The Generalized Activating Function

One of the fundamental differences between cardiac muscle and nerve fibers is that heart cells are electrically interconnected by low resistance gap junctions, which result in a tissue that behaves as an integrated syncytium. Thus, electrical propagation can occur parallel and also perpendicular to the cardiac fiber direction. Tissue responses to electrical stimuli are no longer adequately described by one-dimensional fiber responses, but rather, by the bidomain model, which describes current flow in the intracellular space, extracellular space, and the cell membrane, together with three potentials defined at every point in space: $e, and vm.21 Generalization of (1) to three dimensions using the bidomain model (see Appendix) results in where ¡3 is the membrane surface-to-volume ratio, G; is the intracellular conductivity tensor relating currents in the x-, y-, and z-directions to the potential gradients along those directions, and S is the generalized activating function*

Note that unlike the original formulation of the activating function f, tissue properties (i.e., Gi) are incorporated into S, because these are just as important in generating virtual electrode effects as are gradients in electric field.

Equation (5) can be interpreted as follows. First, Gi consists of the intracellular conductivities along the three orthogonal axes of a tissue having orthotropic anisotropy, adjusted for fiber angle in the tissue (see the Appendix). Second, S can be rewritten as the sum of two components (see the Appendix) that take the general form,

S = ^^ (gradients of intracellular conductance)(gradients of$e)

+ (intracellular conductances)(second derivatives of$e) (6)

* The term generalized activating function has also been used by Rattay22 to describe an activating function for neurons that incorporate fiber diameter, intracellular resistance, and membrane capacitance together with the second derivative of extracellular surface potential along the fiber axis.

Extracellular potential «

Transmembrane potential

Figure 2: Schematic showing the relationships among $e, G;, S, and vm. See text for description or alternatively,

S = ^^ (gradients of intracellular conductance)(components of applied field)

+ (intracellular conductances)(gradients of applied field), (7)

k where the gradient of $e is taken to be the applied field, and the index k in the sums is varied over the three orthogonal directions. Equation (7) tells us that sources can result either from the extracellular field weighted by spatial gradients of intracellular conductivity (the first term), or by spatial gradients of the extracellular field weighted by the intracellular conductivities (the second term). Thus, in the absence of spatial variations in intracellular conductivity, a uniform field cannot excite the tissue.

The relationships among $e, G;, S, and vm are summarized in Fig. 2. S, the generalized activating function, is the stimulus source that acts on the tissue to produce changes in transmembrane potential vm. S is determined by the spatial distributions of extracellular potential, $e, and intracellular conductivities, G;. According to (5), the generalized activating function depends on the actual spatial profile of the extracellular potential, which may differ somewhat from that associated with the applied field. For example, the presence of the cardiac fibers will perturb the applied electric field, but because their diameters are small compared with the typical distance to the electrode, this effect is relatively minor.23 More significantly, the extracellular potential distribution will be perturbed by the transmembrane currents that flow in response to the developing vm (dashed pathway in Fig. 2). Thus, the actual profile of $e (and thus, S and vm) should take into account the influence of vm on $e, whereas with the concept of the generalized activating function, $e is assumed to be known, and an approximate solution for vm is obtained by disregarding the dashed path of Fig. 2. Because the accuracy of the approximate solution is a critical issue in the use of the generalized activation function, the spatial profiles of vm for both the exact and approximate solutions will be compared in some of the examples that follow.

Extracellular potential «

Figure 2: Schematic showing the relationships among $e, G;, S, and vm. See text for description

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