Pflfigers Archiv
PflfigersArch. 366, 233-240 (1976)
EuropeanJournal of Physiology
9 by Springer-Verlag 1976
Analysis of the Dynamic Responses of Deefferented Primary Muscle Spindle Endings to Ramp Stretch* * * UWE W I N D H O R S T , J O R G SCHMIDT, and JUSTUS MEYER-LOHMANN*** PhysiologischesInstitut der Universit~tG6ttingen, Lehrstuhl II, Humboldtallee7, D-3400 G6ttingen, Federal Republic of Germany
Summary. A detailed analysis of the complex dynamic response of 8 deefferented primary muscle spindle endings to ramp stretches of the extensor digitorum longus muscle (EDL) was made in anaesthetized cats. The analysis was based on Lennerstand's linear muscle spindle model, in which the dynamic peak, i.e. the peak frequency at the end of the dynamic phase of a ramp stretch, is assumed to consist of three components: a position response, a slow velocity response, and a quick velocity response. The components of the dynamic peaks analysed were fairly well correlated with the static behaviour of the primary muscle spindle endings, the latter, in turn, being regarded as an indicator of the spindles' location in the non-homogeneous extrafusal muscle. The results thus provide a satisfactory explanation of the high correlations between the static and gross dynamic behaviour of deefferented primary muscle spindle endings of the EDL previously reported from our laboratory. Key words: Primary muscle spindle endings - Dynamic behaviour - Position, slow velocity and quick velocity responses.
INTRODUCTION Primary muscle spindle endings are known to reflect the non-linear mechanics of the extrafusal muscle rather accurately in their response to static muscle exten* A preliminary report of some of the results (Henatsch et al., 1974) was presented at the Symposiumon MuscleSpindlesheld by the Anatomical Society of Great Britain and Ireland at the University of Durham, England, April 4 6, 1974. ** Supported by grants from the Deutsche Forschungsgemeinschaft (Schwerpunktprogramm "Rezeptorphysiologie"and SFB 33: "Nervensystemund biologischeInformation", G6ttingen). *** To whom offprintrequests should be sent.
sions (Meyer-Lohmann et al., 1974). In the present paper we shall be concerned with the question as to how far the extrafusal non-homogeneity influences the responses of deefferented primary endings to dynamic muscle ramp stretches. For an answer, it was necessary to choose a model which might provide a simple description and definition of what the dynamic response of an ending is like. Much evidence of the non-linear behaviour of muscle spindle endings themselves has been accumulated during the last few years (Griisser and Thiele, 1968; Matthews and Stein, 1969; Eysel and Grfisser, 1970; Poppele and Chen, 1972; Hasan and Houk, 1975a, b). However, a linear muscle spindle model, proposed by Lennerstrand (1968) and Lennerstrand and Thoden (1968) would seem to provide a sufficiently accurate representation of the dynamic behaviour of primary endings, particularly to large length changes of muscle spindles. This model forms a suitable basis for dealing with non-linear properties arising from sources other than the muscle spindle itself, such as the non-homogeneous extrafusal muscle (Meyer-Lohmann et al., 1974; Windhorst et al., 1975). In the present study, the different components of the complex response of primary endings of the extensor digitorum longus muscle (EDL) to dynamic ramp stretches have been analysed in accordance with this spindle model. These components are the "position response", the "slow velocity response" and the "quick velocity response". The theoretical description and analysis of these components are detailed more extensively than is usual in an experimental paper in order to demonstrate the important and interesting features more comprehensively. Against this background, it will be shown that the components of the dynamic response to ramp stretches differ from spindle to spindle, and that these differences can be derived, in principle, from the regional extrafusal events during stretch.
234
Pfitigers Arch. 366 (1976)
METHODS Materials and Experimental Procedure. The analysis is based on
experimental findings included in a previous study (Windhorst et al., 1975), the data having been obtained from 8 d eefferented primary spindle endings of the extensor digitorum longus muscle (EDL) in anaesthetized cats. Details of the experimental procedure and data acquisition have been described elsewhere (Meyer-Lohmann et al., 1974; Windhorst et al., 1975). The parameters of ramp stretches of the EDL muscle were: constant amplitudes of 10 mm from the same zero length at various rates ranging from 0.5 - 70 ram/s, corresponding to rise times (A t) of the ramp ranging from 2 0 - 0 . 1 4 s. The resulting instantaneous peak frequency of primary spindle endings at the end of the dynamic phase of ramp stretch is defined as the "dynamic peak", Pd (Windhorst et al., 1975), and forms the basis of the analysis.
Theoretical Basis of Analysis The initial considerations are concerned with the i n p u t - o u t p u t relationships of a spindle which is assumed to be isolated. According to Lennerstrand (1968) and Lennerstrand and Thoden (1968), the instantaneous discharge frequency of deefferented primary spindle endings in response to linear increment in muscle spindle length comprises three superimposed components: one "position response", proportional to the length increment, and two velocity r e s p o n s e s - t h e "quick velocity response" and the "slow velocity response". The velocity responses can be described as responses of first order lag systems to the appropriate inputs. This means that When at time t = 0 the spindle is submitted to a ramp stretch, each velocity response rises from zero to a frequency proportional to the enforced velocity, v~, with an exponentially retarded time course. The two velocity responses have different parameters and can be described mathematically as follows: f~ (t) = aq. v~. [1 - - e x p ( - t / T ~ ) l
(l.l)
f~ (t) = a,. v~. [1 - exp ( - t/T,)]
(1.2)
where fq = f~ = aq, a~ = v~ = t = T~ = T, =
quick velocity response, (instantaneous frequency in s -~) slow velocity response, spindle constants, velocity of spindle length change, time elapsed from t = 0, time constant of the quick velocity response, time constant of the slow velocity response.
It should be noted that these equations describe response components of isolated muscle spindles. Tq is considered to be very short: according to Lennerstrand (1968), there is an almost immediate rise of the quick velocity response after onset of stretch. Thus, although its absolute value is unknown, Tq is certainly much smaller than the time A t from onset to offset of the dynamic stretch phase under our experimental conditions (see Methods). As we are concerned with the quick velocity response at the end of the dynamic phase of ramp stretch, i.e. after A t, Eq. (1.1) can be replaced by fq ~ aq. v~.
(l.la)
The slow velocity response is considered to be governed by a much longer time constant, T~, of yet unknown value (Lennerstrand simply stated T~ > 0.5 s). Hence the time required for the dynamic phase of stretch, A t, may well be shorter than T~, so that at the end of the dynamic phase, the slow velocity response may not have reached its final value [a~. v~ in Eq. (1.2)]. In this case (i.e. A t ~ T~), a simple approximation for the slow velocity response at the end of the dynamic stretch phase can be derived as follows : the initial part of
the response curve described by Eq. (1.2) can be approximated by a tangent with a slope of df~ i dt
a~- v~ to ~- O = - - T s
If the velocity response is prevented from attaining its final value, a, 9 v~, by stopping the movement after A t ~ T~, then its value at this time is approximately equal to the value of the tangent at this time. Thus: fs(At) ~
as " Vs
G
(2)
.At.
Since v~ - A t = A l, it follows from Eq. (2):
L(At)~ a , dr. T,
(3)
We thus obtain the seemingly contradictory result that a "velocity" response may be proportional to displacement only. Although this conclusion is substantiated by Leunerstrand's (1968) experimental findings, it must be remembered that the approximation of Eq. (3) only applies when A t ~ T~, and becomes more and more inexact the more A t approaches T~ or even exceeds it. In general, therefore, it is advisable to use the more exact Eq. (1.2) in order to describe the slow velocity response, an experimental justification for which will be given beiow. The total response of a primary ending to ramp stretch can now be constructed by superimposing the three components, two of which are described by Eqs. (1.1a) and (1.2). When applying this description to the dynamic peak Pd, it must be taken into account, however, that Pd is the response of a spindle in situ to steps of velocity, v, applied to the entire muscle. This implies a change in the spindle constants aq and as to cq and c~, respectively, when formulating the dependency of Pd on v : Pd (V) = p + C~" V + C~" V" [ 1 - exp (--At/Ts)]
(4)
where v = velocity of muscle length change, p = position response at the end of the dynamic phase of ramp stretch. With regard to cq and c, it should be noticed that these factors constitute the points where the non-linearity of the extrafusal muscle enters into the dynamic behaviour of primary endings. This will be explained in more detail below. Here it is sufficient to mention that c~ and cs are products of the respective true spindle constants and a non-linear function describing the conversion of "muscle velocity", v, into "spindle velocity", v~. Such functions are dependent on the length of the muscle and the location of the spindle within it. In the present experiments, in which the final ex tension of the muscle was always the same, cq and c~ are constant for each individual spindle.
RESULTS
Quick Velocity Response to Ramp Stretches Under our experimental conditions, the static components of the dynamic peaks [p in Eq. (4)] should be constant. The same applies to the slow velocity response [third term in Eq. (4)] at higher ranges of velocity in which A t ~ Ts. Under these circumstances the approximation of Eq. (3) is valid and can now be re-formulated as es
fs (A t) ~ ~ -
A L = const.
(3 a)
where A L = length increment of the entire muscle.
U. Windhorst et al. : Components of Dynamic Responses of Primary Muscle Spindle Endings
Thus, at higher velocities of stretch, the dynamic peaks, Pd, should vary linearly with velocity and this linear relationship should be determined only by the quick velocity response [second term in Eq. (4)]. In Figure i the dynamic peaks of a single EDL primary muscle spindle ending are plotted against the velocity of ramp stretch of the EDL muscle. As would be expected, the relationship can be properly approximated in the higher velocity range by a computed linear regression line (solid line). In the example shown in Figure 1, a value of about 13 mm/s was somewhat arbitrarily adopted as the lower limit of the "linear" velocity range and only points referring to velocities exceeding this value were employed for the cornputation of the regression line. The hatched area depicts the quick velocity component in the dynamic peaks for the upper range of ramp velocities. The base line of the hatched area is determined by the intercept of the regression line on the ordinate (v = 0).
Slow Velocity Response to Ramp Stretches The approximation for the slow velocity response, f~ (A t), in Eq. (3 a) is applicable only for relatively high velocities of stretch, i.e., when A t < its; the nearer A t approaches, or the more it exceeds Ts, the more fs (A t) drops below the approximated value. This "drop" should be reflected as a departure from linearity in the plots of dynamic peaks against velocity. This is indeed the case, as can be seen in Figure 1. The deviation increases with increasing At, i.e. decreasing v. The wellknown non-linearity in the low velocity range of such plots as in Figure I (cf. Schfifer and Sch/ifer, 1969) is thus simply explained. The dependence of the slow velocity response on velocity in its lower range can be demonstrated more clearly by subtracting the computed quick velocity responses (Fig. 1) from the measured dynamic peaks. The results of the application of this procedure [labelled h (v)] to the data from two muscle spindles are shown in Figure 2A and B. [The linear coordinates have been maintained because the points were expected not to obey a simple exponential relationship, cf. Eq. (4).] At high velocities, the slow velocity response superimposed on a virtually constant position 'response, asymptotically approaches a constant value equal to the intercept value, I, obtained by extrapolating the regression line to zero (cf. Fig. 1). As discussed above, the increasing deviation from constancy with decreasing velocity suggests the influence of a finite time constant, Ts. The slow velocity response thus shows a characteristic "velocity dependence, especially in the low velocity range. This finding is not in complete agreement with Lennerstrand's
235
~ec-1 7060-
~ 50oY -~ ~0~.35.~._o 30-
.~11~H !! L;~iiii.....". ................
i!
=o 20"10-
"O
0
0
~0
210 30
~0
510
60
70mm/s
velocity of stretch(v) Calculation of the quick velocity response. The calculations Fig. 1. were made for spindle no. 4 in Table 1, the regression line being computed by means of a least squares analysis; rxy denotes the correlation coefficient
results (1968), perhaps on account of methodic differences. In many plots like that of Figure 2B the plotted points, even for the lowest velocities, lie appreciably above the position response which is assumed to be equal to the steady state discharge frequency at 10 mm (horizontal dashed line). The slow velocity responses of many primary endings thus exhibit a very steep increase when the velocity is raised from zero to only a small value (of about I mm/s), whereas the further increase is much less steep and ultimately becomes zero. In fact, however, this high "sensitivity" for low velocities is due to the long durations of stretch (see above). A Method for Estimating the Time Constant of the Slow Velocity Response. The approach developed so far [Eq. (4)] can be extended to provide an estimate of the time constant, T~. This can be achieved by fitting a theoretical curve to experimental points like those in Figure 2. That value of T~ yielding the closest fit can be regarded as the best estimate. For this procedure, the value of e~ is necessary and it can be substituted by solving Eq. (3a) appropriately, f f p is then assumed to be equal to the steady state discharge, fs (A t) in Eq. (3a) is thus the difference between the intercept/and the steady state discharge (cf. Fig. 2). These computations have been performed for a number of spindles. The details, however, are somewhat complicated and beyond the scope of this paper. As a preliminary result it can be stated that T~ is of the order of several seconds.
Calculation of the Slow Velocity Response Extrapolation of a regression line, as exemplified in Figure 1, yields an intercept on the ordinate which is composed of the static component, p, and the slow velocity response for relatively high velocities of stretch (A t ~ T~). The quick velocity response, how-
Pflfigers Arch. 366 (I976)
236 sec-T 80-
J
~. 70-
I = 74.1
9
J i
~-o 60II
average steady state frequency - - - 49.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . at 10 ram e x t e n s i o n { n = 10 )
50-
0-
1'0
0
B
l's
~ / ~ , _ - ~ - -
==/s
2'o
~ ~?~.1- - - - . - -
100-
90-
.~ 8 0 -
~-~ 70-
i
II
60-
50-
-
-/.6.6
average steady state frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . at l O m m e x t e n s i o n {n=25}
40-
0' velocity
110 of s t r e t c h ( v )
ll5
ever, is reduced to zero in this intercept. The static component in the dynamic peak is assumed to be equal to the steady state response in the static phase of ramp stretch (i.e. to the sum of the spontaneous activity, As~, and the static response, R~). The slow velocity response can thus be calculated by subtracting the steady state response from the intercept value on the ordinate obtained as shown in Figure 1. The results for 8 E D L spindles are compiled in Table 1. In dealing with the question of extrafusal influences on slow velocity responses, it would seem worthwhile investigating to what extent these responses are correlated with the static responses, R~, because the latter give some indication of the spindles' location within the muscle. These two responses would be proportional for each individual spindle, since both are proportional to the spindle's length increment. This proportionality should also apply to different spindles, however, if their sensing properties are fairly constant.
m m /s
2'o
Fig. 2A and B Calculation of the slow velocity response. The calculations were made for spindles no. I and 5 in Table 1; the average steady state frequencyat 10 mm extensionwas calculated from recordings obtained 10- 15 s after the end of the dynamic phase of ramp stretch. I refers to the intercept value obtained by extrapolation as shown in Figure 1. N.B. In comparison with Figure 1, the range of velocityis much reduced
With regard to the slow velocity responses, these sensing properties are reflected by the constants as and T~ whose ratio (i.e. a~/Ts) forms the factor of proportionality in Eq. (3). In Figure 3, the computed slow velocity responses from Table 1 are plotted against the static responses to 10 mm muscle extension. The reasonably high correlation coefficient and low error probability would appear to justify two conclusions: firstly, the linear relationship between slow velocity responses, (for A t ~ Ts) and static responses, R~, is due to non-homogeneous extrafusal mechanics influencing these two responses in similar ways; and secondly, the sensing properties of EDL muscle spindles are relatively homogeneous, or alternatively, but less likely, they vary systematically with extraneous factors such as extrafusal muscle properties.
Velocity of Spindle Stretch and Extrafusal Mechanics. When considering a similar quantitative relationship between quick velocity responses and static responses,
237
U. Windhorst et al. : Components of Dynamic: Responses of Primary Muscle Spindle Endings Table 1.
Components of muscle spindle responses (in s -~) to 10 mm ramp stretches at high velocities (A t ~ T~)
Spindle No.
1
2
3
4
5
6
7
8
I = Asp + Rs + svr p = Asp + R~ svr = I - p
74.1 49.4 24.7 29.6
54.6 31.0 23.6 12.7
65.8 36.3 29.5 27.8
35.1 23.2 11.9 6.5
103.1 46.6 56.5 46.6
70.7 54.8 15.9 19.6
117.2 76.6 40.6 70.6
63.5 29.7 33.8 29.0
R~ I
= intercept value (cf., Fig. l);
Asp = spontaneous activity;
R~ = average static response to i0 mm muscle stretch; svr = slow veIocity response (A t ~ Ts);
p
- static component.
sec-1
70" ,., 60" u~
II
~50" m iiJ
~, 40-
~' 30>
rxy=0.77 2CC
PflfigersArch. 366, 233-240 (1976)
EuropeanJournal of Physiology
9 by Springer-Verlag 1976
Analysis of the Dynamic Responses of Deefferented Primary Muscle Spindle Endings to Ramp Stretch* * * UWE W I N D H O R S T , J O R G SCHMIDT, and JUSTUS MEYER-LOHMANN*** PhysiologischesInstitut der Universit~tG6ttingen, Lehrstuhl II, Humboldtallee7, D-3400 G6ttingen, Federal Republic of Germany
Summary. A detailed analysis of the complex dynamic response of 8 deefferented primary muscle spindle endings to ramp stretches of the extensor digitorum longus muscle (EDL) was made in anaesthetized cats. The analysis was based on Lennerstand's linear muscle spindle model, in which the dynamic peak, i.e. the peak frequency at the end of the dynamic phase of a ramp stretch, is assumed to consist of three components: a position response, a slow velocity response, and a quick velocity response. The components of the dynamic peaks analysed were fairly well correlated with the static behaviour of the primary muscle spindle endings, the latter, in turn, being regarded as an indicator of the spindles' location in the non-homogeneous extrafusal muscle. The results thus provide a satisfactory explanation of the high correlations between the static and gross dynamic behaviour of deefferented primary muscle spindle endings of the EDL previously reported from our laboratory. Key words: Primary muscle spindle endings - Dynamic behaviour - Position, slow velocity and quick velocity responses.
INTRODUCTION Primary muscle spindle endings are known to reflect the non-linear mechanics of the extrafusal muscle rather accurately in their response to static muscle exten* A preliminary report of some of the results (Henatsch et al., 1974) was presented at the Symposiumon MuscleSpindlesheld by the Anatomical Society of Great Britain and Ireland at the University of Durham, England, April 4 6, 1974. ** Supported by grants from the Deutsche Forschungsgemeinschaft (Schwerpunktprogramm "Rezeptorphysiologie"and SFB 33: "Nervensystemund biologischeInformation", G6ttingen). *** To whom offprintrequests should be sent.
sions (Meyer-Lohmann et al., 1974). In the present paper we shall be concerned with the question as to how far the extrafusal non-homogeneity influences the responses of deefferented primary endings to dynamic muscle ramp stretches. For an answer, it was necessary to choose a model which might provide a simple description and definition of what the dynamic response of an ending is like. Much evidence of the non-linear behaviour of muscle spindle endings themselves has been accumulated during the last few years (Griisser and Thiele, 1968; Matthews and Stein, 1969; Eysel and Grfisser, 1970; Poppele and Chen, 1972; Hasan and Houk, 1975a, b). However, a linear muscle spindle model, proposed by Lennerstrand (1968) and Lennerstrand and Thoden (1968) would seem to provide a sufficiently accurate representation of the dynamic behaviour of primary endings, particularly to large length changes of muscle spindles. This model forms a suitable basis for dealing with non-linear properties arising from sources other than the muscle spindle itself, such as the non-homogeneous extrafusal muscle (Meyer-Lohmann et al., 1974; Windhorst et al., 1975). In the present study, the different components of the complex response of primary endings of the extensor digitorum longus muscle (EDL) to dynamic ramp stretches have been analysed in accordance with this spindle model. These components are the "position response", the "slow velocity response" and the "quick velocity response". The theoretical description and analysis of these components are detailed more extensively than is usual in an experimental paper in order to demonstrate the important and interesting features more comprehensively. Against this background, it will be shown that the components of the dynamic response to ramp stretches differ from spindle to spindle, and that these differences can be derived, in principle, from the regional extrafusal events during stretch.
234
Pfitigers Arch. 366 (1976)
METHODS Materials and Experimental Procedure. The analysis is based on
experimental findings included in a previous study (Windhorst et al., 1975), the data having been obtained from 8 d eefferented primary spindle endings of the extensor digitorum longus muscle (EDL) in anaesthetized cats. Details of the experimental procedure and data acquisition have been described elsewhere (Meyer-Lohmann et al., 1974; Windhorst et al., 1975). The parameters of ramp stretches of the EDL muscle were: constant amplitudes of 10 mm from the same zero length at various rates ranging from 0.5 - 70 ram/s, corresponding to rise times (A t) of the ramp ranging from 2 0 - 0 . 1 4 s. The resulting instantaneous peak frequency of primary spindle endings at the end of the dynamic phase of ramp stretch is defined as the "dynamic peak", Pd (Windhorst et al., 1975), and forms the basis of the analysis.
Theoretical Basis of Analysis The initial considerations are concerned with the i n p u t - o u t p u t relationships of a spindle which is assumed to be isolated. According to Lennerstrand (1968) and Lennerstrand and Thoden (1968), the instantaneous discharge frequency of deefferented primary spindle endings in response to linear increment in muscle spindle length comprises three superimposed components: one "position response", proportional to the length increment, and two velocity r e s p o n s e s - t h e "quick velocity response" and the "slow velocity response". The velocity responses can be described as responses of first order lag systems to the appropriate inputs. This means that When at time t = 0 the spindle is submitted to a ramp stretch, each velocity response rises from zero to a frequency proportional to the enforced velocity, v~, with an exponentially retarded time course. The two velocity responses have different parameters and can be described mathematically as follows: f~ (t) = aq. v~. [1 - - e x p ( - t / T ~ ) l
(l.l)
f~ (t) = a,. v~. [1 - exp ( - t/T,)]
(1.2)
where fq = f~ = aq, a~ = v~ = t = T~ = T, =
quick velocity response, (instantaneous frequency in s -~) slow velocity response, spindle constants, velocity of spindle length change, time elapsed from t = 0, time constant of the quick velocity response, time constant of the slow velocity response.
It should be noted that these equations describe response components of isolated muscle spindles. Tq is considered to be very short: according to Lennerstrand (1968), there is an almost immediate rise of the quick velocity response after onset of stretch. Thus, although its absolute value is unknown, Tq is certainly much smaller than the time A t from onset to offset of the dynamic stretch phase under our experimental conditions (see Methods). As we are concerned with the quick velocity response at the end of the dynamic phase of ramp stretch, i.e. after A t, Eq. (1.1) can be replaced by fq ~ aq. v~.
(l.la)
The slow velocity response is considered to be governed by a much longer time constant, T~, of yet unknown value (Lennerstrand simply stated T~ > 0.5 s). Hence the time required for the dynamic phase of stretch, A t, may well be shorter than T~, so that at the end of the dynamic phase, the slow velocity response may not have reached its final value [a~. v~ in Eq. (1.2)]. In this case (i.e. A t ~ T~), a simple approximation for the slow velocity response at the end of the dynamic stretch phase can be derived as follows : the initial part of
the response curve described by Eq. (1.2) can be approximated by a tangent with a slope of df~ i dt
a~- v~ to ~- O = - - T s
If the velocity response is prevented from attaining its final value, a, 9 v~, by stopping the movement after A t ~ T~, then its value at this time is approximately equal to the value of the tangent at this time. Thus: fs(At) ~
as " Vs
G
(2)
.At.
Since v~ - A t = A l, it follows from Eq. (2):
L(At)~ a , dr. T,
(3)
We thus obtain the seemingly contradictory result that a "velocity" response may be proportional to displacement only. Although this conclusion is substantiated by Leunerstrand's (1968) experimental findings, it must be remembered that the approximation of Eq. (3) only applies when A t ~ T~, and becomes more and more inexact the more A t approaches T~ or even exceeds it. In general, therefore, it is advisable to use the more exact Eq. (1.2) in order to describe the slow velocity response, an experimental justification for which will be given beiow. The total response of a primary ending to ramp stretch can now be constructed by superimposing the three components, two of which are described by Eqs. (1.1a) and (1.2). When applying this description to the dynamic peak Pd, it must be taken into account, however, that Pd is the response of a spindle in situ to steps of velocity, v, applied to the entire muscle. This implies a change in the spindle constants aq and as to cq and c~, respectively, when formulating the dependency of Pd on v : Pd (V) = p + C~" V + C~" V" [ 1 - exp (--At/Ts)]
(4)
where v = velocity of muscle length change, p = position response at the end of the dynamic phase of ramp stretch. With regard to cq and c, it should be noticed that these factors constitute the points where the non-linearity of the extrafusal muscle enters into the dynamic behaviour of primary endings. This will be explained in more detail below. Here it is sufficient to mention that c~ and cs are products of the respective true spindle constants and a non-linear function describing the conversion of "muscle velocity", v, into "spindle velocity", v~. Such functions are dependent on the length of the muscle and the location of the spindle within it. In the present experiments, in which the final ex tension of the muscle was always the same, cq and c~ are constant for each individual spindle.
RESULTS
Quick Velocity Response to Ramp Stretches Under our experimental conditions, the static components of the dynamic peaks [p in Eq. (4)] should be constant. The same applies to the slow velocity response [third term in Eq. (4)] at higher ranges of velocity in which A t ~ Ts. Under these circumstances the approximation of Eq. (3) is valid and can now be re-formulated as es
fs (A t) ~ ~ -
A L = const.
(3 a)
where A L = length increment of the entire muscle.
U. Windhorst et al. : Components of Dynamic Responses of Primary Muscle Spindle Endings
Thus, at higher velocities of stretch, the dynamic peaks, Pd, should vary linearly with velocity and this linear relationship should be determined only by the quick velocity response [second term in Eq. (4)]. In Figure i the dynamic peaks of a single EDL primary muscle spindle ending are plotted against the velocity of ramp stretch of the EDL muscle. As would be expected, the relationship can be properly approximated in the higher velocity range by a computed linear regression line (solid line). In the example shown in Figure 1, a value of about 13 mm/s was somewhat arbitrarily adopted as the lower limit of the "linear" velocity range and only points referring to velocities exceeding this value were employed for the cornputation of the regression line. The hatched area depicts the quick velocity component in the dynamic peaks for the upper range of ramp velocities. The base line of the hatched area is determined by the intercept of the regression line on the ordinate (v = 0).
Slow Velocity Response to Ramp Stretches The approximation for the slow velocity response, f~ (A t), in Eq. (3 a) is applicable only for relatively high velocities of stretch, i.e., when A t < its; the nearer A t approaches, or the more it exceeds Ts, the more fs (A t) drops below the approximated value. This "drop" should be reflected as a departure from linearity in the plots of dynamic peaks against velocity. This is indeed the case, as can be seen in Figure 1. The deviation increases with increasing At, i.e. decreasing v. The wellknown non-linearity in the low velocity range of such plots as in Figure I (cf. Schfifer and Sch/ifer, 1969) is thus simply explained. The dependence of the slow velocity response on velocity in its lower range can be demonstrated more clearly by subtracting the computed quick velocity responses (Fig. 1) from the measured dynamic peaks. The results of the application of this procedure [labelled h (v)] to the data from two muscle spindles are shown in Figure 2A and B. [The linear coordinates have been maintained because the points were expected not to obey a simple exponential relationship, cf. Eq. (4).] At high velocities, the slow velocity response superimposed on a virtually constant position 'response, asymptotically approaches a constant value equal to the intercept value, I, obtained by extrapolating the regression line to zero (cf. Fig. 1). As discussed above, the increasing deviation from constancy with decreasing velocity suggests the influence of a finite time constant, Ts. The slow velocity response thus shows a characteristic "velocity dependence, especially in the low velocity range. This finding is not in complete agreement with Lennerstrand's
235
~ec-1 7060-
~ 50oY -~ ~0~.35.~._o 30-
.~11~H !! L;~iiii.....". ................
i!
=o 20"10-
"O
0
0
~0
210 30
~0
510
60
70mm/s
velocity of stretch(v) Calculation of the quick velocity response. The calculations Fig. 1. were made for spindle no. 4 in Table 1, the regression line being computed by means of a least squares analysis; rxy denotes the correlation coefficient
results (1968), perhaps on account of methodic differences. In many plots like that of Figure 2B the plotted points, even for the lowest velocities, lie appreciably above the position response which is assumed to be equal to the steady state discharge frequency at 10 mm (horizontal dashed line). The slow velocity responses of many primary endings thus exhibit a very steep increase when the velocity is raised from zero to only a small value (of about I mm/s), whereas the further increase is much less steep and ultimately becomes zero. In fact, however, this high "sensitivity" for low velocities is due to the long durations of stretch (see above). A Method for Estimating the Time Constant of the Slow Velocity Response. The approach developed so far [Eq. (4)] can be extended to provide an estimate of the time constant, T~. This can be achieved by fitting a theoretical curve to experimental points like those in Figure 2. That value of T~ yielding the closest fit can be regarded as the best estimate. For this procedure, the value of e~ is necessary and it can be substituted by solving Eq. (3a) appropriately, f f p is then assumed to be equal to the steady state discharge, fs (A t) in Eq. (3a) is thus the difference between the intercept/and the steady state discharge (cf. Fig. 2). These computations have been performed for a number of spindles. The details, however, are somewhat complicated and beyond the scope of this paper. As a preliminary result it can be stated that T~ is of the order of several seconds.
Calculation of the Slow Velocity Response Extrapolation of a regression line, as exemplified in Figure 1, yields an intercept on the ordinate which is composed of the static component, p, and the slow velocity response for relatively high velocities of stretch (A t ~ T~). The quick velocity response, how-
Pflfigers Arch. 366 (I976)
236 sec-T 80-
J
~. 70-
I = 74.1
9
J i
~-o 60II
average steady state frequency - - - 49.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . at 10 ram e x t e n s i o n { n = 10 )
50-
0-
1'0
0
B
l's
~ / ~ , _ - ~ - -
==/s
2'o
~ ~?~.1- - - - . - -
100-
90-
.~ 8 0 -
~-~ 70-
i
II
60-
50-
-
-/.6.6
average steady state frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . at l O m m e x t e n s i o n {n=25}
40-
0' velocity
110 of s t r e t c h ( v )
ll5
ever, is reduced to zero in this intercept. The static component in the dynamic peak is assumed to be equal to the steady state response in the static phase of ramp stretch (i.e. to the sum of the spontaneous activity, As~, and the static response, R~). The slow velocity response can thus be calculated by subtracting the steady state response from the intercept value on the ordinate obtained as shown in Figure 1. The results for 8 E D L spindles are compiled in Table 1. In dealing with the question of extrafusal influences on slow velocity responses, it would seem worthwhile investigating to what extent these responses are correlated with the static responses, R~, because the latter give some indication of the spindles' location within the muscle. These two responses would be proportional for each individual spindle, since both are proportional to the spindle's length increment. This proportionality should also apply to different spindles, however, if their sensing properties are fairly constant.
m m /s
2'o
Fig. 2A and B Calculation of the slow velocity response. The calculations were made for spindles no. I and 5 in Table 1; the average steady state frequencyat 10 mm extensionwas calculated from recordings obtained 10- 15 s after the end of the dynamic phase of ramp stretch. I refers to the intercept value obtained by extrapolation as shown in Figure 1. N.B. In comparison with Figure 1, the range of velocityis much reduced
With regard to the slow velocity responses, these sensing properties are reflected by the constants as and T~ whose ratio (i.e. a~/Ts) forms the factor of proportionality in Eq. (3). In Figure 3, the computed slow velocity responses from Table 1 are plotted against the static responses to 10 mm muscle extension. The reasonably high correlation coefficient and low error probability would appear to justify two conclusions: firstly, the linear relationship between slow velocity responses, (for A t ~ Ts) and static responses, R~, is due to non-homogeneous extrafusal mechanics influencing these two responses in similar ways; and secondly, the sensing properties of EDL muscle spindles are relatively homogeneous, or alternatively, but less likely, they vary systematically with extraneous factors such as extrafusal muscle properties.
Velocity of Spindle Stretch and Extrafusal Mechanics. When considering a similar quantitative relationship between quick velocity responses and static responses,
237
U. Windhorst et al. : Components of Dynamic: Responses of Primary Muscle Spindle Endings Table 1.
Components of muscle spindle responses (in s -~) to 10 mm ramp stretches at high velocities (A t ~ T~)
Spindle No.
1
2
3
4
5
6
7
8
I = Asp + Rs + svr p = Asp + R~ svr = I - p
74.1 49.4 24.7 29.6
54.6 31.0 23.6 12.7
65.8 36.3 29.5 27.8
35.1 23.2 11.9 6.5
103.1 46.6 56.5 46.6
70.7 54.8 15.9 19.6
117.2 76.6 40.6 70.6
63.5 29.7 33.8 29.0
R~ I
= intercept value (cf., Fig. l);
Asp = spontaneous activity;
R~ = average static response to i0 mm muscle stretch; svr = slow veIocity response (A t ~ Ts);
p
- static component.
sec-1
70" ,., 60" u~
II
~50" m iiJ
~, 40-
~' 30>
rxy=0.77 2CC
