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.

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Low-frequency electrical stimulation is used to treat some drug-resistant forms of epilepsy. Despite the effectiveness of the method in suppressing seizures, there is a considerable risk of side effects. An optogenetic approach allows the targeting of specific populations of neurons, which can increase the effectiveness and safety of low-frequency stimulation. In our study, we tested the efficacy of the suppression of ictal activity in entorhinal cortex slices in a 4-aminopyridine model with three variants of low-frequency light stimulation (LFLS): (1) activation of excitatory and inhibitory neurons (on Thy1-ChR2-YFP mice), (2) activation of inhibitory interneurons only (on PV-Cre mice after virus injection with channelrhodopsin2 gene), and (3) hyperpolarization of excitatory neurons (on Wistar rats after virus injection with archaerhodopsin gene). Only in the first variant did simultaneous LFLS of excitatory and inhibitory neurons replace ictal activity with interictal activity. We suggest that LFLS caused changes in the concentration gradients of K+ and Na+ cations across the neuron membrane, which activated Na-K pumping. According to the mathematical modeling, the increase in Na-K pump activity in neurons induced by LFLS led to an antiepileptic effect. Thus, a less specific and generalized optogenetic effect on entorhinal cortex neurons was more effective in suppressing ictal activity in the 4-aminopyridine model.