two independent datasets recorded in different laboratories and species, the model consistently outperformed a linear autoregressive baseline across all animals, with the advantage growing at longer prediction horizons. The eigenvalue decomposition of the learned Koopman matrix revealed three oscillatory modes at approximately 6.3, 9.5, and 13\,Hz within the sharp-wave passband, stable across independently trained models and both datasets. This three-mode structure was not recoverable by variance-based decomposition methods applied to the same data, indicating that the multi-step training loss selects for dynamical persistence rather than spectral energy. A modal amplitude analysis further showed that the three modes are not uniformly engaged during sharp-wave ripple events, suggesting distinct dynamical roles rather than a redundant frequency decomposition. Future directions are also discussed, including extensions to forced ripple dynamics and applications to closed-loop neuromodulation.