21.1 INTRODUCTION
There are a growing number of documented scientific therapeutic uses of electricity (Seligman, 1982). In most cases, although the mechanism is not always clearly understood, it appears to arise as a consequence of the depolarization or hyperpolarization of excitable cell membranes resulting from the applied currents. Other mechanisms that appear to be sometimes involved include thermal (heating) and neurohumoral effects.
Functional electric stimulation is a very straightforward application for the therapeutic use of electricity. Another area in which electrotherapy may be applied is in the electric stimulation of cardiac tissue, including cardiac pacemakers and cardiac defibrillation. These topics are discussed in the next two chapters.
Since the electric stimulation of biological tissues requires the use of electrodes, any practical study should include consideration of electrodes and electrode-tissue interaction. The mechanical properties of electrodes are important particularly with respect to implants whose lifetime is measured in years. Since the flow of electricity from the electrode (where electrons carry the charges) into the tissue (where ions carry the charges) may involve an electrochemical reaction, this area must be carefully studied as well. Consequently, several sections of this chapter are devoted to these electrode characteristics.
(21.1)
where ri = axial intracellular resistance per internodal length [kW/l] ri = intracellular resistivity [kW·cm] (chosen as 0.1 kW·cm) l = internodal length [cm] di = axon diameter (internal myelin diameter) [cm]
Fig. 21.1 A point current stimulus, Ia, lies at a distance h from a single myelinated nerve fiber. A node of Ranvier is assumed to be aligned with the stimulating monopole. The internodal distance l is related to the outer diameter of the myelin, do (i.e., the fiber diameter, as shown) where l = 100·do.
Fig. 21.2 Electric model of a myelinated fiber stimulated by a point current source of strength Ia. The node directly beneath the source is labeled 0 and its membrane is modeled by Frankenhauser-Huxley (F-H) equations. Lateral nodes are assumed to be subthreshold and represented by parallel resistance and capacitance (where Rm, Cm are the total nodal lumped resistance and capacitance per nodal area). The total intracellular internodal resistance is ri.
This formulation, in effect, considers the fiber itself to have little influence on the applied field (i.e., the secondary sources at the fiber surface set up a very weak field compared to Fo arising from Ia and described by Equation 21.2). If one notes the very small extracellular fields generated by nerve impulses propagating on isolated fibers (Plonsey, 1974), then the extracellular potentials from the secondary sources (which are smaller than action potential sources) should certainly be negligible.
The second term of Equation 21.3 expresses the difference of the intracellular axial current entering the nth node (i.e., (Vi,n-1 - Vi,n)/ri) minus that leaving this node (i.e., (Vi,n - Vi,n+1)/ri); this difference has been set equal to the (outward) transmembrane current of the nth node (right hand side of Equation 21.3) which correctly satisfies the conservation of current. Except at n = 0 (the central node governed by the Frankenhauser-Huxley equations) we assume that subthreshold conditions are elsewhere in effect and the ionic current is given by
Using the definition of transmembrane voltage Vn = Vi,n - Vo,n in Equation 21.3 gives
where Equation 21.4 has been substituted for Ii,n. When n = 0, however, one gets
where n is the nodal width, and the ionic currents are found from the Frankenhauser-Huxley equations. The extracellular nodal potentials in Equations 21.5 and 21.6 are found from Equation 21.2, and constitute the forcing function. We assume that the stimulating current is switched on at t = 0 so that Vm,n = 0 for all n at t = 0; the stimulating current is assumed to remain on for a specified time interval. For stimuli that are subthreshold at all nodes (including the central node), Equation 21.5 alone is sufficient to describe the response and may be applied also at n = 0.
McNeal assumed that the potential on the extracellular side of the nodal membrane was fixed by the stimulating field. Since the latter is a point source (i.e., a monopole), the stimulating (applied) potential field Fo (see Equation 8.7) is
(21.2)
where Fo = stimulating (applied, extracellular) potential field [mV] Ia = applied current [µA] so = extracellular conductivity of the medium [kW·cm] r = distance from any node to the point source [cm] On the basis of the network described in Figure 21.2 and the applied field of Equation 21.2 one can determine the response to step currents of varying strengths (up to that required for excitation at node 0). The equations that must be solved are based on Kirchhoff's laws. Letting subscript i denote intracellular, o extracellular, and n the index denoting a specific node (where the central node is designated 0), we have for the transmembrane current Im per nodal area at the nth node
(21.3)
where Im, n = transmembrane current per nodal area at the nth node Ii, n = transmembrane ionic current per nodal area at the nth node Vm, n = transmembrane voltage at the nth node Cm = membrane capacitance per nodal area at the nth node ri = axial intracellular resistance per internodal length
(21.4)
where Rm, n = transmembrane resistance per nodal area (constant)
(21.5)
(21.6) The solution of Equation 21.5 (for subthreshold conditions) or Equation 21.5 and 21.6 (for near-threshold conditions) at n = 0 requires temporal discretization and the solution of the resulting system of equations by iteration or by matrix techniques. The steady-state response to the stimulus diminishes rapidly with increasing values of n so that only a finite number of nodes (i.e., equations) need to be considered. One wishes to use the smallest number; McNeal (1976) explored the use of a total of 11, 21, and 31 nodes and found that the response to a 1 ms pulse was within 0.2 % accuracy with only 11 nodes.
The response to a subthreshold (at all nodes) current step evaluated at the central node and its four neighbors is described in Figure 21.3. These were calculated from a total of 31 nodes using the same parameters as were chosen by McNeal (see Table 21.1); the results appear to be identical to those obtained by McNeal with 11 nodes. Since the stimulus monopole was assumed to be cathodal, current must leave from the closest node(s) while, to conserve current, current must enter lateral nodes. This accounts for hyperpolarization of nodes ±2, ±3, ±4 . . . and depolarization of the central node. The behavior of node ±1 changes as the membrane capacitances charge up. Initially it is hyperpolarized, but its steady-state potential (response to a pulse of very long duration) is a depolarization. If the stimulus were anodal, then the signs of all responses in Figure 21.3 would be reversed; the initial largest depolarization occurs at node ±1, but after around 15 ms the largest depolarization is at ±2. Thus excitation resulting from a stimulus duration of t 
15 ms and that for t >15 ms occur at different nodes!
| Symbol | Parameter | Value |
| ri | axoplasm resistivity | 0.11 kW·cm |
| ro | extracellular resistivity | 0.3 kW·cm |
| Cm | nodal membrane capacitance/unit area | 2.0 µF/cm2 |
| Gm | nodal membrane conductance/unit area | 0.4 mS/cm² |
| n | nodal gap width | 0.5 µm |
| l / do | ratio of internode spacing to fiber diameter | 100 |
| di / do | ratio of axon diameter (internal myelin) to fiber diameter | 0.7 |
| Rmn | nodal membrane resistance = 1/(Gmpdin) | 29.9 MW |
| Cmn | nodal membrane capacitance = Cmpdin | 2.2 pF |
| ril | internodal resistance = 4ri l /(pdi2) | 14.3 MW |
Figure 21.3 Response at central node (n = 0) and adjoining four nodes to a point current source located 1 mm from the fiber and excited with a current step of 0.1 mA. The geometry is described in Figure 21.1. The fiber diameter is 20 µm, and the internodal spacing is 2 mm. Other parameters are given in Table 21.1.
Returning to the case of cathodal stimulation, it is clear that for a transthreshold stimulus, activation occurs at node 0. A determination of the quantitative values of current strength and duration to achieve activation requires (at least) the use of Equation 21.6 to describe the central node. McNeal was able to calculate that for a 100 ms pulse the threshold current is 0.23 mA. By determinination of the corresponding threshold current for a series of different pulse durations, an experimental strength-duration curve (reproduced in Figure 21.4) was constructed. This curve describes conditions under which excitation is just achieved for the specific case of a 20 µm fiber with a point source at a distance of 1 mm. Under the idealized conditions described in Chapter 3, we expect this curve to have a slope of -1 for small pulse duration, but this is not the case in an experimental curve. The reason for this discrepancy is that the experiment is affected by accommodation, distributed fiber structure, and source geometry.
The dependence of excitability on the fiber diameter of a nerve is important since it describes how a nerve bundle (consisting of fibers of different diameter) will respond to a stimulus (monopole in this case). An examination of Equations 21.5 and 21.6 shows that changes in fiber diameter do affect the solution only through its link with the internodal spacing, l. Figure 21.5 gives the threshold current for a monopole at a distance of 1 mm from a fiber whose diameter do satisfies (2 µm do 25 µm) where the pulse duration is 100 µs. A 10-fold decrease in diameter (from 20 to 2 µm) affects ri and Rm proportionately (each increases by 10). As a rough approximation the induced transmembrane voltage at node 0 is the same fraction of the applied potential difference between nodes zero and 1. This potential decreases quite considerably for the 2 µm diameter nerve versus the 20 µm diameter one since l is 10 times greater in the latter case. Accordingly the excitability is expected to decrease for the nerve of lower diameter. Figure 21.5 confirms this effect through the simulation. Note that at the larger diameters the slope is around -½, meaning that the threshold is approximately inversely proportional to the square root of the fiber diameter.
Figure 21.4 Log-log plot of a strength duration that just produces activation of 20 µm diameter myelinated fiber from a point source of current 1 mm distant. The geometry is described in Figure 21.1, and the fiber electrical properties are tabulated in Table 21.1. (From McNeal, 1976.)
21.3 STIMULATION OF AN UNMYELINATED AXON
The response of a single unmyelinated fiber to a stimulating field can be found through the same type of simulation described for the myelinated fiber. In fact, to obtain numerical solutions it is necessary to discretize the axial coordinate (x) into elements Dx and a network somewhat similar to that considered in Figure 21.2 results. One obtains an expression for the transmembrane current at the nth element, from the cable equations, as
The ionic currents are those of sodium, potassium, and chloride ions, as found from Hodgkin and Huxley (1952) and so on.
(Note that had we assumed steady-state conditions, the capacitive term could be set equal to zero, whereas if linear conditions could be assumed, then imI = Vm,n/Rm.) If, now, we replace Vi,n by Vm,n + Vo,n and reorganize, we have
which is essentially the same as Equation 21.6 except that the spacing between elements in Equation 21.10, namely Dx, may be much smaller than the spacing in Equation 21.6, namely l. Of course, the internodal spacing is set by the myelinated histology while Dx is at our disposal.
and note that for anodal excitation, depolarization results only for | x |
Figure 21.6 The cathodal applied field Vo(x) along a fiber oriented along x due to a point source, and
Figure 21.7 Current-distance relationship for unmyelinated fibers. Excitation occurs for points lying in the shaded region. For cathodal stimulation, a minimum distance arises at the point where anodal block prevents the escape of the action impulse. For anodal stimulation, block does not occur; thus there is no lower limit on the source-fiber distance. The inner scales are for a fiber diameter of 9.6 µm, and the outer for a diameter of 38.4 µm. (An examination of Equations 21.10 and 21.13 shows that scaling the excitation with respect to both current strength, source-fiber distance, and fiber diameter leaves the solution unchanged.) (From Rattay, 1987.)
and the dissolution of the electrode results. For an excessively cathodic condition the result is
and increasingly negative electrode may result in
In either case a consequent rise in pH results which can produce tissue damage.
Figure 21.10 The electrode-tissue interface includes a series ohmic resistance Rs, a series capacitance Cs, and a Faradaic impedance Z. The capacitance represents a double layer that arises at the metal-electrolyte interface, whereas the Faradaic impedance includes several physical and chemical processes that may occur.
(21.7)
ri = the intracellular axial resistance per unit length. Approximating the second derivative in Equation 21.7 by second differences gives us
(21.8)
Vi,n = intracellular potential at the nth element, etc. The desired equation is found by equating the transmembrane current evaluated from Equation 21.8 (which is determined by the fiber structure) with that demanded by the intrinsic nonlinear membrane properties:
(21.9)
cm = membrane capacitance per unit length imI = ionic component of the membrane current per unit length
(21.10) The forcing function in Equation 21.10 is essentially 
2Vo/x2. Depending on the spatial behavior of Vo and the length l, the forcing function in Equation 21.6 is also given approximately by 2Vo/x2. We have noted that Vo is normally the field of the stimulating electrodes in the absence of the stimulated fiber. That is, for a single isolated fiber its (secondary) effect on the axial field in which it is placed is, ordinarily, negligible, as discussed earlier. So 2Vo/x2 can be evaluated from the field of the stimulating electrodes alone. Clearly, the response to a stimulating field is contained in this second-derivative behavior, designated the "activating function" by Rattay
(1986). In fact, for the linear behavior that necessarily precedes excitation, Equation 21.10 may be replaced by the following equation (an approximation that improves as Dx 
0, hence less satisfactory for myelinated fibers where Dx = l )
(21.11)
rm = membrane resistance per unit length (resistance times length) ri = intracellular axial resistance per unit length We consider the response, Vm,n, to a stimulating field, Vo, which is turned on at t = 0 (i.e., a step function). Activation is possible if Vm,n/t > 0, and this condition in turn seems to depend on 2Vo/x2 > 0. Rattay assumed this relationship to be valid and designated 2Vo/x2 as an "activating function." An application, given by Rattay, is described below. However, the reader should note that Vm,n is not directly dependent on 2Vo/x2 but, rather, arises from it only through the solution of Equation 21.11. But this solution also depends on the homogeneous solution plus the boundary conditions. That the outcome of the same forcing function can vary in this way suggests that the interpretation of 2Vo/x2 as an activating function is not assured.
Figure 21.6 shows the applied field Vo(x) along an unmyelinated fiber for a point source that is 1 mm distant, and the corresponding activating function for both an anodal and a cathodal condition. For a negative (cathodal) current of -290 µA, threshold conditions are just reached at the fiber element lying at the foot of the perpendicular from the monopole (which is at the location of the positive peak of the plotted 2Vo/x2 curve). For a positive (anodal) current source, 1450 µA is required just to reach excitation; this is seen in the curve of the corresponding activating function (d2Vo/dx2) that results. The fivefold increase in current compensates for the lateral peaks in the activating function, being only one-fifth of the central peak. The location of the excitation region can be found from Equation 21.2 written as
(21.12)
h = distance from the point source to the fiber x = axial distance along fiber so = conductance of the extracellular medium We examine the lateral behavior of
(21.13) 
h / 
2 (hyperpolarization is produced for | x | h / 2). Conversely, activation from cathodal stimulation is confined to the region | x | h / 2, while hyperpolarization is produced laterally.
For cathodal stimulation and for increasing stimulus intensity, a threshold level is reached that results in activation. A continued increase in stimulus intensity is followed by little change in behavior until a level some eight times threshold is reached. At this point the hyperpolarization lateral to the active region can block the emerging propagating impulse. Rattay (1987) investigated this behavior quantitatively by solving Equation 21.10 with the ionic currents evaluated from the Hodgkin-Huxley equations and with the point source field of Equation 21.12. The results are plotted in Figure 21.7 for an unmyelinated fiber of 9.6 µm diameter (inner scales) and 38.4 µm diameter (outer scales).
2Vo(x)/x2 for both cathodal and anodal stimuli. The cathodal threshold current of -290 µA initiates activation at x = 0 which corresponds to the peak of the "activating function" 2Vo(x)/x2. For anodal conditions the stimulus for threshold activation is 1450 µA and the corresponding activating function is shown. The fivefold increase in current is required to bring the corresponding (anodal) positive peak of 2Vo/x2 to the same level it has under threshold cathodal conditions. Note that excitation takes place at x lateral to the point 0. (From Rattay, 1986.)Figure 21.7 may be explained as follows. For a cathodal stimulus of -4 mA and the smaller-diameter fiber (use inner coordinate scale), excitation occurs (cross-hatched region) if the source-field distance rages from 0.73 to 1.55 mm (following line (A)) whereas for the larger fiber, excitation requires the source-field distance to lie between 0.94 and 2.44 mm (following line (B)). The spherical insets describe the relative minimum and maximum radii for excitation of the smaller fiber (A) and larger fiber (
Fe Fe2+ + 2e¯(2.14)
2H2O + 2e¯ H2 + 2OH¯(2.15)
O2 + H2O + 2e¯ OH¯ + O2H¯(2.16) 
Operating an electrode in its linear (capacitive) range is desirable since it is entirely reversible and results in neither tissue damage or electrode dissolution. To do so requires that the electrode be positioned as close as possible to the target nerve (so that the maximum required current will be minimal) and, if possible, to have a large area for a low current density. Finally the electrode material should be chosen to have maximum capacitance (other things being the same). A roughened surface appears to result in a decrease in microscopic current density without affecting the overall size of the electrode. The values of charge storage appears to increase from 0.5 to 2.0 µC/mm2, yet remain in the reversible region for platinum and stainless steel materials.
If the stimulating current is unipolar, then even though one pulse causes the storage of charge within the reversible range, subseguent pulses will eventually cause the operating point to move out of the reversible region. The reasons are that under the usual operating conditions accumulation of charge takes place but there is inadequate time for the charge to leak away. Thus practical stimulation conditions require the use of biphasic pulses. In fact, these must be carefully balanced to avoid an otherwise slow buildup of charge. A series capacitance will ensure the absence of DC components (Talonen et al., 1983).
