In this research, nearly 100 channels (electrodes) of SEEG data from nine patients hospitalized with refractory focal epilepsy at the Sanbo Brain Hospital of Capital Medical University were visually analyzed and extensively exploited to validate our results independently. The Ethics Committee approved the SMBC of Capital Medical University’s study technique, and a negotiated informed special dispensation was prepared for all participants. This information cannot be shared with the general public due to hospital regulations. Furthermore, the sampling frequency was set at 512 Hz and the average monitoring time was 1 week. Interictal, preictal, seizure, and post-ictal periods were analyzed to determine the network features of people with epilepsy. For each patient, we devised criteria for categorizing the various seizure stages, as indicated in Table 1. These criteria were based on the clinical seizures recognized by physicians.

The data for the control group in this study were collected from the EEG Motion/Image Dataset public dataset (Schalk et al., 2004), which includes over 1,500 1- and 2-min EEG recordings from 109 volunteers with 64 electrodes. The volunteers were required to perform a variety of motion and visual tasks. We selected nine data points from healthy individuals from which we extracted segments of resting or task-state EEG to compare with patients.

Usually, Pearson correlation coefficients are employed to establish functional networks for analyzing resting-state brain network alterations in healthy individuals. However, compared to FMRI data, the SEEG data employed in this work includes more complex dynamic changes in transients due to its temporal and spatial features. Consequently, this research establishes an effective network based on the statistical method proposed in Quiroga et al. (2002), which may improve the assessment of the information flow in the brain network of patients with epilepsy.

We build a dynamic and efficient network by estimating the synchronization and directionality of developing information flow to represent pathological epilepsy. Therefore, it is essential to define events for time series. Here, the events are defined as the local maximums in the time series, i.e.,

where t_k is the event occurrence time, M = 10h = 30 are the two control parameters shaping the event.

Let the event sequences of time series x¹ (n) and x² (n) (total step length L) be tr1(r=1,…,m1) and ts2(s=1,…,m2), where m₁ (≪ L) and m₂ (≪ L) are the total number of event occurrences in x¹ (n) and x² (n), respectively. The time difference between event occurrences in various time series is then used to determine the causality and synchronization of events. In the time delay scale τ, if an event is recorded in the sequence x¹ (n) immediately after an event occurs in the sequence x² (n), then it is considered that the intensity of the causal effect of x¹ (n) on x² (n) will be enhanced by one step. Conversely, x² (n) is one step more causality for x¹ (n). In addition, regardless of the sequence of events in x¹ (n) and x² (n), as long as two related events are observed to occur close enough, they are considered a simultaneous occurrence, and the amount of synchronization is correspondingly increased by one step.

Remember sequence x² (n) for x¹ (n) event-causal e^τ(x¹|x²), the size of representative sequence x² (n) for x¹ (n) the causal role of strength. The global time delay,

is used to limit the smallest time interval between two adjacent events, where τ_rs is the local time delay for each pair of adjacent (temporally closest) events (r, s), Er⁢sτ=1/2⁢ is such set to prevent from double counting for the two simultaneous events. On the contrary, e^τ(x²|x¹) quantifies the causality from x¹ (n) to x² (n).

Based on e^τ(x¹|x²) and e^τ(x²|x¹), we then define the degree of simultaneity Q^τ and causality q^τ for x¹(n) and x²(n) as follows:

Both are normalized to 0 ≤ Q^τ ≤ 1, −1 ≤ q^τ ≤ 1. Q^τ = 1 when and only when all events in both signals occur together. q^τ = 1 when and only when all events from x¹(n) act for all events from x² (n).

Second, to examine the evolution trend of synchronization and causality between the two sequences over time, Q^τ and q^τ are evaluated for each time step, while their cumulative consequences are investigated over time. We define q^τ(n) as Equation(5),

where n = 1, 2,…,L is the time point within the series, and Θ is the step function, i.e., Θ(x) = 1 when x > 0 and Θ(x) = 0 when x ≤ 0. q^τ(n) could be thought of as a random sequence; when the event in x¹ causes the event in x² to occur, it increases by one step, and vice versa, it diminishes by one step. The synchronization of progressive events is defined by Q^τ(n) in Equation (5), where Q^τ(n) improves by one step if a pair of events in x¹ and x² occur within the period τ and remains nearly constant otherwise.

The rate of change of synchronization at time point n is calculated by averaging the synchronization expansion throughout Δn steps (Δn = 5120 in the current calculation) using dQ^τ(n) in Equation (6),

where Δn₁ and Δn₂ correspond to the number of events in x¹ and x² in the band [n−Δn, n], respectively. Similarly, we might define the rate of change of the causal level at time point n as choosing to follow dq^τ (n) in Equation (6), where dQ^τ (n) > 0 and dq^τ (n) > 0 represent a positive increment of synchronization and causality within Δn steps, respectively, and <0 represents a negative escalation within Δn steps.

Specifically, the synchronization growth rate dQ^τ (n) and the causal level change rate dq^τ (n) at time point n are determined for any two nodes i and j in the network whose relevant time series are xⁱ and x^j. The magnitude of the weighted directed action of node i on node j at time point n is represented by a_ij(n), which also contains data on the synchronization level between the two nodes, as described by Equation (7),

where the amplification factor γ = 1000 is considered in this research. Furthermore, it is assumed that there are no self-connecting rings in the network, i.e., a_ii (n) = 0, i = 1, 2,…,n₀, where n₀ is the maximum number of nodes in the network.

Wang Rong et al. (Wang et al., 2021) proposed that the activation and combination of various structural modal interactions in the eigenmodes lead to generating multiple dynamical modes in the system by nature. Moreover, in this study, to discover the dynamical oscillation reorganization law of the directed weighted network, we propose a new hierarchical eigenmode analysis method that includes examining complex numbers. In particular, the directed weighted network and functional connection matrix in Figure 1 is an epitome of the causal effects network calculation results. Since the functional connection matrix is asymmetric, the resulting eigenvalues and eigenvectors must include complex conjugate numbers. It is well known that the eigenvalues on the exact number field domain represent the magnitude of the stretching transformation, whereas the eigenvectors represent the direction. In comparison, the complex domain adds a rotational transformation. Under the transformation of polar coordinates, a+bi becomes r(cosθ + i_*sinθ), where r=a2+b2 represents the stretching quantity, and θ=a⁢r⁢c⁢t⁢a⁢n⁢ba represents the rotation. Additionally, the complex eigenvector component m + ni is multiplied with the corresponding eigenvalue a + bi, representing the real rotation of the basis vector φ and the stretching transform q (Hitzer, 2002):

On this foundation, we consider the modal lengths of the eigenvalues and eigenvectors that correspond precisely to the magnitude of the change in a stretch, which makes more physical sense than merely considering the real part.

Immediately following, we provide the specific modal analytical method. First, the eigenvalues are sorted by mode length from largest to smallest Λi2(i=1,…,N) and the corresponding eigenvectors are rearranged. In order to depict the actual global level of slow oscillations in the brain, the first eigenvector corresponding to the largest eigenvalue modal length was not divided at the first level. The global slow oscillations can be expressed as:

where the parameters and variations can be explained later.

Level 2 eigenvector separated brain regions into three sub-modules: balance, strong oscillation, and weak oscillation. After the entire module was divided into three parts, the latter oscillatory deformation was much higher than the former. (AVG + SD, AVG + 2_*SD) represents a balanced oscillatory deformation while cognitive patterns are switched more flexibly, and cognition is at a greater level (complex planes ② and ⑤ in Figure 1). (AVG + 2_*SD, + ∞) represent a strong oscillatory deformation with significant feature differences and shifts. Moreover, the greater the amplitude of the oscillation, the more it may negatively impair cognition (complex planes ③ and ⑥ in Figure 1). (0,AVG + SD) denotes a weakening of the oscillation level and a slowing of cognition, tending toward a mode that occurs when the oscillation gets suppressed (complex planes ① and ④ in Figure 1). Where AVG and SDrepresent the set’s mean and standard deviation, respectively, this set is made up of the modal lengths of all components of the whole eigenvector. Nonetheless, it is challenging to quantify the degree of rotation in measures of complex numbers. In this part, we do not distinguish between the effect of the rotation angle on the magnitude of oscillations at each level; further explanation will be given in the Section “4. Discussion.”

As the sequence of functional patterns rises, the FC network modularizes until it reaches a state of high modularity. In this process, the case from layer 2 to layer N represents the deformation intensity of the strong local oscillation, expressed as

where M_i is the number of modules within layer i, which is weighted as Hi=Λi2⁢MiN(i=1,…,N). N is the number of regions in total. Nevertheless, the heterogeneous structure of module sizes causes variations in the evaluation of separation and integration parameters; hence, similar to Wang et al. (2021), we also introduce the correction factor pi=∑j|mj-NMi|/N, where m_j(j = 1,…,M_i) is the module size.

In addition, we set H_sb = H_in−H_se: if H_sb < 0, the brain’s functional network is biased toward a state of strong local oscillatory deformation, which has a facilitative effect; if H_sb > 0, which implies that the network is biased toward a state under which slow global oscillations cooperate, the structure of the entire network is stable; if H_sb = 0, a sub-stable balance is reached between strong local oscillations and slow global oscillations, and the contribution of the original network to strong local oscillations is roughly equal to the suppression of strong local oscillations by slow global oscillations. Every H_sb value is normalized to the range [−1, 1]. The closer H_sb is to –1, the greater the strong local oscillations and the more excited the proto-network state. The closer H_sb is to 1, the higher the degree of slow global oscillation and the greater the suppression of strong local oscillations.

In conclusion, causality and synchronization were used to assess the strength of directional propagation of information flow between two epilepsy network nodes to construct a weighted epilepsy effective network. Meanwhile, we fed SEEG into this network and analyzed the output functional connection matrix. Using a modular hierarchical analysis of the eigenmodes, a dynamic description of the brain network’s strong local oscillations versus slow global oscillations in the temporal and spatial dimensions of the patient’s seizures were constructed. Ultimately, the dynamic reconfiguration of network oscillations may be the source of cognitive impairment produced by epilepsy. Figure 2 shows the main flowchart of the whole research.

We discovered that epileptic seizures damage the metastable balance between local strong and slow global oscillations in large-scale networks. Specifically, we investigated the H_sb metrics of healthy individuals, which reflect the metastable balance between strong local oscillations and slow global oscillations. Figure 3 shows the fluctuation of the H_sb indication for nine healthy controls over time. We can observe that the H_sb fluctuates evenly around 0 and is primarily stable around 0. This indicates that healthy brain networks have balanced oscillatory states and switch between strong local oscillations and slow global oscillations more frequently, hence maintaining regular cognitive functional activity. The brain network’s facilitation of strong local oscillations plays off against the suppression of strong local oscillations by slow global oscillations, yet the network is in balance overall. For comparison, we analyzed the variation of brain network states in epileptic patients.

Patient 1 had the exact opposite features compared to healthy controls, as shown in Figure 4. The strong local oscillation index H_se and the slow global oscillation index H_in exhibited rapid, irregular fluctuations within the range [−1, 1]. Similarly, the balance index H_sb exhibited the same alterations (Figure 4), which seemed very distinctive from healthy individuals. The primary data implies that the disruptive impact of seizures on the metastable balance of strong local oscillations and slow global oscillations is rather severe. In addition, both strong local oscillations and slow global oscillations in the functional network were enhanced after a period of seizure. In particular, the degree of enhancement is much greater for strong local oscillations than for slow global oscillations, and it recovers to pre-seizure levels after a seizure. As a result, changing strong local oscillations and slow global oscillations in the functional network may be a pathogenic mechanism that leads to cognitive impairment. In contrast, the interictal phase is associated with a decrease in the flexibility of both strong local oscillations and slow global oscillations. We assume that not only is there cognitive impairment during seizures but also a general divergence between the configuration of the patient’s network’s functional structure and the course of cognitive activity in healthy individuals.

H_sb changes in the form of the distribution of peak nuclear density may be a sign of cognitive impairment caused by seizures. To further validate our conclusion, we investigated the dynamics of strong local oscillations against slow global oscillations throughout various seizure phases for each patient. Figure 5 displays the seizure periods in the H_sb kernel density distribution results for each of the three patients. This study demonstrates a high degree of similarity across state transitions, showing that the dynamic hierarchy of oscillatory reorganization of brain networks varies considerably. The states are divided into four groups corresponding to Figure 5’s four colors.

During the interictal phase (Figure 5, first column), the peaks were single-peaked or equally distributed, showing normal cognitive function and the ability to transition between local strong and slow global oscillations. During the pre-ictal phase, the distribution of strong local oscillations and slow global oscillations showed a bimodal pattern with independent peaks, suggesting reduced switching flexibility and good cognitive condition by epileptic waves (Figure 5, second column). Immediately after and most notably during the seizure period, the network was significantly biased toward strong local oscillations, exhibiting a single-peaked concentration pattern and accumulating negative values, suggesting almost complete cognitive loss of the brain during seizures (Figure 5, third column). Cognitive activity returned to normal in the late seizure phase, when strong local oscillations reverted to a bimodal pattern or uniform distribution with slow global oscillations, similar to the interictal phase (Figure 5, fourth column).

In contrast, the oscillatory patterns of the functional network are often diffused and distributed in states of strong local oscillations and weak global oscillations. Subsequently, seizures cause them to be concentrated in hyper-intense strong local oscillations states, gradually dispersed following the seizure. The increased concentration of network oscillatory patterns during seizures may be substantially responsible for the start of cognitive impairment associated with epilepsy. However, a fascinating phenomenon was discovered: this decrease in regularity was not absolute. In addition, the peak distribution of network oscillation patterns for a particular patient during a specific seizure period was random. However, seizure variability is not entirely random since the overall trend of attack change remained consistent between patients. Because of differences in seizure intensity, duration, and brain network regions, the impact of these specificities on the results is insignificant.

In order to highlight how changes in the state of strong local oscillation and slow global oscillation vary between patient episodes, we estimated the change in the balance state between local strong and slow global oscillations at the peak for eight patients. The x-axis coordinate value of the H_sb distribution’s maximum density (blue dashed line in Figure 6) was drastically decreased and restored not only after a seizure but also happened many times. This analysis suggests that seizures cause a significant increase in the number of strong local oscillations in large-scale brain networks and that the inhibitory influence of slow global oscillations on strong local oscillations has almost completely disappeared at this time. However, after a seizure, protective mechanisms in the brain may prevent the loss of cognitive overload. The observed pattern of rapid increases followed by declines in concentration was commonly seen in patients, validating the generalizability of the concentration as mentioned above fluctuations. Again, based on patient-specificity, it is worthwhile to investigate that the indicated characteristics vary in time of commencement and degree of severity and may occur during either phase of the episode.

Given that high levels of strong local oscillations are expected throughout a vast network of distinct patients, we can also anticipate that network features will not limit these measures. To assess this, we also identified smaller, more specialized (non-epileptogenic) local networks for each patient. According to clinical case reports, each local (non)epileptogenic network included 64 loci with (non)epileptiform discharges during the interictal and ictal phases. In addition, H_sb plunge intervals and the exact time and number of high-level, locally intense oscillations were evaluated.

Figure 7 demonstrates that despite variations in the size and type of the patient’s functional brain networks, the sudden interval of H_sb of each patient is the same, and the number of high-level localized strong oscillations is identical or varies by one. Notably, the sudden interval of H_sb was more frequent and more prolonged in the localized epileptogenic network. Thus, whereas seizures considerably impact the interplay between strong local oscillations and slow global oscillations within the localized epileptogenic network, high levels of strong local oscillations do not significantly impair cognitive performance within the brain network.

Let us consider the emergence of high levels of locally strong oscillations to represent an extreme cognitive state. It makes sense to consider the relationship between the period during which high levels of locally strong oscillations emerge and when they become clinically apparent. We discovered that the moment of creation of high-level, local, strong oscillations was random among the various patients and network types examined in this research (Figure 7). In principle, the onset of high-level strong local oscillations should correspond with the clinically characterized onset of a seizure or the number six. However, high-level local oscillations are more prevalent during the interictal or pre-ictal phase and less commonly during the late phase. Therefore, can high-level strong local oscillations be utilized as a biomarker to identify seizures, and is this information clinically relevant? Cognitive impairment brought on by high-level strong local oscillations before seizures? This needs additional investigation.

Few research has compared local non-epileptogenic networks with epileptogenic networks, another original and novel aspect of our study. First, we discovered that seizures also impact the local non-epileptogenic brain network. Although seizures are far less common in this network than in the localized epileptogenic network, this network is likely to be more intense. Second, periods with high levels of strong local oscillations in the local non-epileptogenic network were much closer to those in the global network (Figure 7). In conclusion, our findings indicate that networks of varying sizes exhibit significant levels of robust local oscillations during epileptic seizures. In response to the severe oscillatory reconfiguration of large-scale brain networks, the brain often develops pathological protective mechanisms against cognitive impairment.