Surrogate analysis determines the dynamics in the time series: whether it is governed by stochastic or deterministic process 282930.

The surrogate data has been generated by Random Shuffled (RS) surrogate method, in which the signals were shuffled randomly so that the probability distribution is same but the temporal correlations are destroyed 282932. D_corr which gives us a measure of the complexity has been estimated for both the original and the surrogate data of the data sets A, B, C, D, and E respectively. Fig 1(a) shows that the D_corr increases with the same trend for both the original [Fig 1(b)] and the surrogate data [Fig 1(c)] for the normal persons with eyes open [Set A]. A similar trend is observed in the case of the persons with eyes closed, i.e., for Set B [Fig 2(a)]. This shows that the brain activity of a normal person is stochastic in nature agreeing with the analysis by Andrzejak, et al. 1. For an epileptic patient we found a different behavior during seizure free intervals and during seizure activity. The EEG signals, at seizure free state, recorded both from the hippocampal formation of the opposite hemisphere of the brain [Fig 2(b) Set C] and within the epileptogenic zone [Fig 2(c) Set D] show almost same trend in the increase in D_corr for the surrogate data as well as for original data, except for a small separation at higher embedding dimensions, which may be due to a high dimensionality of the system. During seizure activity [Fig 2(d) Set E], D_corr saturates with embedding dimension indicating low dimensional deterministic dynamics and these results agrees well with previous analyses 117. For sets A-D, as there is no saturation in D_corr at higher embedding dimension [Fig 1(a) and Figs 2(b)–(d)] and hence it is difficult to estimate actual D_corr. In Ref 1D_corr was computed based on quasiscaling regions, but such an estimation is very much dependent on the variations of the time-frequency-energy characteristics rather than any nonlinear dynamics. Hence this may be inadequate to characterize epilepsy or diseased brain states for clinical application 33.

As we have observed from the surrogate analysis that nonlinear dynamics is responsible for epileptic patients during seizure, we have compared the probability distribution function (PDF) of a normal case and an epileptic person during seizure. The PDF for sets A and E have been shown in Figs 3(a) and 3(b) respectively. Fig 3(a) shows that for a normal healthy person with eyes open, the PDF is Gaussian in nature, whereas for epileptic patients during seizure, it is nongaussian [Fig 3(b)] signifying an intermittent nonlinear effect. But for other three cases this feature is not so clear. So we feel that the PDF may also be useful to differentiate a brain activity of an epileptic patient during seizure from other state.

Since one of the major emphasis of epilepsy investigation is to predict their occurrence, it is necessary to know how the data is correlated. We have carried out a study of the Hurst exponent (H) which has been estimated using Rescaled range analysis (R/S). This method was proposed by Hurst and well established by Mandelbrot, and Wallis 31. For a given set of data series, R/S is defined as 3134:

Here , where , S²(n), and n are respectively the mean, variance, and time lag of the signal. The expected value of the R/S scales like cn^H as n → ∞, where H is called the Hurst exponent, and can be estimated from the slope of typical plot vs lag (n). For a given signal, we divided the data into nonoverlapping blocks of equal length and R/S has been calculated using the Equation 1 and the average value of R/S has been plotted as a function of lag in a log - log plot as shown in Fig 4 and estimated H from the slope of the curve. For random data H = 0.5, while H > 0.5 for the data with long range correlations, and H < 0.5 indicates the presence of long-range anticorrelation or antipersistency in the data.

The estimated average Hurst exponent (<H >) with an error bar of 100 epochs for all the five EEG data sets (viz. A-E) have been shown in Fig 5. The solid box and black dot show the <H > for sets A and B respectively. For a normal person with eyes open (set A), the average H (<H >) ≈ 0.47 whereas, for the data set B, i.e., for a normal person with his eyes closed, <H > ≈ 0.41. The <H > ≈ 0.5 for a normal man with eyes open indicates that the signals are uncorrelated over long time scales signifying stochasticity of the normal brain. But with eyes closed state, decreases in H (<H > ≈ 0.41) may be due to the imposition of some extra constraint, which may influence the system towards an antipersistent state. The <H > for epileptic patients are shown by up triangle for set C; star for set D; and down triangle for set E respectively. <H > ≈ 0.34 and 0.29 for the EEGs recorded at the hippocampal formation and epileptic zone for the seizure free intervals of epileptic patients respectively and during seizure (data set E), we get the lowest H (<H > ≈ 0.19). The H for epileptic patient during seizure and seizure free intervals show anticorrelation which may be due to epileptiform discharges during seizure free intervals indicating that a large discharge is always followed by a small one. The physiology behind the epileptiform discharge is due to the chronic dysfunction or "defect" in the epileptic brain, i.e., the epileptic brain is not normal even during seizure free time 35. Though the hippocampus was nonepileptogenic for these subjects, its H is still less than a normal person which may be due to its participation in secondary, nonautonomous epileptic processes initiated by the epileptic zone 1. The wide dispersion in H for the EEG recorded from the epileptic zone in seizure free intervals (star) indicates that the epileptiform discharges are intermittent probably due to the chronic presence of abnormal epileptogenic tissues 3637. EEG recording during seizure may not be economical and hence it may be better to locate epileptic zone by recording EEG during seizure free intervals. These analyses show the possibility of detecting the onset of the seizure state from the time dependent Hurst exponent estimated during the transition from normal to seizure state 35. Fig 6 shows the Hurst exponents for all the epochs of five different sets that have been discussed above.