update article

article/article.tex

@@ -277,22 +277,27 @@
   % 0-мерная модель
   \subsection*{Temporal aspects of activity in the center of epileptic discharge generation}
 
-  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.
+  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].
+  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.
+  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].
+  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).
+  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.
 
-  Ictal (ID) and interictal events (IID) were reproduced. IDs were 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. As seen, when the potassium concentration reaches a certain threshold level, short spontaneous, interictal-like discharges begin to occur, which are united in an one cluster constituting an ictal discharge. Sodium concentration increases during and ID. 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) and the burst terminates. The observed activity is similar to that reproduced with the original Epileptor-2 model [PLOS CB]. The proposed here extended models show the same characteristic features of epileptic discharges. The dynamics of IDs were subject to oscillations of the extracellular potassium and intracellular sodium ionic concentrations.
-  As shown [PLOS CB], IDs discharges are quasiperiodic oscillations, which consist of clusters – short bursts (SBs) similar to interictal discharges (IIDs) (Fig. \ref{fig:diff_K_point_center}A,\ref{fig:syn_K_point_center}A).  Bursts are spontaneous large-amplitude oscillations.
+  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.
 
 
 
   % Модель 1
-  \subsection*{Model 1: Diffusion}
+  \subsection*{Spatial aspects in Model 1: Diffusion}
   In the case of the diffusion mechanism simulations were conducted without Eq.(\ref{eqn:phi}), and with Eq.(\ref{eqn:theta}) set to $\theta = \varphi$. %The diffusion was considered to be equal in both spatial directions $x$ and $y$.
 
   The model's behaviour at the central point is qualitatively similar to the spatially homogeneous case, with only quantitative differences.
   %The observations from multiple points allows us to calculate additional characteristics of the model such as wave's velocity and length. We start from the point's description of the model, continue with a comparison of measurements at multiple points and finally look at the spatial picture of the resulted potassium wave.
-  The quasi-periodic spontaneous IDs occur with a period about 90 seconds (Fig \ref{fig:diff_K_point_center_period}). Each ID is characterized by a high rate of activity for about 20 seconds (Fig \ref{fig:diff_K_point_center}), and consists of SBs resembling IIDs (Fig \ref{fig:diff_K_point_center}A). $[K]_o$ dynamics determines the time length of an ID. As soon as its slow increase reaches a certain maximum level (around 4mM), an ID begins, and $[K]_o$ begins to increase rapidly, because of intensive potassium extrusion through potassium voltage-gated and glutamatergic channels 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]. It keeps increasing until it reaches the maximum at the end of an ID and slowly decays to the original concentration before the next ID.
-  %Both the increase and the decrease of the sodium concentration are approximately linear.
-  The $Na^+/K^+$ pump peaks at the end of an ID. Its activity remains high until the baseline potassium concentration is restored.
+  The quasi-periodic spontaneous IDs occur with a period about 90 seconds (Fig \ref{fig:diff_K_point_center_period}). Each ID is characterized by a high rate of activity for about 20 seconds (Fig \ref{fig:diff_K_point_center}), and consists of SBs resembling IIDs (Fig \ref{fig:diff_K_point_center}A). $[K]_o$ dynamics determines the time length of an ID. As soon as $[K]_o$  reaches a certain maximum level (around 4mM), an ID begins. Initiated at the center, the IF  forms a radial wave, which spreads across the entire cortical domain (Fig. \ref{fig:diff_K_board}). This wave is vizualized for the  field of the extracellular potassium concentration. $[K]_o$ rapidly increases at the front of the wave and more gradually decreases since activation of the Na-K pump by high sodium concentration, thus constituting the rear phase of the wave. $[K]_o$ finally returning to its baseline and even below.
+  When the baseline potassium concentration is restored, a new ID occurs.
+
+  The wave profile remains approximately the same during its propagation, as seen from comparison of $[K]_o$ evolution at two sites (Fig. \ref{fig:diff_K_points}), at the center and the periphery, shown in Fig. \ref{fig:diff_K_board}. The pulses of $[K]_o$ correspond to a single ID, formed as a cluster of IID-like discharges seen in the voltage plot in Fig. \ref{fig:diff_V_points}. The ID duration is approximately the same at the two sites, however the patterns of IIDs are different, remaining the bursting character of activity. The amplitude of the $[K]_o$ pulses varies within a couple of millimoles (Fig. \ref{fig:diff_K_points}).
+
   Velocity of the first K wave is about $0.047 mm/s$. The second wave is faster with velocity about $0.07mm/s$.
 
   %TODO! The similar picture can be seen in \cite{Whalen2018}.
@@ -340,7 +345,7 @@
   \end{figure}
 
   % Модель 2
-  \subsection*{Model 2: Axo-dendritic spread}
+  \subsection*{Spatial aspects in Model 2: Axo-dendritic spread}
   Velocity of the first K wave is about $0.15 mm/s$. The second wave is faster with velocity about $0.21mm/s$.
 
   \label{Results_Model2}