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\section*{Introduction}
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- 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.
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+ 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.
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\section*{Materials and methods}
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- The previous implementation of Epileptor-2 \cite{Chizhov2018} didn't consider any spatial propagation. By introducing the diffusion of $[K]_o$ and splitting general population's firing rate $\nu$ on soma's $\nu$ and presynaptic's $\varphi$ firing rates we have extended the model for the two dimensional case. Below we represent the model with these updates. Please note that variables in equations declared at the left, before the equation sign, are functions of time $(t)$ and two-dimensional space $(x,y)$.
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+ The previous implementation of Epileptor-2 \cite{Chizhov2018} didn't consider any spatial propagation. It's primary focus was a biophysical interpretation of its governing variables during the pathological states of brain activity \cite{Bazhenov2004, Kager2000, Cressman2009}. For that the implementation included ionic dynamics in 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.
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+ 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.
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+ 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.
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\begin{equation}
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\label{eqn:K}
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