Elsevier

Neuroscience Research

Volume 156, July 2020, Pages 178-187
Neuroscience Research

Mathematical structures for epilepsy: High-frequency oscillation and interictal epileptic slow (red slow)

https://doi.org/10.1016/j.neures.2019.11.008Get rights and content

Highlights

  • Wideband ECoG recorded brain wave data of two patients of epilepsy was analyzed.

  • The main term of the power spectrum obeys the power law distribution.

  • Circle map type dynamical system is observed within seizure onset time segments at epileptic foci with HFO.

  • Interictal epileptic slow (red slow) phenomena is detected by residue of power spectrum.

Abstract

In the present study, we attempted to characterize two characteristic features within the dynamic behavior of wideband electrocorticography data, which were recorded as the brain waves of epilepsy, comprising high-frequency oscillations (HFOs) and interictal epileptic slow (red slow). The results of power spectrum and nonlinear time series analysis indicate that, on one hand, HFOs at epileptic focus are characterized by one-dimensional dynamical systems in ictal onset time segments at an epileptic focus for two patients’ datasets; on the other hand, an interictal epileptic slow is characterized by the residue of power spectrum. The results suggest that the degree of freedom of the brain dynamics during epileptic seizure with HFO degenerates to low-dimensional dynamics; hence, the interictal epileptic slow as the precursors of the seizure onset can be detected simply from interictal brain wave data for the dataset of one patient. Therefore, our results are essential to understand the brain dynamics in epilepsy.

Introduction

Approximately 1 % of population ratio is diagnosed with epilepsy, which is a common disease. Therapy for refractory patients is essential to identify their epileptic foci by electroencephalographic (EEG) or electrocorticography (ECoG) analyses. Once seizures onset, extreme oscillations are recorded at epileptic foci, which subsequently propagate to the whole brain. The seizure onset can be characterized using a wideband EEG or ECoG analyses, co-occurrences with direct current (DC) shift, and high-frequency oscillation (HFO) that ranges from 80 Hz to 500 Hz on epileptic foci (Imamura et al., 2011; Ikeda et al., 1996, 1999; Kanazawa et al., 2015).

Given this information regarding epilepsy, it is challenging to understand the brain dynamics in epilepsy observed as a brain wave using wideband EEG or ECoG from the viewpoint of dynamical system. Mathematical models that describe the behavior of the brain dynamics should be proposed as dynamical systems. A mathematical model based on an ordinary differential equation system with six variables is proposed (Jirsa et al., 2014), and the model describes DC shift as extremely slow dynamics and HFO as a pair of slow and fast dynamics. The model is established in a heuristic style; that is, first a simple model was proposed and modified to fit the given brain dynamics as brain wave data such as DC shift and HFO. In contrast, we follow the opposite approach; that is, our model is directly obtained from the brain dynamics using standard methods of time series analyses. In our approach, because brain wave data varies with time, the brain wave data is considered as time series data, and we conducted power spectral and nonlinear time series analyses. Once we identify the probability distribution of power spectra, the dynamics of the time series can be assumed. Furthermore, using the nonlinear time series analysis, the dynamical system that determines the time evolution of the time series can be developed (Kantz and Schreiber, 2004). The details of the theoretical aspects of the method are described by Takens (1981) and Sauer et al. (1991) and the application aspects by Kantz and Schreiber (2004). As an application in the field of medicine, the method was used to distinguish the dynamics of pulsation recorded from a patient of disease (Tsuda et al., 1992). Furthermore, as an application to epilepsy, the differences of dynamical systems during seizures and interictal states, the dimension of attractors and the largest Lyapunov exponent were estimated by Babloyantz and Destexhe (1986). Dimension of attractor was also studied by Yuan et al. (2008) using entropy method. Unlike the previous studies, we determined a type of dynamical system from the datasets.

From these perspectives, the objective of the study is to characterize the difference between the behavior of the brain dynamics represented as brain waves during ictal onset and during inter-ictal cases, based on the wideband ECoG-recorded data of two refractory epilepsy patients, patient one and patient two. Our results are summarized in the following three parts. First, using a power spectrum analysis, we determined that the power spectrum obeys the power law distribution with exponent approximately -2. From the results of our analyses, we can assume that the dynamics is approximated by fractional Brownian motion, that is a generalized class of Brownian motion (Mandelbrot, 1982; Peitgen and Saupe, 1988). Moreover, anomalous HFOs are identified on epileptic foci during ictal onset. Second, using nonlinear time series analysis, a family of circle-map type one-dimensional dynamical system (Devaney, 1989) is presented on the anomalous HFO. These results indicate that the degree of freedom of brain activity is reduced to low-dimensional dynamics during certain time intervals of ictal onset at epileptic foci, and the dynamical system characterizes the anomalous HFOs. The dynamical system was obtained from the datasets of the brain waves and expressed using elementary functions. Although in usual studies on nonlinear time series analysis only statistical exponents are estimated from the time series data, we further found the model equation itself. In this respect, the present results are distinguished from those of previous studies. Finally, interictal epileptic slow shift, i.e., red slow (Inoue et al., 2019), is characterized by the residue of the power spectrum, which is a residue in the linear regression of the power spectrum in log-log scale. Interictal epileptic slow shift is observed in epileptic foci of patient one as the precursors of seizure onset (Inoue et al., 2019), which can be accurately detected using our system. This means that the residue of power spectrum in log-log scale helps us identify epileptic foci only from interictal time segments of ECoG-recorded datasets and that the method reduces the burden of patients.

Section snippets

Dataset and method

In the study we analyze wideband ECoG-recorded brain wave data (Imamura et al., 2011; Ikeda et al., 1996), which is stored with MATLAB format file. For patient one (Pt1), who has right temporal lobe epilepsy, brain waves were recorded using fifty subdural electrodes by ECoG. Epileptic foci are recognized around channel B13 (33rd electrode) and D01 (47th electrode). A target dataset comprises 5,563,000 time steps (2768 s), seizure onset at 3,430,641 time steps (1715.3 s), and seizure end at

Main results

Our results are summarized in the following three propositions:

Proposition 1

For patients one, the main term of the power spectrum at the electrodes near epileptic foci around ictal onset time segments obeys the power law distribution with exponent, which is approximately -2. Moreover, the contribution of HFO in ictal onset time intervals are observed as a peak at approximately 120 Hz.

Proposition 2

For both patients one and two, a family of one-dimensional discrete dynamical system named circle map is observed, just

Discussion

We analyzed the wideband ECoG-recorded brain wave data on epileptic foci of Pt1 and Pt2. Using the power spectral analysis, differences between ictal and interictal time segments are described and summarized in Proposition 1. During ictal onset time segment, when HFO is observed, a circle-map-like dynamical system describes its difference data using embedding method as in Proposition 2.

Propositions 1 and 2 suggest that within the interictal time segment during the Brownian motion is a model to

Declaration of Competing Interest

Masao Matsuhashi and Akio Ikeda: Department of Epilepsy, Movement Disorders and Physiology, Kyoto University is the Industry-Academia Collaboration Courses, supported by Eisai Co., Ltd., Nihon Kohden Corporation, Otsuka Pharmaceutical Co., Ltd. and UCB Japan Co. Ltd.

Acknowledgments

This study was supported by JSPS KAKENHI Grant Numbers 15K21731, 18H04929, 18H05323, and 19H03574. This work was also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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