### Abstract

Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates (r, θ, ϕ). The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

Original language | English (US) |
---|---|

Article number | 9 |

Journal | Journal of Mathematical Neuroscience |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2016 |

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### Keywords

- Analytic modeling
- Bidomain

### ASJC Scopus subject areas

- Neuroscience (miscellaneous)

### Cite this

*Journal of Mathematical Neuroscience*,

*6*(1), [9]. https://doi.org/10.1186/s13408-016-0041-1

**Analytic Modeling of Neural Tissue : I. A Spherical Bidomain.** / Schwartz, Benjamin L.; Chauhan, Munish; Sadleir, Rosalind.

Research output: Contribution to journal › Article

*Journal of Mathematical Neuroscience*, vol. 6, no. 1, 9. https://doi.org/10.1186/s13408-016-0041-1

}

TY - JOUR

T1 - Analytic Modeling of Neural Tissue

T2 - I. A Spherical Bidomain

AU - Schwartz, Benjamin L.

AU - Chauhan, Munish

AU - Sadleir, Rosalind

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates (r, θ, ϕ). The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

AB - Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates (r, θ, ϕ). The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

KW - Analytic modeling

KW - Bidomain

UR - http://www.scopus.com/inward/record.url?scp=84986630234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84986630234&partnerID=8YFLogxK

U2 - 10.1186/s13408-016-0041-1

DO - 10.1186/s13408-016-0041-1

M3 - Article

VL - 6

JO - Journal of Mathematical Neuroscience

JF - Journal of Mathematical Neuroscience

SN - 2190-8567

IS - 1

M1 - 9

ER -