Baroreceptor responses derived from a fundamental concept

Annals of BiomedicalEngineering. Vol. 16, pp. 429-443, 1988

Printed in the USA.All rights reserved.

0090-6964/88 $3.00 + .00 Copyright 9 1988PergamonPress plc

BARORECEPTOR RESPONSES DERIVED FROM A FUNDAMENTAL CONCEPT M o n a F. T a h e r , A l b e r t B . P . Cecchini, M a r k A . Allen, S h e r i f R. G o b r a n , R o b e r t C. G o r m a n , B r i a n L. G u t h r i e , Kyle A . L i n g e n f e l t e r , Sina Y. R a b b a n y , P h i l i p M. R o l c h i g o , Julius M e l b i n , a n d A b r a h a m N o o r d e r g r a a f (Received 12/3/87)

Cardiovascular Studies Unit, Departments of Bioengineering and Animal Biology, University of Pennsylvania, Philadelphia

A model is presented that relates the change in baroreceptorfiring rate to a step change in blood pressure. This relationship is nonlinear since the alteration in rate of firing depends on the current rate of firing. It is shown that this simple relationship embodies all currently established baroreceptor response modes. The model needs refinement to allow f o r effects arising from the properties o f the tissue matrix in which the receptors are embedded. Further analysis is precluded at present owing to paucity o f quantitative experimental data. K e y w o r d s - Baroreceptor, Model o f baroreceptor firing rate.

INTRODUCTION An interpretation of mammalian systemic arterial pressure regulation was provided by Hering (20,21) when he described the reflex function of carotid sinus receptors. Hering demonstrated that stimulation of the carotid sinus in the dog caused bradycardia (20) and lowering of peripheral resistance (21). These works identified an afferent pathway to the central nervous system and efferent pathways to the heart and peripheral circulation. Experimental studies have focused on the relationship between afferent nerve activity and blood pressure, on the relationship between pressure in the isolated carotid artery and that in the remaining systemic arterial system, and related effects such as on heart rate and peripheral resistance. In addition, models have been formulated to describe the observed relations. None of these models, however, encompass the wide range of responses currently attributed to baroreceptor control, or suggest why hypertension occurs in the presence of apparently normal baroreceptors. Supported in part by NIH grants HL 10,330, HL 22,223, HL 24,966, and HL 31,480. This study developed from a class project in a graduate Biocontrol course taught by Abraham Noordergraaf. Address correspondenceto Mona F. Taher, Cardiovascular Studies Unit, Departments of Bioengineering and Animal Biology, University of Pennsylvania, Philadelphia, PA 19104. 429

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In this paper, we develop a model that is intimately related to documented receptor behavior and demonstrate that, on the basis of this model, currently established phenomena can be interpreted. EXPERIMENTAL BACKGROUND

Bronk and Stella (4,5) were the first to document observations of baroreceptor cell activity. Figure 1A illustrates an example of their findings which indicates that firing rate varies as pressure varies. Using rabbits and a single fiber preparation, carotid sinus baroreceptor activity was obtained simultaneously with mean blood pressure. Bronk and Stella demonstrated that, for a single baroreceptor, the rate o f afferent impulses increased with arterial pressure commencing at a threshold pressure and reaching a saturation rate of 120-140 impulses per second (Fig. 1B). They also demonstrated that a receptor could discharge at pressures below this threshold provided that the rate of change in pressure was sufficiently rapid. Baroreceptor saturation (maximum firing rate) was confirmed by Landgren (26) as well as by others. Using the isolated carotid sinus and applying pressure steps, Landgren (27) developed a sigmoidal pressure-impulse rate response curve (Fig. 1C). Angell-James (2) studied the effects o f gradually increasing aortic pressure in rabbits. She reported widely varying threshold pressure with a mean value of 50.8 mm Hg, and essentially a linear relation between pressure and firing rate from threshold to saturation. Similar to threshold pressures, pressures at which saturation occurred were also observed to differ greatly (2,15). Focusing on the dynamic response of single baroreceptors to step changes in pressure, Landgren (26) found that firing rate decayed with time, reaching a near steadystate in about 500 seconds (Fig. 1D). This detailed the adaptation that Bronk and Stella (4,5) had observed. In response to a stepwise decrease in pressure, firing may cease for some seconds, followed by a recovery of firing rate commensurate with the

FIGURE 1A. Bronk and Stella (4) demonstrated that baroreceptor discharge rate varies with varying blood pressure. Mean blood pressure is 105 mm Hg. Protrusions in the lower boundary mark 1/5 second.

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newly established pressure level (4,5,7). Firing rate associated with two stepwise changes in pressure is reproduced in Fig. 1E. Asymmetry, or a difference in response at the same pressure level for rising and falling pressure, was first noted by Bronk and Stella (4,5). Clynes (11) attributed this to "unidirectional rate sensitivity" o f the baroreceptors. Franz (14) concluded from experimental evidence that baroreceptors respond to both increase and decrease in

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FIGURE 1C. The sigmoidal response curve of impulse frequency (n| of the baroreceptor unit as a function of the intrasinus blood pressure (P) as reported by Landgren (26). n is determined by counting the impulses elicited between 0.25 and 0.75 second after the pressure rise (adapted from Landgren).

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pressure. Franz et al. (15,16) later demonstrated that baroreceptor response to small amplitude square waves in pressure becomes more asymmetric with longer square waves and more rapid pressure drops. The asymmetric response to a symmetric triangular pressure input plots as a hys-

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teresis loop. The enclosed area of the loop was found by Clark (10) to depend on frequency. This was confirmed by Coleridge, Coleridge, Kaufman, and Dangel in 1981 (12) (Fig. 1F). McCubbin, Green, and Page (28) coined the term resetting to describe the observed shift in threshold toward higher pressure for hypertensive subjects. These authors also found that the longer the period of hypertension, the slower is the return to normal with restoration of normotension, this being attributed to adaptation. Subsequent investigators observed resetting in two directions (i.e., as a shift of the sigmoidal pressure response curve to a lower pressure range in hypotension, or to a higher pressure range in hypertension) (Fig. 1G) (22). Resetting was found to occur in established hypertension (29), in early hypertension (6,38) and with acute changes of arterial pressure (23). Resetting was generally found to be reversible. Many experimenters have sought to quantify the time required for baroreceptors to reset. Experimental protocol differed widely and some did not accommodate detection of short reset times, if any, while others did not accommodate observation of long reset times. A number of investigators noted multiple reset times in each experiment. To the extent that the reported experimental data make it possible, characteristic response times (defined as the time in which the response to a step change in pressure is reduced to e -~ (i.e., to about one-third of the initial response) were extracted.

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FIGURE 1F. The asymmetric response of baroreceptors to increasing and decreasing pressure plotted in the form of a hysteresis loop by Coleridge e t al. (12). Mean impulse frequency is plotted at intervals of 10 mm Hg. The open circle indicates baroreceptor impulse activity at the mean aortic pressure of 100 mm Hg. The dots and the continuous line indicate baroreceptor response to an increase in pressure above the baseline; the dots and the dotted line indicate the response to a decrease in the pressure below the baseline. Crosses indicate the changes in activity as pressure was returned from its highest and lowest values to the original baseline of 1 0 0 mm Hg. (Reproduced by permission.)

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Results are listed in Table 1, together with weighting factors, when available, where multiple response times were observed. The table also lists reported maximum firing rate nma• The experimental literature also hints at longer response time, sometimes approaching months. Since these could not be quantified they are not included in Table 1. Peterson (33) proposed that resetting stems from viscoelastic arterial wall properties. Krieger (24) recently observed in rats that the time required for resetting coincides with the time creep takes to adjust the vessel lumen to a new pressure level. Franz (14) concluded that adaptation can be described in terms of viscoelastic relaxation processes in the vessel wall. Other investigators argued that adaptation, resetting, etc. find their origin primarily in the membrane properties of the baroreceptor (7,18,19,31,43). Baroreceptor response can be modified by agents that alter membrane electrophysiological properties (e.g., 7,19). In addition, resetting may, in part, become irreversible which is attributed to the formation of lesions during chronic hypertension (3). Considerable discussion ensued as to whether stress or strain of the sinus wall caused by intrasinus pressure is responsible for baroreceptor firing. In 1949, Hauss, Kreuziger, and Asteroth (17) demonstrated that deformation of the sinus wall, not stress, provided the stimulus. After embedding the sinus in a plaster cast, an increase in intrasinus pressure no longer elicited a response. Charlton and Baertschi (9) investigated whether baroreceptors respond to changes in aortic flow. They perturbed normal aortic flow in the rat using a 1 Hz sinusoidal pump. Fiber activity was found to increase with pump instroke and to decrease with pump outstroke, primarily during late systole. In their experiments pressure varied as well, in synchrony with piston movement.

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The response of baroreceptors is further complicated by the finding that myelinated or nonmyelinated axons exhibit different characteristics (7). REVIEW OF A V A I L A B L E MODELS

Landgren (26) described baroreceptor adaptation to a step change in sinus pressure by: n - no = kl t -k2

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for t > 0, where n = firing frequency, t = time, na = adapted firing frequency and kl > 0 and 0 < k2 < 1 are constants. An equivalent formulation for piecewise ranges in which t and In t are approximately linearly related, is (la)

n - n~ = k e - ' / ~

where k and z are positive constants. Robinson and Sleight (35) described the adaptive response of single fibers to a stepwise change in pressure as II : al e-t/r~ + a2 e-t/T2 4- a 3

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where n denotes firing frequency, al, a2, a 3 are weighting factors, and r~, 72 are time constants. Warner (41) proposed a baroreceptor model that relates firing rate to pressure. This incorporated both threshold and sensitivity to change o f rate of pressure, such that do n = kx ( p - P 0 ) + k2 --r

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where p = pressure, Po = threshold pressure, and kl, k2 are constants. This linear model exhibits neither saturation nor adaptation. Later, Warner (42) modified his model by assigning different weights to the positive and negative rate of pressure change to allow for asymmetric response, i.e.

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Poitras, Pantelakis, Marble, Dwyer, and Barnett (34) included adaptation, described by two time constants, and threshold and saturation in such a way that threshold and saturation are independent of adaptation. In 1972, Srinivasan and Nudelman (40) proposed a complex mathematical model described by two nonlinear, ordinary differential equations with eight free parameters. This model exhibits threshold, adaptation and asymmetry, but not saturation. These models manifest some or all of the features of nonlinearity, threshold and saturation phenomena, adaptation, and asymmetric response. In c o m m o n amongst them is the separate incorporation of each of the properties implying that they arise independently. That such is not the case was demonstrated by Cecchini, Tiplitz, Melbin, and N o o r d e r g r a a f (8). Using Eq. 2 as the descriptor for baroreceptor response to a single step in pressure in the linear range (i.e., assuming adaptation), these investigators showed that responses then also display frequency dependent hysteresis, and asymmetry in the sense of Eq. 4. We show here that generalization of this assumption develops the entire range of established baroreceptor properties, thereby yielding a unified model. UNIFIED M O D E L

The change in firing rate An is assumed to respond to a stepwise change in pressure Ap as

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The coefficients al, a2, and a3 are weighting factors that determine the relative importance of the three exponential functions. The response An to a pressure step decreases with time at a rate defined by the characteristic times T1, z2, and 73. To compute the response to a time-varying pressure signal, this signal m a y be thought of as represented by a succession of small pressure increments and decrements Ap (Heaviside theorem). For the initial value of the firing rate n(to), An is calculated as a function of time for the first Ap. Consequently, An (tl) is known at the instant that the next Ap occurs. This permits updating o f the value of n to the current firing rate n (to) + An (tl). The response to the second Ap is then calculated and added to that of the first, allowing for the time interval between the occurrence of the first and second pressure step, etc. For a periodic p h e n o m e n o n the response m a y be computed for a number of periods until the response has reached a steady state. This steady state is independent of the value of the initial firing rate, while the transient part yields the adaptation f r o m one state to another. In the summation, the factor containing the sine plays a significant role. It specifies how the response An to a pressure step depends on the current firing rate n. This factor approaches zero when n is close to zero or close to the m a x i m u m firing rate nm~. When n = 89 the factor equals 1, its m a x i m u m value. Hence, the response to a step is maximal when the firing rate, just prior to its application, equals half the m a x i m u m firing rate of that receptor. The model in Eq. 6 does not contain an explicit threshold. In essence, it states in analytical f o r m what m a n y have observed as the response to a single pressure step.

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T E S T OF M O D E L The parameter values appearing in Eq. 6 were selected with Table 1 as guide. They are: al = 0.25, a2 = 0.4, a3 = 1 (mm H g s) -1, rl = 0.1, r2 = 100, r3 = 1000 s, nma~ = 120 s -I . The proposed model was implemented on a computer and firing rates computed for a number o f pressure curves. The computed firing rate in response to a normal blood pressure pulse is shown in Fig. 2A. Comparison to Fig. 1A demonstrates a firing pattern akin to classical experimental findings of Bronk and Stella (4). Figure 2B illustrates computed firing rate in response to a symmetric triangular pressure input and exhibits a number of features established by experimental observations. The ascending pressure response manifests a threshold, a linear portion, and saturation as in Fig. 1, panels B and C. Ascending and descending responses together exhibit hysteresis, or asymmetry, as in Fig. 1F. In agreement with early observations by Bronk and Stella (4,5), Eq. 6 predicts that at pressures below the threshold, firing may be elicited provided this pressure increases sufficiently rapid. The computed response to a step increase in pressure, followed by a step decrease in pressure of the same magnitude is displayed in Fig. 2C. In response to the step increase, firing rate increases rapidly, then decays exponentially, as in Fig. 1D (adaptation). The step decrease in pressure causes firing to cease for a period; firing resumes subsequently. The pattern in Fig. 2C is similar to the one illustrated in Fig. 1E. Since the response to a constant pressure decays with time, resetting is also embodied in Eq. 6. This is illustrated in Fig. 2D, where computed response curves are displayed for hypotensive, normotensive, and hypertensive pressure levels. A hypotensive pressure level shifts the response curve to lower pressure values compared

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FIGURE 2A. Firing rate computed from equation (6) in response to a blood pressure pulse as in Fig. 1A.

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to a normotensive response, as observed in Fig. 1G. A hypertensive pressure level shifts the curve to higher pressure values. DISCUSSION Equation 6 contains seven constants (3a's, 3r's and t/max). The typical computed response patterns illustrated in this report persist when the numerical values used here are altered significantly or when the number of constants is drastically increased or

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PRESSURE m,nal FIGURE 2D. Calculated response curves to a ramp input pressure for hypotensive, normotensive and hypertensive pressure levels. The magnitude of the pressure range is 1 6 0 mm Hg in all cases and the mean pressures are 65, 92, and 1 2 2 mm Hg, respectively.

decreased (to a minimum of one a, one z, and t/max). Such changes cause quantitative, but not qualitative alterations. Thus, the use of three exponential functions in Eq. 6 is illustrative and does not imply that baroreceptors necessarily exhibit three rates of adaptation. Such details cannot be concluded from presently available experimental data. The numerical values of the constants may be sensitive to metabolic and other effects. When the value of the longest time constant (r3 of Eq. 6 in the computed responses of this report) is long compared to the duration of data collection, the term a3 e x p ( - t / r 3 ) is approximately constant. A number of investigators reported a constant term in the firing rate, such as in Eq. 2. The value of such a (nearly) constant term is determined by the history of the procedures executed prior to the experimental observation shown. It is c o m m o n practice not to report such preceeding procedures. As a consequence, different constant firing rates m a y be shown at the beginning of reported experimental findings. In addition, values of pressures at which threshold and saturation are observed to occur may vary widely, as has been observed experimentally. Equation 6 predicts that the firing rate response to a constant pressure upon which is superimposed a sinusoidal pressure, eventually deteriorates to bursts of spikes roughly coincident with the positive half of the sine wave. The exact f o r m of this response depends on the values of the constants in Eq. 6. The constant component of the pressure is not represented in the steady state firing rate. While firing rate response is more sensitive to rapid as opposed to slow pressure changes, there is nothing to suggest that baroreceptors e m b o d y a set point in their firing rate. Hence, baroreceptor control of blood pressure lacks reference to a mean blood pressure level. The shift of the response curve to higher pressures in hypertension and to lower pressures in hypotension (Figs. 1G, 2D) supports the model's prediction that discharge rate is not dependent on a constant pressure component after a sufficiently long time (i.e., after the exponentials in Eq. 6 have decayed to zero).

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441

If resetting is not exclusively due to fiber damage, and there is at present n o firm evidence that it is, the model suggests that resetting can be attributed to adaptation in which the term with the longest time constant (z3 in Eq. 6) displays its influence. (Terms with shorter time constants vanish earlier.) The model of Eq. 6 may also include the small changes in firing rate observed by Charlton and Baertschi (9) in response to induced modulations in aortic flow since their data show concomitant modulations in pressure. Consequently, their suggestion that baroreceptors respond to flow is open to question. In this model it is implied that the arterial wall is a linear, purely elastic material with a constant radius to wall thickness ratio. More realistically, if the baroreceptors form an integral part of a viscoelastic tissue matrix, responses may be modified in several ways. As pressure increases, the radius to wall thickness ratio tends to increase as the quantities involved change in opposite directions. This will tend to increase wall stress more at a high pressure level than at a low pressure level for the same increase in pressure. Whether wall strain increases also, depends on the established dependance of wall stiffness on pressure level (32). The combined effect may cause the response curves of Fig. 2D to display different slopes (gains). Since the wall material is viscoelastic, alteration in pressure will be accompanied by creep phenomena. It has been found that a number of time constants is involved in vessel wall creep (32). Both creep and the exponentials in Eq. 6 may affect measured responses to alteration of pressure. Owing to the discrete nature of the appearance of spikes, investigators have to select a time interval to measure firing rate subsequent to, for example, a step increase in pressure. The exponentials in Eq. 6 indicate that the measured response will be smaller with later placement of this time interval, while creep will tend to increase the response. Though the effect of creep may escape the attention of the experimenter in an experiment involving a step change in pressure, its effect may be more conspicious in measuring the response to a symmetric triangular pressure course. The response computed from Eq. 6 and depicted in Fig. 2B shows a closed hysteresis loop. However, if creep effects persist beyond the time of the last measurement of firing rate (i.e., at the initial pressure), the measured rate will exhibit a positive value and the loop will be open. This phenomenon may have influenced the results reproduced in Fig. 1F. CONCLUSION A model is presented that brings the well-established baroreceptor responses under the umbrella of a single, experimentally observed response. Its behavior implies that baroreceptors respond well to rapid changes in arterial pressure but not to slowly developing hypertension. The proposed model needs further refinement. Such refinement requires new experimental data. Sensors other than baroreceptors may be amenable to the same description. REFERENCES 1. Aars, H. Aortic baroreceptor activity in normal and hypertensive rabbits. Acta Physiol. Scand. 72:298-309; 1968. 2. Angell-James, J.E. Studies of the impulse activity in baroreceptor fibres from an isolated aortic arch preparation of the rabbit. J. Physiol. (Lond.) 169:51-52 P; 1968.

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3. Angell-James, J.E. Characteristics of single aortic and right subclavian baroreceptor fiber activity in rabbits with chronic renal hypertension. Circ. Res. 32:149-161; 1973. 4. Bronk, D.W.; Stella, G. Afferent impulses in the carotid sinus nerve. I. The relation of the discharge from single end organs to arterial blood pressure. J. Cell. Comp. Physiol. 1:113-130; 1932. 5. Bronk, D.W.; Stella, G. The response to steady pressures of single end organs in the isolated carotid sinus. Am. J. Physiol. 110:708-714; 1935. 6. Brown, A.M.; Saum, W.R.; Tuley, F.H. A comparison of aortic baroreceptor discharge in normotensive and spontaneously hypertensive rats. Circ. Res. 39:488-496; 1976. 7. Brown, A. Receptors under pressure: An update on baroreceptors. Circ. Res. 46:1-10; 1980. 8. Cecchini, A.B.P.; Tiplitz, K.L.A.; Melbin, J.; Noordergraaf, A. Baroreceptor activity related to cell properties. Proceedings 35th ACEMB Philadelphia, PA, p. 20, 1982. 9. Charlton, J.D.; Baertschi, A.J. Responses of aortic baroreceptors to changes of aortic blood flow and pressure in rat. Am. J. Physiol. 242:H520-H525; 1982. 10. Clark, W. Static and dynamic characteristics of carotid sinus baroreceptors. Ph.D. Dissertation, University of Michigan, 1968. 11. Clynes, M. Unidirectional rate sensitivity, a biocybernetic law of reflex and humoral systems as physiological channels of control and communication. Ann. NY Acad. Sci. 92:946-969; 1961. 12. Coleridge, H.M.; Coleridge, J.C.G.; Kaufman, M.P.; Dangel, A. Operational sensitivity and acute resetting of aortic baroreceptors in dogs. Circ. Res. 48:676-684; 1981. 13. Dorward, P.K.; Andersen, M.C.; Burke, S.L.; Oliver, J.R.; Korner, P.I. Rapid resetting of the aortic baroreceptors in the rabbit and its implications for short-term and longer-term reflex control. Circ. Res. 50:428-439; 1982. 14. Franz, G.N. Nonlinear rate sensitivity of the carotid sinus reflex as a consequence of static and dynamic nonlinearities in baroreceptor behavior. Ann. NY Acad Sci. 156:811-824; 1969. 15. Franz, G.N.; Scher, A.M.; Ito, C.S. Small signal characteristics of carotid sinus baroreceptor of rabbits. J. Appl. Physiol. 30:527-535; 1971. 16. Franz, G.N. On blood pressure control. The Physiol. 17:73-86; 1974. 17. Hauss, W.H.; Kreuziger, H.; Asteroth, H. Uber die Reizung der Pressorezeptoren im sinus caroticus beim Hund T.Z. Kreislforsch 38:28-33; 1949. 18. Heesch, C.M.; Thames, M.D.; Abboud, F.M. Acute resetting of carotid sinus baroreceptors. I. Dissociation between discharge and wall changes. Am. J. Physiol. 247:H824-H832; 1984. 19. Heesch, C.M.; Abboud, F.M.; Thames, M.D. Acute resetting of carotid sinus baroreceptors II. Possible involvement of electrogenic Na + pump. Am. J. Physiol. 247:H833-H839; 1984. 20. Hering, H.E. Die Aenderung des Herzschlagzahl durch Aenderung des arteriellen Blutdruckes erfolgt auf reflectorischem Wege. Pflu. Arch. Ges. Physiol. 206:721-723; 1924. 21. Hering, H.E. Der Sinus caroticus an der Ursprungsstelle der Carotis interna als Ausgangsort eines hemmenden Herzreflexes und eines depressorischen Gefassreflexes. Munch. med. Wochnschr. 71:701-704; 1924. 22. Igler, F.O.; Donegan, J.H.; Hoo, K.C.; Korns, M.E.; Kampine, J.M. Chronic localized hypotension and resetting of carotid sinus baroreceptors. Circ. Res. 49:649-654; 1981. 23. Krieger, E.M. Time course of baroreceptor resetting in acute hypertension. Am. J. Physiol. 218:486-490; 1970. 24. Krieger, E.M. Aortic diastolic caliber changes as a determinant for complete aortic baroreceptor resetting. Fed. Proc. 46:41-45; 1987. 25. Kunze, D.L. Rapid resetting of the carotid baroreceptor reflex in the cat. Am. J. Physiol. 241:H802-H806; 1981. 26. Landgren, S. On the excitation mechanism of the carotid baroreceptors. Acta Physiol. Scand. 26:1-34; 1952. 27. Landgren, S. The baroreceptor activity in the carotid sinus nerve and the distensibility of the sinus wall. Acta Physiol. Scand. 26:35-56; 1952. 28. McCubbin, J.W.P.; Green, J.H.; Page, I.H. Baroreceptor function in chronic renal hypertension. Circ. Res. 4:205-210; 1956. 29. McCubbin, J.W.P. Carotid sinus participation in experimental renal hypertension. Circulation 17:791-797; 1958. 30. Munch, P.A.; Andersen, M.C.; Brown, A.M. Rapid resetting of aortic baroreceptors in vitro. Am. J. Physiol. 24:H672-H680; 1983. 31. Munch, P.A.; Brown, A.M. Role of vessel wall in acute resetting of aortic baroreceptors. Am. J. Physiol. 17:H843-H852; 1985.

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NOMENCLATURE current firing rate a d a p t e d firing rate m a x i m u m firing rate //max An = change in firing rate in response to a stepfunction in pressure p = b l o o d pressure Ap = stepwise change in pressure t = time 7time constant a l , a 2 , a 3 = weighting factors a, b, c, k, kl, kz,Po, G, H = constants n /'/a