update article

by Anton

article/article.tex

@@ -120,17 +120,19 @@
   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.
 
 
-  \section*{Introduction}
-  The spread of activity through cortical circuits has been studied in experiments by means of electrical registrations and optical imaging \cite{Trevelyan2006, Trevelyan2007, Smith2016}. Experiments show slow propagation of an ictal wavefront and fast spread of discharges behind the front \cite{Smith2016}. A speed of the ictal wavefront is similar in different electrophysiological \cite{Trevelyan2006, Trevelyan2007} and imaging studies \cite{Wenzel2017}, it is about tenths of millimeters per second (0.6 mm/s in \cite{Wenzel2017}). The mechanism is still an open question \cite{Smith2016}. Some models that considers spatial propagation suggest that the excitability of the tissue surrounding the seizure core may play a determining role in the seizure onset pattern \cite{Wang2017}. Whereas the generation of interictal discharges is modelled in the conditions of impaired but fixed ionic concentrations \cite{Chizhov2017}, the dynamics of ictal discharges and the excitability of the cortical tissue is hypothesized to be governed by the ionic dynamics. Generally, a computational approach to this issue requires a biophysical consideration of the neuronal population interactions in the conditions of changing ionic concentrations of sodium, potassium, chloride and calcium ions inside and outside the neurons and glial cells. This problem is quite complex and computationally expensive. The most well-elaborated biophysical models considers either a single neuron \cite{Wei2014} or a network \cite{Bazhenov2004, Krishnan2011} without a spatial structure. Thus, the consideration of spatial propagation requires a reduced but biophysically detailed model able to reproduce ictal discharges. Recently, we have proposed a spatially concentrated biophysical model of ictal and interictal discharges \cite{Chizhov2018}, called Epileptor-2 after the known abstract model Epileptor \cite{Jirsa2014}. Our model might be extended to the spatially distributed case. As shown, the major role in excitability belongs to the extracellular potassium concentration. Recently, the spatial patterns of the extracellular potassium distribution have been registered by means of nanoparticle based technique \cite{Mller2018}. The wavefront of potassium elevation from a seizure in 4-AP (4-apinopyridyne that strengthens synaptic connections) based model of cortical epilepsy spreads with a speed about tenths of millimeters per second. In the present work, the Epileptor-2 model is extended by introducing the diffusion equation for the potassium concentration.
+  \section{Introduction}
+  Epilepsy is characterized by repeated seizures associated with abnormal intense electrical neural discharges. Despite ongoing studies, the mechanisms of generation and propagation of these discharges are not yet fully understood. Understanding of these mechanism is important for medical treatment development and also for mathematical modeling as an explanatory example of a regime of strong synchronization of neuronal activity, that is simpler than normal functioning.
 
+  The spread of activity through cortical circuits has been studied in experiments by means of electrical registrations and optical imaging \cite{Trevelyan2006, Trevelyan2007, Smith2016}. Experiments show slow propagation of an ictal wavefront and fast spread of discharges behind the front \cite{Smith2016}. A speed of the ictal wavefront is similar in different electrophysiological \cite{Trevelyan2006, Trevelyan2007} and imaging studies \cite{Wenzel2017}, it is about tenths of millimeters per second (0.6 mm/s in \cite{Wenzel2017}). The mechanism is still an open question \cite{Smith2016}. Some models that considers spatial propagation suggest that the excitability of the tissue surrounding the seizure core may play a determining role in the seizure onset pattern \cite{Wang2017}. Whereas the generation of interictal discharges is modelled in the conditions of impaired but fixed ionic concentrations \cite{Chizhov2017}, the dynamics of ictal discharges and the excitability of the cortical tissue is hypothesized to be governed by the ionic dynamics. Generally, a computational approach to this issue requires a biophysical consideration of the neuronal population interactions in the conditions of changing ionic concentrations of sodium, potassium, chloride and calcium ions inside and outside the neurons and glial cells. This problem is quite complex and computationally expensive. The most well-elaborated biophysical models considers either a single neuron \cite{Wei2014} or a network \cite{Bazhenov2004, Krishnan2011} without a spatial structure. Thus, the consideration of spatial propagation requires a reduced but biophysically detailed model able to reproduce ictal discharges. Recently, we have proposed a spatially concentrated biophysical model of ictal and interictal discharges \cite{Chizhov2018}, called Epileptor-2 after the known abstract model Epileptor \cite{Jirsa2014}. Our model might be extended to the spatially distributed case. As shown, the major role in excitability belongs to the extracellular potassium concentration. Recently, the spatial patterns of the extracellular potassium distribution have been registered by means of nanoparticle based technique \cite{Mller2018}. The wavefront of potassium elevation from a seizure in 4-AP (4-aminopyridyne that strengthens synaptic connections) based model of cortical epilepsy spreads with a speed about tenths of millimeters per second. In the present work, the Epileptor-2 model is extended by introducing the diffusion equation for the potassium concentration.
 
-  \section*{Materials and methods}
+
+  \section{Materials and methods}
 
   The previous implementation of Epileptor-2 \cite{Chizhov2018} was restricted to consideration of a spatially homogeneous ativity, thus not considering any spatial propagation. A primary focus of that model was a biophysical interpretation of its governing variables describing the pathological states of brain activity. For that purpose, the ionic dynamics described in earlier studies \cite{Bazhenov2004, Kager2000, Cressman2009} was implemented into a rate-based model for recurrently connected excitatory and inhibitory neuronal populations. Ionic dynamics were comprised of extracellular potassium and intracellular sodium. The firing rate was assumed to be proportional to that of the excitatory population while the inhibitory population has been accounted for implicitly. The firing rate has been described as a rectified sigmoid function of a membrane potential. The membrane potential has been described by Kirchoff’s current conservation law, which was written for a one-compartment neuron. The expressions for the excitatory and inhibitory synaptic currents, the input-output-function, the rate-based equations for the ionic dynamics, etc., are justified in \cite{Chizhov2018}. The short-term synaptic depression is described according to the Tsodyks-Markram model. An adaptive quadratic integrate-and-fire model was used as a model for a representative neuron.
 
   To take into account spatial propagation the model has been expanded with the diffusion of potassium and the spike propagation along neuronal axons. The diffusion term was included into potassium ion dynamics with the diffusion coefficient of the same value as in \cite{Bazhenov2004}. And for the spike propagation a separation of the general population firing rate on soma's and presynaptic's firing rates was considered. In that concern presynaptic's firing rate depends on soma's one and represents its value after axial propagation.
 
-  As in the previous implementation the equations are split on three subsystems that describe: (i) the ionic dynamics, (ii) the neuronal excitability, and (iii) a neuron-observer.
+  As in the previous implementation the equations are split into three subsystems that describe: (i) the ionic dynamics, (ii) the neuronal excitability, and (iii) a neuron-observer.
 
   \begin{equation}
     \label{eqn:K}
@@ -160,7 +162,7 @@
   \end{equation}
   \begin{equation}
     \label{eqn:theta}
-    \theta = \nu \text{ or } \theta = \varphi \tag{5.2}
+    \theta = \nu \text{ (for Model 1)~~~ or ~~~} \theta = \varphi \text{ (for Models 2 and 3)} \tag{5.2}
   \end{equation}
   The choice of $\theta$ in Eq. (\ref{eqn:theta}) depends on the mechanism of spatial propagation. For the case of diffusion it is $\theta = \nu$. For the case of presynaptic firing rate it is $\theta = \varphi$.
   The input current $u$ includes the potassium depolarizing current, the synaptic drive, and the noise $\xi$, respectively:
@@ -265,47 +267,48 @@
   The simulations were performed in the Python 3.6 environment. The Euler-Maruyama explicit numerical scheme was applied for the integration of the stochastic ordinary differential equations. For solving of (\ref{eqn:phi}) were used two methods: Implicit Finite Difference and Iterative Jacobi. The typical value of a time step was 1 ms. The spatial area was represented as a square with length 6 mm and discretization grid of 80x80 (80 cells in each spatial dimension). The results were dependent on the numerical parameter in a similar extent as for different realizations of noise. To preserve the results reproduction the noise seed was fixed at the value 66. The numerical realizations of the model are available from the website \href{https://bitbucket.org/vogdb/epilepsy-potassium-calculation/}{bitbucket.org/vogdb/epilepsy-potassium-calculation/};
 
   % Results and Discussion can be combined.
-  \section*{Results}
+  \section{Results}
 
   % 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.
+  We have 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 of spikes and synaptic currents through axons and dendrites isotropically distributed within the cortex.
   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
-  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.
+  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 consider a square-shaped domain of nervous tissue with a small circular central zone being a source of epileptic discharges. In the central zone, the excitability $G_{syn}$ is increased, setting $G_{syn}/g_L=5$mV$\cdot$s in contrast to $G_{syn}/g_L=1$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 \cite{Chizhov2018}.
-  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 \cite{Chizhov2019}.
+  The two 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}.
+  Ictal (ID) and interictal discharges (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 the discharges.
+  The quasi-periodic spontaneous IDs occur with an interburst interval of order of minutes. Each ID is characterized by a high rate spiking activity lasting about a few tens of seconds and consisting 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 duration of an ID. The ID begins as soon as the slowly increasing $[K]_o$ reaches a certain threshold level. Then, $[K]_o$ increases 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 the ID. After that, $[K]_o$ begins to decrease, finally returning to its baseline and even below. The phase of ID, where concentration approaching the baseline, determines the termination of the ID. The Na-K pump is activated by the elevated intracellular sodium concentration. The sodium concentration increases because of high spiking and glutamatergic synaptic activity during IDs \cite{Chizhov2019}.
   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.
+  The $Na^+/K^+$ pump peaks at the end of the 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.
 
-  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.
+  The proposed here spatially extended models show the same characteristic features of activity in the center of epileptic discharge generation as the spatially homogeneous model. From a mathematical point of view,  the dynamics of IDs is governed by the quasi-periodic oscillations of the extracellular potassium and intracellular sodium ionic concentrations, as show in \cite{Chizhov2018}. These concentrations are described with a slow subsystem of the full system of equations, based on Eq.(\ref{eqn:K}) withouth the diffusion term and Eq.(\ref{eqn:Na}). The IIDs that constitute IDs are  spontaneous large-amplitude oscillations (Fig. \ref{fig:diff_K_point_center}A,\ref{fig:syn_K_point_center}A), governed by a fast subsystem, based on Eqs.(\ref{eqn:V}, \ref{eqn:xD}).
 
 
 
   % Модель 1
-  \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 $y$.
+  \subsection{Spatial aspects in Model 1: Diffusion}
+  In the consideration of the extracellular potassium diffusion-based mechanism of activity spread, simulations were conducted  with the diffusion equation Eq.(\ref{eqn:K}) and Eq.(\ref{eqn:theta}) set to be $\theta = \nu$, thus without Eq.(\ref{eqn:phi}). %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 $[K]_o$  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 quasi-periodic spontaneous IDs occur with an interval about 110-130 seconds (Fig \ref{fig:diff_K_point_center_period}). Each ID is characterized by a high rate of activity for about 17 seconds (Fig \ref{fig:diff_K_point_center}), and consists of SBs resembling IIDs (Fig \ref{fig:diff_K_point_center}A). The potassium threshold for ID initiation is about 4mM. Initiated at the center, the IF  forms a radial wave, which spreads across the entire cortical domain, as visualized for the  field of the extracellular potassium concentration (Fig. \ref{fig:diff_K_board}). $[K]_o$ rapidly increases at the front of the wave and more gradually decreases since the activation of the Na-K pump by increasing sodium concentration, thus constituting the rear phase of the wave. $[K]_o$ finally returns to its baseline and even lower.
+  After its minimum, the potassium concentration slowly increases towards the threshold of another ID initiation.
 
-  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}).
+  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}), located at the center and periphery, and 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. The amplitude of the $[K]_o$ pulses varies within 10$\%$ (Fig. \ref{fig:diff_K_points}).
+  At the same time, the voltage patterns of IIDs are different, reflecting spontaneous bursting character of activity.
 
-  Velocity of the first K wave is about $0.047 mm/s$. The second wave is faster with velocity about $0.07mm/s$.
+  Velocity of the first potassium wave is about $0.045 mm/s$. The second wave initiates at about 149s (Fig. \ref{fig:diff_K_board}). Its velocity is greater, about $0.08mm/s$. The acceleration is due to higher level of $[K]_o$ in front of the second wave, as seen from comparison of the shapes of the first and second IDs detected at the same site S1 (Fig. \ref{fig:diff_K_points}).
 
   %TODO! The similar picture can be seen in \cite{Whalen2018}.
 
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{diff/x40y40_period.png}
-    \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 $x^D$}.
+    \caption{Diffusion mechanism (Model 1). Repeated 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}
@@ -313,7 +316,7 @@
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{diff/x40y40.png}
-    \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 $x^D$}.
+    \caption{Diffusion mechanism (Model 1). 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}
@@ -324,14 +327,14 @@
       \centering
       \captionsetup{width=0.8\linewidth}
       \includegraphics[width=1.0\linewidth]{diff/K_points.png}
-      \caption{Diffusion mechanism. A comparative plot of $[K]_o$ at two different points.}
+      \caption{Diffusion mechanism. A comparative plot of $[K]_o$ at two points S1 and S2 shown in Fig. \ref{fig:diff_K_board}.}
       \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{Diffusion mechanism. A comparative plot of $U$ at two different points.}
+      \caption{Diffusion mechanism (Model 1). A comparative plot of $U$ at the sites S1 and S2.}
       %Fig. 3
       \label{fig:diff_V_points}
     \end{minipage}
@@ -340,7 +343,7 @@
   \begin{figure}
     \centering
     \makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{diff/K_board.png}}
-    \caption{Diffusion mechanism. Potassium concentration patterns during one ID.}
+    \caption{Diffusion mechanism (Model 1). Potassium concentration spatial-temporal patterns during generation of two IDs. Two sites S1 and S2 are remote on 2mm.}
     \label{fig:diff_K_board}
   \end{figure}
 
@@ -348,13 +351,26 @@
   \subsection{Spatial aspects in Model 2: Axo-dendritic spread}
   \label{Results_Model2}
 
-  Velocity of the first K wave is about $0.15 mm/s$. The second wave is faster with velocity about $0.21mm/s$.
+  In the consideration of the synaptic mechanism of activity spread, simulations were conducted with the equation for the axo-dendritic propagation of spiking activity, Eq.(\ref{eqn:phi}) and Eq.(\ref{eqn:theta}) set to be $\theta = \varphi$, thus without Eq.(\ref{eqn:K}).
+
+  The model's behaviour at the central point is also qualitatively similar to the spatially homogeneous case, with only quantitative differences. The quasi-periodic spontaneous IDs occur with an interval about 220 seconds (Fig \ref{fig:syn_K_point_center_period}). Each ID is characterized by a high rate of activity for about 40 seconds (Fig \ref{fig:syn_K_point_center}), and consists of SBs resembling IIDs (Fig \ref{fig:syn_K_point_center}A). The potassium threshold for ID initiation is about 4mM.
+  The spatial propagation is qualitatively similar to that in simulations with Model 1.
+  Initiated at the center, the IF  forms a radial wave, which spreads across the entire cortical domain (Fig. \ref{fig:syn_K_board}).
+  $[K]_o$ rapidly increases at the front of the wave and more gradually decreases since the activation of the Na-K pump due to increasing sodium concentration (Fig \ref{fig:syn_K_point_center}C), thus constituting the rear phase of the wave. $[K]_o$ finally returns to its baseline and even lower (shots at 170s and later in (Fig \ref{fig:syn_K_board}).
+  After its minimum, the potassium concentration slowly increases towards the threshold of another ID initiation since the time moment about 381s.
+
+  Velocity of the first potassium wave is about $0.15 mm/s$. The second moves with roughly the same velocity (Fig. \ref{fig:syn_K_board}).
+
+  The wave profile remains approximately the same during its propagation, as seen from comparison of $[K]_o$ evolution at two sites (Fig. \ref{fig:syn_K_points}), located at the center and periphery, and shown in Fig. \ref{fig:syn_K_board}. The pulses of $[K]_o$, shown in Fig. \ref{fig:syn_K_points}, correspond to a single ID. However, the same ID forms different clusters of IID-like discharges seen in the voltage plot in Fig. \ref{fig:diff_V_points}. These clusters of spike bursts are different at the two sites, because of spontaneous generation of IID-like events.
+
+
+  %Velocity of the first K wave is about $0.15 mm/s$. The second wave is faster with velocity about $0.21mm/s$.
 
 
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{syn/x40y40_period.png}
-    \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 $x^D$}.
+    \caption{Synaptic mechanism  (Model 2). Repeated 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}
@@ -362,7 +378,7 @@
   \begin{figure}
     \centering
     \includegraphics[width=1.0\textwidth]{syn/x40y40.png}
-    \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 $x^D$}.
+    \caption{Synaptic mechanism (Model 2). 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}
@@ -373,14 +389,14 @@
       \centering
       \captionsetup{width=0.8\linewidth}
       \includegraphics[width=1.0\linewidth]{syn/K_points.png}
-      \caption{Synaptic mechanism. A comparative plot of $[K]_o$ at two different points.}
+      \caption{Synaptic mechanism (Model 2). 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{Synaptic mechanism. A comparative plot of $U$ at two different points.}
+      \caption{Synaptic mechanism (Model 2). A comparative plot of $U$ at two different points.}
       \label{fig:syn_V_points}
     \end{minipage}
   \end{figure}
@@ -388,18 +404,23 @@
   \begin{figure}
     \centering
     \makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{syn/K_board.png}}
-    \caption{Synaptic mechanism. Potassium concentration patterns during one ID.}
+    \caption{Synaptic mechanism (Model 2). Potassium concentration spatial-temporal patterns during generation of two IDs.}
     \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.
+  To clarify the spatial character of IID-like event generation and propagation,
+  we have outlined 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), similar to the analysis performed in \cite{Ma2012} (see Fig.6 there). A vertical coordinate -versus-time diagram (Fig. \ref{fig:syn_V_cut}B) demonstrates the effect of rapid propagation of the voltage bursts from the center of the ictal discharge generation zone to its periphery. The propagation of the voltage bursts is almost instantaneous in comparison with that of the potassium wave (Fig.\ref{fig:syn_K_board}). This effect of essentially different speeds of the waves of slow and fast variables is quite consistent with the the experimental observations from \cite{Ma2012}. In their experiment, the wave of an intrinsic optical signal that reflects the cerebral blood volume was much slower than the wave of a voltage-sensitive dye signal. Suggesting that the cerebral blood volume follows by the extracellular potassium concentration, our simulation explains that the slow wave originates as an envelope of the fast voltage bursts and travels with the speed of IDs. The slow waves in the simulation ($0.15 mm/s$) was quite comparable to that in the experiment ($0.45 mm/s$), too.
+
   \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.}
+    \makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{syn/cut_V_at163s.png}}
+    \caption{Synaptic mechanism (Model 2). A.  The membrane potential $V$ at the 163rd second of the simulation. The red bar marks the region of interest (ROI) for spatial-temporal analysis. B. Propagation pattern of the ROI within the interval of 400 ms since the time moment at 163s.}
     \label{fig:syn_V_cut}
   \end{figure}
 
+  Model 2 shows essentially larger waves than Model 1. It is seen from comparison of the domains with high $[K]_o$ in Fig.\ref{fig:syn_K_board} to those in Fig.\ref{fig:diff_K_board}. This is because of the bigger speed and longer duration of IDs in Model 2. The large waves are more consistent with experiments. Also, the speed of the wave in Model 1 is too slow in comparison to measurements (Kutsy et al. 1999; Blume
+  et al. 2001; Trevelyan 2006) which estimate the range of about 0.2-10 mm/s. These facts justify in favor of the Model 2. The potassium diffusion mechanism is too slow to explain the ictal discharge propagation. On the contrary the propagation of spikes and synaptic currents by the axons and dendrites is a sufficient mechanism to explain the ictal discharge propagation.
+
 
   % Комбинированная модель
   \subsection{Model 3: Combination of the two mechanisms of activity spread}