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113 Bioelectricity 113. 1 Neuroelectric Principles Electrical Model for Nerve Excitation 113.2 Bioelectric Events J. Patrick Reilly Origin of Bioelectricity. Law of Stimulation. Recording Action Potentials. The Electrocardiogram(ECG). Electromyograph (EMG). Electroencephalography (EE magnetic(Eddy- L. A. Geddes Current) Stimulation 113.3 Application of Electric and Magnetic Fields in Bone and Soft Tissue Repa C. Polk History. Devices for Bone and Cartilage Repair. Soft Tissue pair and Nerve Regeneration. Mechanisms and Dosimetry 113.1 Neuroelectric Principles Patrick reilly Natural bioelectric processes are responsible for nerve and muscle function. These processes can be affected by xternally applied electric currents that are intentionally introduced through medical devices or unintentional introduced through accidental exposure(electric shock). A thorough treatment of this topic is given in Reilly [1992] Externally applied electric currents can excite nerve and muscle cells. Muscle can be stimulated directly or indirectly through the nerves that enervate the muscle. Thresholds of stimulation of nerve are generally well below thresholds for direct stimulation of muscle. An understanding of neuroelectric principles is a valuable foundation for investigation into both sensory and muscular responses to electrical stimulation. Figure 113. 1 illustrates functional components of sensory and motor(muscle)neurons. The illustrated nerve fibers are myelinated, i.e., covered with a fatty layer of insulation called myelin and having nodes of ranvier where the myelin is absent. The conducting portion of the nerve fiber is a long, hollow structure known as an axon. The axon plus myelin sheath is frequently referred to as a nerve fiber, or neuron. Bundles of neurons are called nerves The body is equipped with a vast array of sensors (receptors)for monitoring its internal and external environment. Electrical stimulation generally involves the somatosensory system, i.e., the system of receptors found in the skin and internal organs. Other specialized receptors include those in the visual and auditory systems and chemical receptors by which neurons communicate with one another. The somatosensory receptors can be classified as mechanoreceptors, thermoreceptors, chemoreceptors, and nociceptors. Numerous specializations of mechanoreceptors respond to specific attributes of mechanical stim- ulation Thermoreceptors are specialized to respond to either heat or cold stimuli. Nociceptors are unresponsive ntil the stimulus reaches the point where tissue damage is imminent and are usually associated with pain. Many nociceptors are responsive to a broad spectrum of noxious levels of mechanical, heat, and chemical stimuli. The muscles are equipped with specialized receptors to monitor and control muscle movement and posture. Figure 113.1 illustrates a pacinian corpuscle, which responds to the onset or termination of a pressure stimulus applied to the skin. C 2000 by CRC Press LLC

© 2000 by CRC Press LLC 113 Bioelectricity 113.1 Neuroelectric Principles Electrical Model for Nerve Excitation 113.2 Bioelectric Events Origin of Bioelectricity • Law of Stimulation • Recording Action Potentials • The Electrocardiogram (ECG) • Electromyography (EMG) • Electroencephalography (EEG) • Magnetic (EddyCurrent) Stimulation 113.3 Application of Electric and Magnetic Fields in Bone and Soft Tissue Repair History • Devices for Bone and Cartilage Repair • Soft Tissue Repair and Nerve Regeneration • Mechanisms and Dosimetry 113.1 Neuroelectric Principles J. Patrick Reilly Natural bioelectric processes are responsible for nerve and muscle function. These processes can be affected by externally applied electric currents that are intentionally introduced through medical devices or unintentionally introduced through accidental exposure (electric shock). A thorough treatment of this topic is given in Reilly [1992]. Externally applied electric currents can excite nerve and muscle cells. Muscle can be stimulated directly or indirectly through the nerves that enervate the muscle. Thresholds of stimulation of nerve are generally well below thresholds for direct stimulation of muscle. An understanding of neuroelectric principles is a valuable foundation for investigation into both sensory and muscular responses to electrical stimulation. Figure 113.1 illustrates functional components of sensory and motor (muscle) neurons. The illustrated nerve fibers are myelinated, i.e., covered with a fatty layer of insulation called myelin and having nodes of Ranvier where the myelin is absent. The conducting portion of the nerve fiber is a long, hollow structure known as an axon. The axon plus myelin sheath is frequently referred to as a nerve fiber, or neuron. Bundles of neurons are called nerves. The body is equipped with a vast array of sensors (receptors) for monitoring its internal and external environment. Electrical stimulation generally involves the somatosensory system, i.e., the system of receptors found in the skin and internal organs. Other specialized receptors include those in the visual and auditory systems and chemical receptors by which neurons communicate with one another. The somatosensory receptors can be classified as mechanoreceptors, thermoreceptors, chemoreceptors, and nociceptors. Numerous specializations of mechanoreceptors respond to specific attributes of mechanical stimulation. Thermoreceptors are specialized to respond to either heat or cold stimuli. Nociceptors are unresponsive until the stimulus reaches the point where tissue damage is imminent and are usually associated with pain. Many nociceptors are responsive to a broad spectrum of noxious levels of mechanical, heat, and chemical stimuli. The muscles are equipped with specialized receptors to monitor and control muscle movement and posture. Figure 113.1 illustrates a pacinian corpuscle, which responds to the onset or termination of a pressure stimulus applied to the skin. J. Patrick Reilly Metatec Associates L. A. Geddes Purdue University C. Polk University of Rhode Island

© 2000 by CRC Press LLC When a sensory receptor is stimulated, it produces a voltage change called a generator potential. The generator potential is graded: if you squeeze a pacinian corpuscle, for example, it produces a voltage; if you squeeze it harder, it produces a greater voltage. The generator potential initiates a sequence of events that leads to a propagating action potential (a “nerve impulse” in common parlance). The functional boundary of the biological cell is a thin (about 10 nm) bimolecular lipid and protein structure called a membrane. Electrochemical forces across the membrane regulate chemical exchange across the cell. The medium within the cell (the plasm) and outside the cell (the interstitial fluid) is composed largely of water containing various ions. The difference in the concentration of ions inside and outside the cell causes an electrochemical force across the cell membrane. The membrane is a semipermeable dielectric that allows some ionic interchange. Under conditions of electrochemical equilibrium (no net force in either direction), the membrane will attain a potential described by the Nernst equation (113.1) where [S]i and [S]o represent the concentrations of ionic substance S inside and outside the cell, R is the universal gas constant, T is absolute temperature, F is the Faraday constant (number of coulombs per mole of charge), and Z is the valence of substance S. Using the values R = 8.31 J/mol K, T = 310 K, F = 96,500 C/mol, and Z = +1 (for a monovalent cation), converting to the base 10 logarithm, and expressing Vm in millivolts, we obtain (113.2) FIGURE 113.1 Functional components of (a) motor and (b) sensory neurons.Arrows indicate the direction of information flow. Signals are propagated across synapses via chemical neurotransmitters and elsewhere by membrane depolarization. Synapses are inside the spinal column. The sizes of the components are drawn on a distorted scale to emphasize various features. V RT FZ S S m o i = ln [ ] [ ] V S S m o i = 61 log [ ] [ ]

In a quiescent state, nerve and muscle cells maintain a membrane potential typically around -60 to-90 mV, with the inside of the cell negative relative to the outside. Two ions that are involved in the electrical response of nerve and muscle are Na+ and K+. The concentration of these ions inside and outside the cell dictates the Nernst potential according to Eq (113. 2). Example concentrations in H M/cm for a nerve axon would be (Nat 50, (Na*.= 460, [K'li=400, and [K*.= 10. The Na* potential is found to be around +60 mV; the K+ potential is found to be somewhat more negative than the resting potential. Obviously, the cell maintains in a state of electrochemical disequilibrium. The energy that maintains this force is derived from the metabolism of the cell-a dead cell will eventually revert to a state of equilibrium. Considering the transmembrane potential (100 mV), and its small thickness(=10 nm), the electric field across the membrane is enormous(=10 MV/m). The membrane is semipermeable; that is, it is a lossy dielectric which allows the passage of certain ions. The ionic permeability varies substantially from one ionic species to another. The ionic channels in the excitable nembrane will vary their permeability in response to the transmembrane potential; this property distinguishes the excitable membrane from the ordinary cellular membrane, and it supports propagation of nerve impulses. The electrodynamics of the excitable membrane of unmyelinated nerves were first described in detail in theOutside Nobel prize work of Hodgkin and Huxley [1952]. This Membrane work was later extended to the myelinated nerve mem- c brane by Frankenhaeuser and Huxley [1964 Figure 113.2 illustrates an electrical model of the Hodgkin-Huxley membrane, which consists of nonlinear conductances for Na* and K+ and a linear leakage ele EK ment. The potential sources shown in the diagram ar the Nernst potentials for the particular ions as given by Eq (113.2). The capacitance term Cm is formed by the dielectric membrane separating the conductive media on FIGURE 113.2 Hodgkin-huxley membrane model either side. The conductances &Na and &x apply to Na* and K+ channels; the conductance gt is a general"leakage"channel that is not specific to any particular ion. The &Na and gx conductivities are highly dependent on the voltage applied across the membrane as described by a set of nonlinear differential equations. When the membrane is in the resting state, &Na < &k, and the membrane potential moves toward the Nernst potential for Nat. In this depolarized state, the membrane is said to be excited. The transition between the resting and excited condition of the membrane occurs rather abruptly when tances vary again, causing the membrane to revert back to its resting potentia the ionic channel conduc- the membrane potential has been depolarized by roughly 15 mV. After excitatie The duration of the excited state lasts roughly 1 ms. The progression of the membrane voltage during the period of excitation and recovery is termed an action potential. After the membrane has been excited, it cannot be reexcited until a recovery period, called the refractory period, has passed Figure 113.3 illustrates the processes that support the propagation of an action potential. Consider that point A on the axon is depolarized. The local depolarization causes ionic transfer between adjacent points on the axon, thus propagating the region of depolarization. If depolarization were initiated from an external electrical source on a resting membrane at point A, an action potential would propagate in both directions away from the site of stimulation. The body's natural condition, however, is to initiate an action potential at the terminus of the axon, which then propagates in only one direction. Electrical model for Nerve excitation FIGURE 113.3 Spread of the depolarization wave Myelinated fibers have much lower thresholds of excitation front along an axon. Depolarization occurring in than unmyelinated fibers. Accordingly, the myelinated fiber region A results in charge transfer from the adjacent is an appropriate choice for electrical stimulation studies. regions e 2000 by CRC Press LLC

© 2000 by CRC Press LLC In a quiescent state, nerve and muscle cells maintain a membrane potential typically around –60 to –90 mV, with the inside of the cell negative relative to the outside. Two ions that are involved in the electrical response of nerve and muscle are Na+ and K+. The concentration of these ions inside and outside the cell dictates the Nernst potential according to Eq. (113.2). Example concentrations in mM/cm3 for a nerve axon would be [Na+]i = 50, [Na+]o = 460, [K+]i = 400, and [K+]o = 10. The Na+ potential is found to be around +60 mV; the K+ potential is found to be somewhat more negative than the resting potential. Obviously, the cell maintains in a state of electrochemical disequilibrium. The energy that maintains this force is derived from the metabolism of the cell—a dead cell will eventually revert to a state of equilibrium. Considering the transmembrane potential (ª100 mV), and its small thickness (ª10 nm), the electric field across the membrane is enormous (ª10 MV/m). The membrane is semipermeable; that is, it is a lossy dielectric which allows the passage of certain ions. The ionic permeability varies substantially from one ionic species to another. The ionic channels in the excitable membrane will vary their permeability in response to the transmembrane potential; this property distinguishes the excitable membrane from the ordinary cellular membrane, and it supports propagation of nerve impulses. The electrodynamics of the excitable membrane of unmyelinated nerves were first described in detail in the Nobel prize work of Hodgkin and Huxley [1952]. This work was later extended to the myelinated nerve membrane by Frankenhaeuser and Huxley [1964]. Figure 113.2 illustrates an electrical model of the Hodgkin-Huxley membrane, which consists of nonlinear conductances for Na+ and K+ and a linear leakage element. The potential sources shown in the diagram are the Nernst potentials for the particular ions as given by Eq. (113.2). The capacitance term Cm is formed by the dielectric membrane separating the conductive media on either side. The conductances gNa and gK apply to Na+ and K+ channels; the conductance gL is a general “leakage” channel that is not specific to any particular ion. The gNa and gK conductivities are highly dependent on the voltage applied across the membrane as described by a set of nonlinear differential equations. When the membrane is in the resting state, gNa << gK, and the membrane potential moves toward the Nernst potential for Na+. In this depolarized state, the membrane is said to be excited. The transition between the resting and excited condition of the membrane occurs rather abruptly when the membrane potential has been depolarized by roughly 15 mV. After excitation, the ionic channel conductances vary again, causing the membrane to revert back to its resting potential. The duration of the excited state lasts roughly 1 ms. The progression of the membrane voltage during the period of excitation and recovery is termed an action potential. After the membrane has been excited, it cannot be reexcited until a recovery period, called the refractory period, has passed. Figure 113.3 illustrates the processes that support the propagation of an action potential. Consider that point A on the axon is depolarized. The local depolarization causes ionic transfer between adjacent points on the axon, thus propagating the region of depolarization. If depolarization were initiated from an external electrical source on a resting membrane at point A, an action potential would propagate in both directions away from the site of stimulation. The body’s natural condition, however, is to initiate an action potential at the terminus of the axon, which then propagates in only one direction. Electrical Model for Nerve Excitation Myelinated fibers have much lower thresholds of excitation than unmyelinated fibers. Accordingly, the myelinated fiber is an appropriate choice for electrical stimulation studies. FIGURE 113.2 Hodgkin-Huxley membrane model. FIGURE 113.3 Spread of the depolarization wave front along an axon. Depolarization occurring in region A results in charge transfer from the adjacent regions

© 2000 by CRC Press LLC Figure 113.4 illustrates an electrical model for myelinated nerve as originally formulated by McNeal [1976]. The myelin internodes are treated as perfect insulators and the nodes as individual circuits consisting of capacitance Cm and an ionic conductance term. The nodes are interconnected through the internal axon medium by conductances Ga. The current flowing in the biological medium creates voltage disturbances Ve,n at the exterior of the nodes. The current emanating from the nth node is the sum of capacitive and ionic currents described by (113.3) where Cm is the membrane capacitance at the node, Vn is the transmembrane potential, Ii,n is the total ionic current, and Vi,n is the internal voltage. In this expression, Vn is taken relative to the resting potential, such that Vn = 0 applies to the membrane resting potential. The ionic current flux is the sum of individual ionic terms (similar to the representation in Fig. 113.4), Ii,n = pdW(JNa + JK + JL + JP) (113.4) where the J terms are ionic current densities as described by a set of nonlinear differential equations developed by Frankenhaeuser and Huxley [1964] for a myelinated nerve membrane. Other relationships are (113.5) Cm = cmpdW (113.6) where d is the axon diameter at the node, ri is the resistivity of the internal axon medium, L is the internodal distance, W is the nodal gap width, and cm is the membrane capacitance per unit area. The relationship between the axon diameter d and the fiber diameter D (including myelin) is d ª 0.7D. The voltage Vn across the membrane is Vn = Vi,n – Ve,n (113.7) where Vi,n and Ve,n are the internal and external nodal voltages with reference to a distant point in the conducting medium outside the axon. Substituting Eq. (113.7) into (113.3) results in FIGURE 113.4 Equivalent circuit model for electrical excitation of myelinated nerve fiber. The membrane conductance Gm is described by nonlinear ionic conductances, similar to the representation in Fig. 113.2. C dV dt m I G V V V n + = i n, a i n, – i n, + i n, + ( – ) 1 1 2 G d L a i = p r 2 4

[G (Vn-1-2Vn Vn++Ven-2Ven Ven+1)-Iil (1138) For application to an unmyelinated fiber, Eq. (113. 8)may be analogously expressed in continuous form +V= (113.9 where V and V are membrane voltage and external voltage, respectively, at longitudal position x. Equation (113.9)can be derived from first principles, or can be obtained from(113. 8)by substituting Cm =CmIdAx Ga=Id/(4p: 4x), Gm= 8dAx, where d is the fiber diameter, Ax is the longitudinal increment, P; is the axoplasm resistivity(in Q2cm)internal to the fiber, cm is capacitance per unit area, and gm is conductance times unit area. Continuous and discrete spatial derivatives are connected by d?v/dx=(Vm-1-2Vn+ Vm+l)/Ax d?v/dx2=(Vm-2Vem+ Vn+1)Ax2; tm is the member time constant given by cm/gm i A is the membrane space constant given by n=(rm/ri)=(dpm/4p: ))2, and pm is the membrane specific resistance(in Q2cm2)An additional relationship is Iin=V/Gm If one treats A as a constant, then(113.9)describes the membrane response only during its sub-threshold (linear)phase. For membrane depolarization approaching the threshold of excitation, membrane conductance of ionic constituents becomes highly nonlinear, as noted above is this nonlinear behavior that leads to nerve excitation The left-hand side of Eq. (113.9)is the so-called cable equation that was developed by Oliver Heaviside over 100 years ago in connection with the analysis of the first transatlantic telegraphy cable. The right-hand side is a driving function due to the external field in the biological medium. For additional information on cable theory as applied to the excitable membrane, the reader is directed to Jack et al. [ 1983] One conclusion that can be drawn from Eqs. (113. 8)and (113.9)is that a second spatial derivative of voltage (or equivalently a first derivative of the electric field) must exist along the long axis of an excitable fiber order to support excitation. Nevertheless, excitation is possible in a locally constant electric field where the iber is terminated or where it bends. The orientation change or the termination creates the equivalent of a spatial derivative of the applied field Stimulation at"ends and bends"can be the dominant mode of excitation In many cases The external voltages in Eq.(113.8)are dependent on the distribution of current within the biological medium. For a point electrode in an isotopic medium, for instance, we can determine these voltages by (113.10) where r, is the distance between the stimulating electrode and the nth node and Pe is the resistivity of the external medium. For a uniform current density flowing in a direction parallel to the fiber axis, the external Ven= vel t eln (113.11) where Vel is a reference voltage at the terminal node, L is the internodal distance, n is the node number, and E is the electric field in the medium. The electric field is related to current density by j= Eo, where o= 1/p is the conductivity of the medium and J is the current density. Since the response of the electrical model is dependent of Vel, we may assume V 1=0 for convenience in Eq (113. 11). The internodal distance L is proportional to fiber diameter D through the relationship L/D= 100. Other fiber diameter relationships are expressed in Eqs. (113.5)and (113.6). Because of these relationships, thresholds of electrical stimulation will vary inversely with fiber diameter. The distribution of myelinated nerve diameters found in human peripheral nerve or skeletal muscle typically ranges from 5 to 20 um. e 2000 by CRC Press LLC

© 2000 by CRC Press LLC (113.8) For application to an unmyelinated fiber, Eq. (113.8) may be analogously expressed in continuous form as (113.9) where V and Ve are membrane voltage and external voltage, respectively, at longitudal position x. Equation (113.9) can be derived from first principles, or can be obtained from (113.8) by substituting Cm = cmpdDx, Ga = pd2 /(4riDx), Gm = gmpdDx, where d is the fiber diameter, Dx is the longitudinal increment, ri is the axoplasm resistivity (in Wcm) internal to the fiber, cm is capacitance per unit area, and gm is conductance times unit area. Continuous and discrete spatial derivatives are connected by ¶2 V/ ¶x2 ª (Vn–1 – 2Vn+ Vn+1)/Dx2 ; ¶2 Ve/¶x2 ª (Ve,n–1 – 2Ve,n + Ve,n+1)/Dx 2 ; tm is the member time constant given by cm/gm; l is the membrane space constant given by l = (rm/ri)1/2 = (drm/4ri))1/2, and rm is the membrane specific resistance (in Wcm2 ). An additional relationship is Ii,n = V/Gm. If one treats l as a constant, then (113.9) describes the membrane response only during its sub-threshold (linear) phase. For membrane depolarization approaching the threshold of excitation, membrane conductance of ionic constituents becomes highly nonlinear, as noted above — it is this nonlinear behavior that leads to nerve excitation. The left-hand side of Eq. (113.9) is the so-called cable equation that was developed by Oliver Heaviside over 100 years ago in connection with the analysis of the first transatlantic telegraphy cable. The right-hand side is a driving function due to the external field in the biological medium. For additional information on cable theory as applied to the excitable membrane, the reader is directed to Jack et al. [1983]. One conclusion that can be drawn from Eqs. (113.8) and (113.9) is that a second spatial derivative of voltage (or equivalently a first derivative of the electric field) must exist along the long axis of an excitable fiber in order to support excitation. Nevertheless, excitation is possible in a locally constant electric field where the fiber is terminated or where it bends. The orientation change or the termination creates the equivalent of a spatial derivative of the applied field. Stimulation at “ends and bends” can be the dominant mode of excitation in many cases. The external voltages in Eq. (113.8) are dependent on the distribution of current within the biological medium. For a point electrode in an isotopic medium, for instance, we can determine these voltages by (113.10) where rn is the distance between the stimulating electrode and the nth node and re is the resistivity of the external medium. For a uniform current density flowing in a direction parallel to the fiber axis, the external voltages are determined by Ve,n = Ve,1 + ELn (113.11) where Ve,1 is a reference voltage at the terminal node, L is the internodal distance, n is the node number, and E is the electric field in the medium. The electric field is related to current density by J = Es, where s = 1/re is the conductivity of the medium and J is the current density. Since the response of the electrical model is independent of Ve,1, we may assume Ve,1 = 0 for convenience in Eq. (113.11). The internodal distance L is proportional to fiber diameter D through the relationship L/D ª 100. Other fiber diameter relationships are expressed in Eqs. (113.5) and (113.6). Because of these relationships, thresholds of electrical stimulation will vary inversely with fiber diameter. The distribution of myelinated nerve diameters found in human peripheral nerve or skeletal muscle typically ranges from 5 to 20 mm. dV dt C GV V V V V V I n m = + +- + a n n n en en en in + + 1 2 2 11 1 [ ( – )– ] – , ,, , t ¶ l ¶ ¶ l ¶ ¶ m V e dt V x V V x – 2 2 2 2 2 2 + = V I r e n e n , = r 4p

(b)I 2IT d)-(f)IT 08 Time(msec) FIGURE 113.5 Response of myelinated nerve model to rectangular monophasic current of 100 ms duration, 20-um diameter fiber, point electrode 2 mm from central node. Solid lines show response at node nearest electrode for three levels of current. I denotes threshold current. Dashed lines show propagated response at next three adjacent nodes for a stimulus at threshold. Source: J. P. Reilly, V. T. Freeman, and W. D. Larkin, "Sensory effects of transient electrical stimulation-Eval uation with a neuroelectric model, "IEEE Trans. Biomed. Eng, vol. BME-32, no. 12, pp. 1001-1011,@ 1985 IEEE. Figure 113.5 illustrates the response of the myelinated nerve model of Fig. 113.4 to a rectangular current stimulus [Reilly et al., 1985]. The example is for a small cathodal electrode that is 2 mm radially distant from a 20-um fiber and directly above a central node. The transmembrane voltage AV is scaled relative to the resting potential. The solid curves show the response at the node nearest the stimulating electrode Response a is for a pulse that is 80% of the threshold current, b is at threshold, and c is 20%above threshold. The threshold stimulus pulse in this example has an amplitude Ir of 0.68 mA Response a is similar to that of a linear network with a parallel resistor and apacitor a charged by a brief current pulse Responses b and c demonstrate the highly nonlinear response of the excitable membrane. The dashed curves in Fig. 113.5 show the membrane response to a threshold stimulus at the three nodes adjacent to the one nearest the stimulating electrode. The time delay implies a propagation velocity of 43 m/s, which is typical of a 20-um fiber. The membrane response seen in curves b through f illustrates the action potential described earlier. The action potential is typically described as an" all-or-nothing"response; that is, its amplitude is not normally graded--either the axon is The threshold current needed for excitation is highly dependent on its duration and waveshape. A common rmat for representing the response of a nerve is through strength-duration curves, i. e, the plot of the threshold of excitation versus the duration of the stimulating current. We can determine the threshold of excitation by "titrating"the stimulus current between a threshold and no-threshold condition Figure 113.6 illustrates strength-duration curves derived from the myelinated nerve model described previ ously under the same conditions applying to Fig. 113.5. Three types of stimulus current apply to Fig. 113.6: a monophasic constant current pulse, a symmetric biphasic rectangular current, and a single cycle of a sine wave The phase duration indicated on the horizontal axis applies to the initial cathodal half cycle for the two biphasic waves. Stimulus magnitude is given in terms of peak current on the right vertical axis and in terms of the charge in a single monophasic phase of the stimulus on the left vertical axis. The charge is computed by Q= It, for the rectangular waveforms and Q=(2/I)It, for the sinusoidal waveforms(I is threshold current and t, is phase e 2000 by CRC Press LLC

© 2000 by CRC Press LLC Figure 113.5 illustrates the response of the myelinated nerve model of Fig. 113.4 to a rectangular current stimulus [Reilly et al., 1985]. The example is for a small cathodal electrode that is 2 mm radially distant from a 20-mm fiber and directly above a central node. The transmembrane voltage DV is scaled relative to the resting potential. The solid curves show the response at the node nearest the stimulating electrode. Response a is for a pulse that is 80% of the threshold current, b is at threshold, and c is 20% above threshold. The threshold stimulus pulse in this example has an amplitude IT of 0.68 mA. Response a is similar to that of a linear network with a parallel resistor and capacitor and charged by a brief current pulse. Responses b and c demonstrate the highly nonlinear response of the excitable membrane. The dashed curves in Fig. 113.5 show the membrane response to a threshold stimulus at the three nodes adjacent to the one nearest the stimulating electrode. The time delay implies a propagation velocity of 43 m/s, which is typical of a 20-mm fiber. The membrane response seen in curves b through f illustrates the action potential described earlier. The action potential is typically described as an “all-or-nothing” response; that is, its amplitude is not normally graded—either the axon is excited, or it is not. The threshold current needed for excitation is highly dependent on its duration and waveshape. A common format for representing the response of a nerve is through strength-duration curves, i.e., the plot of the threshold of excitation versus the duration of the stimulating current. We can determine the threshold of excitation by “titrating” the stimulus current between a threshold and no-threshold condition. Figure 113.6 illustrates strength-duration curves derived from the myelinated nerve model described previously under the same conditions applying to Fig. 113.5. Three types of stimulus current apply to Fig. 113.6: a monophasic constant current pulse, a symmetric biphasic rectangular current, and a single cycle of a sine wave. The phase duration indicated on the horizontal axis applies to the initial cathodal half cycle for the two biphasic waves. Stimulus magnitude is given in terms of peak current on the right vertical axis and in terms of the charge in a single monophasic phase of the stimulus on the left vertical axis. The charge is computed by Q = Itp for the rectangular waveforms and Q = (2/p)Itp for the sinusoidal waveforms (I is threshold current and tp is phase duration). FIGURE 113.5 Response of myelinated nerve model to rectangular monophasic current of 100 ms duration, 20-mm diameter fiber, point electrode 2 mm from central node. Solid lines show response at node nearest electrode for three levels of current. IT denotes threshold current. Dashed lines show propagated response at next three adjacent nodes for a stimulus at threshold. (Source: J. P. Reilly, V. T. Freeman, and W. D. Larkin, “Sensory effects of transient electrical stimulation—Evaluation with a neuroelectric model,” IEEE Trans. Biomed. Eng., vol. BME-32, no. 12, pp. 1001–1011, © 1985 IEEE.)

Monophasic cathodal S TT b-o Biphasic rectangular Sine wave 10 Current Charge 0. k。! 0.01 Stimulus phase duration, tp IGURE 113.6 Strength/duration relationships derived from the myelinated nerve model: current thresholds and charge thresholds for single-pulse monophasic and for single-cycle biphasic stimuli with initial cathodal phase, point electrode 2 mm distant from 20 um fiber. Threshold current refers to the peak of the stimulus waveform. Charge refers to a single phase for biphasic stimuli. Source: J. P. Reilly, V. T. Freeman, and W. D. Larkin, "Sensory effects of transient electrical stimula- tion-Evaluation with a neuroelectric model " IEEE Trans. Biomed. Eng, vol. BME- 32, no 12, Pp. 1001-1011,9 1985 IEEE The solid curve labeled"current"is of the type that is most often represented as a strength-duration curve For this curve, the minimum threshold current occurs for long-stimulus durations and is called the rheobasic current,or simply rheobase. The duration consistent with twice the rheobase is called the chronaxie. The solid curve in Fig. 113.6 labeled"charge"gives the area under the rectangular current pulse. The threshold charge is a minimum for short -duration stimul Mathematical curve fits to the strength-duration curves for monophasic rectangular stimuli are (113.12) 。1 Q t/τ (113.13) -t/te where I is threshold current, Q is threshold charge, I, is the minimum threshold current for long-duration stimuli,Q, is the minimum threshold charge for short-duration stimuli, and t is an experimentally determined strength-duration time constant. It is readily shown that chronaxie =t, In 2=0.693t in this formulation. Values of I, and Q, vary considerably with experimental parameters such as electrode size and location and the size of the neuron. Values of te also vary considerably with experimental conditions: a value around 250 us is typical for both sensory and motor nerve excitation via cutaneous electrodes, and values around 125 us are observed for stimulation of axons by small electrodes. Much longer time constants are associated with direct stimulation of muscle cells e 2000 by CRC Press LLC

© 2000 by CRC Press LLC The solid curve labeled “current” is of the type that is most often represented as a strength-duration curve. For this curve, the minimum threshold current occurs for long-stimulus durations and is called the rheobasic current, or simply rheobase. The duration consistent with twice the rheobase is called the chronaxie. The solid curve in Fig. 113.6 labeled “charge” gives the area under the rectangular current pulse. The threshold charge is a minimum for short-duration stimuli. Mathematical curve fits to the strength-duration curves for monophasic rectangular stimuli are (113.12) and (113.13) where IT is threshold current, QT is threshold charge, Io is the minimum threshold current for long-duration stimuli, Qo is the minimum threshold charge for short-duration stimuli, and te is an experimentally determined strength-duration time constant. It is readily shown that chronaxie = te ln 2 = 0.693te in this formulation. Values of Io and Qo vary considerably with experimental parameters such as electrode size and location and the size of the neuron. Values of te also vary considerably with experimental conditions: a value around 250 ms is typical for both sensory and motor nerve excitation via cutaneous electrodes, and values around 125 ms are observed for stimulation of axons by small electrodes. Much longer time constants are associated with direct stimulation of muscle cells. FIGURE 113.6 Strength/duration relationships derived from the myelinated nerve model: current thresholds and charge thresholds for single-pulse monophasic and for single-cycle biphasic stimuli with initial cathodal phase, point electrode 2 mm distant from 20 mm fiber. Threshold current refers to the peak of the stimulus waveform. Charge refers to a single phase for biphasic stimuli. (Source: J. P. Reilly, V. T. Freeman, and W. D. Larkin, “Sensory effects of transient electrical stimulation—Evaluation with a neuroelectric model,” IEEE Trans. Biomed. Eng., vol. BME-32, no. 12, pp. 1001–1011, © 1985 IEEE.) I I e T o t e = 1 1 – – /t Q Q t e T o e t e = / – – / t t 1

© 2000 by CRC Press LLC The current reversal of a biphasic stimulus can reverse a developing action potential that was elicited by the initial phase. As a result, a biphasic pulse may have a higher threshold than a monophasic pulse as suggested by the biphasic thresholds in Fig. 113.6. The degree of biphasic threshold elevation is magnified as the stimulus duration is reduced. A sinusoidal current is a special case of a biphasic stimulus. Sinusoidal threshold response can be represented by strength-frequency curves, as shown by the solid curves in Fig. 113.7 for the myelinated nerve model. Several experimental curves have been included in the figure; these have been shifted vertically to facilitate comparisons. Notice that the myelinated nerve model predicts a lower threshold for stimulation by a continuous sine wave as compared with a single cycle. The strength-frequency curve follows a U-shaped function, with a minimum at mid frequencies and an upturn at both low and high frequencies. At low frequencies the slow rate of change of the sinusoid prevents the membrane capacitance from building up a depolarizing voltage because membrane capacitance is counteracted by membrane leakage. This process describes the neural property known as accommodation, i.e., the adaptation of a nerve to a slowly varying or constant stimulus. The high-frequency upturn occurs because of the canceling effects of a current reversal on the membrane voltage change. An empirical fit to strengthfrequency curves is It = IoKHKL (113.14) where It is the threshold current,Io is the minimum threshold current, and KH and KL are high- and low-frequency terms, defined, respectively, as (113.15) and FIGURE 113.7 Strength-frequency curves for sinusoidal current stimuli. Dashed curves are from experimental data. Solid curves apply to myelinated nerve model. Experimental curves have been shifted vertically to facilitate comparisons. K f f H e a = Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ 1 – exp – –

i NAM 12 TTT IGURE 113. 8 Model response to continuous sinusoidal stimulation at 500 Hz. The lower panel depicts the response to stimulus current set at threshold level (I)for a single-cycle stimulus. Upper pa responses for stimulation 20 50% above the single-cycle threshold. Source: J. P. Reilly, V. T. Freeman, and W. D. Larkin,"Sensory effects of transient electrical stimulation-Evaluation with a neuroelectric model, "IEEE Trans. Biomed. Eng, voL. BME-32, no. 12, pp. 1001-1011 @1985 IEEE (113.16) where f and fo are constants that determine the points of upturn in the strength-frequency curve at high and low frequencies, respectively. An upper limit of K,s 4.6 is assumed for Eq (113.16)to account for the fact that excitation may be obtained with finite dc currents. An empirical fit of Eqs. (113.15)and(113. 16)to the mylinated nerve model thresholds indicates that a=1. 45 for a single-cycle stimulus and a=0.9 for a continuous stimulu including the size of the electrode, its location on the body, and the location of the stimulated nerve lation, b=0.8 regardless of stimulus duration. The value of I, will depend on various conditions of stimu With continuous sinusoidal stimulation, it is possible to produce a series of action potentials that are phase locked to the individual sinusoidal cycles, as noted in Fig. 113.8. This makes the sinusoidal stimulus much more potent than a single pulse of the same phase duration. This potency is a consequence of the fact that perceived magnitude for neurosensory stimulation and muscle tension for neuromuscular stimulation both increase with the rate of action potential production. Defining Terms Action potential: A propagating change in the conductivity and potential across a nerve cells membrane;a nerve impulse in common parlance Axon: The conducting portion of a nerve fiber-a roughly tubular structure whose wall is composed of the cellular membrane and which is filled with an ionic medium Chronaxie: The minimum duration of a unidirectional square-wave current needed to excite a nerve when he current magnitude is twice rheobase e 2000 by CRC Press LLC

© 2000 by CRC Press LLC (113.16) where fe and fo are constants that determine the points of upturn in the strength-frequency curve at high and low frequencies, respectively. An upper limit of KL £ 4.6 is assumed for Eq. (113.16) to account for the fact that excitation may be obtained with finite dc currents.An empirical fit of Eqs. (113.15) and (113.16) to the mylinated nerve model thresholds indicates that a = 1.45 for a single-cycle stimulus and a = 0.9 for a continuous stimulus; b = 0.8 regardless of stimulus duration. The value of Io will depend on various conditions of stimulation, including the size of the electrode, its location on the body, and the location of the stimulated nerve. With continuous sinusoidal stimulation, it is possible to produce a series of action potentials that are phaselocked to the individual sinusoidal cycles, as noted in Fig. 113.8. This makes the sinusoidal stimulus much more potent than a single pulse of the same phase duration. This potency is a consequence of the fact that perceived magnitude for neurosensory stimulation and muscle tension for neuromuscular stimulation both increase with the rate of action potential production. Defining Terms Action potential: A propagating change in the conductivity and potential across a nerve cell’s membrane; a nerve impulse in common parlance. Axon: The conducting portion of a nerve fiber—a roughly tubular structure whose wall is composed of the cellular membrane and which is filled with an ionic medium. Chronaxie: The minimum duration of a unidirectional square-wave current needed to excite a nerve when the current magnitude is twice rheobase. FIGURE 113.8 Model response to continuous sinusoidal stimulation at 500 Hz. The lower panel depicts the response to a stimulus current set at threshold level (IT)for a single-cycle stimulus. Upper panels show responses for stimulation 20 and 50% above the single-cycle threshold. (Source: J. P. Reilly, V. T. Freeman, and W. D. Larkin, “Sensory effects of transient electrical stimulation—Evaluation with a neuroelectric model,” IEEE Trans. Biomed. Eng., vol. BME-32, no. 12, pp. 1001–1011, © 1985 IEEE.) K f f L o b = Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ 1 – exp – –