We begin with discussing the impact of various rhythms on the temporal changes of statistical moments estimated for the dynamics of 19 brain regions recorded non-invasively with scalp EEG (nEEG) (Niedermeyer and Lopes da Silva, 2005) (Figure 2A). A large fraction of the temporal variability of the nEEG signals’ standard deviation, skewness, and kurtosis can be attributed to the circadian rhythm with a period length at about 24 h. Interestingly, both the rhythm’s intensity and its period length vary for the different recording sites, which would indicate that the dynamics of the different brain regions are differently affected by this rhythm. Similar observations can be made for ultradian rhythms with period lengths at about 12, 8, 6, 4 h, 90, and 60 min but with comparably smaller intensities. Topographically, these rhythms are most pronounced in fronto-central areas, and these areas are known to reflect, for instance, the dynamics of the wake-dependent, or homeostatic component of sleep regulation (Cajochen et al., 2002; Croce et al., 2018). Another important aspect of the rhythms’ impact can be identified from the circadian distribution of nEEG signals’ statistical moments. Within the circadian cycle, the standard deviation attains highest values during the nighttime, as expected (Hjorth, 1970, 1973). Notably, the nEEG signals’ third and fourth statistical moment indicate a clear deviation from Gaussianity for data recorded during the daytime.

We observe a similarly pronounced impact of the circadian rhythm on statistical moments of intracranially recorded brain dynamics (Figure 2B). Remarkably, for the iEEG signals’ standard deviation σ, the 24 h peak in the corresponding periodogram is rather narrow, and one might speculate that this can be related to the narrow spatial sampling of circumscribed brain regions. Apart from rhythms with period lengths at about 12, 8 and 4 h, other ultradian rhythms contribute only to a small extent to the temporal variability of the iEEG signals’ statistical moments. Within the circadian cycle, the standard deviation again attains highest values during the nighttime, while skewness and kurtosis here indicate a clear deviation from Gaussianity mostly independent of the time of day.

It is important to note that the observed deviations from Gaussianity—seen for both nEEG and iEEG signals and that differentially depend on the time of day—may strongly impact on both the design of EEG-based investigations of various physiological and pathophysiological phenomena and the suitability and reliability of various EEG analysis techniques that assume the data to be Gaussian distributed. In the same manner, the observed 1/f-like temporal fluctuations of the EEG signals’ statistical moments (as indicated by the scaling exponents in the range γ ∈ 0.6 , 1.1 ) not only contributes critically to the ongoing debate on the functional significance of scale-free brain dynamics (Bédard et al., 2006; Chialvo, 2010; Beggs and Timme, 2012; Papo, 2013; He, 2014; Papo, 2014; Lehnertz et al., 2017; Uddin, 2020) but also can be regarded a cautionary tale for EEG-based studies of e.g. cycles in epilepsy (Karoly et al., 2018; Baud et al., 2021; Leguia et al., 2021) or of seizure-precursor identification using e.g. the concept of critical slowing down (see Wilkat et al. (2019) and Hagemann et al. (2021) and references therein for a critique). Not taking into account dependencies of statistical characteristics of EEG signals on biological rhythms can lead to erroneous statements about brain dynamics.

We proceed with discussing the impact of the various biological rhythms on the temporal changes of interactions between the dynamics of the sampled brain regions (Figure 3). For both, the phase-based and the amplitude-based estimator for the strength of interaction and independent of the type of recording, we observe—on average—a pronounced impact of the circadian rhythm. Note, however, that interactions are not equally affected by this rhythm; its impact may vary for short-ranged (nearest neighbor brain regions), intermediate-ranged (regions within same hemisphere), and long-ranged (across hemispheres) interactions (cf. Kreuz et al. (2004) and Lehnertz et al. (2017)). Moreover, for some of these interactions we observe additional contributions with period lengths around 17–19 h and around 27–30 h, which lead to either a triplett-like structure together with the circadian peak or to a broadening of that peak in the averaged periodogram.

Different ultradian rhythms appear to impact on temporal changes of the strength of interactions estimated from nEEG signals (Figure 3A) and from iEEG signals (Figure 3B) but with comparably smaller intensities. For the former, we observe contributions at period lengths around 3.5 and 12 h, while for the latter period lengths around 5 and 8 h or around 4 and 7 h can be identified depending on the employed estimator for the strength of interactions (R or ρ) together with an additional contribution at 12 h (cf. Porz et al. (2014)). A differential impact of these biological rhythms on the strength of interactions, estimated with either R or with ρ, can also be observed within the circadian cycle. Independent of the type of recording, the strengths of interactions are comparably higher during the daytime, which would point to a decreased level of synchronization between brain regions during the nighttime (Steriade and Hobson, 1976; Horovitz et al., 2009; Lazar et al., 2015; Mizrahi-Kliger et al., 2018; Nguyen et al., 2018). Interestingly, the daytime amplitude-based contributions (estimated with ρ) appear to be delayed by 3–6 h to the phase-based contributions (estimated with R). We can not yet provide an explanation for such a time delay, but it would need to be taken into account in comparative studies of e.g. functional connectivity or (patho-)physiologic synchronization phenomena.

In line with a number of previous studies (see, e.g., Gong et al. (2003); Stam and De Bruin (2004); Botcharova et al. (2014); Racz et al. (2018); Daffertshofer et al. (2018); Stylianou et al. (2020); La Rocca et al. (2021)), we observe 1/f-like temporal fluctuations of characteristics of brain interactions (as indicated by the scaling exponents γ ∈ 0.77 , 0.94 ). We conjecture that such scale-free-like brain interaction properties can be traced back to specific aspects of the dynamics of brain regions involved in these interactions. We also conjecture that the various biological rhythms impact in a similar manner on the other properties of interactions, namely direction and coupling functions. Not taking into account dependencies of interaction properties on biological rhythms can lead to erroneous statements about brain interactions. At the same token, such dependencies may rank among the main determinants of the repeatedly reported low reproducibility of functional brain imaging studies (e.g. (Calhoun et al., 2014; Hodkinson et al., 2014; Zalesky et al., 2014; Nakamura et al., 2015; Open Science Collaboration, 2015; Preti et al., 2017; Thomas et al., 2018; Facer-Childs et al., 2019; Lurie et al., 2020; Orban et al., 2020; Specht, 2020)).

Eventually, we discuss the impact of the various biological rhythms on temporal changes of exemplary global (clustering coefficient (C ^R and C ^ρ )) and local characteristics (centralities of vertices ( C v R and C v ρ ) and edges ( C e R and C e ρ )) of evolving brain networks (Figure 4). In general, we observe for both EEG recordings a vastly different impact of biological rhythms on network characteristics. Importantly, the impact strongly depends on which estimator (R or ρ) was employed to derive network edges.

The temporal changes of C ^ρ of nEEG-based evolving networks (Figure 4A) are strongly dominated by the circadian rhythm with only minor contributions from ultradian rhythms with period lengths ranging from 1 h to about 14 h. In contrast, for the same networks but with edges derived via R, the circadian rhythm’s impact on C ^R compares to the ones seen for the aforementioned ultradian rhythms. Within the circadian cycle, fluctuations of both C ^R and C ^ρ amount to a few percent.

For the clustering coefficient of iEEG-based evolving networks (Figure 4B), the dependence of the rhythms’ impact on the estimator used to derive edges is less pronounced. The circadian rhythm appears to be decomposed into quartett-like structures with contributions at period lengths 22, 24, 19, and 27 h (descending order of intensities) for C ^R and at 25, 28, 21, and 19 h for C ^ρ . For C ^R , we observe contributions of ultradian rhythms with period lengths at about 15, 12, 8, and 3 h but with comparably small intensities. For C ^ρ , there are contributions with higher intensities and with period lengths at about 17.5, 15, and 16 h (triplett-like structure) as well as at 13, 8.5, 6.5, and 3.8 h. As with the nEEG-based evolving networks, fluctuations within the circadian cycle of clustering coefficients amount to a few percent. Comparable findings were reported for the clustering coefficient as well as for other global characteristics of binary evolving brain networks derived from continuous, long-term, multichannel iEEG (Kuhnert et al., 2010; Geier et al., 2015; Lehnertz et al., 2017) and nEEG data (Mitsis et al., 2020) recorded from a larger group of epilepsy patients. Note that the temporal fluctuations of the clustering coefficients of the nEEG-based and iEEG-based evolving brain networks comply with a power-law with a scaling exponent γ in the range γ ∈ 0.71 , 0.96 ). We are not aware of previous reports on such an observation.

For the nEEG-based evolving networks (Figure 4A), we concentrate on centrality ( C v R and C v ρ ) of vertices associated with a central brain area (cf. Section 2.1) and with occipital brain areas (Kuhnert et al., 2012) as well as on centrality ( C e R and C e ρ ) of edges connecting these brain areas (central—occipital and left occipital—right occipital). Using a centrality-based ranking (Liao et al., 2017) and independent of the estimator employed to derive network edges, we find the occipital vertices to rank higher, on average, than the central vertex. The same holds true for the edge connecting the occipital vertices when compared to the edge connecting the occiptal and the central vertex. Note, that these high-ranking vertices and edges are not the most important (highest rank) network constituents. We observe a strongly pronounced impact of the circadian rhythm on temporal changes of centrality ( C v R and C v ρ ) of the central vertex (which was to be expected, at least to some extent, from our findings in Section 2.1). Apart from a comparably small contribution with a period length at around 12 h, the impact of other ultradian rhythms appears negligible. For temporal changes of centrality of the occipital vertex, however, the rhythms’ impact is strongly reduced ( C v ρ ) or barely detectable ( C v R ) (note the reduced range of the scaling exponent γ ∈ 0.59 , 0.63 ). The differential impact of the biological rhythms on vertex centralities is also reflected within the circadian cycle. We only observe during the daytime substantial alterations (∼ 10%) of the otherwise almost constant centrality of the central vertex. Constancy is also seen for the centrality of the occipital vertex.

Now, for temporal changes of edge centrality ( C e R and C e ρ ), it is surprisingly the edge connecting the occipital vertices for which we observe a strong impact of the circadian rhythm. For the other edge (that connects central and occipital vertices), the dependence of the rhythms’ impact on the estimator used to derive network edges is strongly pronounced. For C e ρ , the rhythms’ impact is barely detectable, and for C e R , we observe contributions with period lengths at 18, 22, and 25 h as triplett-like structures. In addition to comparably strong contributions with a period length at around 12 h, we observe contributions from other ultradian rhythms down to period lengths around 60 min, particularly for temporal changes of C e R for both investigated edges. Within the circadian cycle, the fluctuations of edge centralities associated with these rhythms amount to a few percent.

For the iEEG-based evolving networks (Figure 4B), we concentrate on centrality ( C v R and C v ρ ) of vertices associated with the right mesial temporal lobe (RMTL) and with the left frontal (LF) area (see Section 2.1) as well as on centrality ( C e R and C e ρ ) of edges connecting these brain areas (RMTL—LF and LF—LF). As with the nEEG-based evolving networks, we use a centrality-based ranking to estimate the importance of the chosen vertices and edges. Independent of the estimator employed to derive network edges, we find the RMTL vertex to rank higher, on average, than the LF vertices, and the same holds true for the edge connecting the LF vertices. We also note here, that these high-ranking vertices and edges are not the most important (highest rank) network constituents. Independent of the estimator employed to derive network edges, we observe a strongly pronounced impact of the circadian rhythm on temporal changes of centrality ( C v R and C v ρ ) of the RMTL vertex. For C v R , we find a triplett-like structure with a period length of a main peak at around 24 h as well as less pronounced contributions at 21 and 29 h. For C v ρ , we find a quartett-like structure peaking at around 24 h and with less pronounced contributions at 21, 26, and 29 h. For both centrality estimates, we observe additional contributions with a period length centered around 12 h (again as triplett-/quartett-like structure), and the impact of other ultradian rhythms appears negligible. The impact of the circadian rhythm on temporal changes of centrality of the LF vertex is stronger pronounced for C v R than for C v ρ . We observe again triplett-/quartett-like structures with period lengths that compare to the ones seen for the RMTL vertex. The impact of other ultradian rhythms, including contributions at around 12 h, appears negligible. Within the circadian cycle, particularly C v R from both vertices exhibits stronger fluctuations (reaching up to 20%) from noon until evening, while the fluctuations of C v ρ amount to only a few percent. Geier and Lehnertz (2017) reported comparable observations for other vertex centralities of weighted evolving brain networks derived from continuous, long-term, multichannel iEEG data recorded from a larger group of epilepsy patients.

For temporal changes of edge centrality ( C e R and C e ρ ), we observe a strong impact of the circadian rhythm for the edge connecting the right mesial temporal lobe and with the left frontal area. For the edge connecting vertices within the frontal area, the rhythm’s impact is strongly pronounced for C e ρ and barely detectable for C e R . If detectable, we find triplett-like structures with a period length of the main peak at around 24 h as well as less pronounced contributions around 21 h and around 28 h. For the RMTL—LF edge, there is an additional contribution with a period length at around 12 h (mostly triplett-like structure), and the impact of other ultradian rhythms appears negligible. A 12 h contribution is barely detectable for C e R of the LF—LF edge, and for C e ρ of that edge, we find comparable though less pronounced contributions at period lengths around 8 and 12 h. The impact of other ultradian rhythms appears negligible also for this edge. Within the circadian cycle, fluctuations of C e R of the RMTL—LF edge resemble the ones seen for C v R of the RMTL vertex, albeit with higher amplitudes (reaching up to 40%). Fluctuations of C e R of the LF—LF edge amount to only a few percent. Interestingly, for C e ρ we observe more pronounced fluctuations for the LF—LF edge (reaching up to 20%), while the ones of the RMTL—LF edge amount to only a few percent. As with the clustering coefficient, we note that the temporal fluctuations of the centrality of vertices and edges of the nEEG-based and iEEG-based evolving brain networks comply with a power-law with a scaling exponent γ in the range γ ∈ 0.59 , 1.21 ). We are not aware of previous reports on such an observation.

Not taking into account dependencies of network characteristics on biological rhythms can lead to erroneous statements about evolving functional brain networks.