arXiv subjects

Nikos Logothetis

Publications and source records attributed to Nikos Logothetis.

3 recordsLinked to original sources

arXiv2025

A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems

We present Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework for approximating the Koopman semigroup in stochastic dynamical systems. Unlike existing methods, SDMD explicitly incorporates sampling time into its approximation, ensuring numerical stability and precision. By directly approximating the Koopman semigroup instead of the generator, SDMD avoids computationally expensive matrix exponential computations, which offers a more efficient and practical pathway for analyzing stochastic dynamics. The framework further integrates neural networks to automate basis selection, which reduces the reliance on manual intervention while maintaining computational efficiency. Rigorous theoretical guarantees, including convergence in the large data limit, zero-limit of sampling time, and large dictionary size, establish the method's reliability. Numerical experiments on canonical stochastic systems validate SDMD's effectiveness in approximating eigenvalues and eigenfunctions of the stochastic Koopman operator.

math.DS↗

arXiv2025

ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the \emph{spectral residual} to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator's complete spectral information, a limitation known as the `spectral inclusion' problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the \emph{spectral residual} to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.

cs.LG↗

arXiv2019

Signal detection in extracellular neural ensemble recordings using higher criticism

Information processing in the brain is conducted by a concerted action of multiple neural populations. Gaining insights in the organization and dynamics of such populations can best be studied with broadband intracranial recordings of so-called extracellular field potential, reflecting neuronal spiking as well as mesoscopic activities, such as waves, oscillations, intrinsic large deflections, and multiunit spiking activity. Such signals are critical for our understanding of how neuronal ensembles encode sensory information and how such information is integrated in the large networks underlying cognition. The aforementioned principles are now well accepted, yet the efficacy of extracting information out of the complex neural data, and their employment for improving our understanding of neural networks, critically depends on the mathematical processing steps ranging from simple detection of action potentials in noisy traces - to fitting advanced mathematical models to distinct patterns of the neural signal potentially underlying intra-processing of information, e.g. interneuronal interactions. Here, we present a robust strategy for detecting signals in broadband and noisy time series such as spikes, sharp waves and multi-unit activity data that is solely based on the intrinsic statistical distribution of the recorded data. By using so-called higher criticism - a second-level significance testing procedure comparing the fraction of observed significances to an expected fraction under the global null - we are able to detect small signals in correlated noisy time-series without prior filtering, denoising or data regression. Results demonstrate the efficiency and reliability of the method and versatility over a wide range of experimental conditions and suggest the appropriateness of higher criticism to characterize neuronal dynamics without prior manipulation of the data.

stat.AP↗