[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
Perception (Dalziel, 1972) (Anderson Munson, 1951) SENN model single cycle (Hz) FIGURE 113.7 Strength-frequency curves for sinusoidal current stimuli. Dashed curves are from experimental data. Solid urves apply to myelinated nerve model. Experimental curves have been shifted vertically to facilitate comparisons 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 representee 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 frequencie upturn at both low and high frequencies. At low frequencies the slow rate of change of the sinusoid the membrane capacitance from building up a depolarizing voltage because membrane capacitance is coun- teracted 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 strengt frequency curves is I,=IoKHKL (113.14) where I, is the threshold current, I, is the minimum threshold current, and KHand KLare high-and low-frequency terms,defined, respectively H (113.15) e 2000 by CRC Press LLC
© 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 – –