An extended bidomain framework incorporating multiple cell types

Abstract

The muscular layers within the walls of the gastrointestinal tract contain two distinct cell types, the interstitial cells of Cajal and smooth muscle cells, which together produce rhythmic depolarizations known as slow waves. The bidomain model of tissue-level electrical activity consists of single intracellular and extracellular domains separated by an intervening membrane at all points in space and is therefore unable to adequately describe the presence of two distinct cell types in its conventional form. Here, an extension to the bidomain framework is presented whereby multiple interconnected cell types can be incorporated. Although the derivation is focused on the interactions of the interstitial cells of Cajal and smooth muscle cells, the conceptual framework can be more generally applied. Simulations demonstrating the feasibility of the proposed model are also presented.

Figure 1

Schematic of the extended bidomain framework for a single control volume. Within the control volume, the membrane area available for ion exchange is separated into three components, one for the interaction between the ICC and the extracellular space, one for the interaction between the SMC and the extracellular space, and the third for the interaction between the ICC and the SMC. Currents are defined in the body text with the exception of $I_{\text{diff}}^{\text{ICC}}$, $I_{\text{diff}}^{\text{SMC}}$, and $I_{\text{diff}}^{\text{EXT}}$, which represent $\mathrm{\nabla }·\left({\sigma }_{\text{i}}^{\text{ICC}}\mathrm{\nabla }{\varphi }_{\text{i}}^{\text{ICC}}\right)$, $\mathrm{\nabla }·\left({\sigma }_{\text{i}}^{\text{SMC}}\mathrm{\nabla }{\varphi }_{\text{i}}^{\text{SMC}}\right)$, and $\mathrm{\nabla }·\left({\sigma }_{\text{e}}\mathrm{\nabla }{\varphi }_{\text{e}}\right)$, respectively.

Figure 2

Current injection into an extended bidomain framework, showing the unstimulated ICC membrane potential at the intrinsic frequency of 2.9 cpm (gray dashed lines), the ICC membrane potential at the stimulated frequency of 3.3 cpm (solid black lines), and the SMC membrane potential at 3.3 cpm (solid gray lines). (a) The result of injecting a periodic stimulus at 3.3 cpm into an ICC via $I_{\text{stim}}^{\text{ICC}}$. (b) The result of injecting a stimulus into the extracellular space via $I_{\text{stim}}^{\text{EXT}}$. Stimuli in both a and b are able to produce stable entrainment.

Figure 3

(a–c) Spatiotemporal plots of voltages $V_{\text{m}}^{\text{ICC}}$, $V_{\text{m}}^{\text{SMC}}$, and ϕ_e, respectively. In each case, the distance along the cable is plotted on the vertical axis, and time is shown on the horizontal axis. The proximal end of the cable is depicted as a distance of zero and the distal end as a distance of 100. Propagation along the cable is therefore observed by activity that moves to the right in time and distally (upward) in space. Here, the extracellular potential at the proximal end of the cable has been grounded to a value of 0 mV for all time. (d) The three voltages described in a–c as a function of time from a location halfway (50 mm) along the cable.