In a seminal analysis of external stimulation of unmyelinated nerve axons, Rattay15 advanced the idea of an activating function that determines the excitation of the axon. In a slight modification of his original work, the nerve fiber can be represented as a cylindrical, one-dimensional cable located a distance z0 beneath the surface of a semi-infinite three-dimensional slab of tissue (Fig. 1a). An electrode (cathode) is placed on the surface, and its location is defined as the origin of the coordinate system.
The electrical representation for the fiber is shown in Fig. 1b. Unlike the conventional form of the core-conductor model used to describe a one-dimensional fiber lying in one-dimensional space,17 the extracellular potential $e for the one-dimensional fiber lying in three-dimensional space is considered to be imposed and is an input forcing function that produces a change in transmembrane potential, vm. Thus, as derived in the Appendix, dvm 1 d 2vm 1
where f is the activating function,15 defined to be d2$ d f (X't) = ^ = — dXEx' (2)
and the electric field Ex is the negative gradient of $e. Thus, gradients in extracellular electric field (second derivatives of extracellular potential) directed along the axis of the fiber act as virtual sources that drive the electrical cable. Apart from its mathematical definition as a source, the activating function also gives insight into the initial polarization pattern of the fiber.18 This is because prior to the onset of the stimulus, vm, its spatial derivatives, and Iion are zero. Thus, (1) reduces to
Hence, the polarity of the initial response of the fiber follows that of f.
For the fiber located a distance z0 from the point electrode as shown in Fig. 1a, it is readily seen that the equipotential lines are not uniformly parallel to the fiber axis. This implies that there exists a gradient in electric field along the fiber axis, and a nonzero activating function will develop. $e and f are plotted in Fig. 1c for the case where z0 = A (the fiber space constant), and their mathematical expressions are given in the Appendix. Directly under the cathode in the region \x\ < %/2A/2, the activating function is greater than 0 (and can be called a virtual cathode), which from (3) leads to membrane depolarization and activation, whereas outside this region the activating function is less than 0 (a virtual anode), leading to membrane hyperpolarization. Figure 1d shows the response h of the fiber to an intracellular point source of current applied at x = 0, and thus the vm response of the fiber (Fig. 1e) to f is the convolution of f and h (see the Appendix). As with f, vm is positive and decays with distance along x in the region directly under the electrode (neighborhood of x = 0), similar to the response of a one-dimensional core-conductor model to an extracellular cathode or intracellular anode (Fig. 1d).17 However, in contrast to the one-dimensional
Figure 1: Point stimulation of a one-dimensional fiber lying in a semi-infinite, three-dimensional volume conductor. (a) Schematic of electrode location with respect to fiber. Equipotential surfaces are indicated by the circular lines. (b) Equivalent circuit for fiber membrane. The extracellular potential $e is imposed on the fiber by the potential gradient in the volume conductor. This acts to drive current im across the cell membrane and Ii along the intracellular pathway, resulting in a gradient in , which if not matched to the gradient in $e produces a gradient in transmembrane potential vm(= — $e). Intracellular resistance ri is assumed to be constant in this situation. (c) Plot of $e (dashed line) on the fiber surface and of the activating function f (solid line), equal to the second spatial derivative of $e. The function f has been normalized to its maximum value. When the point electrode is a cathode (as is the case here), the activating function has a positive value (virtual cathode) in the center flanked by negative values (virtual anodes) on either side. (d) Plot of h, the steady-state vm response to an intracellular point source of current at x = 0, normalized to its maximum value. (e) Convolution of f and h, giving the steady-state vm response to the stimulus current of (a). The vm response has been normalized to its maximum value core-conductor model, there exist side lobes of opposite polarity (membrane hyperpolar-ization) that are generated by the virtual anodes shown in Fig. 1c. These lobes may be significant during stimulation with a physical cathode because they can cause conduction block of the electrical impulse. Conversely, during stimulation with an extracellular anode, the side lobes become positive in sign and at sufficiently large intensities can lead to fiber excitation. In the cardiac literature, the regions of opposite polarity were initially described by Hoshi and Matsuda19 and Bonke.20
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