Scaling of noisy fluctuations in complex networks and applications to network prediction.

We study the collective dynamics of oscillator-network systems in the presence of noise. By focusing on the time-averaged fluctuation of dynamical variable of interest about the mean field, we discover a scaling law relating the average fluctuation to the node degree. The scaling law is quite robust as it holds for a variety of network topologies and node dynamics. Analyses and numerical support for different types of networks and node dynamics are provided. We also point out an immediate application of the scaling law: predicting complex networks based on time series only, and we articulate how this can be done.

Detection of seizure rhythmicity by recurrences.

Epileptic seizures show a certain degree of rhythmicity, a feature of heuristic and practical interest. In this paper, we introduce a simple model of this type of behavior, and suggest a measure for detecting and quantifying it. To evaluate our method, we develop a set of test segments that incorporate rhythmicity features, and present results from the application of this measure to test segments. We then analyze electrocorticogram segments containing seizures, and present two examples. Finally, we discuss the similarity of our method to techniques for detecting unstable periodic orbits in chaotic time series.

Correlation dimension and integral do not predict epileptic seizures.

Reports in the literature have indicated potential value of the correlation integral and dimension for prediction of epileptic seizures up to several minutes before electrographic onset. We apply these measures to over 2000 total hours of continuous electrocortiogram, taken from 20 patients with epilepsy, examine their sensitivity to quantifiable properties such as the signal amplitude and autocorrelation, and investigate the influence of embedding and filtering strategies on their performance. The results are compared against those obtained from surrogate time series. Our conclusion is that neither the correlation dimension nor the correlation integral has predictive power for seizures.

Accumulated energy revisited.

OBJECTIVE: To examine the seizure prediction and detection abilities of the accumulated energy on multi-center data submitted to the First International Collaborative Workshop on Seizure Prediction. METHODS: The accumulated energy (AE), windowed average power, and FHS seizure detection algorithm were applied to a single channel of ECoG data taken from the data sets contributed to the workshop. The FHS seizure detection algorithm was used to perform automated scoring of the data in order to locate subclinical events not picked up by the centers where the data was collected. The results were analyzed retrospectively, comparing the behavior of the accumulated energy and windowed average power on segments containing seizures to interictal segments. RESULTS: Accumulated energy curves showed no divergence from interictal curves prior to seizure. Distinctive or clear increases in the AE slope occurred sometime at or after electrographic seizure onset for some seizures. Similarly, the windowed average power showed no consistent increases in broadband energy prior to seizures. However, both methods may have detection ability for some seizures. CONCLUSIONS: The accumulated energy did not appear to have predictive abilities for these data sets. Some detection ability was apparent. SIGNIFICANCE: In data unsorted by sleep/wake state, no seizure prediction was evident. The lack of prediction calls into question the existence of a preictal state as previously claimed in the literature using this method.

Controlled test for predictive power of Lyapunov exponents: their inability to predict epileptic seizures.

Lyapunov exponents are a set of fundamental dynamical invariants characterizing a system's sensitive dependence on initial conditions. For more than a decade, it has been claimed that the exponents computed from electroencephalogram (EEG) or electrocorticogram (ECoG) signals can be used for prediction of epileptic seizures minutes or even tens of minutes in advance. The purpose of this paper is to examine the predictive power of Lyapunov exponents. Three approaches are employed. (1) We present qualitative arguments suggesting that the Lyapunov exponents generally are not useful for seizure prediction. (2) We construct a two-dimensional, nonstationary chaotic map with a parameter slowly varying in a range containing a crisis, and test whether this critical event can be predicted by monitoring the evolution of finite-time Lyapunov exponents. This can thus be regarded as a "control test" for the claimed predictive power of the exponents for seizure. We find that two major obstacles arise in this application: statistical fluctuations of the Lyapunov exponents due to finite time computation and noise from the time series. We show that increasing the amount of data in a moving window will not improve the exponents' detective power for characteristic system changes, and that the presence of small noise can ruin completely the predictive power of the exponents. (3) We report negative results obtained from ECoG signals recorded from patients with epilepsy. All these indicate firmly that, the use of Lyapunov exponents for seizure prediction is practically impossible as the brain dynamical system generating the ECoG signals is more complicated than low-dimensional chaotic systems, and is noisy.

Inability of Lyapunov exponents to predict epileptic seizures.

It has been claimed that Lyapunov exponents computed from electroencephalogram or electrocorticogram (ECoG) time series are useful for early prediction of epileptic seizures. We show, by utilizing a paradigmatic chaotic system, that there are two major obstacles that can fundamentally hinder the predictive power of Lyapunov exponents computed from time series: finite-time statistical fluctuations and noise. A case study with an ECoG signal recorded from a patient with epilepsy is presented.

Correlation-dimension and autocorrelation fluctuations in epileptic seizure dynamics.

We focus on an anomalous scaling region in correlation integral [C(epsilon)] analysis of electrocorticogram in epilepsy patients. We find that epileptic seizures typically are accompanied by wide fluctuations in the slope of this scaling region. An explanation, based on analyzing the interplay between the autocorrelation and C(epsilon), is provided for these fluctuations. This anomalous slope appears to be a sensitive measure for tracking (but not predicting) seizures.