update article

Abstract
More details to Materials and Methods
Introduction

article/article.bib

@@ -50,4 +50,174 @@
   author = {E.M. Izhikevich},
   title = {Simple model of spiking neurons},
   journal = {{IEEE} Transactions on Neural Networks}
-},
+},
+@article{Trevelyan2006,
+  doi = {10.1523/jneurosci.2787-06.2006},
+  url = {https://doi.org/10.1523/jneurosci.2787-06.2006},
+  year  = {2006},
+  month = {nov},
+  publisher = {Society for Neuroscience},
+  volume = {26},
+  number = {48},
+  pages = {12447--12455},
+  author = {A. J. Trevelyan and D. Sussillo and B. O. Watson and R. Yuste},
+  title = {Modular Propagation of Epileptiform Activity: Evidence for an Inhibitory Veto in Neocortex},
+  journal = {Journal of Neuroscience}
+},
+@article{Trevelyan2007,
+  doi = {10.1523/jneurosci.0145-07.2007},
+  url = {https://doi.org/10.1523/jneurosci.0145-07.2007},
+  year  = {2007},
+  month = {mar},
+  publisher = {Society for Neuroscience},
+  volume = {27},
+  number = {13},
+  pages = {3383--3387},
+  author = {A. J. Trevelyan and D. Sussillo and R. Yuste},
+  title = {Feedforward Inhibition Contributes to the Control of Epileptiform Propagation Speed},
+  journal = {Journal of Neuroscience}
+},
+@article{Smith2016,
+  doi = {10.1038/ncomms11098},
+  url = {https://doi.org/10.1038/ncomms11098},
+  year  = {2016},
+  month = {mar},
+  publisher = {Springer Nature},
+  volume = {7},
+  number = {1},
+  author = {Elliot H. Smith and Jyun-you Liou and Tyler S. Davis and Edward M. Merricks and Spencer S. Kellis and Shennan A. Weiss and Bradley Greger and Paul A. House and Guy M. McKhann II and Robert R. Goodman and Ronald G. Emerson and Lisa M. Bateman and Andrew J. Trevelyan and Catherine A. Schevon},
+  title = {The ictal wavefront is the spatiotemporal source of discharges during spontaneous human seizures},
+  journal = {Nature Communications}
+},
+@article{Wenzel2017,
+  doi = {10.1016/j.celrep.2017.05.090},
+  url = {https://doi.org/10.1016/j.celrep.2017.05.090},
+  year  = {2017},
+  month = {jun},
+  publisher = {Elsevier {BV}},
+  volume = {19},
+  number = {13},
+  pages = {2681--2693},
+  author = {Michael Wenzel and Jordan P. Hamm and Darcy S. Peterka and Rafael Yuste},
+  title = {Reliable and Elastic Propagation of Cortical Seizures In~Vivo},
+  journal = {Cell Reports}
+},
+@article{Wang2017,
+  doi = {10.1371/journal.pcbi.1005475},
+  url = {https://doi.org/10.1371/journal.pcbi.1005475},
+  year  = {2017},
+  month = {may},
+  publisher = {Public Library of Science ({PLoS})},
+  volume = {13},
+  number = {5},
+  pages = {e1005475},
+  author = {Yujiang Wang and Andrew J Trevelyan and Antonio Valentin and Gonzalo Alarcon and Peter N Taylor and Marcus Kaiser},
+  editor = {William W Lytton},
+  title = {Mechanisms underlying different onset patterns of focal seizures},
+  journal = {{PLOS} Computational Biology}
+},
+@article{Chizhov2017,
+  doi = {10.1371/journal.pone.0185752},
+  url = {https://doi.org/10.1371/journal.pone.0185752},
+  year  = {2017},
+  month = {oct},
+  publisher = {Public Library of Science ({PLoS})},
+  volume = {12},
+  number = {10},
+  pages = {e0185752},
+  author = {Anton V. Chizhov and Dmitry V. Amakhin and Aleksey V. Zaitsev},
+  editor = {Gennady Cymbalyuk},
+  title = {Computational model of interictal discharges triggered by interneurons},
+  journal = {{PLOS} {ONE}}
+},
+@article{Wei2014,
+  doi = {10.1152/jn.00541.2013},
+  url = {https://doi.org/10.1152/jn.00541.2013},
+  year  = {2014},
+  month = {jul},
+  publisher = {American Physiological Society},
+  volume = {112},
+  number = {2},
+  pages = {213--223},
+  author = {Yina Wei and Ghanim Ullah and Justin Ingram and Steven J. Schiff},
+  title = {Oxygen and seizure dynamics: {II}. Computational modeling},
+  journal = {Journal of Neurophysiology}
+},
+@article{Bazhenov2004,
+  doi = {10.1152/jn.00529.2003},
+  url = {https://doi.org/10.1152/jn.00529.2003},
+  year  = {2004},
+  month = {aug},
+  publisher = {American Physiological Society},
+  volume = {92},
+  number = {2},
+  pages = {1116--1132},
+  author = {M. Bazhenov and I. Timofeev and M. Steriade and T. J. Sejnowski},
+  title = {Potassium Model for Slow (2-3 Hz) In Vivo Neocortical Paroxysmal Oscillations},
+  journal = {Journal of Neurophysiology}
+},
+@article{Krishnan2011,
+  doi = {10.1523/jneurosci.6200-10.2011},
+  url = {https://doi.org/10.1523/jneurosci.6200-10.2011},
+  year  = {2011},
+  month = {jun},
+  publisher = {Society for Neuroscience},
+  volume = {31},
+  number = {24},
+  pages = {8870--8882},
+  author = {G. P. Krishnan and M. Bazhenov},
+  title = {Ionic Dynamics Mediate Spontaneous Termination of Seizures and Postictal Depression State},
+  journal = {Journal of Neuroscience}
+},
+@article{Jirsa2014,
+  doi = {10.1093/brain/awu133},
+  url = {https://doi.org/10.1093/brain/awu133},
+  year  = {2014},
+  month = {jun},
+  publisher = {Oxford University Press ({OUP})},
+  volume = {137},
+  number = {8},
+  pages = {2210--2230},
+  author = {Viktor K. Jirsa and William C. Stacey and Pascale P. Quilichini and Anton I. Ivanov and Christophe Bernard},
+  title = {On the nature of seizure dynamics},
+  journal = {Brain}
+},
+@article{Mller2018,
+  doi = {10.1002/adfm.201704598},
+  url = {https://doi.org/10.1002/adfm.201704598},
+  year  = {2018},
+  month = {jan},
+  publisher = {Wiley},
+  volume = {28},
+  number = {9},
+  pages = {1704598},
+  author = {Bernhard J. M\"{u}ller and Alexander V. Zhdanov and Sergey M. Borisov and Tara Foley and Irina A. Okkelman and Vassiliy Tsytsarev and Qinggong Tang and Reha S. Erzurumlu and Yu Chen and Haijiang Zhang and Claudio Toncelli and Ingo Klimant and Dmitri B. Papkovsky and Ruslan I. Dmitriev},
+  title = {Nanoparticle-Based Fluoroionophore for Analysis of Potassium Ion Dynamics in 3D Tissue Models and In Vivo},
+  journal = {Advanced Functional Materials}
+},
+@article{Kager2000,
+  doi = {10.1152/jn.2000.84.1.495},
+  url = {https://doi.org/10.1152/jn.2000.84.1.495},
+  year  = {2000},
+  month = {jul},
+  publisher = {American Physiological Society},
+  volume = {84},
+  number = {1},
+  pages = {495--512},
+  author = {H. Kager and W. J. Wadman and G. G. Somjen},
+  title = {Simulated Seizures and Spreading Depression in a Neuron Model Incorporating Interstitial Space and Ion Concentrations},
+  journal = {Journal of Neurophysiology}
+},
+@article{Cressman2009,
+  doi = {10.1007/s10827-008-0132-4},
+  url = {https://doi.org/10.1007/s10827-008-0132-4},
+  year  = {2009},
+  month = {jan},
+  publisher = {Springer Nature},
+  volume = {26},
+  number = {2},
+  pages = {159--170},
+  author = {John R. Cressman and Ghanim Ullah and Jokubas Ziburkus and Steven J. Schiff and Ernest Barreto},
+  title = {The influence of sodium and potassium dynamics on excitability,  seizures,  and the stability of persistent states: I. Single neuron dynamics},
+  journal = {Journal of Computational Neuroscience}
+}

article/article.tex

@@ -31,6 +31,9 @@
 % array package and thick rules for tables
 \usepackage{array}
 
+% package to indent first paragraphs
+\usepackage{indentfirst}
+
 % create \thickcline for thick horizontal lines of variable length
 \newlength\savedwidth
 \newcommand\thickcline[1]{%
@@ -113,11 +116,11 @@
   \section*{Abstract}
   Should be in Abstract: First the motivation. What task is actual and unsolved. Further methods, results and conclusion.
 
-  The model of Epileptor-2 brings a new hypothesis about the mechanism of interictal and ictal discharges (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 diffusion of potassium and spike propagation along neurons axons from a soma to a presynaptic terminal.
+  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*{Materials and methods}
@@ -140,6 +143,8 @@
     \label{eqn:xD}
     \frac{dx^D}{dt} = \frac{1-x^D}{\tau_D} - \delta_{x} x^D\varphi
   \end{equation}
+
+  $[K]_o$ and $[Na]_i$ are extracellular potassium concentration and intraneuronal sodium concentration respectively. $V$ is the membrane depolarization. $x^D$ is the synaptic resource. $\theta$ is the firing rate of the excitatory population for the chosen mechanism of propagation. It equals either $\nu$ or $\varphi$ where $\nu$ is the firing rate of the excitatory population on it's somas and $\varphi$ is the firing rate of the excitatory population on it's presynapses. Inhibitory population firing rate is assumed to be proportional to $\nu$. The dynamics described by these equations is driven by $\nu$, which is calculated with a sigmoidal input-output function where $[x]_+$ is equal to $x$ for the positive argument and $0$ otherwise.
   \begin{equation}
     \label{eqn:nu}
     \nu = \nu_{max}\Bigg[\frac{2}{1+\exp\Big[-2(V(t)-V_{th})/k_v\Big]}-1\Bigg]_{+}
@@ -150,20 +155,25 @@
   \end{equation}
   \begin{equation}
     \label{eqn:theta}
-    \theta = 0.33\varphi + 0.66\nu \tag{5.2}
+    \theta = \nu \text{ or } \theta = \varphi \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:
   \begin{equation}
     \label{eqn:uu_diff}
     u = g_{K,leak}(V_K - V^0_K) + G_{syn}\varphi(x^D-0.5) + \sigma\xi
   \end{equation}
+  The potassium reversal potential is obtained from the ion concentrations via the Nernst equation:
   \begin{equation}
     \label{eqn:V_K}
     V_K = 26.6mV\ln\biggl(\frac{[K]_o}{130mM}\biggr)
   \end{equation}
+  The $Na^+/K^+$ pump current is taken from \cite{Cressman2009} in the form:
   \begin{equation}
     \label{eqn:I_pump}
     I_{pump} = \frac{\rho}{\big(1+\exp(3.5-[K]_o)\big)\big(1+\exp\big(({25-[Na]_i})/{3}\big)\big)}
   \end{equation}
+  The representative neuron is modeled with an adaptive quadratic integrate-and-fire neuron \cite{Izhikevich2003}. The equations for the membrane potential $U$ and the adaptation current $w$ are as follows:
   \begin{equation}
     \label{eqn:U}
     C_U\frac{dU}{dt} = g_U(U-U_1)(U-U_2) - w + u + I_{a}
@@ -177,13 +187,8 @@
     \text{if } U > V^T \text{ then } U = V_{reset}, w = w + \delta w
   \end{equation}
 
-  % Описание смысла уравнений, хотя бы в виде текста между формулами из статьи Chizhov//Neurocomputing-2018.
-
-  (\ref{eqn:K}) and (\ref{eqn:Na}) are ionic dynamics of extracellular potassium concentration $[K]_o(t,x,y)$ and intraneuronal sodium concentration $[Na]_i(t,x,y)$ respectively; (\ref{eqn:V}) is the membrane depolarization equation $V(t,x,y)$; (\ref{eqn:xD}) is the synaptic resource $x^D(t,x,y)$; (\ref{eqn:nu}) is the firing rate of the excitatory population on it's somas $\nu(t,x,y)$; (\ref{eqn:phi}) is the firing rate of the excitatory population on it's presynapses $\varphi(t,x,y)$; (\ref{eqn:theta}) is the combined firing rate of the excitatory population $\theta(t,x,y)$; (\ref{eqn:uu_diff}) is the input current $u(t,x,y)$; (\ref{eqn:V_K}) is the potassium reversal potential $V_K(t,x,y)$; (\ref{eqn:I_pump}) is the $Na^+/K^+$ pump current $I_{pump}(t,x,y)$;
-  (\ref{eqn:U}), (\ref{eqn:w}), (\ref{eqn:U_reset}) are the representative neuron membrane depolarization $U(t,x,y)$ which is an adaptive integrate-and-fire (AIF) model \cite{Izhikevich2003}.
-
   %\subsubsection{The extensions to the original Epileptor-2 model}
-  This new spatially extended version of the model includes the diffusion term $D_K\big(\partial^2{[K]_o}/\partial{x^2} + \partial^2{[K]_o}/\partial{y^2}\big)$ in the equation for $[K]_o$ (\ref{eqn:K}) and the equation for the presynaptic firing rate (\ref{eqn:phi}).
+  This new spatially extended version of the model includes the diffusion term $D_K\big(\partial^2{[K]_o}/\partial{x^2} + \partial^2{[K]_o}/\partial{y^2}\big)$ in the equation for $[K]_o$ (\ref{eqn:K}) and the equation for the presynaptic firing rate (\ref{eqn:phi}), (\ref{eqn:theta}).
   The eq.(\ref{eqn:phi}) implies that the presynaptic firing rate $\varphi$ is obtained as a convolution of the somatic firing rate $\nu$ with the exponentially decaying kernel $\exp(-r/\lambda)$, where $r$ is the distance between pre- and postsynaptic neurons and $\lambda$ is the characteristic length of the connectivity profile.
 
   Alternatively, the eq.(\ref{eqn:phi}) can be derived from the mean field equations proposed in \cite{Jirsa1996}, \cite{Robinson1997} in assumption that the axonal delay is negligible. In our consideration, the axonal delay is much smaller than the membrane time constant which is the time scale in the Epileptor-2 model, thus it is to be neglected.
@@ -251,6 +256,8 @@
 
              \hline
   \end{tabular}
+  \subsection*{Modeling and analysis}
+  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}
@@ -267,7 +274,7 @@
 
   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 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.
 
 
 
@@ -275,7 +282,7 @@
 
   % 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 $\theta = \varphi$.
 
 
   \begin{figure}