Anton Chizhov edited untitled.tex  almost 8 years ago

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External potassium was modeled in \cite{Krishnan_2011}. Glial uptake was modeled in \cite{Kager_2006}. To simplify the consideration, we proposed the following phenomenological model. Potassium currents through neuronal membranes are most pronounced during the spiking activity. That is why, approximately these currents are just proportional to the firing rate of principal neurons, $\nu(t)$. $[K^+]_e$ is then buffered by glial cells. Concequently, the equation for the rate of potassium concentration change due to glia should look like the refractory equation, as well as the associated extracellular potential $\varphi_G$:  \begin{equation} \label{e3}  \tau {\varphi_G(t) \over dt } = - \varphi_G(t) + c k_2  ~\nu(t), \end{equation}  where $\tau$ is the characteristic time constant; $c$ $k_2$  is a coefficient. The time constant cumulatively express the kinetics of neuronal and glial voltage-gated channels, as well as the scales of glial intracellular processes evoked by neuronal activity.  \subsection {Cumulative LFP model}  The contributions of potassium currents through mebranes of both neurons and glial cells are additive. Thus taking into account e.(\ref{e1}), we get  \begin{equation} \label{e4}  \varphi(t) = k_N k_3  \bigl(V_d(t)-V_s(t)\bigr) + \varphi_G(t) \end{equation}  where $k_N={p}/({2\, $k_3={p}/({2\,  \sigma \, r_{i} })$. A set of equations (\ref{e3}) and (\ref{e4}) determines the LFP model. According to these equations, LFP is governed by the somatic and dendritic membrane voltages as well as the firing rate of principal neurons. The model might be matched to experimental data by fitting three coefficients, $k_N$, $\tau$ $k_1$, $k_2$, $k_3$  and $c$. $\tau$.