Entropy Measures in Neural Signals

Abstract

Entropy measures have been widely used in analyzing neural signals from micro- to macroscales for the normal or abnormal brain assessment. As it is unrealistic to systematic analysis in all the information entropy-based indices in different application areas within limited chapters, we mainly focus on the comparison of the capability of 12 entropy indices derived from electroencephalogram (EEG) for monitoring depth of anesthesia (DoA) and detecting the burst suppression pattern (BSP), in anesthesia induced by GABAergic agents.

Twelve indices were included: response entropy (RE) and state entropy (SE); three wavelet entropy (WE) measures (Shannon WE (SWE), Tsallis WE (TWE), and Renyi WE (RWE)); Hilbert–Huang spectral entropy (HHSE); approximate entropy (ApEn); sample entropy (SampEn); Fuzzy entropy (FuzzyEn); and three permutation entropy (PE) measures (Shannon PE (SPE), Tsallis PE (TPE), and Renyi PE (RPE)). Two EEG data sets recorded from sevoflurane-induced and isoflurane-induced anesthesia, respectively, were selected to assess the capability of each entropy index in DoA monitoring and BSP detection. To validate the effectiveness of these entropy algorithms, pharmacokinetic/pharmacodynamic (PK/PD) modeling and prediction probability (P _k ) analysis were applied.

All the entropy indices could track the changes of anesthesia states. Three PE measures outperformed the other entropy indices, with less baseline variability, higher coefficient of determination (R ²) and prediction probability, and RPE performed best; while ApEn and SampEn discriminated BSP best.

Each entropy index has its advantages and disadvantages in estimating DoA. Overall, it is suggested that the RPE index has a superior performance. Investigating the advantages and disadvantages of these entropy indices could help improve current clinical indices for monitoring DoA.

Appendices

Appendices

Each entropy index contains several parameters, which can severely impact the output of its index. Therefore, it is very important to select the appropriate parameters. In anesthesia researches, there are many methods to select parameters including the interindividual variations, eg, the relationship with drug effect-site concentration obtained from PK/PD and prediction probability (Li etal. 2012, [88]). In this study, the method of selecting the parameters is based on indices’ performance in distinguishing different anesthesia states and interindividual variations. Fifty data sets in awake, deep anesthesia, and recovery states from 19 patients are selected, respectively. The RE and SE algorithms applied in the study adopts the Datex–Ohmeda S/5™ entropy module (Viertiö‐Oja etal. 2004). The PE’s parameter selection is based on our previous work (Li etal. 2012, 2010). The details of other entropy algorithm’s parameters are discussed as follows. All the results are given by mean ± standard deviation. The blue, red, and green colors represent the awake state, deep anesthesia state, and recovery state, respectively.

8.1.1 Appendix A: WE

There are three types of WE measures (SWE, TWE, RWE) considered in this study. The parameters include basis functions, the data length N, the Tsallis entropy parameter q, and the Renyi entropy parameter a. The basis functions and data length selection are based on the SWE. First, several common basis functions including Haar, Daubechies, Coiflets, Symlets, and Biorthogonal wavelet families were selected. The result is shown in Fig. 8.14a–e. N is assumed to be 1000. As can be seen, only the SWE based on the biorthgonal basis function can completely separate anesthesia state (red color) from awake (blue color), and recovery states (green color), without overlap. However, there are some basic functions in the biorthgonal family. Figure 8.14e–g show the results obtained by bior2.2, bior3.3, and bior4.4. It can be seen that the SWE achieved by bior3.3 not only distinguishes anesthesia states from non-anesthesia but also differentiates between awake and recovery states, especially when the number of layers is 2 and 3. So in this study, the bior3.3 was chosen as the wavelet basis function, and the number of layers was 3. Then, based on them, the selection of N is given in Fig. 8.14h. N ranges from 500 to 3000 points with the step of 500 points under the sample of 100 Hz. The figure shows that when N ≥ 1000, there is no significant difference in WE for each state. So N = 1000 (10 s) was selected to calculate the WE. Furthermore, based on the parameters of basis function and N, the selections of q in TWE and a in RWE are given in Fig. 8.14i–l.

Fig. 8.14

The changes of WE in different anesthesia with different parameters. (a–g) The change of SWE with different wavelet basis functions: (a) haar, (b) db2, (c) coif2, (d) sym2, (e) bior2.2, (f) bior3.3, and (g) bior4.4. The horizontal axis shows the number of layers (n) decomposed by corresponding wavelet function. The vertical axis shows SWE values. (h) The changes of SWE with different N. (i–j) The changes of TWE with $0<q<1$ (i) and $q>1$ (j). (k–l) The changes of RWE with $0<a<1$ (k) and $a>1$ (l)

8.1.2 Appendix B: HHSE

In order to choose an appropriate data length N in HHSE algorithm at the sample rate of 100 Hz, a series of N were used to calculate the HHSE in different anesthesia states. The result is shown in Fig. 8.15. All values of N could distinguish different anesthesia states. And when the data length was equal to or greater than 1000, the HHSE value would be nearly invariable with the changes of N. So N = 1000 (10 s) was selected in this study.

Fig. 8.15

The change of HHSE with different N at different anesthesia states. N is from 500 to 3000 stepped by 500 points

8.1.3 Appendix C: ApEn

Figure 8.16a–b, respectively, give the result of ApEn over different r in different anesthesia states with N = 1000, m = 2, and m = 3. With increasing r, the ApEn values in awake state and recovery state increased and then decreased, while it monotonously changes with r in deep anesthesia state. Both figures show that when r is 0.2 or 0.25 of SD, the difference between deep anesthesia and other states is larger. Considering that the r in ApEn is also used to suppress the noise, its value is chosen as small as possible. The r was chosen as 0.2 of SD. Figure 8.16c shows the selection of m with N = 1000 and r = 0.2*SD. It can be seen that ApEn nearly doesn’t change with m. Meanwhile when m is 2, the interindividual variations of ShEn is smaller. With r = 0.2*SD and m = 2, the selection of N is shown in Fig. 8.16d. The ApEn was very dependent on N, and it increased with enlarging N, as well as its interindividual variations, especially in awake and recovery states. The difference between awake and anesthesia state also became larger, but when N was greater than 1000, the difference was not obvious. Therefore, r = 0.2*SD, m = 2, and N = 1000 were selected in this study, which is consistent with the study (Bruhn etal. 2000) through different methods to choose parameter was used.

Fig. 8.16

The changes of ApEn with different parameters at different anesthesia states. (a) m = 2, N = 1000; (b) m = 3, N = 1000;(c) r = 0.2*SD, N = 1000; (c) r = 0.2*SD, m = 2

8.1.4 Appendix D: SampEn

Figure 8.17a–b, respectively, show the changes of SampEn over different r in different anesthesia states, with N = 1000, m = 2 and 3. The SampEn values monotonously decrease with increasing r in all states. The difference between awake state and deep anesthesia was obvious, but not between awake and recovery state. There is much overlap between them. Considering the interindividual variations, r = 0.2 is better. Figure 8.17c shows the changes of SampEn over different m with r = 0.2, N = 1000. There is no significant difference in different m values. For simplicity, m = 2 was selected. The selection of N is given in Fig. 8.17d. The SampEn values with r = 0.2, m = 2 were almost invariable as N increased. This implies that SampEn values are not dependent on N. Finally, N = 1000 (10 s at the sample rate of 100 Hz), m = 2, and r = 0.2*SD were selected in this study.

Fig. 8.17

The changes of SampEn with different parameters at different anesthesia states. (a) m=2, N=1000; (b) m=3, N=1000; (c) r=0.2*SD, N=1000; (c) r=0.2*SD, m=2

8.1.5 Appendix E: FuzzyEn

Figure 8.18 gives the changes of FuzzyEn with different parameters in different anesthesia states. Accordingly, we selected r = 0.2, m = 2, and N = 1000 for the computation of FuzzyEn in this study.

Fig. 8.18

The changes of FuzzyEn with different parameters at different anesthesia states. (a) m = 2, N = 1000; (b) m = 3, N = 1000;(c) r = 0.2*SD, N = 1000; (c) r = 0.2*SD, m = 2

8.1.6 Appendix F: TPE and RPE

The embedded dimension m, lag τ, and data length N had been discussed in our previous study (Li etal. 2008a, 2010, 2012). It is suggested that $m=6$ and $\tau =1$ are suitable for the sevoflurane DoA monitoring, and $m=3$ $\tau =2$ is better for isoflurane analysis. So, for the parameters of TPE and RPE, we only considered the embedded dimension of 3 and 6. For the sevoflurane, using $\tau =1$ and isoflurane is $\tau =2$. Figure 8.19a–b are the TPE of three anesthesia states at the $0<q<1$ and $q>1$, respectively, the $m=3$. Figure 8.19c–d show the TPE of $m=6$ at $0<q<1$ and $q>1$, respectively. It can be seen that in $m=6$, $q=0.9$ has a better performance in TPE. Figure 8.19e–h are the RPE measure of three anesthesia states similar as Fig. 8.19a–d. We select the $m=6$ and $a=2$ for RPE calculation.

Fig. 8.19

The changes of TPE and RPE with different parameters at different anesthesia states. (a–d) The TPE measures with different parameters: (a) $m=3$, $0<q<1$; (b) $m=3$, $q>1$; (c) $m=6$, $0<q<1$; and (d) $m=6$, $q>1$. (e–h) The RPE measures with different parameters: (e) $m=3$, $0<a<1$; (f) $m=3$, $a>1$; (g) $m=6$, $0<a<1$; and (h) $m=6$, $a>1$