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@@ -272,18 +272,18 @@
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In order to distinguish between the two mechanisms, we have considered two different models of spatial propagation. The both models generalize the recently proposed spatially homogeneous model of epileptic activity "Epileptor-2" \cite{Chizhov2018}. The first model supplies Epileptor-2 with the diffusion term in the equation for the extracellular potassium concentration, Eq.\ref{eqn:K}. Below we refer to this model as "the diffusion model", or Model 1. The second model adds to the system the equation of spiking activity spread, Eq.\ref{eqn:phi}, which connects presynaptic to somatic firing rates by assuming an exponentially decaying profile of connectivity with some characteristic length. Below we refer to this model as "the synaptic model", or Model 2.
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% Постановка задачи для расчётов
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- Our simulations are aimed to reproduce the spatial-temporal patterns of activity of the cortical neural tissue during the generation of epileptic ictal discharges after local application of the proepileptic agent 4-AP [Tsytsarev]. In our models, we considered a square-shaped domain of nervous tissue with a small circular central zone with increased excitability $G_{syn}$, being a source of epileptic discharges. In the central zone $G_{syn}/g_L=5$mV$\cdot$s in the center and $G_{syn}/g_L=1$mV$\cdot$s at the periphery.
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+ Our simulations are aimed to reproduce the spatial-temporal patterns of activity of the cortical neural tissue during the generation of epileptic ictal discharges after local application of the proepileptic agent 4-AP \cite{Mller2018}. In our models, we considered a square-shaped domain of nervous tissue with a small circular central zone with increased excitability $G_{syn}$, being a source of epileptic discharges. In the central zone $G_{syn}/g_L=5$mV$\cdot$s in the center and $G_{syn}/g_L=1$mV$\cdot$s at the periphery.
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% 0-мерная модель
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\subsection*{Temporal aspects of activity in the center of epileptic discharge generation}
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- The both spatially distributed models and the original spatially homogeneous model Epileptor-2 show similar patterns of activity in the center of epileptic discharge generation[PLOS CB].
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+ The both spatially distributed models and the original spatially homogeneous model Epileptor-2 show similar patterns of activity in the center of epileptic discharge generation \cite{Chizhov2018}.
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Ictal (ID) and interictal events (IID) are reproduced. IDs are represented as clusters of spike bursts, and IIDs as bursts. The membrane potential of the representative neuron (Fig.\ref{fig:diff_K_point_center_period}A,\ref{fig:diff_K_point_center}A,\ref{fig:syn_K_point_center_period}A,\ref{fig:syn_K_point_center}A) and the concentrations of potassium and sodium ions (Fig. \ref{fig:diff_K_point_center_period}C,\ref{fig:syn_K_point_center_period}C) reflect the spontaneous occurrence of discharges.
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- The quasi-periodic spontaneous IDs occur with a period of order of minutes. Each ID is characterized by a high rate of activity for about a few tens of seconds, and consists of short bursts that resemble IIDs (Fig \ref{fig:diff_K_point_center}A, \ref{fig:syn_K_point_center}A), i.e. the interictal-like discharges are united in an one cluster constituting an ictal discharge. $[K]_o$ dynamics determines the onset and the time length of an ID. As soon as its slow increase reaches a certain threshold level, an ID begins, and $[K]_o$ begins to increase rapidly, because of intensive potassium extrusion through potassium voltage-gated and glutamatergic channels that are active during the ID. $[K]_o$ grows until it is balanced by the Na-K pump. The peak of $[K]_o$ takes place at the middle of an ID. After that, $[K]_o$ begins to decrease, finally returning to its baseline and even below. The phase of an ID, where concentration approaching the baseline, defines the termination of the ID. The Na-K pump is activated by the elevated intracellular sodium concentration. The sodium concentration begins to increase because of high spiking and glutamatergic synaptic activity during IDs [Chizhov et al. PLOS One 2019].
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+ The quasi-periodic spontaneous IDs occur with a period of order of minutes. Each ID is characterized by a high rate of activity for about a few tens of seconds, and consists of short bursts that resemble IIDs (Fig \ref{fig:diff_K_point_center}A, \ref{fig:syn_K_point_center}A), i.e. the interictal-like discharges are united in an one cluster constituting an ictal discharge. $[K]_o$ dynamics determines the onset and the time length of an ID. As soon as its slow increase reaches a certain threshold level, an ID begins, and $[K]_o$ begins to increase rapidly, because of intensive potassium extrusion through potassium voltage-gated and glutamatergic channels that are active during the ID. $[K]_o$ grows until it is balanced by the Na-K pump. The peak of $[K]_o$ takes place at the middle of an ID. After that, $[K]_o$ begins to decrease, finally returning to its baseline and even below. The phase of an ID, where concentration approaching the baseline, defines the termination of the ID. The Na-K pump is activated by the elevated intracellular sodium concentration. The sodium concentration begins to increase because of high spiking and glutamatergic synaptic activity during IDs \cite{Chizhov2019}.
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When a certain high level of the intracellular sodium concentration is reached, the potassium-sodium pump activates (Fig. \ref{fig:diff_K_point_center_period}C,\ref{fig:syn_K_point_center_period}C).
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The $Na^+/K^+$ pump peaks at the end of an ID. Its activity remains high until the baseline potassium concentration is restored. The burst terminates. The sodium concentration slowly decays to the original concentration before the next ID.
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- Because the proposed here extended models show the same characteristic features of epileptic discharges, their behaviour in the center of the epileptic discharge generation can be explained in the terms of oscillations, similar to the spatially homogeneous model. The dynamics of IDs is governed by the quasiperiodic oscillations of the extracellular potassium and intracellular sodium ionic concentrations, which constitute a slow subsystem of the full model [PLOS CB]. The IDs consist of clusters – short bursts (Fig. \ref{fig:diff_K_point_center}A,\ref{fig:syn_K_point_center}A), which are spontaneous large-amplitude oscillations.
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+ Because the proposed here extended models show the same characteristic features of epileptic discharges, their behaviour in the center of the epileptic discharge generation can be explained in the terms of oscillations, similar to the spatially homogeneous model. The dynamics of IDs is governed by the quasiperiodic oscillations of the extracellular potassium and intracellular sodium ionic concentrations, which constitute a slow subsystem of the full model \cite{Chizhov2018}. The IDs consist of clusters – short bursts (Fig. \ref{fig:diff_K_point_center}A,\ref{fig:syn_K_point_center}A), which are spontaneous large-amplitude oscillations.
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@@ -345,10 +345,12 @@
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\end{figure}
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% Модель 2
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- \subsection*{Spatial aspects in Model 2: Axo-dendritic spread}
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+ \subsection{Spatial aspects in Model 2: Axo-dendritic spread}
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+ \label{Results_Model2}
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+
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Velocity of the first K wave is about $0.15 mm/s$. The second wave is faster with velocity about $0.21mm/s$.
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- \label{Results_Model2}
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+
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\begin{figure}
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\centering
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\includegraphics[width=1.0\textwidth]{syn/x40y40_period.png}
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