Minimal level of detail needed to model extracellular potentials originating from activity of interacting neuronal populations is an important but still open question. Whereas in regimes with moderate ionic concentration changes the neuronal contribution into the local field potential in a layered cortical or hippocampal tissue can be evaluated as the difference between the somatic and dendritic membrane potentials of pyramidal neurons, hyper-synchronous states of activity reguires taking into account a contribution of glia. We propose an effective consideration of the glial contribution that is reduced to a low-pass filtered neuronal firing rate. Our formalism is applicable to the study of cortical activity using two-compartment neuronal population models.

In experimental neuroscience, most electric signals are picked up from the electrolytic solution that constitutes extracellular (interneuronal) space, yet the origin of these potentials is incompletely understood. Our study touches the related open questions: - What level of detail is needed to model extracellular electrical phenomena, and which formalism can be employed? - How valid is the cable equation, or need slow concentration changes of the major ionic components involved be modeled explicitly? - How can LFP calculations be incorporated in simulation software? - Can single-compartmental model neurons generate local field potentials at all?

Interpretation of extracellular potentials is unclear citeBuzhaki et al. 2012, citeBedard et al. 2006, citeBedard and Destexhe 2011, citeBedard and Destexhe 2013, citeBedard and Destexhe 2014 even in the case of synchronous neuronal epileptiform events the relationship between simultaneously registered extracellular and intracellular signals is non-trivial citeAlvarado-Rojas et al. 2015. Assuming an homogeneous and isotropic extracellular space (but see citeRanck and BeMent 1965), an extracellular potential is commonly described by the Poisson equation with macroscopically averaged transmembrane currents as distributed sources citeNicholson 1973, citeNicholson and Freeman 1975 or by the Laplace equation with the transmembrane currents distributed along all membrane surfaces citeLogothetis 2007, citeLinden et al. 2011, citeSchomburg et al. 2012, citeGold et al. 2006, citeEinevoll et al. 2013, citeIbarz et al. 2010. If neuronal activity is modeled by multiple compartment neurons, then calculation of the local field potential (LFP) with the Poisson equation is straightforward. However, in most cases biophysically detailed mean-field models also deals with somatic signals of neuronal activity exclusively, and the question of how to estimate the LFP from these somatic signals is unclear citeChizhov 2014, citeBaladron et al. 2012. A simple formula linking intracellular parameters of neuronal activity to extracellular field potential has been derived in citeChizhov et al. 2015, neglecting ionic dynamics and glial contributions.

In our reduced consideration of the LFP interpretation problem we assume a layer of homogeneously distributed neurons and glial cell. As shown in (Chizhov 2015), in the first approximation, the relationship between local field potential (LFP) and intracellular signals in layered neural tissue is governed by the balance of postsynaptic, leak, capacity and trans-dendritic currents. This contribution is approximated by a proportionality of the LFP to the dendritic and somatic voltage difference. In the second approximation, the currents during action potentials contributes into the LFP, giving an additional term proportional to the neuronal firing rate. However, this model does not take into consideration the contributions of glia and slow extracellular ionic concentration changes. Taking this into account might be especially important in cases of rather intensive synchronized neuronal activity such as epileptic discharge. In the resent paper, we assert that the next important factor (third-order approximation) is the change of the extracellular potassium concentration, \([K^{+}]_{e}\). The increase of \([K^{+}]_{e}\) in time scales of tens and hundreds of milliseconds is caused by ionic flows through neuronal membranes, mainly due to voltage gated potassium channels, compared to slow effect of active and passive transporters. The increase is then buffered by glial cells, which effectively filter the impact of the transneuronal potassium flow (Wallraff 2006). The glial membrane potassium currents contribute into the LFP, in addition to the neuronal membrane currents. As a result the LFP model has two terms.

According to (Chizhov 2015), the LFP at the somatic level, \(\varphi_{N}(t)\), is determined by the neuronal membrane voltage distribution and the firing activity as follows:

\begin{equation}
\label{e1}\varphi_{N}(t)=\frac{p}{2\,\sigma\,r_{i}}\bigl{(}V_{d}(t)-V_{s}(t)\bigr{)}+k_{1}~{}\nu(t)\\
\end{equation}

where \(V_{s}(t)\) and \(V_{d}(t)\) are the somatic and dendritic neuronal membrane potentials; neurons are homogeneously distributed with the density \(p\) per area; \(r_{i}\) is the specific intracellular resistivity; \(\sigma\) is the mean conductivity of extracellular medium, assumed to be a constant; \(\nu(t)\) is the firing rate; \(k_{1}\) is a coefficient.

The glial contribution into LFP, \(\varphi_{G}(t)\), is proportial to the glial membrane currents of potassium ions, thus can be approximated as follows:

\begin{equation}
\label{e2}\varphi_{G}(t)=k_{G}\biggl{[}{d[K^{+}]_{e}\over dt}\biggr{]}_{Glia},\\
\end{equation}

where \(\Bigl{[}{d[K^{+}]_{e}\over dt}\Bigr{]}_{Glia}\) is the potassium concentration change due to glia; \(k_{G}\) is a coefficient.

External potassium was modeled in (Krishnan 2011). Glial uptake was modeled in (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)+k_{2}~{}\nu(t),\\
\end{equation}

where \(\tau\) is the characteristic time constant; \(k_{2}\) 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.

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_{1}~{}\nu(t)+k_{3}~{}\bigl{(}V_{d}(t)-V_{s}(t)\bigr{)}+\varphi_{G}(t)\\
\end{equation}

where \(k_{3}={p}/({2\,\sigma\,r_{i}})\). 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 \(V_{s}(t)\) and \(V_{d}(t)\) as well as the firing rate of principal neurons \(\nu(t)\). The model might be matched to experimental data by fitting four coefficients, \(k_{1}\), \(k_{2}\), \(k_{3}\) and \(\tau\).

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