Cable equation for general geometry

The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.

Figure 1

Cable with general geometry. The vectors $\widehat{T},\widehat{N},\widehat{B}$ are shown in two different points on the curve $\vec{\gamma }$

Figure 2

Voltage for the cylindric cable. Parameter values used for simulations correspond to realistic dendritic parameters as in Ref. [14]: $c_M=1\mathrm{mF}/{\mathrm{cm}}^2,r_M=3000\mathrm{\Omega }{\mathrm{cm}}^2,r_L=100\mathrm{\Omega }\mathrm{cm},R_0={10}^{-4}\mathrm{cm}.$ The initial condition is given by (47).

Figure 3

(a) Cable with radius variable (38). (b) Voltage (39). Parameter values used for simulations correspond to realistic dendritic parameters as in Ref. [14]: $c_M=1\mathrm{mF}/{\mathrm{cm}}^2,r_M=3000\mathrm{\Omega }{\mathrm{cm}}^2,r_L=100\mathrm{\Omega }\mathrm{cm},R_0={10}^{-4}\mathrm{cm},V_0=1\mathrm{mV},l_0=1\mathrm{cm}.$

Figure 4

(a) Cable with geometry (48). (b) Cable with geometry (50). (c) Maximum voltage for a cable with radius (50) at time $t=1000\mathrm{s}$ vs the parameter ${\alpha }_3$. The voltage changes only if ${\alpha }_3>50$, a values for nonrealistic cable geometries with swellings. Parameter values used for simulations correspond to realistic dendritic parameters as in Ref. [14]: $c_M=1\mathrm{mF}/{\mathrm{cm}}^2,r_M=3000\mathrm{\Omega }{\mathrm{cm}}^2,r_L=100\mathrm{\Omega }\mathrm{cm},R_0={10}^{-4}\mathrm{cm}.$ The initial condition is given by (47).

Figure 5

(a) Cable with geometry (51). (b) Voltage for the cable with a Gaussian swelling, radius (51). (c) Voltage vs $t$ in $s=0$ with different values for ${\alpha }_3$. (d) Voltage vs $s$ at time $t=1000$ s for different values of ${\alpha }_3.$ Parameter values used for simulations correspond to realistic dendritic parameters as in Ref. [14]: $c_M=1\mathrm{mF}/{\mathrm{cm}}^2,r_M=3000\mathrm{\Omega }{\mathrm{cm}}^2,r_L=100\mathrm{\Omega }\mathrm{cm},R_0={10}^{-4}\mathrm{cm}.$ The initial condition is given by (47).

Figure 6

(a) Cable with geometry (52). (b) Voltage for the cable with a Gaussian train swellings, radius (52). (c) Voltage vs $t$ in $s=0$ with different values for ${\alpha }_3$. (d) Voltage vs $s$ at time $t=1000$ s for different values of ${\alpha }_3.$ Parameter values used for simulations correspond to realistic dendritic parameters as in Ref. [14]: $c_M=1\mathrm{mF}/{\mathrm{cm}}^2,r_M=3000\mathrm{\Omega }{\mathrm{cm}}^2,r_L=100\mathrm{\Omega }\mathrm{cm},R_0={10}^{-4}\mathrm{cm}.$ The initial condition is given by (47).

Figure 7

(a) Cable with geometry (53). (b) Voltage for the cable with the radius (53). (c) Voltage vs $t$ in $s=0$ with different values for ${\alpha }_3$. (d) Voltage vs $s$ at time $t=1000$ s for different values of ${\alpha }_3.$ Parameter values used for simulations correspond to realistic dendritic parameters as in Ref. [14]: $c_M=1\mathrm{mF}/{\mathrm{cm}}^2,r_M=3000\mathrm{\Omega }{\mathrm{cm}}^2,r_L=100\mathrm{\Omega }\mathrm{cm},R_0={10}^{-4}\mathrm{cm}.$ The initial condition is given by (47).