Data-driven Koopman operator approach for computational neuroscience
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Data-driven Koopman operator approach for computational neuroscience Natasza Marrouch1
· Joanna Slawinska2 · Dimitrios Giannakis3 · Heather L. Read4
© The Author(s) 2019
Abstract This article presents a novel, nonlinear, data-driven signal processing method, which can help neuroscience researchers visualize and understand complex dynamical patterns in both time and space. Specifically, we present applications of a Koopman operator approach for eigendecomposition of electrophysiological signals into orthogonal, coherent components and examine their associated spatiotemporal dynamics. This approach thus provides enhanced capabilities over conventional computational neuroscience tools restricted to analyzing signals in either the time or space domains. This is achieved via machine learning and kernel methods for data-driven approximation of skew-product dynamical systems. The approximations successfully converge to theoretical values in the limit of long embedding windows. First, we describe the method, then using electrocorticographic (ECoG) data from a mismatch negativity experiment, we extract time-separable frequencies without bandpass filtering or prior selection of wavelet features. Finally, we discuss in detail two of the extracted components, Beta (∼ 13 Hz) and high Gamma (∼ 50 Hz) frequencies, and explore the spatiotemporal dynamics of high- and low- frequency components. Keywords Koopman operator · Spectral decomposition · Nonlinear · Spatiotemporal dynamics · ECoG · Brain · Mismatch negativity Mathematics Subject Classification (2010) 37M10 · 37M25 · 58C40 · 30C40 · 37N25 · 47A35 · 92C55 This article is an expanded version of research presented at the 2018 International Joint Conference on Neural Networks in Rio de Janeiro, Brazil. N. M. acknowledges support from the CT Institute for the Brain and Cognitive Sciences Graduate Summer Fellowship and the University of Connecticut Department of Psychological Sciences’ Maurice L. Farber Endowment. J. S. acknowledges support from NSF grants 1551489 and 1842538. D. G. acknowledges support from ONR YIP grant N00014-16-1-2649, NSF grant DMS-1521775, and DARPA grant HR0011-16-C-0116. H. R. acknowledges support from NSF grant 1355065, NIH DC015138 01, and the University of Connecticut Brain Computer Interface Core. Natasza Marrouch
[email protected]
Extended author information available on the last page of the article.
N. Marrouch et al.
1 Introduction Mammalian brains exhibit different oscillations of electrical current. These oscillations are commonly divided into 5–8 bands based on their frequency. The oscillations vary from slow, e.g., Delta (1–4 Hz) and Theta (4–8 Hz), to medium like Alpha (8–12 Hz), to quickly oscillating modes such as Beta (12–30 Hz) and Gamma (30–80 Hz). Electrophysiologic recordings, which allow researchers to capture distinct oscillations, are one of the most affordable and readily-accessible sources of data on brain activation patterns. Such recordings from multi-channel electrode arrays capture data with temporal resolution
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