Abstract
On the Nature of Seizure Dynamics.
Jirsa VK, Stacey WC, Quilichini PP, Ivanov AI, Bernard C. Brain 2014;137(pt 8):2210–2230.
Seizures can occur spontaneously and in a recurrent manner, which defines epilepsy; or they can be induced in a normal brain under a variety of conditions in most neuronal networks and species from flies to humans. Such universality raises the possibility that invariant properties exist that characterize seizures under different physiological and pathological conditions. Here, we analysed seizure dynamics mathematically and established a taxonomy of seizures based on first principles. For the predominant seizure class we developed a generic model called Epileptor. As an experimental model system, we used ictal-like discharges induced in vitro in mouse hippocampi. We show that only five state variables linked by integral-differential equations are sufficient to describe the onset, time course and offset of ictal-like discharges as well as their recurrence. Two state variables are responsible for generating rapid discharges (fast time scale), two for spike and wave events (intermediate time scale) and one for the control of time course, including the alternation between ‘normal’ and ictal periods (slow time scale). We propose that normal and ictal activities coexist: a separatrix acts as a barrier (or seizure threshold) between these states. Seizure onset is reached upon the collision of normal brain trajectories with the separatrix. We show theoretically and experimentally how a system can be pushed toward seizure under a wide variety of conditions. Within our experimental model, the onset and offset of ictal-like discharges are well-defined mathematical events: a saddle-node and homoclinic bifurcation, respectively. These bifurcations necessitate a baseline shift at onset and a logarithmic scaling of interspike intervals at offset. These predictions were not only confirmed in our in vitro experiments, but also for focal seizures recorded in different syndromes, brain regions and species (humans and zebrafish). Finally, we identified several possible biophysical parameters contributing to the five state variables in our model system. We show that these parameters apply to specific experimental conditions and propose that there exists a wide array of possible biophysical mechanisms for seizure genesis, while preserving central invariant properties. Epileptor and the seizure taxonomy will guide future modeling and translational research by identifying universal rules governing the initiation and termination of seizures and predicting the conditions necessary for those transitions.
Commentary
Dynamical systems theory is the mathematical study of physical systems whose state changes over time. It has found many predictive and diagnostic applications in physics, environmental science, biology, and medicine. Inspired by these successful applications, a number of research efforts have sought to use dynamical systems theory to identify the causes of the brain's transition to the epileptic seizure state and predict consistently when seizures will occur in patients. For example, investigations of simplified macroscopic corticothalamic models have led to the identification of specific patterns of activity where the model loses stability and produces seizure-like events that share qualitative features with scalp EEG recordings.1–3 In addition, studies of cortical neural mass models,4 which represent brain function at the scale of macroscopic ensembles of neurons, have been used to describe ordered sequences of qualitatively different oscillatory regimes that strongly resemble clinical recordings from patients with temporal lobe epilepsy.5 The general aim in such studies is to understand which patterns of epileptic activity are “the same” in a mathematical sense and to classify all seizure-like events in a unified framework.3 Yet what has been missing is experimental evidence beyond specific brain regions and small populations of human patients to answer the question of whether there are general, fundamental properties of seizure dynamics that hold across multiple patients, species, and brain regions.
Jirsa and collaborators set out to categorically address this grand but rather thorny question by means of a new theoretical framework called Epileptor. They first identified the “building blocks” of seizure dynamics to be two major activities that were consistently observed in seizures: an increase in fast frequency oscillations and spike-and-wave complexes. They then analyzed seizure-like events from in vitro recordings in the mouse hippocampus, in a zebrafish model of hyperthermia-induced seizures, and in EEG recordings of human patients, and then presented a minimal dynamic model with five variables describing the onset, time-course, and offset of the seizure activity. Fast oscillations are described by one coupled pair of variables in the model, and large amplitude spikes followed by long-lasting wave components are described by a second pair of variables. One additional variable acts on a very slow time scale and guides the whole system between and throughout seizure events, which is necessary to reflect the recurrence of seizure events. The Epileptor model is sufficient to describe transitions from a normal state to a seizure state and back. These transitions are termed “bifurcations” in dynamical systems theory, and by identifying all possible conditions for the bifurcations to occur, Jirsa and colleagues were able to build a mathematical taxonomy of seizure-like events and identify two types of bifurcations as the predominant candidates for seizure onset and offset. From this analysis, they made two predictions: 1) seizure onset can only occur in the presence of a DC shift of the field potential, and 2) the interspike intervals show a logarithmic scaling as they approach the seizure offset. The first of these predictions was confirmed in direct current recordings of seizures in isolated mouse whole hippocampus. The second prediction was confirmed by analysis of interspike intervals in mouse hippocampus, in zebrafish recordings, and in human patients.
In the Epileptor model, normal brain function and seizures are two regimes of activity that are separated by divergent trajectories of the inhibitory and excitatory pairs of variables. The boundary that separates the two regimes is known as the separatrix in dynamical systems theory, and can be related to the seizure threshold in epileptic brains.6 The authors argued that the region near the separatrix is analogous to the hypothesized “preictal state.”7 The Epileptor framework then makes it possible to know the distance from the separatrix and explore various conditions that will push the system to cross the boundary into the seizure state. Jirsa and colleagues conducted several experiments to validate model predictions of how seizures start. They first provoked seizures by electrical stimulation and found a refractory period after the seizure event, which was also present in the model. Next, they explored the contribution of synaptic noise to seizures. Introducing noise in the model caused seizure onset earlier than anticipated in the theoretical analysis. A subsequent experiment demonstrated that increasing synaptic noise to the hippocampus via the septum was sufficient to trigger seizure events upon reaching a critical threshold.
The Epileptor model is an abstract and very general description of seizure dynamics that is not meant to replace detailed analysis of the underlying physiological conditions. Recognizing the need for such analysis, the authors developed a strategy to identify these conditions experimentally, while acknowledging that this strategy may only apply to their experimental preparation of a mouse hippocampus in low Mg2+ conditions. They linked the slow time-scale variable in the model with levels of extracellular potassium, oxygen, and intracellular ATP use but did not identify a biophysical variable that changed during the interictal period to drive the system to the seizure threshold. A further model prediction was that the excitatory pair of variables should be active during spike-and-wave complexes with fast oscillations occurring only during the wave part of the event. The authors validated this latter prediction with experimental evidence that suggested that glutamatergic and GABAergic cells contributed to fast discharges and spike-and-wave events, respectively, and that GABAergic neurons fired action potentials during spike-and-wave events, stopped firing during the fast discharge, and resumed firing when spike-and-wave events occurred during seizure-like activity.
In conclusion, Jirsa and collaborators have shown that it is possible to develop a formalization of the invariant features of the neuronal interactions in the preictal state and during seizures, and they have laid the groundwork for a universal predictive theory of seizure dynamics in large-scale models of the brain. The Epileptor model and the taxonomy of seizures are valuable contributions toward the mathematical identification of preictal states and could serve as a guide for the development of methods for seizure intervention. However, several challenges are still outstanding. First is the identification of biophysical processes that exhibit dynamics of the type encoded in the model, which would be an important step towards Epileptor-inspired experimental and clinical interventions focused on modulating brain dynamics away from the seizure boundary. An intriguing possibility in this regard is to utilize existing highly realistic, biological data-driven computational models of neuronal networks8 to identify key biophysical variables that underlie the dynamical transitions described by the Epileptor model. Another challenge is to extend the Epileptor to represent the interactions between intrinsic neuronal dynamics and network activity so that it can be used to study the spatial and temporal evolution of seizures between connected brain regions. Whatever the future will bring for the Epileptor, the work by Jirsa and colleagues vividly illustrates the witty maxim about mathematics that “no matter how determinedly its practitioners ignore the world, they consistently produce the best tools for understanding it.”9