Dynamics of intracranial electroencephalographic recordings from epilepsy patients using univariate and bivariate recurrence networks

Recently Andrezejak et al. combined the randomness and nonlinear independence test with iterative amplitude adjusted Fourier transform (iAAFT) surrogates to distinguish between the dynamics of seizure-free intracranial electroencephalographic (EEG) signals recorded from epileptogenic (focal) and nonepileptogenic (nonfocal) brain areas of epileptic patients. However, stationarity is a part of the null hypothesis for iAAFT surrogates and thus nonstationarity can violate the null hypothesis. In this work we first propose the application of the randomness and nonlinear independence test based on recurrence network measures to distinguish between the dynamics of focal and nonfocal EEG signals. Furthermore, we combine these tests with both iAAFT and truncated Fourier transform (TFT) surrogate methods, which also preserves the nonstationarity of the original data in the surrogates along with its linear structure. Our results indicate that focal EEG signals exhibit an increased degree of structural complexity and interdependency compared to nonfocal EEG signals. In general, we find higher rejections for randomness and nonlinear independence tests for focal EEG signals compared to nonfocal EEG signals. In particular, the univariate recurrence network measures, the average clustering coefficient $\mathcal{C}$ and assortativity $\mathcal{R}$, and the bivariate recurrence network measure, the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$, can successfully distinguish between the focal and nonfocal EEG signals, even when the analysis is restricted to nonstationary signals, irrespective of the type of surrogates used. On the other hand, we find that the univariate recurrence network measures, the average path length $\mathcal{L}$, and the average betweenness centrality $\text{BC}$ fail to distinguish between the focal and nonfocal EEG signals when iAAFT surrogates are used. However, these two measures can distinguish between focal and nonfocal EEG signals when TFT surrogates are used for nonstationary signals. We also report an improvement in the performance of nonlinear prediction error $N$ and nonlinear interdependence measure $L$ used by Andrezejak et al., when TFT surrogates are used for nonstationary EEG signals. We also find that the outcome of the nonlinear independence test based on the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$ is independent of the outcome of the randomness test based on the average clustering coefficient $\mathcal{C}$. Thus, the univariate and bivariate recurrence network measures provide independent information regarding the dynamics of the focal and nonfocal EEG signals. In conclusion, recurrence network analysis combined with nonstationary surrogates can be applied to derive reliable biomarkers to distinguish between epileptogenic and nonepileptogenic brain areas using EEG signals.

Figure 1

Impact of varying $f_c$ on the global behavior of a surrogate signal. (a) 10 sec of an exemplary surrogate signal (dashed red line) and original EEG signal (solid blue line) at $f_c=0$ which is equivalent to iAAFT surrogate. Figures 1–1(h) as in Fig. 1 but for $f_c=0.01–0.07$ respectively. Note that for display purposes, in each case the surrogate signal (dashed red line) has been shifted below the original EEG signal (solid blue line).

Figure 2

Impact of varying $f_c$ on the local structure of a surrogate signal. (a) Exemplary surrogate signal (dashed red line) superimposed on an EEG signal (solid blue line) at $f_c=0$ which is equivalent to iAAFT surrogate. Figures 2–2(h) as in Fig. 2 but for $f_c=0.01–0.07$ respectively.

Figure 3

Contrast between the rejection probabilities of focal $p^F$(red) and nonfocal $p^N$(green) EEG signals as given by the randomness test based on the average clustering coefficient $\mathcal{C}$ for all the signals (a), stationary signals (b), and nonstationary signals (c). Rows 2–5, as in (a)–(c), but for assortativity $\mathcal{R}$, the average path length $\mathcal{L}$, the average betweenness centrality $\text{BC}$, and nonlinear prediction error $N$ respectively. The bars indicate $95\%$ confidence intervals, the dashed horizontal line indicates significance levels of the tests (0.05), $D$ values indicate the relative difference used to compare probabilities with $95\%$ confidence intervals given in parentheses [5].

Figure 4

(a) Contrast between the rejection probabilities of focal $p^F$(red) and nonfocal $p^N$(green) EEG signals as given by the randomness test combined with TFT surrogates based on the average clustering coefficient $\mathcal{C}$ for nonstationary signals, from left to right, at $f_c=0$, 0.01, 0.02, 0.03, 0.04, 0.05. (b)–(e) As in (a), but for assortativity $\mathcal{R}$, the average path length $\mathcal{L}$, the average betweenness centrality $\text{BC}$, and nonlinear prediction error $N$ respectively. The bars indicate $95\%$ confidence intervals, the dashed horizontal line indicates significance levels of the tests (0.05), $D$ values indicate the relative difference used to compare probabilities with $95\%$ confidence intervals given in parentheses [5].

Figure 5

(a) Contrast between the rejection probabilities of focal $p^F$(red) and nonfocal $p^N$(green) EEG signals as given by the nonlinear independence test based on the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$ for all the signals (a), stationary signals (b), and nonstationary signals (c). (d)–(f)As in (a)–(c) but for the nonlinear independence test based on nonlinear interdependence measure $L$ described in [5]. Other elements in the figure are as in Fig. 3.

Figure 6

Contrast between the rejection probabilities of focal $p^F$(red) and nonfocal $p^N$(green) EEG signals as given by the nonlinear independence test combined with TFT surrogates based on the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$ (left) and the nonlinear interdependence measure $L$ (right). The bars indicate $95\%$ confidence intervals, the dashed horizontal line indicates significance levels of the tests (0.05), $D$ values indicate the relative difference used to compare probabilities with $95\%$ confidence intervals given in parentheses [5]. Other elements in the figure are as in Fig. 3.

Figure 7

(a) Impact of the outcome of the randomness tests on the nonlinear independence test for the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$ when one of the randomness tests is rejected (left), neither of the randomness tests is rejected (middle), and both of the randomness tests are rejected (right). (b) As in (a) but for the nonlinear interdependence measure $L$ and nonlinear prediction error $N$. Other elements in the figure are as in Fig. 3.

Figure 8

Outcome of the randomness test based on the average clustering coefficient $\mathcal{C}$ does not impact the outcome of nonlinear independence test based on the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$. The rejection probabilities shown are for shuffled signal pairs that are independent of each other. Other elements in the figure are as in Fig. 3.

Figure 9

Impact of the outcome of the nonlinear independence test based on the average cross-clustering coefficient ${\mathcal{C}}_{\mathrm{cross}}$ on the outcome of randomness test based on the average clustering coefficient $\mathcal{C}$. Probability that neither randomness test is rejected (a), one of the randomness tests is rejected (b), and both randomness tests are rejected (c). Other elements in the figure are as in Fig. 3.