Minimal model of interictal and ictal discharges “Epileptor-2”

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.


Illustrations to the mechanisms of IDs and IIDs.
Below we illustrate the contribution of each of the elements of the mechanisms of simulated IDs and IIDs. The schematic of the mechanisms is given in Fig.1B in the main text and reproduced in Fig. A with the all the transitions (arrows) numbered. According to the arrows in Fig. A, the main processes in the mechanisms of ID and IID generation are as follows: 1. High potassium concentration leads to recurrent excitation. A low potassium concentration results in a trivial, silent resting state of the system, which turns to IID generation regime when the bath concentration changes from 3 to 8.5 mM (Fig. B). At the interval between 50 and 100 ms a cluster of SBs is observed with significant modulation of , similar to an ID, which then turns into irregular SBs generation with low modulation of the concentrations, i.e. to the regime with IID generation.     . If the density of the Na-K-pumps is low, then the IDs might last longer or transfer to non-interruptive spiking regime (Fig. I). Summarizing, all stages of the IID and ID generation mechanisms have been illustrated by the simulations described above. The roles of key factors have been clarified.

Disinhibition as a model of epilepsy.
Our model of epilepsy, considered in the paper, implies both mechanismsdisinhibition and an increase of extracellular potassium concentration by means of elevated potassium concentration in a bath. This model is close to the models that we used in our experiments ( The partial disinhibition is set by the coefficient 0.5.

Depolarization block.
The depolarization block (DB) is sometimes observed in intracellular recordings during IDs, mainly in the interneurons. However, this factor has not been included in our Epileptor-2 model. The reasoning for this is following. First, the depolarization block is not a frequent effect in our recordings (Amakhin et al. // Front. Cell. Neurosc. 2016; Chizhov et al. // PloS One 2017). Second, the block of interneurons might lead to a temporal lack of inhibition during IDs, i.e., provides a pro-epileptic effect. At the same time, there are several pro-epileptic factors that support the ID generation, but a challenge to modeling is to reveal those factors that terminate each of IIDs and IDs. That is why, for the sake of simplicity, DB is excluded from the main model. However, in order to demonstrate its effect, here we modify eq.(6) in the following way.
By this way, we imply that if the total current that drives interneurons exceeds the threshold DB u , then the interneurons are blocked by the depolarization and do no longer contribute into the total current ) (t u . The formula (6') converges to (6) for infinitely large DB u .

An alternative model of a neuron-observer.
In the paper, we have described a quadratic integrate-and-fire (QIF) model as a model of neuron-observer. Here we compare QIF with adaptive QIF models (Fig. M).
The addition of a second variable, w , allows for the inclusion of an adaptation and captures the bursting behavior of neurons (Izhikevich // IEEE Transactions on Neural Networks 2003). The equations for the membrane potential ) (t U and the adaptation current ) (t w are as follows: