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A three-dimensional mathematical model for the signal propagation on a neuron's membrane.

ABSTRACT: 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.

SUBMITTER: Xylouris K

PROVIDER: S-EPMC4505111 | biostudies-literature | 2015

REPOSITORIES: biostudies-literature

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A three-dimensional mathematical model for the signal propagation on a neuron's membrane.

Xylouris Konstantinos K Wittum Gabriel G

Frontiers in computational neuroscience 20150717

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 extrac ...[more]

PMID: 26236230

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Project description:Cell migration is essential in animal development, homeostasis, and disease progression, but many questions remain unanswered about how this process is controlled. While many kinds of individual cell movements have been characterized, less effort has been directed towards understanding how clusters of cells migrate collectively through heterogeneous, cellular environments. To explore this, we have focused on the migration of the border cells during Drosophila egg development. In this case, a cluster of different cell types coalesce and traverse as a group between large cells, called nurse cells, in the center of the egg chamber. We have developed a new model for this collective cell migration based on the forces of adhesion, repulsion, migration and stochastic fluctuation to generate the movement of discrete cells. We implement the model using Identical Math Cells, or IMCs. IMCs can each represent one biological cell of the system, or can be aggregated using increased adhesion forces to model the dynamics of larger biological cells. The domain of interest is filled with IMCs, each assigned specific biophysical properties to mimic a diversity of cell types. Using this system, we have successfully simulated the migration of the border cell cluster through an environment filled with larger cells, which represent nurse cells. Interestingly, our simulations suggest that the forces utilized in this model are sufficient to produce behaviors of the cluster that are observed in vivo, such as rotation. Our framework was developed to capture a heterogeneous cell population, and our implementation strategy allows for diverse, but precise, initial position specification over a three- dimensional domain. Therefore, we believe that this model will be useful for not only examining aspects of Drosophila oogenesis, but also for modeling other two or three-dimensional systems that have multiple cell types and where investigating the forces between cells is of interest.

Dataset Information

A three-dimensional mathematical model for the signal propagation on a neuron's membrane.

Publications

A three-dimensional mathematical model for the signal propagation on a neuron's membrane.

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