update article

article/article.tex

@@ -114,7 +114,8 @@
   \end{flushleft}
 
   \section*{Abstract}
-  Should be in Abstract: First the motivation. What task is actual and unsolved. Further methods, results and conclusion.
+  %Should be in Abstract: First the motivation. What task is actual and unsolved. Further methods, results and conclusion.
+
 
   Mechanisms of epileptic discharge generation and spread are not well known. Recently proposed simple biophysical model Epileptor-2 of interictal (IID) and ictal (ID) discharges brings a new hypothesis about the principle of IIDs and IDs. It states that IDs originate due to the elevated extracellular potassium concentration. To further verify the hypothesis we extend Epileptor-2 to the case of two-dimensional spatial propagation. We take into account two different mechanisms of propagation. The first is diffusion of potassium, and the second is spike propagation along neurons axons from a soma to a presynaptic terminal. The results clearly show that both mechanisms produce similar activity as the zero-dimensional case, and differ to each other in the spatial velocity and the length of epileptic discharge wave.
 
@@ -268,38 +269,47 @@
 
   % Механизмы и модели
   We hypothesized two different mechanisms of the spatial propagation of epileptic activity. The first mechanism is based on the diffusion of potassium in the extracellular medium. The second is determined by the spread along the cortical layers of spikes in axons and synaptic currents in dendrites.
-  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 by the diffusion term in the equation for the extracellular potassium concentration, Eq.\ref{eqn:K}. 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 the connectivity with some characteristic length.
+  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.
 
   % Постановка задачи для расчётов
-  According to these models, we simulated the activity in a square cortical domain, where a circular central zone was set to be a source of epileptic discharges, having an increased value of the excitation parameter $G_{syn}$.
+  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.
 
   % 0-мерная модель
-  \subsection{Temporal aspects of activity in the center of epileptic discharge generation}
+  \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.
 
-  Ictal (ID) and interictal events (IID) were reproduced. IDs were represented as clusters of spike bursts, and IIDs as bursts of spikes where bursts are spontaneous large-amplitude oscillations. The dynamics of events were subject to oscillations of the extracellular potassium and intracellular sodium ionic concentrations. Lets review the results of diffusion first.
-
-
+  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.
 
 
 
   % Модель 1
   \subsection*{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 resulted model's behaviour at the point is consistent with the homogeneous case, and 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.
+  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 central point of the domain shows a population activity with periodic spontaneous IDs (Fig \ref{fig:diff_K_point_center_period}). The period is about 90 seconds. 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), the increase velocity jumps to a higher value and gives a start to an ID. The velocity keeps growing until it is balanced by the Na-K-pump. After that the velocity decreasing and the potassium concentration starts returning to its baseline. The concentration approaching the baseline defines the ID's termination. The peak of $[K]_o$ takes place at the middle of an ID, and the form of $[K]_o$ wave is symmetric about this middle. The $Na^+/K^+$ pump in its turn peaks at the end of an ID. The form of the pump's wave is shifted towards the ID's end as its activity keeps increasing until the original potassium concentration is restored. The sodium concentration starts its increase together with the jump of the potassium concentration velocity. After that 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 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.
+  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}.
 
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{diff/x40y40_period.png}
-    \caption{Periodic activity of series of IDs recorded in the center of the cortical domain(Fig.\ref{fig:diff_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    \caption{Diffusion mechanism. Periodic activity of series of IDs recorded in the center of the cortical domain(Fig.\ref{fig:diff_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    %Fig. 1
     \label{fig:diff_K_point_center_period}
   \end{figure}
+
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{diff/x40y40.png}
-    \caption{An ictal discharge recorded in the center of the cortical domain (Fig.\ref{fig:diff_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    \caption{Diffusion mechanism. An ictal discharge recorded in the center of the cortical domain (Fig.\ref{fig:diff_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    %Fig. 2
     \label{fig:diff_K_point_center}
   \end{figure}
 
@@ -309,14 +319,15 @@
       \centering
       \captionsetup{width=0.8\linewidth}
       \includegraphics[width=1.0\linewidth]{diff/K_points.png}
-      \caption{A comparative plot of $[K]_o$ at two different points}
+      \caption{Diffusion mechanism. A comparative plot of $[K]_o$ at two different points.}
       \label{fig:diff_K_points}
     \end{minipage}\hfill
     \begin{minipage}{0.5\linewidth}
       \centering
       \captionsetup{width=0.8\linewidth}
       \includegraphics[width=1.0\linewidth]{diff/U_points.png}
-      \caption{A comparative plot of $U$ at two different points}
+      \caption{Diffusion mechanism. A comparative plot of $U$ at two different points.}
+      %Fig. 3
       \label{fig:diff_V_points}
     \end{minipage}
   \end{figure}
@@ -324,23 +335,28 @@
   \begin{figure}
     \centering
     \makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{diff/K_board.png}}
-    \caption{Potassium concentration patterns during one ID}
+    \caption{Diffusion mechanism. Potassium concentration patterns during one ID.}
     \label{fig:diff_K_board}
   \end{figure}
 
   % Модель 2
   \subsection*{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}
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{syn/x40y40_period.png}
-    \caption{Periodic activity of series of IDs recorded in the center of the cortical domain(Fig.\ref{fig:syn_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    \caption{Synaptic mechanism. Periodic activity of series of IDs recorded in the center of the cortical domain(Fig.\ref{fig:syn_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    %Fig. 6
     \label{fig:syn_K_point_center_period}
   \end{figure}
+
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{syn/x40y40.png}
-    \caption{An ictal discharge recorded in the center of the cortical domain (Fig.\ref{fig:syn_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    \caption{Synaptic mechanism. An ictal discharge recorded in the center of the cortical domain (Fig.\ref{fig:syn_K_board}). \textbf{A}, the representative neuron membrane depolarization $U$. \textbf{B}, the total input current $u$. \textbf{C}, The ionic concentrations $[K]_o$ and $[Na]_i$, and the $Na^+/K^+$ pump current $I_{pump}$. \textbf{D}, the somatic firing rate $\nu$ and the synaptic resource $x^D$}.
+    %Fig. 7
     \label{fig:syn_K_point_center}
   \end{figure}
 
@@ -350,14 +366,14 @@
       \centering
       \captionsetup{width=0.8\linewidth}
       \includegraphics[width=1.0\linewidth]{syn/K_points.png}
-      \caption{A comparative plot of $[K]_o$ at two different points}
+      \caption{Synaptic mechanism. A comparative plot of $[K]_o$ at two different points.}
       \label{fig:syn_K_points}
     \end{minipage}\hfill
     \begin{minipage}{0.5\linewidth}
       \centering
       \captionsetup{width=0.8\linewidth}
       \includegraphics[width=1.0\linewidth]{syn/U_points.png}
-      \caption{A comparative plot of $U$ at two different points}
+      \caption{Synaptic mechanism. A comparative plot of $U$ at two different points.}
       \label{fig:syn_V_points}
     \end{minipage}
   \end{figure}
@@ -365,10 +381,19 @@
   \begin{figure}
     \centering
     \makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{syn/K_board.png}}
-    \caption{Potassium concentration patterns during one ID}
+    \caption{Synaptic mechanism. Potassium concentration patterns during one ID.}
     \label{fig:syn_K_board}
   \end{figure}
 
+  As in \cite{Ma2012} we can outline a region of interest (ROI) in the form of a vertical strip across the center of the simulation area, (Fig. \ref{fig:syn_V_cut}A). A time evolvent of the strip (Fig. \ref{fig:syn_V_cut}B) demonstrates the similar effect of rapid membrane voltage propagation from the onset of the ictal zone to the periphery part of the simulation area outside of the 4-AP applicationto.
+  \begin{figure}
+    \centering
+    \makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{syn/cut_V_at63s.png}}
+    \caption{Synaptic mechanism. A. The red bar ROI(region of interest) of the membrane potential $V$ at the 63rd second of the simulation. B. Propagation pattern of the ROI at the first 400 ms of the 63rd second.}
+    \label{fig:syn_V_cut}
+  \end{figure}
+
+
   % Комбинированная модель
   \subsection{Model 3: Combination of the two mechanisms of activity spread}