A unified physiological framework of transitions between seizures, sustained ictal activity and depolarization block at the single neuron level

Abstract

The majority of seizures recorded in humans and experimental animal models can be described by a generic phenomenological mathematical model, The Epileptor. In this model, seizure-like events (SLEs) are driven by a slow variable and occur via saddle node (SN) and homoclinic bifurcations at seizure onset and offset, respectively. Here we investigated SLEs at the single cell level using a biophysically relevant neuron model including a slow/fast system of four equations. The two equations for the slow subsystem describe ion concentration variations and the two equations of the fast subsystem delineate the electrophysiological activities of the neuron. Using extracellular K+ as a slow variable, we report that SLEs with SN/homoclinic bifurcations can readily occur at the single cell level when extracellular K+ reaches a critical value. In patients and experimental models, seizures can also evolve into status epilepticus (SE) and depolarization block (DB), activities which are also parts of the dynamic repertoire of the Epileptor. Increasing extracellular concentration of K+ in the model to values found during experimental SE and DB, we show that SE-like events and DB can also occur at the single cell level. Thus, seizures, SE and DB, which have been first identified as network events, can exist in a unified framework of a biophysical model at the single neuron level and exhibit similar dynamics as observed in the Epileptor.

Modeling seizures: From single neurons to networks

“…Recent progress in seizure modeling has brought a new understanding of this complex phenomenon. At the level of biophysical interactions, such as variations in ionic concentrations [8,10,23], it is possible to create direct correspondences with experimentally measurable physical quantities. It is possible to make predictions and design intervention strategies targeting known biophysical pathways.…”

“…Following the same general strategy, a different reduced model has been proposed [8]. It offers a unified framework for spiking, burst, seizure, status epilepticus, and depolarization block.…”

“…Building on this work, we propose here to highlight some recent works in this field. Numerous approaches emerged, from phenomenological aspects of dynamics to detailed biophysical models, from the subcellular scale to the large scale of whole brain regions [7][8][9][10][11][12]. We discuss the interest of each of these different approaches to show how all contribute to a better understanding of the mechanisms underlying seizures.…”

Mean-field approximation of network of biophysical neurons driven by conductance-based ion exchange

“…The concentrations of potassium, sodium, and chlorine in the intracellular and extracellular space along with the active transport pump (N a + /K + pump) in the cell membrane of neurons generate input currents to a neuron cell that drive the electrical behavior of a single neuron in terms of its membrane potential. The ion exchange mechanism in the cellular microenvironment, including local diffusion, glial buffering, ion pumps, and ion channels, has been mathematically modeled based on conductance-based ion dynamics to reflect the resting state and seizure behaviors in single neurons ( [21], [44], [45]). The mechanism of ion exchange in the intracellular and extracellular space of the neuronal membrane is represented schematically in Figure 1.…”

“…The Nernst equation was used to couple the membrane potential of the neuron with the concentrations of the ionic currents. This mechanism gives rise to a slow-fast dynamical system in which the membrane potential (V ) and potassium conductance gating variable (n) constitute the fast subsystem and the slow subsystem is represented in terms of the variation in the intracellular potassium concentration (∆[K + ] i ) and extracellular potassium buffering by the external bath ([K + ] g ) (in (1)); where input currents due to different ionic substances and pump are represented as follows: ( [21], [45]):…”