Abstract
Seizures occur in a recurrent manner with intermittent states of interictal and ictal discharges (IIDs and IDs). The transitions to and from IDs are determined by a set of processes, including synaptic interaction and ionic dynamics. Although mathematical models of separate types of epileptic discharges have been developed, modeling the transitions between states remains a challenge. A simple generic mathematical model of seizure dynamics (Epileptor) has recently been proposed by Jirsa et al. (2014); however, it is formulated in terms of abstract variables. In this paper, a minimal population-type model of IIDs and IDs is proposed that is as simple to use as the Epileptor, but the suggested model attributes physical meaning to the variables. The model is expressed in ordinary differential equations for extracellular potassium and intracellular sodium concentrations, membrane potential, and short-term synaptic depression variables. A quadratic integrate-and-fire model driven by the population input current is used to reproduce spike trains in a representative neuron. In simulations, potassium accumulation governs the transition from the silent state to the state of an ID. Each ID is composed of clustered IID-like events. The sodium accumulates during discharge and activates the sodium-potassium pump, which terminates the ID by restoring the potassium gradient and thus polarizing the neuronal membranes. The whole-cell and cell-attached recordings of a 4-AP-based in vitro model of epilepsy confirmed the primary model assumptions and predictions. The mathematical analysis revealed that the IID-like events are large-amplitude stochastic oscillations, which in the case of ID generation are controlled by slow oscillations of ionic concentrations. The IDs originate in the conditions of elevated potassium concentrations in a bath solution via a saddle-node-on-invariant-circle-like bifurcation for a non-smooth dynamical system. By providing a minimal biophysical description of ionic dynamics and network interactions, the model may serve as a hierarchical base from a simple to more complex modeling of seizures.
Highlights
A simple canonical mathematical model of epileptic discharges has been proposed by V
The proposed model consists of three subsystems that describe: (i) the ionic dynamics, (ii) the neuronal excitability, and (iii) a neuron-observer (Fig 1A)
Only the main variables are introduced, which are as follows: [K]o and [Na]i represent extracellular potassium and intraneuronal sodium concentrations, respectively; V(t) is the membrane depolarization; xD(t) is the synaptic resource; ν(t) is the firing rate of an excitatory population; and an inhibitory population firing rate is assumed to be proportional to ν(t)
Summary
A simple canonical mathematical model of epileptic discharges has been proposed by V. The model can be expressed in a set of six ordinary differential equations [2]. The set of variables chosen to describe the epileptic state transitions along with the interpretation of the terms included in the equations constitute the primary model predictions. The model distinguishes between main variables: two ionic concentrations of the extracellular potassium and intracellular sodium, a membrane potential, and a synaptic resource. The selection of these variables as well as the forms of the governing equations with experimental observations taken from both the experiments and the literature are justified and described in detail
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