(Received 10 November 1964)

Transcription

1 42 J. Phy8iol. (1965), 179, pp With 9 text-fgure Printed in Great Britain A MODEL ILLUSTRATING SOME ASPECTS OF MUSCLE SPINDLE PHYSIOLOGY BY A. J. BULLER From the Sherrington School of Physiology, St Thomas' Hospital, London, S.E. 1 and Department of Physiology, King's College, London, W.C. 2 (Received 1 November 1964) Katz (195a) first suggested that the apparently random sequence of nerve impulses discharged from a frog muscle spindle lying within a slightly stretched muscle might be due either to molecular agitation in the mechanical receptor substance or to ionic noise in the terminal nerve membrane. In either case it would be expected that some irregularity in the sensory discharge would also be present when the muscle containing the spindle was subjected to more intense stretch. Under such conditions the increased rate of discharge from the sensory ending would make the fluctuation in impulse frequency less obvious, but accurate analysis should indicate an approximately constant source producing the irregularity. This expectation was tested experimentally by Buller, Nicholls & Str6m (1953) with frog muscle spindles. The static discharge patterns of muscle spindles subjected to varying degrees of stretch were analysed and it was found that for mean firing frequencies ranging from 7 to 163 impulses per second there was an approximately constant fluctuation of impulse frequency, having a standard deviation of 3-5 imp./sec. The small dots of Fig. 9, copied from Fig. 3 of Buller et al. (1953), show a plot on double logarithmic co-ordinates of the mean interval between successive impulses and the standard deviation, measured in msec, at that mean interval. A regression line fitted for intervals of less than 16 msec was found to have a slope of 2-3. Since this value is close to 2 it implies that the standard deviation of impulse interval is proportional to the square of the mean interval, and certain suggestions were put forward to account for this relation. However, as was pointed out by A. F. Huxley (personal communication, 1952) these suggestions presupposed certain conditions at the sensory terminations which could not be directly verified. It was therefore decided to build a model of the muscle spindle, the model behaving in the manner suggested in the paper of Buller et al., and to compare the results obtained from the model with those obtained from the frog muscle spindle.

2 A MUSCLE SPINDLE MODEL 43 The design of the model was based on the assumption that two mechanisms operate at the sensory nerve terminal. The first produces a completely regular discharge at a, frequency directly proportional to the degree of depolarization (ED) of the sensory terminal, between the limits of threshold depolarization (EO) and maximal depolarization (Emax) (cf. Katz, 195b; Buller et al. 1953). The second mechanism gives rise to random fluctuations in the value of the membrane potential of the sensory terminal, irrespective of whether the degree of polarization is greater or less than EO. This paper reports the results obtained with the spindle model. The majority of the experiments were made in and a preliminary report has already been published (Buller, 1959). The results obtained suggest that some of the interpretations of the earlier biological experiments (Buller et al. 1953) may need revision. METHODS A block diagram of the model is shown in Fig. 1 A. The two rectangles shown on the right represent the noise generator, and initially may be ignored. The remainder of the model is responsible for the generation of a regular sequence of electrical pulses, each pulse representing a nerve impulse, the frequency of which is linearly related to an imposed trigger voltage (E) which falls from a starting value (Emax) until it reaches either a lower limit of E. or the imposed trigger voltage ED (if ED > E,, see Fig. 1B). In the former case the value of E then remains at EO until the imposed voltage is raised above EO; in the second case the resetting circuit is actuated when E = ED, the wave form generator being rapidly reset by E rising to Emax, and after an interval of 6 msec the cycle is repeated. In this sequence the hyperbolic wave form represents the recovery of excitability in the sensory nerve terminal following the generation of an impulse, and the level of the imposed voltage (ED) represents the extent of the depolarization of the sensory termination due to stretch (Katz, 195b). The hyperbolic shape of the recovery curve accounts for the linear relation between the rate of firing and depolarization (within the limits EO < ED < Emax). The resetting time of the model is 6 msec, and this sets an upper limit to the firing frequency of approximately 16 imp./sec. This rate corresponds to the highest spindle discharge rates analysed by Buller et al. (1953). In addition the value of Eo was made variable. This allows the lowest regular firing rate to be altered at will. Thus, in the absence of added noise the model fires regularly with a highest frequency (fmrax) of 16 imp./sec. and a lowest frequency (fo) which may be varied at will from 4 imp./sec. to 5 imp./sec. The ability to alter EO means that the falling hyperbolic wave form does not approach its final value asymptotically but abruptly terminates its decay at a set voltage (E), as illustrated in Fig. 1 B. As EO was varied so also was the lowest regular firing frequency (f) which may be specified as the discharge frequency in the absence of noise when ED = E. The two boxes on the right of Fig. 1A represent the noise source. This consists of an RCA 931 photomultiplier run under conditions of constant low-intensity illumination. The random fluctuations of voltage occurring at the anode provide a source of approximately white noise, the characteristics of which may be altered by changing its upper frequency limit, and/or by varying the shaping circuits. The upper frequency limit may be varied between 1 and 1 kc/s (see Fig. 2A). The noise may be shaped to match that expected with membrane time constants of between 5 x 1-8 and 5 x 1-5 sec. The theoretical amplitude 26-2

3 44 A. J. BULLER frequency spectra for membrane time constants between 5 x 1O-3 and 5 x 1O-5 sec were calculated according to the equation (Fatt & Katz, 1952) a = V/[1/(1 +47T2f2tm2)+ IIV(+47T2f2tm2)], (1) where f = frequency (c/s), tm = membrane time constant (sec). Wave form Shaping generator Mixer circuits Resetting Amplitude White circuit 4j comparator noise source Emax--- Imposed voltage A o~~~~~~~~~~~~~~~e Timne B Fig. 1. (A) Block diagram of the model. For sequence of operation see text. (B) Diagram (not to scale) of the voltage wave form used to represent recovery of excitability following an impulse. Emax = maximum voltage developed at the instant of generating an impulse. This voltage then falls with time either to EO, a lower limit, or to ED (if ED > EO) which is determined by the imposed trigger voltage and may have any value from to Emaz. The value of EO may be varied within the limits indicated by the two horizontal interrupted lines.

4 A MUSCLE SPINDLE MODEL 45 Figure 2 B compares the theoretical curve of the amplitude frequency relation for a membrane time constant of 5 x 1- sec with observed points obtained in the model with an appropriate filter setting. The individual characteristics of the noise, namely the r.m.s. value, the upper frequency limit and the amplitude frequency spectrum, could be varied independently. The mean squared voltage of the shaped noise was measured by means of a vacuum thermocouple and the noise voltage resistively mixed with the hyperbolic wave form (see Fig. 1A). In this way the effects of alteration of individual noise parameters on the firing frequency and the time intervals between successive impulses could be studied. A '.. I I IA, k L.! k.1 it I - I,. I i ir r - I I a I r I VP I IC B x *B -8 - XC) 8 t6 g 4_ Cd 1z C Frequency 1, Fig. 2. (A) Single sweep of each of two of the varieties of noise used in the model. On the left the noise has an upper frequency limit of 1 kc/s, on the right an upper frequency limit of 1 kc/s. Time marker, 2 c/s. (B) Relation between mean squared voltage per cycle band width (ordinates) and frequency (abscissae). Full line represents the theoretical expectation for nerve membrane having a time constant of 5 x 1-3 sec (see eqn. (2)), crosses show the relation achieved in the model. The mean squared voltage at different frequencies has been plotted as a fraction of its value at a frequency of 1 c/s.

5 46 A. J. BULLER The intervals between successive impulses from the model were recorded on moving film, using a method of time coding described previously (Buller & Styles, 1958). Although different from the method of interval measurement used by Buller et al. (1953) the results obtained by the two methods agreed well. Experimental procedure The model was always allowed to 'warm up' for some hours before records were taken. The noise voltage was then reduced to zero and a series of paired readings (mean impulse interval and imposed voltage) were taken with the model firing regularly. The noise voltage was then increased and observations made of the impulse discharge pattern with variations of the imposed voltage (ED), the threshold voltage (EO) and the noise parameters. Statistical technique The techniques used were essentially the same as those employed by Buller et al. (1953). The mean interval and standard deviation of pools of twenty consecutive intervals were calculated in msec. From these values the standard deviation expressed in impulses per second was found by taking half the difference between the reciprocals of (mean interval + S.D.) and (mean interval - S.D.). Since accommodation does not occur in the pulse pattern from the model as it does in the discharge pattern from muscle spindles, the validity of the standard deviation values could have been enhanced by increasing the size of the pools, but it was thought desirable to adhere as closely as possible to the methods employed in the earlier paper. However, on occasions pools of twenty intervals were grouped as in Table 1 of Buller et al. (1953) in order to facilitate plotting. A note ofthe number of intervals analysed is given in the description of each graph. The fitting of regression lines was accomplished by the usual method of least squares. RESULTS Regular firing In the absence of added noise and with Eo at its lowest value the model fired regularly at rates ranging from 4 imp./sec to 16 imp./sec (mean intervals msec), the rate being approximately linearly related to ED (the imposed voltage). With increasing values of EO, the lowest obtainable firing frequency increased, but still with the frequency linearly related to ED within the limits EO < ED < Emax (see Fig. 3). Without added noise the model did not fire at all when ED < EO. Random firing By substituting a short time constant R.C. network (time constant =.1 msec) for the wave form generator (see Fig. 1A) a very rapid voltage rundown from the starting voltage to EO could be achieved. If ED was set to a value less than E, the model would not fire in the absence of noise; however, with an input of noise to the mixer circuit, the model fired irregularly, the initiation of each impulse being due to a fluctuation of noise voltage which exceeded the difference between the trigger voltage (ED) and the threshold voltage (EO). At mean firing intervals which were very long compared with the discharge time of the R.C. network, the frequency distribution curve of intervals approached an exponential (see

6 A MUSCLE SPINDLE MODEL 47 Fig. 4A), a characteristic of a random time series (Fatt & Katz, 1952; Parratt, 1961). As a corollary, the number (n) of intervals shorter than a given time (t) plotted against t (see Fig. 4B) yielded a curve fitting closely the theoretical expectation: y = N[L-exp (-t/t)], (2) where N = total number of observed intervals, T = mean interval (sec) _ o~15 E Frequency (imp./sec) Fig. 3. Relation between the imposed voltage (ED) and the regular firing frequency from the model in the absence of added noise. In this experiment EO was adjusted to give a lowest regular firing frequency of 9 imp./sec. Mixture of regular firing and random element The addition of noise to a regular firing pattern decreased the mean interval between successive pulses and introduced a variability in the intervals between successive pulses. The decrease of mean interval was noted at all firing frequencies, and with all possible variations in the noise characteristics. It was, however, less marked at high frequencies of firing and with noise having a low (1 kc/s) cut-off frequency. The greater the r.m.s. amplitude of the added noise voltage the greater the increase in the mean firing frequency. Increase in the r.m.s. value of the noise also increased the interval variability. A plot of the impulse irregularity expressed as a standard deviation in impulses per second against the r.m.s. value of the added noise is shown in Fig. 5. A straight line has been drawn through the experimental points, though there seems no theoretical reason to expect an exact linear relation. The result illustrated was obtained with the added

7 48 A. J. BULLER noise having an upper frequency limit of 1 kc/s and shaped to mimic the calculated theoretical spectrum for a membrane time constant of5 x 1-3 sec. To a first approximation, provided that the r.m.s. value of the noise was kept constant, variations of the noise cut-off frequency and/or the noise shaping within the limits specified above did not alter the amount of interval variability. When noise was added to a regular firing pattern the lowest possible firing rate was decreased. Thus with ED less than EO the model still fired 25 A -4 6 z Interval (sec) B 3 cz $4 2 E- H Interval (sec) Fig. 4. A. Distribution curve of intervals between successive pulses from the model when the latter was adjusted to produce a very rapid rundown from the starting voltage to E, and ED (the imposed voltage) was set to a value less than EO. Mean interval 34-6 msec. Total number of intervals measured, 385. B. A plot of the total number of intervals shorter than a given time t plotted against t, from the data ofthe same experiment. The full line is the theoretical expectation for a random time series.

8 A MUSCLE SPINDLE MODEL 49 if a noise peak was sufficiently large. It follows from the definition above that if the firing rate (f) was less than the lowest regular firing rate (f ) the discharge was being produced solely by noise transients. As stated above, in the observed discharge patterns from frog muscle spindles, the degree of frequency irregularity was approximately constant, with a S.D. of 3.5 imp./sec. For this reason the added r.m.s. noise voltage in the experiments now to be described was always adjusted to give a S.D. of between 2-5 and 3-5 imp./sec at a mean firing frequency of approximately 2 imp./sec Standard deviation (imp./sec) Fig. 5. Relation between the r.m.s. value of the added noise (ordinates) and the standard deviation of frequency expressed in imp./sec (abscissae). In this experiment the noise had an upper frequency limit of 1 kc/s, and was shaped to match a membrane time constant of 5 x 1-3 sec. With the noise so adjusted, and EO at its minimum value, the effect of variation of ED was studied. The result of one such experiment (upper frequency limit 1 kc/s, membrane time constant 5 x 1-3 sec) is illustrated by the crosses in Fig. 6. The dots represent experimental results from the frog spindle, taken from Buller et at. (1953). The slope of the regression line drawn through the spindle results approximates to 2- (1.97 for the points illustrated), while the slope of the regression line through the model results was found to be Contrary to expectations (cf. Buller et at. 1953) it was found that variation of the noise band width and/or shaping (while keeping the r.m.s. value constant) altered the slope of the model regression line hardly at all. For example, the value of the slope with the type of noise illustrated in Fig. 2A (left) was 1-56, and using the noise of Fig. 2A (right) it was For all the noise variations investigated the slope limits were 1-56 and 1-62.

9 41 A. J. BULLER Although, when the complete range of mean intervals illustrated in Fig. 6 was considered, there was an obvious difference of slope between the results from the model and those from the muscle spindle, it was noted that for intervals less than approximately 5 msec the difference in slope was very much less. In fact, the slope of the regression line drawn through the muscle spindle results with mean intervals less than 5 msec was found to be It therefore appeared that the deviation between the two regression lines of Fig. 6 was due dominantly to a divergence in the results for mean intervals longer than 5 msec. 5 _ 2 *,,( 5 / / 2 A^ /.5 _ *5 'I~~ -2 2 ll Interval (msec) Fig. 6. Relation between mean interval in msec (abscissae) and standard deviation in msec (ordinates) plotted on a double logarithmic scale. The crosses represent results from the model with fo = 4 imp./sec, each cross being derived from the analysis of 5 intervals (for explanation, see text). The dots, which represent results from the frog muscle spindle, are taken from Table 1 of Buller et at. (1953), each dot representing the analysis of between 2 and 1 intervals. The slope of the regression line through the crosses is 1 56 and ofthe regression line through the dots For this reason, the effect of varying Eo in the model was studied. As explained above, for each value of Eo there is an appropriate lowest regular firing frequency (fo). It was found that for all settings of Eo firing frequencies (f) greater than the appropriate value of fo fell on the model regression line of Fig. 6. However, when the frequency (f) was slightly

10 A MUSCLE SPINDLE MODEL 411 less than the minimum regular firing frequency (fg) it was noted that the S.D. of intervals increased to a greater extent for a given increase of mean interval than was predicted by the slope of 1 6. This was because the probability of an excessively long interval increased as the value of f fell below f, The apparent reason for this is illustrated diagrammatically in Fig. 7. With ED > EO (left-hand diagram) the course of the hyperbola is across ED and firing (which actually occurs when the two curves first meet) is virtually certain over a short interval of time. With ED < EO (righthand diagram) the probability of an early firing due to a noise transient is approximately the same as in the previous diagram, but since the hyperbola does not cross ED the probability of a longer interval is increased. The occasional occurrence of a very long interval-virtually impossible in the left-hand figure-increases the S.D. of the interval above that for the same mean firing frequency with conditions as in the left-hand figure. ED AD EO< D Fig. 7. Diagram to illustrate the altered probability of firing (which occurs when E = ED) with alteration in the setting of ED. A. ED is set above E. The hyperbola on which the noise is superimposed must fall to the value ED and firing (not illustrated) occurs when E = ED. B. ED is set below E. The hyperbola does not itself fall to the value ED but downward-going noise transients may still cause firing if E = ED- The same effect may be observed by considering the histogram of intervals. Figure 8A illustrates two of the distribution curves of intervals obtained from the frog muscle spindle (Buller et al. 1953). Figure 8B illustrates distribution curves obtained with the model. In the left-hand column are shown curves with a mean interval of approximately 34 msec, while in the right-hand column are curves having a mean interval of approximately 11 msec. The upper left-hand histogram of Fig. 8B was obtained with f = 4 imp./sec. It will be seen that the distribution curve is approximately normal. The lower left-hand curve of Fig. 8B was obtained with f = 1 imp./sec. It is seen that, while the mean interval is very similar, the distribution curve of intervals is definitely skewed (resembling the left-hand illustration of Fig. 8A) and the S.D. has almost doubled. The upper right-hand curve of Fig. 8B was obtained withf = 1 imp. sec while the lower right-hand curve with fo = 2 imp./sec. Again it is

11 412 4A. J. BULLER A m = 31 S.D. = 23 I m = 17 S.D. = m = 355 S.D. = m = 337 S.D. = m = 116 S.D. = T m = 11 S.D. = 57 B Fig. 8. A. Histogram of intervals from the frog muscle spindle; on the left for a mean interval of 31 msec (34 intervals), on the right for a mean interval of 17 msec (36 intervals). B. Histogram of intervals from the model. On the left two distribution curves with mean intervals of 355 and 337 msec, respectively. The upper curve was taken withfo = 4 imp./sec, the lower curve withfo = 1 imp./sec. On the right two histograms with mean intervals of 116 and 11 msec, respectively. The upper curve was withfo = 1 imp./sec, the lower curve withf = 2 imp./sec. (Each histogram from the analysis of 3 intervals.) M = mean interval (msec), S.D. = standard deviation (msec). In all histograms the ordinate represents the percentage of the total number of intervals and the abscissa the percentage of the mean interval.

12 A MUSCLE SPINDLE MODEL 413 seen that when f > f, the distribution curve is symmetrical, while with f slightly less than fg the curve becomes skewed and the S.D. rises. When f <' fg the distribution tends to an exponential and the S.D. tends to equal the mean, both characteristics of purely random processes (Parratt, 1961). By varying the value offo and plotting the standard deviation, measured in milliseconds against the mean interval measured in milliseconds over the complete range of discharge frequencies as in Fig. 9, a series of curves was obtained which departed significantly from the slope of 1-6 at mean 1C 5 C.) Co C3 Co CI) i Interval (msec) Fig. 9. Relation between mean interval in msec (abscissae) and standard deviation in msec (ordinates) plotted on a double logarithmic scale. The small dots are copied from the frog muscle spindle results of Buller et al. (1953, Fig. 3), each dot representing a pool of twenty intervals. The large dots, through which the smooth curve is drawn, are results obtained from the model with f = 17 imp./sec. Each large dot represents the analysis of 5 intervals. intervals proportional to the setting off. The 'best fit' for the frog muscle spindle results (shown in Fig. 9 by the small dots) was obtained with E. adjusted to give a value of fo = imp./sec. The large dots of Fig. 9 through which a smooth curve is drawn were obtained from the model when the latter was adjusted to fo = 17 imp./sec and the noise set to have an upper frequency limit of 1 kes and shaped to mimic a membrane time constant of 5 x 1-3. The agreement is good.

13 414 A. J. BULLER DISCUSSION The design of the model described makes certain assumptions as to the nature of the events occurring at the sensory nerve terminals of the frog muscle spindle. It assumes that the biological mechanism responsible for the generation of a firing rate proportional to depolarization is functionally independent of the source of random fluctuations, the latter serving only to modify the regular discharge pattern. It further assumes that the former mechanism has a lowest firing rate of approximately 16 imp./sec, and that the impulse discharge rates lower than this value which may be observed from frog spindles are due solely to noise transients superimposed upon some level of (subthreshold) depolarization. This latter assumption gains some support from the demonstration by Lippold, Nicholls & Redfearn (196) and. C. J. Lippold (personal communication) that the sensory discharge from the primary sensory ending of the cat muscle spindle remains regular down to its lowest firing frequency. This lowest frequency is typically about 2 imp./sec. The lowerfrequency irregular discharges observed in frog spindles have no counterpart in the behaviour of cat spindles. If the two assumptions made above are valid, the model described, which can accurately mimic the results obtained from the frog muscle spindle, illuminates some of the issues raised in the discussion of the previous paper (Buller et al. 1953). There it was suggested that alterations in both the r.m.s. value of the noise and in the frequency spectrum of the noise, which must result from stretching and depolarizing the sensory terminals, might lead to alterations in the degree of impulse interval irregularity. The model confirms that with an increase in the r.m.s. value of any type of noise the irregularity in discharge increases (see Fig. 5), but it also illustrates that the standard deviation is not particularly sensitive to changes in r.m.s. voltage, and the 1% change in r.m.s. voltage suggested in the previous paper would alter the standard deviation by less than 1%. Even less marked were the changes in irregularity observed with alterations in the frequency spectrum of the noise. Since the range of spectra used in the present experiments include the values most likely to occur during stretch of the sensory endings, it may be concluded that the alterations of both the r.m.s. value and the frequency spectrum of the noise which occur at the nerve terminal are probably of second order magnitude and unlikely to modify the irregularity appreciably. The model in fact suggests that throughout the range of muscle spindle discharge frequencies studied by Buller et al. (1953) the noise may be considered as remaining constant in character, but that the effects produced

14 A MUSCLE SPINDLE MODEL 415 by it depend upon the simultaneous firing frequency of the postulated regular spindle mechanism. When the spindle mechanism is firing as a result of a degree ofdepolarization sufficient to exceed its threshold value (ED > EO) the noise modifies the discharge pattern and leads to an irregularity in the discharge. When plotted in the manner of Fig. 6 or 9 this irregularity yields a slope of approximately 1-6. When the spindle mechanism is not being fired because the degree of depolarization is just less than that required to exceed threshold (ED < EJ), impulses are generated only by noise transients. As pointed out above, at such mean intervals the standard deviation of the irregularity is higher than predicted by the slope of 1 6 due to the occurrence of occasional, excessively long intervals. This leads to an upward trend in the curve of Fig. 9 at mean intervals of 5-6 msec. When the degree of depolarization at the nerve terminal is very much less than the threshold value of the spindle mechanism (ED < EO) firing takes place only when a very large noise transient occurs. With decreasing degrees of depolarization the firing becomes less and less frequent and the interval sequence approaches that of a random distribution. The slope of the curve of Fig. 9 approaches the limiting value of 1F. If these observations and interpretations are correct the slope of 2-4 found by Buller et al. for all intervals less than 16 msec appears to have been fortuitous and of no functional significance. The experiments described here can throw no light on the nature of the generator of the random fluctuations. If values for transverse membrane resistance, membrane capacitance and specific resistivity of axoplasm comparable to those for other non-medullated fibres are used appropriate scaling suggests that a noise voltage sufficient to produce the degree of irregularity observed might be generated across the surface membrane of a fibre -5-1 ct in diameter. Electron microscope studies of the sensory terminations in the frog muscle spindle (Katz, 196) have revealed terminations of 15,t, which are of the order of size required. It would be interesting to know if cat spindles lack these finest terminations. If they do, the absence of any irregular firing by such spindles which was described above might be explained. However, it would seem at the present time that there is insufficient evidence as to the physical and electrical characteristics of terminal nerve fibres to warrant further speculation as to the exact nature of the noise source.

15 416 A. J. BULLER SUMMARY 1. A model has been constructed to elucidate the disrupting effect of random fluctuations upon a regular time series. 2. It is suggested that comparable regular firing and random fluctuation generators exist at the sensory terminals of frog muscle spindles. 3. As a result of experiments with the model new interpretations are suggested for some of the biological observations on the irregularity of discharge from muscle spindles. I would like to thank sincerely Professor B. Katz for a most helpful criticism of the first draft of this paper, and Mr Peter Styles of the Electronics Department, St Thomas' Hospital. for continuous assistance in the building of the model. REFERENCES BULLER, A. J. (1959). A model to illustrate the effects of noise at a sensory nerve terminal. J. Phy8iol. 147, 62P. BULLER, A. J. & STYLES, P. (1958). A simple method of time coding for use with continuous film records. J. Phy8iol. 141, 21 P. BULLER, A. J., NICHOLLS,J. G. & STROM, G. (1953). Spontaneousfluctuationsofexcitability in the muscle spindle of the frog. J. Physiol. 122, FATT, P. & KATZ, B. (1952). Spontaneous activity at nerve endings. J. Physiol. 117, KATZ, B. (195a). Action potentials from a sensory nerve ending. J. Physiol. 111, KATZ, B. (195b). Depolarization of sensory terminals and the initiation of impulses in the muscle spindle. J. Physiol. 111, KATZ, B. (196). Sensory terminations in the muscle spindle of the frog. J. Physiol. 152, 13P. LIPPOLD,. C. J., NICHOLLS, J. G. & REDFEARN, J. W. T. (196). Electrical and mechanical factors in the adaptation of a mammalian muscle spindle. J. Physiol. 153, PARRATT, L. G. (1961). Probability and Experimental Errors in Science. New York: John Wiley and Sons.