The Generalized Activating Function

Abstract

Electrical field stimulation of the heart occurs during pacing and defibrillation. In cardiac tissue, the generalized activating function is the mathematical forcing function that drives changes in cellular transmembrane potential in response to field stimulation. It is the underlying principle giving rise to virtual electrodes near a physical electrode or throughout the tissue. The pattern and strength of the virtual electrodes is the result of a complex interaction between the spatial patterns of local electric field and local tissue properties of conductivity. This review defines the generalized activating function and illustrates its utility in a number of examples.

Appendix

For a one-dimensional, linear fiber, the membrane current i _m per unit length contains two components: the capacitive current i _C and total ionic current i _ion (units of mA/cm):

$$ {i}_{\mathrm{m}}={i}_{\mathrm{C}}+{i}_{\mathrm{ion}}={c}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}+{i}_{\mathrm{ion}} $$

(7.8)

where c _m is the length-specific membrane capacitance (units of F/cm). Along the intracellular pathway, intracellular current I _i (units of mA) is related to i _m and to the intracellular potential Φ_i (units of mV) according to,

$$ \frac{\partial {I}_{\mathrm{i}}}{\partial x}=-{i}_{\mathrm{m}} $$

(7.9)

$$ {r}_{\mathrm{i}}{I}_{\mathrm{i}}=-\frac{\partial {\Phi}_{\mathrm{i}}}{\partial x} $$

(7.10)

where r _i is intracellular resistance per unit length (units of Ω/cm). Combining (7.9) and (7.10), utilizing the relation Φ_i = v _m + Φ_e, where v _m is the change in transmembrane potential (units of mV), and rearranging terms produces,

$$ {i}_{\mathrm{i}\mathrm{on}}+{c}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}-\frac{1}{r_{\mathrm{i}}}\frac{\partial^2{v}_{\mathrm{m}}}{\partial {x}^2}=\frac{1}{r_{\mathrm{i}}}f, $$

(7.11)

where f is the so-called activating function [22], defined to be

$$ f\left(x,t\right)={\left.\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {x}^2}\right|}_{\mathrm{fiber}\ \mathrm{surface}}. $$

(7.12)

(units of mV/cm²). Prior to the onset of the stimulus, v _m, its spatial derivatives and I _ion are zero. Thus, (7.11) reduces to,

$$ {c}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}=\frac{1}{r_{\mathrm{i}}}f. $$

(7.13)

The polarity of the initial response of the fiber follows that of f and defines regions of virtual cathodes and anodes . Note, however, that over time as charge diffuses through the tissue, the spatial polarity of v _m will no longer mirror that of f (compare Fig. 7.1c with Fig. 7.1e).

For the extracellular cathodal point source of Fig. 7.1a, Φ_e in the semi-infinite volume conductor is twice that for an infinite medium having conductivity σ _e:

$$ {\Phi}_{\mathrm{e}}=-\frac{I_{\mathrm{e}}}{2{\pi \sigma}_{\mathrm{e}}r}=-\frac{I_{\mathrm{e}}}{2{\pi \sigma}_{\mathrm{e}}\sqrt{x^2+{z}_0^2}} $$

(7.14)

and the activating function is therefore,

$$ f={\left.\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {x}^2}\right|}_{z={z}_0}=-\frac{I_{\mathrm{e}}}{2{\pi \sigma}_{\mathrm{e}}}\frac{2{x}^2-{z}_0^2}{{\left({x}^2+{z}_0^2\right)}^{5/2}}. $$

(7.15)

The activating function is greater than 0 in the region $\mid x\mid <\sqrt{2}{z}_0/2$, which from (7.13) leads to membrane depolarization, and is less than 0 for $\mid x\mid >\sqrt{2}{z}_0/2$, which leads to membrane hyperpolarization.

If the fiber possesses homogeneous properties, is infinitely long, and has a passive membrane with resistance R _m (units of Ω-cm) so that I _ion = v _m /R _m, (7.11) can be solved by convolving f with the response h of the fiber to a unitary point source of intracellular⁴ current (i.e., the spatial impulse response) [31, 37, 38],

$$ {v}_{\mathrm{m}}(x)=\int \frac{1}{r_{\mathrm{i}}}f\left(\xi \right)h\left(x-\xi \right)\mathrm{d}\xi, $$

(7.16)

where h is well known [24] and in steady-state has the form of an exponentially decaying function

$$ h=\frac{r_{\mathrm{i}}\uplambda}{2}{I}_0{\mathrm{e}}^{-\mid x\mid /\uplambda} $$

(7.17)

with a space constant $\uplambda =\sqrt{r_{\mathrm{m}}/{r}_{\mathrm{i}}}$, and I ₀ = 1. Under transient conditions, the general time-varying response to a step input of current applied at x = 0 should be used instead of (7.17). Equations (7.17) and (7.16) are plotted in Figs. 7.1d and 7.1e, respectively, for the case where z ₀ = λ.

Generalization of (7.11) to three dimensions involves the use of the bidomain equations [32]

$$ \nabla \cdot \left({\mathrm{G}}_{\mathrm{i}}\nabla {\Phi}_{\mathrm{i}}\right)=\beta \left({I}_{\mathrm{i}\mathrm{on}}+{C}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}\right), $$

(7.18)

$$ \nabla \cdot \left({\mathrm{G}}_{\mathrm{e}}\nabla {\Phi}_{\mathrm{e}}\right)=-\beta \left({I}_{\mathrm{ion}}+{C}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}\right), $$

(7.19)

where Φ_i, Φ_e, and v _m are the extracellular, intracellular, and transmembrane potentials (units of mV), respectively, G _i and G _e are the intracellular and extracellular conductivity tensors (units of S/cm), β is the surface membrane area-to-volume ratio (units of cm⁻¹), and I _ion and C _m are defined as in the one-dimensional fiber except that they are now membrane area-specific densities rather than length-specific parameters (units of mA/cm² and F/cm², respectively). Analogous to the case of the one-dimensional fiber, substituting the relation Φ_i = v _m + Φ_e into (7.18) and rearranging terms produces [32],

$$ \beta \left({I}_{\mathrm{i}\mathrm{on}}+{C}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}\right)-\nabla \cdot \left({\mathbf{G}}_{\mathrm{i}}\nabla {v}_{\mathrm{m}}\right)=S, $$

(7.20)

where G _i is the intracellular conductivity tensor relating currents in the x, y, and z directions to the potential gradients along those directions,

$$ {\mathbf{G}}_{\mathrm{i}}=\left\lfloor \begin{array}{ccc}{g}_x& {g}_{xy}& {g}_{xz}\\ {}{g}_{yx}& {g}_y& {g}_{yz}\\ {}{g}_{zx}& {g}_{zy}& {g}_z\end{array}\right\rfloor $$

(7.21)

and S is the generalized activating function (units of mA/cm³)

$$ S=\nabla \cdot \left({\mathbf{G}}_{\mathrm{i}}\nabla {\Phi}_{\mathrm{e}}\right). $$

(7.22)

Equations (7.21) and (7.22) are written in their most general form, but can be understood more readily under some simplifying conditions. First, G _i is just a rotation of the conductivity tensor G _f in the fiber coordinate system

$$ {\mathbf{G}}_{\mathrm{i}}={\mathbf{A}\mathbf{G}}_{\mathrm{f}}{\mathbf{A}}^{\mathrm{T}}, $$

(7.23)

where

$$ {\mathbf{G}}_{\mathrm{f}}=\left[\begin{array}{ccc}{g}_1& 0& 0\\ {}0& {g}_{\mathrm{t}}& 0\\ {}0& 0& {g}_{\mathrm{u}}\end{array}\right]. $$

(7.24)

The parameters g _l, g _t, and g _u are the conductivities along the fiber axis and the two principal axes perpendicular to the fiber axis, respectively, and A is the rotation tensor [32]. Thus, G _i consists of the conductivities of a tissue having orthotropic anisotropy, adjusted for fiber angle in the tissue.

Next, S can be written as the sum of two components [35],

$$ S=-\left(\nabla \cdot {\mathbf{G}}_{\mathrm{i}}\right)\cdot \mathbf{E}-{\mathbf{G}}_{\mathrm{i}}:\nabla \mathbf{E}, $$

(7.25)

where the electric field E = −∇Φ_e. This equation 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). When fully expanded in component form for the two-dimensional case, (7.25) becomes,

$$ S=\left(\begin{array}{c}\frac{\partial {g}_x^{\mathrm{i}}}{\partial x}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial x}+\frac{\partial {g}_{yx}^{\mathrm{i}}}{\partial x}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial y}+\frac{\partial {g}_{xy}^{\mathrm{i}}}{\partial y}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial x}+\frac{\partial {g}_y^{\mathrm{i}}}{\partial y}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial y}\\ {}+{g}_x^{\mathrm{i}}\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {x}^2}+\left({g}_{xy}^{\mathrm{i}}+{g}_{yx}^{\mathrm{i}}\right)\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial x\partial y}+{g}_y^{\mathrm{i}}\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {y}^2}\end{array}\right). $$

(7.26)

Other variants of (7.26) can be obtained starting with different combinations of the bidomain equations . Rewriting (7.19) as,

$$ -\beta \left({I}_{\mathrm{ion}}+{C}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}\right)=S $$

(7.27)

gives for S terms that depend on extracellular conductivities [42],

$$ S=\left(\begin{array}{c}\frac{\partial {g}_x^{\mathrm{e}}}{\partial x}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial x}+\frac{\partial {g}_{xy}^{\mathrm{e}}}{\partial x}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial y}+\frac{\partial {g}_{yx}^{\mathrm{e}}}{\partial y}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial x}+\frac{\partial {g}_y^{\mathrm{e}}}{\partial y}\frac{\partial {\Phi}_{\mathrm{e}}}{\partial y}\\ {}+{g}_x^{\mathrm{e}}\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {x}^2}+\left({g}_{xy}^{\mathrm{e}}+{g}_{yx}^{\mathrm{e}}\right)\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial x\partial y}+{g}_y^{\mathrm{e}}\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {y}^2}\end{array}\right). $$

(7.28)

The relative influence of the first four terms (electric field) versus the latter three terms (electric field gradient) in (7.28) has been studied experimentally in rabbit heart [42]. Alternatively, combining (7.18) and (7.19) by multiplying (7.18) by ${g}_y^{\mathrm{e}}/\left({g}_y^{\mathrm{i}}+{g}_y^{\mathrm{e}}\right)$ and (7.19) by ${g}_y^{\mathrm{i}}/\left({g}_y^{\mathrm{i}}+{g}_y^{\mathrm{e}}\right)$, subtracting the latter from the former, and utilizing the relation Φ_i = v _m + Φ_e yields,

$$ \beta \left({I}_{\mathrm{i}\mathrm{on}}+{C}_{\mathrm{m}}\frac{\partial {v}_{\mathrm{m}}}{\partial t}\right)-\frac{g_y^{\mathrm{i}}}{g_y^{\mathrm{i}}+{g}_y^{\mathrm{e}}}\nabla \cdot \left({\mathrm{G}}_{\mathrm{i}}\nabla {v}_{\mathrm{m}}\right)=S $$

(7.29)

and gives for S terms that depend on both intra- and extracellular conductivities [43],

$$ {\displaystyle \begin{array}{c}S=\frac{1}{g_y^{\mathrm{i}}+{g}_y^{\mathrm{e}}}\left(\left[{g}_y^{\mathrm{e}}\left(\frac{\partial {g}_x^{\mathrm{i}}}{\partial x}+\frac{\partial {g}_{yx}^{\mathrm{i}}}{\partial y}\right)-{g}_y^{\mathrm{i}}\left(\frac{\partial {g}_x^{\mathrm{e}}}{\partial x}+\frac{\partial {g}_{yx}^{\mathrm{e}}}{\partial y}\right)\right]\right.\frac{\partial {\Phi}_{\mathrm{e}}}{\partial x}\\ {}+\left.\left[{g}_y^{\mathrm{e}}\left(\frac{\partial {g}_y^{\mathrm{i}}}{\partial y}+\frac{\partial {g}_{xy}^{\mathrm{i}}}{\partial x}\right)-{g}_y^{\mathrm{i}}\left(\frac{\partial {g}_y^{\mathrm{e}}}{\partial y}+\frac{\partial {g}_{xy}^{\mathrm{e}}}{\partial x}\right)\right]\frac{\partial {\Phi}_{\mathrm{e}}}{\partial y}\right)\\ {}+\frac{g_x^{\mathrm{i}}{g}_y^{\mathrm{e}}-{g}_x^{\mathrm{e}}{g}_y^{\mathrm{i}}}{g_y^{\mathrm{i}}+{g}_y^{\mathrm{e}}}\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial {x}^2}+2\frac{g_{xy}^{\mathrm{i}}{g}_y^{\mathrm{e}}-{g}_{xy}^{\mathrm{e}}{g}_y^{\mathrm{i}}}{g_y^{\mathrm{i}}+{g}_y^{\mathrm{e}}}\frac{\partial^2{\Phi}_{\mathrm{e}}}{\partial x\partial y}.\end{array}} $$

(7.30)

Just as with (7.26), S in (7.28) and (7.30) consists of the sum of terms containing first and second derivatives of Φ_e weighted by first derivatives of conductivities or conductivities, respectively.

According to (7.22), the generalized activating function is determined by the actual spatial distribution of extracellular potential, which as seen in the example of Fig. 7.9 is not necessarily determined solely by the applied electric field. To begin with, the presence of the cardiac fibers and their effect on the applied field need to be accounted for [45]. However, given that the fiber diameter is small compared with the typical distance to the electrode, such effects will be relatively minor [31]. More significantly, the extracellular potential distribution will be perturbed even further by the transmembrane currents that flow in response to the developing v _m (dashed pathway in Fig. 7.2). This is accounted for by (7.19). Rigorously speaking, the exact solution for v _m (and Φ_e) must satisfy both (7.20) and (7.19), whereas with the concept of the generalized activating function Φ_e is assumed to be known, and an approximate solution for v _m is obtained by using just (7.20) alone.