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Edited by: Tobias A. Mattei, Ohio State University, USA

Reviewed by: Tobias A. Mattei, Ohio State University, USA; Paul Rapp, Uniformed Services University of the Health Sciences, USA

*Correspondence: Jianbo Gao, Institute of Complexity Science and Big Data Technology, Guangxi University, 100 Daxue Road, Nanning 530005, China e-mail:

This article was submitted to the journal Frontiers in Computational Neuroscience.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Epilepsy is a relatively common brain disorder which may be very debilitating. Currently, determination of epileptic seizures often involves tedious, time-consuming visual inspection of electroencephalography (EEG) data by medical experts. To better monitor seizures and make medications more effective, we propose a recurrence time based approach to characterize brain electrical activity. Recurrence times have a number of distinguished properties that make it very effective for forewarning epileptic seizures as well as studying propagation of seizures: (1) recurrence times amount to periods of periodic signals, (2) recurrence times are closely related to information dimension, Lyapunov exponent, and Kolmogorov entropy of chaotic signals, (3) recurrence times embody Shannon and Renyi entropies of random fields, and (4) recurrence times can readily detect bifurcation-like transitions in dynamical systems. In particular, property (4) dictates that unlike many other non-linear methods, recurrence time method does not require the EEG data be chaotic and/or stationary. Moreover, the method only contains a few parameters that are largely signal-independent, and hence, is very easy to use. The method is also very fast—it is fast enough to on-line process multi-channel EEG data with a typical PC. Therefore, it has the potential to be an excellent candidate for real-time monitoring of epileptic seizures in a clinical setting.

Epilepsy is a relatively common brain disorder which may be very debilitating. It affects approximately 1% of the world population (Jallon,

In the past several decades, considerable efforts have been made to detect/predict seizures through non-linear analysis of EEGs (Kanz and Schreiber,

Note that most of the proposed methods assume that EEG signals are chaotic and stationary. As a result, they tend to have performances that are signal- and patient-dependent due to the noisy and non-stationary nature of the EEG within and across patients. In addition, they are computationally expensive. Consequentially, studies of epilepsy still heavily involve visual inspection of multi-channel EEG signals by medical experts. Visual inspection of long (e.g., tens of hours or days) EEG data is, however, tedious, time-consuming, and in-efficient. Therefore, it is important to develop new non-linear seizure monitoring approaches.

In this paper, we explore recurrence time based analysis of EEG (Gao,

When developing a new method, it is important to compare its performance with that of existing methods. For seizure detection, such a task has been greatly simplified by our recent studies (Gao et al.,

_{small−ε}, (B) λ_{large−ε}, (C) Lyapunov exponent, (D) correlation entropy, (E) correlation dimension, and (F) the Hurst parameter obtained using DFA.

The remainder of the paper is organized as follows. In section 2, we describe the data used here and the recurrence time method and the STLmax method for seizure detection. In section 3, we compare the performance of the recurrence time and STLmax method for seizure detection, as well as study seizure propagation. In section 4, we make a few concluding remarks.

In this section, we first describe EEG data used here, then describe the recurrence time method and the short-time Lyapunov exponent (STLmax) method.

The EEG signals analyzed here are human EEG. They were recorded intracranially with approved clinical equipment by the Shands hospital at the University of Florida, with a sampling frequency of 200 Hz. Figure

Intracranial EEG is also called depth EEG, and is considered less contaminated by noise or motion artifacts. However, the clinical equipment used to measure the data has a pre-set, unadjustable maximal amplitude, which is around 5300 μV. This causes clipping of the signals when the signal amplitude is higher than this threshold. This is often the case during seizure episodes, especially for certain electrodes. To a certain extent, clipping complicates seizure detection, since certain seizure signatures may not be captured by the measuring equipment. However, we did not apply any filtering or conditioning methods to preprocess the raw EEG signals when we use our recurrence time method. The good results presented below thus suggest that the method is very reliable.

Altogether we have data of seven patients. The total duration of the measurement for each patient was up to about 3 days, as shown in the 2nd column of Table

_{2} and the STLmax method for seven patients' data

P92 | 35 | 7 | 100 | 0.09 | 100 | 0.00 |

P93 | 64 | 23 | 78 | 0.02 | 78 | 0.02 |

P148 | 76 | 17 | 58 | 0.07 | 76 | 0.00 |

P185 | 47 | 19 | 73 | 0.02 | 89 | 0.04 |

P40 | 5.3 | 1 | 100 | 0.00 | 100 | 0.00 |

P256 | 4.5 | 1 | 100 | 0.00 | 100 | 0.00 |

P130 | 5.7 | 2 | 50 | 0.18 | 100 | 0.00 |

The method involves first partitioning a long EEG signal into (overlapping or non-overlapping) blocks of data sets of short length

Let us first define the recurrence time of the 2nd type. Suppose we are given a scalar time series {_{i} = [_{i}, _{0} on the reconstructed trajectory, and consider recurrences to its neighborhood of radius _{r}(_{0}) = {_{0}∥ ≤ _{0}, and have been used by all other chaos theory-based non-linear methods.

_{r}(_{0})

Let us be more precise mathematically. We denote the recurrence points of the 1st type by _{1} = {_{t1}, _{t2}, …, _{ti} …}, and the corresponding Poincare recurrence time of the 1st type by {_{1}(_{i + 1} − _{i}, _{1}(_{1} (which can be easily achieved by monitoring whether the recurrence times of the first type are one or not). Let us denote the remaining set by _{2} = {_{t′1}, _{t′2}, …, _{t′i} …}. _{2} then defines a time sequence {_{2}(_{i + 1} − _{i},

_{2}(_{2}(_{2} (Gao et al., _{2}(_{2}(_{1} by a simple scaling law (Gao, _{r}(_{0}), thus contribute dimension 0, or form a smooth curve inside _{r}(_{0}), thus contribute dimension 1. These properties make the recurrence time based method very versatile and powerful in detecting signal transitions.

We now explain how the mean recurrence time of the 2nd type can be computed. We simply evaluate this quantity for every reference point in a window, then take the mean of those times. Such calculation is carried out for all the data subsets, resulting in a curve which describes how

We observe from Figure

Since there are altogether four parameters involved, namely, the embedding dimension

The basic idea is to compute the largest positive Lyapunov exponent for each window's EEG signal using the Wolf et al.'s algorithm (Wolf et al.,

To apply the Wolf et al.'s algorithm (Wolf et al., _{i} = [_{j} = [_{K} is usually taken as the nearest neighbor of _{1}. That is, _{j} and _{1}. When time evolves, the distance between _{i} and _{j} also changes. Let the spacing between the two trajectories at time _{i} and _{i + 1} be _{i} and _{i + 1}, respectively. Assuming _{i + 1} ~ _{i}^{λ1 (ti + 1 − ti)}, the rate of divergence of the trajectory, λ_{1}, over a time interval of _{i + 1} − _{i} is then

To ensure that the separation between the two trajectories is always small, when _{i + 1} exceeds certain threshold value, it has to be renormalized: a new point in the direction of the vector of _{i + 1} is picked up so that _{i + 1} is very small compared to the size of the attractor. After

Note that this algorithm assumes but does not verify exponential divergence. In fact, the algorithm can yield a positive value of λ_{1} for any type of noisy process so long as all the distances involved are small. The reason for this is that when _{i} is small, evolution would move _{i} to the most probable spacing, which is typically much larger than _{i}. Then, _{i + 1}, being in the middle step of this evolution, will also be larger than _{i}; therefore, a quantity calculated based on Equation (3) will be positive. This argument makes it clear that the algorithm cannot distinguish chaos from noise. In other words, even if the algorithm returns a positive λ_{1} from EEG data, one cannot conclude that the data are chaotic.

It is worth noting that in practice, to simplify implementation of the algorithm, one may replace the renormalization procedure described above by requiring that _{i + 1} is constructed whenever _{i + 1} = _{i} +

As we pointed out earlier, the method contains four parameters: the embedding dimension

First, we consider the window length

^{−4}), (2000, 4, 4, 2^{−4}), (2000, 4, 4, 2^{−3}), (2000, 3, 4, 2^{−4}), (2000, 4, 2, 2^{−4}), and (2000, 4, 6, 2^{−4}), respectively.

Next, we consider the size

Finally, we consider the embedding parameters. As is well known, the embedding parameters critically control the geometrical structure formed by the constructed vectors. Because of this feature, optimal embedding is a critical issue, especially when geometrical or dynamical quantities of the dynamics are concerned, such as the fractal dimension, Lyapunov exponents, and Kolmogorov entropy. For an in-depth discussion of this issue, we refer to Gao et al. (

To summarize, the recurrence method is much less sensitive to the parameters when compared with other non-linear methods, where embedding and other parameters have to be chosen carefully, and have to be specifically adapted to each patient for good results. For our recurrence time method, however, we have used the same parameter combination (^{−4}) for all seven patients' data.

To illustrate the idea, we shall arbitrarily pick up three channels of EEG data,^{1}

To more systematically compare the performance of the two methods in detecting seizures, we have computed positive detection (or equivalently, sensitivity) and false alarm per hour for the two methods. Positive detection is defined as the ratio between the number of seizures correctly detected and the total number of seizures. The false alarm per hour is simply the number of falsely detected seizures divided by the total time period. Table

The recurrence time method is very fast. With an ordinary PC (CPU speed less than 2 GHz), computation of

Formation and propagation of epileptic seizures in the brain is an outstanding example of complex spatial-temporal pattern formations. One of the most desirable ways of studying these problems is to understand how and when information flows from one region of the system to other regions. To resolve this issue, it is critical to accurately providing timing information for interesting events occurring in the system. With the exact timing information, one can then use concepts such as cross correlation and cross spectrum, mutual information, or measures from chaos theory, such as related to cross recurrence plots, to more fully characterize the spatial–temporal patterns. Recurrence time method can effectively provide such a timing information. To illustrate this point, we have shown in Figure

Motivated by developing a non-linear method without the limitations of assuming that EEG signals are chaotic and stationary, we have proposed a recurrence time based approach to characterize brain electrical activity. The method is very easy to use, as it only contains a few parameters that are largely signal-independent. It very accurately detects epileptic seizures from EEG signals. Most critically, the method is very fast—it is fast enough to real-time on-line process multi-channel EEG data with a typical PC. Therefore, it has the potential to be an excellent candidate for real-time monitoring of epileptic seizures in a clinical setting.

The recurrence time method is also able to accurately give the timing information critical for understanding seizure propagation. Therefore, it may help characterize epilepsy type, lateralization and seizure classification (Holmes,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{1}In fact, the three chosen channels EEG data may not correspond to where a seizure was localized. This further indicates the robustness of our method.