Abstract

Patient stratification according to seizure types is a pivotal step in the diagnosis and treatment of epilepsy. Great progress has been made in the understanding of the pathology and physiological substrates of epilepsy, leading to formal classification schemes based on empirical clinical features. However, much remains elusive regarding the mechanisms underlying seizures and the heterogeneity of its manifestations, which formal classification schemes attempt to capture. In the pursuit of a fundamental theory of seizure dynamics, dynamical systems theory provides the mathematical framework for a mechanistic understanding for the underlying dynamical processes. Within this framework, the need for a classification of seizures based on dynamics becomes evident. The diverse types of seizure onset and offset patterns are shown to be captured mathematically by different types of bifurcations that exhibit canonical invariant dynamical features. A seizure can then be classified into a corresponding “dynamotype” consisting of its onset and offset pair of bifurcations. Sixteen dynamotypes arise as combinations of the simplest relevant bifurcations, resulting in an organizational taxonomy of seizure dynamics (TSD). The current chapter provides a brief introduction into the mathematical theory upon which this taxonomy is built before presenting the different seizure dynamotypes. It then describes how TSD can be implemented as a classification scheme for the analysis of clinical stereo electroencephalographic data, along with sample results that summarize the observed prevalence of the different dynamotypes in a sample clinical dataset. The chapter ends with a discussion of the clinical ramifications of TSD and its implications for seizure control.

Introduction

The formal classification of epilepsy and seizure types aims at identifying unique diagnostic entities with etiologic, therapeutic, and prognostic implications that can guide clinical strategies and discovery of fundamental mechanisms. A first modern classification was put forward in the 1960s (Gastaut, 1969) and continued to be revised and improved, up to our day, with the advancement of our understanding of relevant pathophysiological mechanisms (Fisher et al., 2017). Still, in their latest position paper, the International League Against Epilepsy (ILAE) chose to use the phrase “operational classification,” “because it is impossible at this time to base a classification fully on the science of epilepsy” (Fisher et al., 2017). The classification is thus mainly based on observations of phenotypes in the form of clinical symptoms arising during the seizure, for example, the extent of anatomical involvement (focal or generalized onset), whether awareness is impaired or not, and the presence of motor or non-motor symptoms. In addition, investigations into the genotype of specific epilepsies (McGovern et al., 2013; Epi4K Consortium et al., 2013) aim at matching genetic deficits with pathological mechanisms for devising better informed therapeutics. However, despite the great variability in the possible underlying pathological or anatomical conditions of epilepsy, which is reflected in the various categories of the formal classification schemes, the electrophysiological signals of different seizures display remarkable similarity. For example, biologically distinct epileptogenic lesions were observed to share intracranial EEG seizure-onset patterns that were empirically categorized into a small finite number of distinct types (Perucca et al., 2014; Lagarde et al., 2019). Moreover, different types of observed seizure onset patterns were found to have nontrivial relationships with underlying pathology (Perucca et al., 2014), surgical outcome, (Jiménez-Jiménez et al., 2015; Lagarde et al., 2016) and sudden unexpected death in epilepsy (Rajakulendran and Nashef, 2015). This phenomenon of “many-to-one” and “one-to-many” characteristics of complex systems generates mechanistic degeneracy and has been named one of the key obstacles to neuroscientific progress (Frégnac, 2017).

That the dynamic signatures of seizures are informative as such should come as no surprise, given that the basic definition of a seizure is as follows: “a transient occurrence of signs and/or symptoms due to abnormal excessive or synchronous neuronal activity in the brain” (Fisher et al., 2017). That is, a seizure is intrinsically a dynamic phenomenon. This motivates the need for characterizing what will be referred to as the “dynamotype” of a seizure with the aim of complementing current classification schemes that mainly rely on phenotype and genotype specifications. The term “dynamotype” was recently coined “to describe a seizure’s composite, observable, dynamic characteristics in electrophysiological recordings comprising seizure onset and offset” (Saggio et al., 2020). Building on fundamentals of dynamical systems theory, practical and objective metrics were devised to allow a classification of seizures based on dynamics. Specifically, the metrics categorize seizure onset and offset into standard types of dynamical transitions, referred to as “bifurcation” types in technical mathematical jargon. Then, analogously to how proton number and electronic configuration allows the arrangement of chemical elements in the periodic table, the uncovered possible bifurcation types, at onset and offset, organize seizures into a tabular taxonomy of seizure dynamics (TSD) that paves the way for a better understanding of the underlying principles governing seizure genesis and termination.

A Taxonomy of Seizure Dynamics

The Dynamotypes

A Primer on Planar Bifurcations

It is common in dynamical systems theory to represent and visualize the state of a system as a point in an abstract mathematical space, called state space (or sometimes phase space for historical reasons), in which the coordinates of that point are the values of the interacting dependent variables that vary in time. Then, the behavior of the system can be visualized as the trajectory that such a point will trace in the state space as the system evolves in time. The global flow of the dynamics will be governed by the existence or absence of repellers and attractors, such as equilibrium points, limit cycles (periodic solutions), strange attractors, or other high dimensional invariant manifolds¹; an attractor will pull trajectories toward, it while a repeller will push them away. The direction of the flow along trajectories is dictated by the instantaneous value of the rates of change of the interacting variables, which are usually expressed in the form of ordinary differential equations that relate the time evolution of the variables to each other and their derivatives in time. The corresponding mathematical object is referred to as Structured Flow on Manifold (SFM) and can arise in dynamic neuronal networks under a range of conditions such as symmetry breaking in the connectivity (Pillai and Jirsa, 2017; Jirsa, 2008; Jirsa and Sheheitli, 2022). The simplest systems in which oscillations can occur are two-dimensional (2D) systems for which the phase space is a 2D plane with two axes corresponding to the two independent variables describing the instantaneous state of the system. A time series of an observable of the system resulting from a physical measurement can then be thought of as a lower dimensional projection of the underlying dynamical variables. A stable quiescent steady state can be then represented as a fixed point in the plane which can attract trajectories, such that the system can settle in that quiescent state after an initial transient and, if slightly perturbed, will tend to return to that fixed point. Whereas an attracting oscillatory state will correspond to a closed curve (referred to as a limit cycle) that nearby trajectories will tend to approach and circulate around. In the case of the coexistence of two such stable attractors (fixed point and limit cycle), the plane of all possible states will be partitioned by what are called separatrices, which are trajectories that play the role of boundaries between the basins of attraction of the attractors, that is, the sets of states from which the system will tend to move toward a corresponding attractor (see Fig. 18–1).

Figure 18–1.

Schematic of sample planar bifurcations and corresponding evolution of attractors and separatrices in phase space (a solid dot represents a stable fixed point, curves are sample trajectories in phase space, a closed curve is a stable limit cycle, arrow (more...)

As system parameters are varied, the relative location of attractors and separatrices can evolve; and for critical values, collisions may occur, resulting in bifurcations through which the number and/or stability of the attractors may change. It has been argued that modulation mechanisms can manifest as a slow variable that shifts system parameters such as the system moves into the seizure through a bifurcation that abolishes the quiescent state in favor of the oscillatory state and then, with further slow evolution, the return to quiescence occurs through another bifurcation type that causes the reversed effect (Jirsa et al., 2014). From formal bifurcation theory (Kuznetsov, 2013), we know that there are four different types of planar bifurcations through which a system can transition from quiescence to oscillations (onset) and four types through which it can transition back from oscillations to quiescence (offset). Each of these types of bifurcations is characterized by a set of invariant properties that systems of different nature will share and exhibit when undergoing the corresponding bifurcation. Those properties are not expected to cover the full global dynamics of the system; instead, they are descriptive of the local dynamics in phase space, around the bifurcation point. Invariant properties can be the way the amplitude and frequency change in time (i.e., their dynamics, not their absolute value), as well as the presence of a shift in the baseline of the signal generated by the process. Particularly, in the context of electroencephalographic signals, frequency is that of successive spikes generated during the seizure, where a spike refers to a prominent fast transient in the amplitude of the signal. We now list and briefly describe these onset and offset bifurcations and their corresponding invariant properties, which are summarized in Figure 18–2.

Figure 18–2.

Scaling-laws of bifurcations. Six bifurcations are responsible for the transition from rest to seizure and vice-versa. For each bifurcation we report: name and abbreviation; an example of timeseries; whether the amplitude or frequency of the oscillations (more...)

Footnotes

1.

“Manifold,” here, refers to the mathematical term; informally, it can be defined as an abstract mathematical subspace in which every point has a neighborhood which resembles Euclidean space, although the global structure of the subspace may be more complicated.

Onset Types

The scenarios corresponding to the four bifurcations, through which a system can transition from a quiescent state to oscillations as parameters are varied, can be summarized as follows:

1.

Saddle node bifurcation (SN): (example I—onset, Fig. 18–3) Before the bifurcation, a stable fixed point coexists with a stable limit cycle and an unstable fixed point, referred to as a saddle point, which attracts trajectories along certain directions but repels them along other directions. The saddle point lies on the intersection of the separatrices between the stable fixed point and the limit cycle. As the parameter is varied, the saddle point approaches the stable fixed point until it collides with it as the two coincide at the bifurcation point and then disappear as the bifurcation point is crossed. After the bifurcation, trajectories are attracted toward the only remaining stable attractor, which is the stable limit cycle, and circulate around it. The resulting time series would correspond to an oscillation that is about a center of a different value than that corresponding to the stable fixed point that existed before the bifurcation. This manifests as a shift in the baseline of the signal accompanied by an abrupt change in magnitude and frequency.

2.

Supercritical Hopf bifurcation (SupH): (example II—onset Fig. 18–4) A stable equilibrium loses stability at the bifurcation point and a small amplitude limit cycle appears around it; as the parameter is further increased, the emerging limit cycle grows, and its amplitude increases in proportion to the square root of the distance from the bifurcation point.

3.

Saddle node on an invariant circle (SNIC): (Fig. 18–5) The scenario here is similar to SN, with the distinction that before the bifurcation, instead of a limit cycle, an invariant circle exists on which the stable fixed point and the saddle point originally lie. As the parameter is varied, the fixed point and the saddle point approach each other and coalesce at the bifurcation. Afterward, the invariant circle becomes a stable limit cycle which all trajectories approach and circulate, manifesting as oscillations. The limit cycle evolves in shape as the parameter is further increased such that the frequency of the resulting oscillations increases proportionally to the square root of the distance from the bifurcation point (that distance is captured by the difference between the instantaneous value of the parameter and its critical value at the bifurcation point).

4.

Subcritical Hopf bifurcation (SubH): (Fig. 18–5) Before the bifurcation, an unstable limit cycle exists inside a larger stable limit cycle, and both share a center which is a stable fixed point. As the parameter is varied, the unstable limit cycle shrinks gradually until it disappears as it collapses onto the fixed point at the bifurcation point. Afterward, the fixed point loses stability and trajectories are attracted toward the other remaining stable limit cycle. This also manifests as an abrupt change in amplitude and frequency, though without any baseline shift since the limit cycle that persists after the bifurcation has at its center what used to be the original stable fixed point.

Figure 18–3.

Example I; a sample observed dynamotype; Scale bar: 10 s. first row shows a sample time series of an analyzed DC-coupled SEEG recording (SN onset characterized by DC shift at onset); second row shows the same signal after high-pass filtering; third row (more...)

Figure 18–4.

Example II; another sample dynamotype; Scale bar: 10 s. first row shows a sample time series of an analyzed DC-coupled SEEG recording; second row shows the same signal after high-pass filtering; third row displays in the left column the corresponding (more...)

Figure 18–5.

SNIC (upper row) and SubH (lower row) bifurcations in phase space (the parameter increases from left to right, a solid dot represents a stable fixed point, a hollow dot represents an unstable fixed point, a solid circle is a stable limit cycle, a dashed (more...)

Offset Types

Two of the bifurcations that could lead to seizure onset can also occur in reverse and lead to seizure offset. These two are the SNIC and the SupH:

1.

A seizure offset via a SNIC bifurcation will display a slowing down of the frequency of the spiking, with a square-root scaling, as the system approaches the bifurcation point where a stable fixed point and a saddle node are born on the limit cycle, which gets transformed into an invariant circle instead; the spiking stops as system trajectories return to be attracted to the stable fixed point instead of circulating in a limit cycle oscillation.

2.

A seizure offset via SupH bifurcation will display a square-root-scaled decreasing amplitude of oscillation as the limit cycle shrinks on route to disappearing at the bifurcation point where its fixed-point center becomes stable and attracts system trajectories back to quiescence.

There are two additional bifurcation types through which a system can move from spiking to quiescence, corresponding to the following scenarios:

3.

Fold limit cycle bifurcation (FLC): (example I—offset; Fig. 18–3) The spiking activity corresponds to a stable limit cycle that engulfs a smaller unstable limit cycle with a stable fixed point at the center. As the parameter is varied, the unstable limit cycle expands and approaches the stable limit cycle until the two coalesce at the bifurcation point into a periodic orbit that attracts trajectories from one side but repels them from another, hence referred to as a “fold.” After the bifurcation, the orbit disappears, and the spiking ceases as trajectories are attracted to the stable fixed points.

4.

Saddle-Homoclinic bifurcation (SH): (example II—offset; Fig. 18–4) A stable limit cycle coexists with a saddle point whose stable and unstable manifolds wrap around the limit cycle (here, stable/unstable manifold refers to the trajectory along the direction through which the system approaches/moves away from the saddle point). As the parameter is varied, the limit cycle expands and approaches the stable and unstable manifold of the saddle, and the frequency of the corresponding spiking oscillations decreases with logarithmic scaling. At the bifurcation point, the limit cycle collides and coincides with the stable and unstable manifolds of the saddle; the resulting invariant trajectory from and to the saddle point is referred to as a homoclinic orbit and has an infinite period (flow along the orbit gets slower and slower as it gets closer and closer to the saddle, such that it never really reaches the saddle point). As the bifurcation point is crossed, the spiking ceases as the homoclinic orbit breaks, and the cycle disappears. The resulting time series will display a shift in its baseline as trajectories move away from and no longer circulate around the unstable fixed point that used to be the center of the limit cycle.

Dynamotype Identification Using Canonical Features

The specific amplitude and frequency variation profiles that accompany the different bifurcation types are canonical features that can be used as fingerprints to classify the underlying bifurcation in a seizure. This can be done through the analysis of the time evolution of the signal amplitude and its interspike intervals (ISIs). Particularly, the distinctive features are the presence or absence of the following changes: a DC (baseline) shift, an increase or decrease in the amplitude or frequency of spiking, and whether that occurs abruptly or with a scaling law (square root or logarithmic). These canonical features are the basis of various methods for seizure classification (e.g., an automated algorithm, visual inspection by expert), which provide convergent results (Saggio et al., 2020). Some quantitative aspects can be difficult to discern visually, such as the presence of a logarithmic or square root scaling law; however, the presence/absence of a DC shift along with the general trends of ISI and amplitude of spikes can be easily determined through visual analysis by human expert reviewers. In fact, visual classification by human reviewers was found to be more reliable than the automated algorithm for the analysis of noisy clinical data. The simple visual classification scheme is summarized in Table 18–1; the procedure starts with the expert clinician designating the candidate regions of seizure onset and then the analysis is performed on the time series of the electrode corresponding to the candidate region with the highest signal amplitude at seizure onset. The classification of the corresponding seizure signal then entails observing the ISI and amplitude evolution plots of the first and last 10 spikes of the seizure and determining whether the trends were scaling to zero, constant, or arbitrary, and if there was a DC shift. “Scaling to zero” amplitude was defined as steadily decreasing to less than three times the background level near T = 0 (the onset or offset time), while for ISI it refers to steadily larger ISI near T = 0, with the last two ISI being more than 50% larger than the mean ISI of the prior 10 seconds.

Table 18–1

Visual Classification System.

Here, “+/–DC” refers to signals recorded with DC coupled/uncoupled hardware; for seizures recorded with DC uncoupled hardware, lack of information about presence/absence of a baseline shift makes it difficult to distinguish between SN and SubH at onset as well as SH and SNIC at offset. Full elaboration on the data processing details, method implementation, and validation can be found in Saggio et al. (2020).

Application to Clinical Data

Summary of Observed Dynamotypes

With four possible onset bifurcations and four possible offset bifurcations, the taxonomy predicts a total of 16 dynamotypes, 12 of which were identified in the analyzed clinical data (taking into consideration that several dynamotypes could not be distinguished due to absence of DC measurement). The predominantly identified onset types were the SN and SubH, while SH-SNIC and FLC were the predominant offset types. The SupH and SNIC onsets were less commonly identified, accounting for the four dynamotypes that were not encountered in the analyzed data set. Figure 18–6 displays a summary of the obtained results. A more thoroughly detailed report on the prevalence of the various dynamotypes in the analyzed data can be found in Saggio et al. (2020).

Figure 18–6.

Taxonomy of Seizure Dynamics (TSD).(I) Final results for all onset and offset bifurcations tested. All four bifurcation types were present. Ambiguous seizure offsets were included in the analysis as FLC. (II) Final taxonomy of the 39 patients with onset (more...)

It is worth noting that the authors also performed a stratification of patients by their onset and offset bifurcations and examined the prevalence of different dynamotypes across available metadata features such as gender, age, pathology, and seizure localization (Saggio et al., 2020). No significant correlations were found, emphasizing that classification of seizures based on dynamics serves to capture nonredundant characteristics that go beyond classical seizure classification criteria.

Implications for the Clinic

The mathematical framework that the TSD puts forth has a great conceptual added value to efforts of seizure characterization. Specifically, classification of seizures based on dynamics brings forward a novel perspective: seizure dynamics are not strictly determined by phenotype, genotype, pathology, or location of anatomical substrate but also by “local” dynamics that posit the system in an instantaneous location on the seizure map. This is reflected by two nontrivial results: that seizures of different patients with different types of focal pathologies can be classified into similar dynamotypes, and that individual patients may exhibit seizures of a multitude of dynamotypes.

A legitimate question then calls for investigation: Do different onset/offset bifurcation types mirror distinct underlying biophysical mechanisms? Could insight into that shed light on reasons of drug resistance? That is, if a patient’s repertoire of seizures includes multiple dynamotypes, emerging through different biophysical processes, how could one medication be expected to suffice for full seizure control? With the thorough unfolding of the seizure map, the full range of potential seizure onset and offset activity is uncovered, paving the way for a better mechanistic understanding of seizure evolution dynamics. Within this framework, numerous clinical ambiguities can start to be addressed; particularly those concerning mechanisms of seizure onset and offset in a way that would enable distinguishing inter- or preictal spiking from that at seizure initiation as well as determining instantaneous level of seizure risk through a measure of distance to bifurcation point, and consequently assessing treatment efficacy by probing the variation in that distance from the seizure threshold.

The classification of seizures based on dynamics brings forth the nontrivial fact that individual patients can display seizures of different dynamotypes; this should be greatly consequential for the design of medical devices targeting seizure detection which often assume that a patient’s seizures would have similar dynamical properties over time (Morrell, 2011; Cook et al., 2013). In addition, various therapeutic strategies for seizure control, such as with electrical stimulation, rely on proper system identification that captures the dynamical properties of responses to external stimuli. It is evident that the characterization of seizures based on dynamotypes will be greatly informative and decisive here; building on the extensive studies on neuronal bursting systems that embody the same bifurcations that constitute the TSD, it is predicted that systems poised near different bifurcation types can possess drastically different sensitivity to stimulation. For example, the SubH onset type acts as a resonator, requiring fine tuning of stimulus frequency for oscillations to occur, whereas the SN onset type acts as an integrator instead, with sensitivity to whether the stimulus is excitatory or inhibitory. The variability in the properties of the dynamical responses of systems poised near different bifurcation types will directly manifest in the synchronization properties of these systems when coupled to each other; considering that different regions can be poised near different bifurcations at seizure onset, characterization of dynamotypes can provide better understanding of the dynamical mechanisms underlying the propagation of seizures across different region in the brain. These realizations can play a great informative role in the context of personalized brain modeling efforts of clinical translational relevance (Jirsa et al., 2017), providing proper mathematical constraints that can render models faithful to underlying local dynamical processes.

TSD is concerned with dynamical properties of neural tissue. The spatial extent of the tissue is not considered in the framework; thus, no statement about the spatiotemporal properties of seizure activity (such as seizure propagation) can be made. However, spatiotemporal phenomena may mask dynamotypes. For instance, propagating seizure fronts arising from a SN bifurcation can mimic in the encephalographic recordings of a stationary sensor the temporal characteristics of a subcritical Hopf bifurcation. Detailed full brain modeling may address these challenges and is in full development with first applications at the horizon.

A main underlying hypothesis that has been assumed by the presented framework is the presence of two timescales in seizure evolution dynamics. The first is a fast timescale on the order of the ictal (spiking) duration, and the second is a slow timescale on the order of the interictal duration (between seizures). Particularly, a critical role is played by a slow endogenous parameter that moves the system across the seizure map, driving it into and out of different types of bifurcations and manifesting in seizures of different dynamotypes. This promisingly calls for research and clinical efforts to identify the relevant physiological variables that can be feeding into the slow parameter predicted by the mathematical modeling framework. In this context, given the phenomenological nature of the modeling approach, it is crucial to recognize that each parameter of the map should be considered as a representation of a manifold of physiological variables that cooperate to produce a particular change in the system. This is reflected in the fact that different patients with different pathologies exhibit common dynamotypes, as different combinations of physiological factors lead to the emergence of similar low-dimensional canonical dynamics. Because of degeneracy (different parameter configurations can produce the same output), it is not possible to provide biological intuitions regarding the model parameters. Given the slow timescale at which the changes in the bifurcation parameters are presumed to occur, biological candidates evolving on a slow timescale are less numerous. They may neurochemical substances, including hormones and neuromodulators. However, the model has a predictive value. Biological parameters important for the system’s dynamics must behave as those of the model. The slow modulation of seizure probability on the timescale of hours, days, months, or years is a well-known medical fact, with potential identified modulating factors linked to the circadian, multidien, or menstrual rhythms, for example (Karoly et al., 2021). Knowledge of the mathematical structure of the seizure map sets the landscape for further investigations on the possible slow physiological correlates that underly the navigation parameter; the identification, measurement, and possibly manipulation of these correlates would open exciting avenues for the possibility of detecting when a patient is heading toward unsafe regions on the seizure map and devising strategies that efficiently reroute the system into safety.

Conclusion

In summary, despite the heterogeneity of biological underlying conditions that can lead to seizures, the electrophysiological signatures of seizures display a remarkably limited set of dynamical behavior that can be captured and described using elementary mathematical relations. The core dynamical features uncovered through dynamical systems analysis of the mathematical composition of seizures are often obscured by the high individual variability of the signals due to noise and complex spiking morphology. Classical clinical interpretation of seizure dynamics often uses simple observation to single out certain frequency or spiking morphology features which were found to be useful in certain contexts, such as for identifying primary generalized seizures, but falling short in others, such as within the context of focal seizures. Interestingly, low voltage fast activity (Wetjen et al., 2009) and focal DC shifts (Ikeda et al., 1999) were found to be indicative of the identity of the seizure focus; both features correspond to common dynamotypes observed in the analyzed data (Saggio et al., 2020). TSD presents a framework to formalize such efforts aiming at interpretation of seizure dynamics and takes it a step further; it provides a seizure map that unravels the relationships between different types of dynamics and exposes the branching structure of possible routes of seizure evolution from which the likelihood of prevalence of the different dynamotypes can be inferred.

Finally, the classical operational seizure classification scheme captures a patient’s symptoms and associated pathological or anatomical substrates. Complementarily, the classification of seizures based on dynamics captures the behavior of the seizure itself. The accompanying TSD is a first principled approach through which a hierarchical organization of seizure dynamotypes emerges, indexed by simple objective metrics that are grounded in rigorous mathematical theory. Combining the two frameworks of classification should lead to better patient stratification and ultimately improved diagnosis and treatment strategies.

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