^{*}

Edited by: Tobias Alecio Mattei, Brain & Spine Center - InvisionHealth - Kenmore Mercy Hospital, USA

Reviewed by: Ingo Bojak, University of Reading, UK; Le Wang, Boston University, USA; Xin Tian, Tianjin Medical University, China; Mikhail Katkov, Weizmann Institute of Science, Israel

*Correspondence: Konstantinos Xylouris, Department of Simulation and Modeling, Faculty of Informatics, Goethe Center for Scientific Computing, Goethe University Frankfurt, Kettenhofweg 139, Frankfurt am Main 60325, Germany

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution and reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In order to be able to examine the extracellular potential's influence on network activity and to better understand dipole properties of the extracellular potential, we present and analyze a three-dimensional formulation of the cable equation which facilitates numeric simulations. When the neuron's intra- and extracellular space is assumed to be purely resistive (i.e., no free charges), the balance law of electric fluxes leads to the Laplace equation for the distribution of the intra- and extracellular potential. Moreover, the flux across the neuron's membrane is continuous. This observation already delivers the three dimensional cable equation. The coupling of the intra- and extracellular potential over the membrane is not trivial. Here, we present a continuous extension of the extracellular potential to the intracellular space and combine the resulting equation with the intracellular problem. This approach makes the system numerically accessible. On the basis of the assumed pure resistive intra- and extracellular spaces, we conclude that a cell's out-flux balances out completely. As a consequence neurons do not own any current monopoles. We present a rigorous analysis with spherical harmonics for the extracellular potential by approximating the neuron's geometry to a sphere. Furthermore, we show with first numeric simulations on idealized circumstances that the extracellular potential can have a decisive effect on network activity through ephaptic interactions.

The membrane potential belongs to the most important quantities of a neuron. Its function of time and space describes neuronal activity. It is a voltage across the membrane defined by the difference between the intra- and extracellular potential.

Since the neuron is embedded in ionic milieus, potential gradients in the off-membrane spaces result in electric fluxes, which are conserved according to the first principles. This conservation law is the basis of the standard cable equation which describes the unfolding and propagation of an action potential (Rall,

The resulting extracellular potentials can be theoretically computed with the line source method (Holt and Koch,

These extracellular potentials in turn can be exploited to examine ephaptic feedbacks on other neurons (Holt and Koch,

The goal of the current paper is to develop and implement an integrated three-dimensional model which synchronously captures both quantities, the membrane potential and the extracellular potential, during activity and which uses the neuron's geometry as it is instead of reducing it to cylindric compartments. The aim of such a model is to deepen the knowledge in signal processing and to carry out simulations on small networks of realistic neurons while having all these influences in action.

The work of Voßen et al. (

The study of Xylouris et al. (

This paper introduces a completely new coupling of the unknowns. Therein, the defining equation for the membrane potential contains its own spacial differential operator. For the first time, we could carry out simulations on three-dimensionally resolved ideal neurons and on a small network of cells. This description, furthermore, allows for a proof that the extracellular potential distributes in the extracellular space like a current multipole. It will show that the only current monopole for a neuron exists at rest.

Let Ω_{in} and Ω_{out} be domains in ℝ^{3} denoting the neuron's intra- and extracellular space, respectively, and ^{3}. Let _{in}, Φ_{out}, and _{m} be the intra-, extracellular, and membrane potential, respectively. Φ will represent either Φ_{in} or Φ_{out}.

The quantities σ_{in} and σ_{out} denote the intra- and extracellular conductivities, respectively. The normal _{in → out} is the normal on the membrane Γ pointing from the intracellular space to the extracellular. We will need this quantities in order to define the fluxes. For the active transmembrane flux, we will just consider the Hodgkin–Huxley model for the sake of a simpler writing. There we have the sodium conductivity _{L}. The quantities _{L} denote the reversal potentials of the indexed ions. The gating parameters

Considering the non-membrane conductivity (≈ 3_{in} and Ω_{out}). Indeed, this is the basis of the derivation for the three dimensional cable equation. In addition, we will assume to have time invariant magnetic fields (

The constants ϵ_{0} and ϵ are the dielectricities in vacuum and material, respectively.

Because of flux continuity, the flux across the membrane is continuous and must correspond to the flux emerging from the membrane dynamics [denoted with _{all}(_{m})]. Hence,

With this boundary condition in mind, we arrive at the three-dimensional cable equation (Figure

The flux _{all} contains all fluxes passing the membrane. Considering just the Hodgkin–Huxley model and some additional stimulus, it looks like:

Since it is possible to have different dynamics on each region of the neuronal membrane, we furthermore introduce the following δ-functions

We define

The synaptic activity is simply modeled with the aid of a modified Heaviside function _{m}|_{pre},_{m}|_{pre} is the membrane potential at the presynaptic terminal.

Then the refined total transmembrane current has the form:

_{all}. Within _{all} all transmembrane currents are accumulated: capacitive, channel, any stimulation or synaptic currents.

The three dimensional cable equation (Equations 6–8) is a non-symmetric system (Φ_{in} does not couple with Φ_{out} the same way as Φ_{out} with Φ_{in}) of PDEs which couples two Laplace equations in the intra- and extracellular space with the transmembrane flux. This flux depends on the membrane potential. One difficulty in solving this system is the coupling of the membrane potential, which lives on a lower dimensional manifold, with the quantities, which live in full space. Since the discretization of this system is carried out with the help of integrals, the lower dimensional quantity cannot be measured the same way as the quantities in space (because the space integrals do not see it at all). In order to get rid of this particularity, we will extend the membrane potential, which is defined by the difference between the intra- and extracellular potential (_{m} = Φ_{in} − Φ_{out}) on the membrane, to the intracellular space. To that end, we extend the extracellular potential to the intracellular space and combine its extension with the intracellular potential equation. So, we arrive at a problem for the membrane potential in the intracellular space.

Because _{m} = Φ_{in} − Φ_{out} on the membrane Γ, we will extend Φ_{out} to the intracellular space continuously so that the following identity holds. Let this extension be denoted with

At this point we have some freedom to choose the right hand side of the extracellular potential extension equation. We choose it to be zero. Then it can be easily combined with the intracellular problem (Equation 7), which is a Lapalcian, too. We have

Thus, instead of solving the system (Equations 6–8) we solve (Figure

For referencing reasons, we will call the additional current, which is considered in the boundary condition of the membrane potential equation (Equation 19), as ephaptic current

In space, we discretize this system (Equations 17–19) with the finite volume method (Versteeg and Malalasekera,

Similarly to the finite element method, we discretize the domain Ω with volume elements, for example tetrahedrals, whose edge points and edges form the grid Ω_{h}, and we approximate the unknown functions (in our case _{m},Φ_{out}, and _{j}(_{h} = _{h} =

For the finite volume method, we need to construct a so called dual grid, which arises from the domain discretization and which is used in order to discretize the differential space operators. We call the elements of the dual grid control volumes. The volume elements of the dual grid are defined by the edge points which correspond to the barycenters of the initial tetrahedrals and the barycenters of its sides and edges. By this construction, we create as many control volumes as we have nodes in the grid Ω_{h}. Let _{k} be the control volume of the _{k} is a polyhedron and _{j}(

The intracellular problem (Equation 7) is a Laplace problem with a Neumann boundary. We referred this to the approximation of purely resistive non-membrane spaces (i.e., the intra- and extracellular space do not contain any free charges). Thus, the driving force of the intracellular potential is given by its Neumann-flux on the boundary (i.e., the membrane). Now, integrating the Laplace equation over the whole neuron and applying Gauß's theorem yields an important constrain for the transmembrane currents: The fluxes are balanced out over the whole membrane at each point of time!

There are at least two important implications of this situation. First, an influx at some point of the membrane, necessarily leads to an out-flux at some other point of the membrane with the same total amount of current. Moreover, this must happen simultaneously, since otherwise the condition is violated.

Second, the extracellular potential distributes like a multipole in the extracellular space.

Regardless of the neuron's shape, the extracellular potential equation (Equation 17) demonstrates that its only source is the transmembrane flux as expressed through its boundary condition. A current monopole of the extracellular potential would be defined by the overall transmembrane flux. Yet, this flux is always zero as shown before (Equation 29). Thus, there is no monopole component and the extracellular potential distributes in space like a current multipole. To get some quantitative idea of its distribution, we approximate the neuron's geometry to a sphere. Then, we are able to express the extracellular potential with a generalized Fourier series of spherical harmonics.

Let Ω_{in} = _{R} be a sphere with radius _{R} its boundary. The spherical harmonics _{out} is concretized by the coefficients _{lm}. These are determined by the transmembrane flux _{all}(_{m}):

Especially, we obtain for the first coefficient _{00} which corresponds to the potential of a monopole:

NEURON (Hines and Carnevale,

We use proMesh (Reiter,

This test domain we now use in order to first verify the the correct implementation of our discretization schema and second in order to see that we indeed obtain almost identical solutions in comparison with those produced by NEURON.

First is obtained, if the computed solution converges as the computational grid fineness is increased. In order to assess the second point, we have to compare the one dimensional solution of NEURON with the three-dimensional solution of our model. By construction of the one dimensional cable equation, each quantity, although computed on every point of a line, actually represents a volumetric quantity. Thus, the one-dimensional model assumes for all quantities to be radial symmetric and iso-potential on cross-sections of a three-dimensional cylinder. Considering this particularity, we can blow up the solution of NEURON to a three-dimensional solution and compare it with the solution of our model or we compare NEURON's solution with our solution recorded on the cylinder axis. For the sake of simplicity, we use the second way considering that its difference with the volumetric comparison is just the factor of the cross-section area.

Because for three dimensional numeric computations, domains have to be discretized, even simple cylinders never correspond to ideal cylinders, which, however, are the basis of the one-dimensional model. Thus, we will always expect small quantitative differences in such a comparison and, therefore, we are already satisfied to evaluate the differences with NEURON with the aid of an Euclidean integral norm
_{mNEURON} and the solution computed at refinement level _{mLevel x}, over the interval [0,

Concerning the numeric convergence at grid refinement, we computed the solution on our cylinder, composed by a tetrahedral grid, at two levels of refinement and observed the desired convergence (Figure

The solution between the standard cable equation and the three dimensional model are qualitatively undistinguishable (Figure

_{mLevel 0} |
0.2002 |

_{mLevel 1} |
0.1174 |

_{mLevel 2} |
0.1174 |

_{mNEURON} and the solution computed on refinement level x, denoted with V_{mLevel x} is very small. This implies that qualitative characteristics like propagation speed, signal width as well are very similar. The small differences measured here can be explained with the nature of the three-dimensional model which automatically considers the extracellular potential in the signal processing and which works with discretized and finite domains (in this case: cylinders are supposed to be ideal and infinite for the standard cable equation)

However as regards the emerging of the action potential (Table

With a computationally quite demanding simulation, we also solve the Equations (17–19) on a more complicated geometry representing four idealized neurons with chemical synapses (Figure

The simulation is demanding, because we have a non-linear time-dependent domain problem in three dimensions. It means we solve several a huge linear systems in each time step within Newton's method. Thereby, the time step to be chosen is constrained by the fast dynamics of the active membrane's gating variables, which in our case is chosen with 10 μs, while we aim to simulate the time period of 14 ms. This means we need to compute the solution for 1400 time steps, which is time demanding despite parallel procedures due to the geometry's complexity.

We constructed the computational domain given by a small network of four neurons with the help of an algorithm developed in Niklas Antes' master thesis (Antes,

As regards the transmembrane current _{all}(_{m}) (Equation 12) for the different cell parts, we just considered passive properties on the dendrites while an active membrane reflecting Hodgkin–Huxley dynamics for the soma as well as for the nodes of Ranvier. On the myelinated sheaths, the transmembrane current _{all}(_{m}) is composed of the first term in Equation (12) only, the capacitive current. Furthermore, two of the cells (cell 1 and cell 4, see Figure

Because we simulate the relatively small time period of 14 ms, we let the synapses work as pre-defined strong post-synaptic current pulses of some nA, which are triggered as soon as the membrane potential at the pre-synapse indicates that an action potential has arrived. This is assumed to happen when the membrane potential at the pre-synapse exceeds the value of 5 mV.

For the sake of simplicity, we choose a constant intra- and extracellular conductivity

We activate the network by stimulating cell number one (see Figure

The model integrates the impact of the extracellular potential into the signal processing. Though its impact is rather small, it still can have a significant effect when combined with the right stimulation at the right time. Action potentials can arise, which otherwise would not show up (Figure

The three-dimensional passive model of Voßen et al. (

For the sake of verifying the correct implementation of this model and because it should deliver similar results as the one-dimensional cable equation for the limit case of long and thin cylinders, we carried out a comparison with NEURON and obtained very good agreement between the two models.

Based on the assumption of charge-free non-membrane spaces -an assumption also used for the derivation of the standard cable equation-, we could provide strong theoretical evidence (to our knowledge for the first time) with the aid of the three-dimensional model that there aren't any current monopoles as the overall out-flux across the membrane balances out. A significant consequence of this behavior is that the leading term of the extracellular potential's multipole expansion vanishes so that it falls in space with higher powers of its distance to the transmembrane current source. In the work of Lindén et al. (

We consider the ability to carry out realistic simulations with the cable equation on three-dimensionally resolved ideal neurons as important step and milestone on the way of refining and generalizing existing models for neuronal activity. This three dimensional model facilitates gaining a better understanding of all the processes involved in the signal processing, especially the influence of the extracellular potential activity on the membrane and the impact of the precise three-dimensional shape of the neuron's geometry. Concerning the ephaptic communication, it would be interesting to further investigate its influence on synchronous firing within networks. The latter point also seems to be very promising since lots of precise experimental geometric data are produced. Questions connecting function with geometry can be directly tackled with this model.

However, there is still a long way to go on this path, as the biggest challenge at the moment for our model is its computational demand. Further algorithmic and computational analysis needs to be invested in order to make applicable cutting edge solvers of linear systems arising from partial differential equations -like algebraic multi grid methods- on highly parallel machines, even on graphic card clusters. As next steps, we want to focus on these improvements.

On the other hand, the computational efficiency is a big advantage for standard one dimensional cable equation. Once we accomplished this efficiency for the three-dimensional model, there are still lots of interesting applications which we wish to address- especially concerning backward modeling with questions like which are the underlying network properties in order to reproduce a given a extracellular potential activity wave.

Furthermore, we see the need of a deeper theoretical analysis of this model with the purpose to provide a mathematical proof that it converges to the standard cable equation for the limit case of infinite cylinders and vanishing extracellular resistivity.

Our long-range purpose is to generalize this model with homogenization and multi-scale techniques so that to be able to simulate the activity of bigger clusters of neuronal networks while also considering the detail in processing on the small scale.

Realized steps on this path will be hopefully items of future publications.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement number 650003 (Human Brain Project).

The Supplementary Material for this article can be found online at: