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Increased gravitational force reveals themechanical, resonant nature of physiological tremorLakie, M; Vernooij, C A; Osler, C J; Stevenson, A T; Scott, Jpr; Reynolds, R F

DOI:10.1113/JP270464

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Citation for published version (Harvard):Lakie, M, Vernooij, CA, Osler, CJ, Stevenson, AT, Scott, J & Reynolds, RF 2015, 'Increased gravitational forcereveals the mechanical, resonant nature of physiological tremor', The Journal of Physiology, vol. 593, no. 19, pp.4411–4422. https://doi.org/10.1113/JP270464

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J Physiol 00.0 (2015) pp 1–12 1

The

Jou

rnal

of

Phys

iolo

gy

Neuroscience Increased gravitational force reveals the mechanical,

resonant nature of physiological tremor

M. Lakie1, C. A. Vernooij1,2, C. J. Osler1,3, A. T. Stevenson4,5, J. P. R. Scott6 and R. F. Reynolds1

1School of Sport, Exercise and Rehabilitation Sciences, University of Birmingham, UK2Institute des Sciences du Mouvement E.J. Marey, Aix-Marseille Universite, Marseille, France3Sport, Outdoor and Exercise Science, Department of Life Sciences, University of Derby, UK4QinetiQ Aircrew Systems, Air Division, QinetiQ, Farnborough, UK5Centre of Human and Aerospace Physiological Sciences, King’s College London, UK6Wyle GmbH, Koln, Germany

Key points

� Physiological hand tremor has a clear peak between 6 and 12 Hz, which has been attributed toboth neural and resonant causes.

� A reduction in tremor frequency produced by adding an inertial mass to the limb has usuallybeen taken as a method to identify the resonant component.

� However, adding mass to a limb also inevitably increases the muscular force required tomaintain the limb’s position against gravity, so ambiguous results have been reported.

� Here we measure hand tremor at different levels of gravitational field strength using a humancentrifuge, thereby increasing the required muscular force to preserve limb position withoutchanging the limb’s inertia.

� By comparing the effect of added mass (inertia + force) versus solely added force upon handacceleration, we conclude that tremor frequency can be almost completely explained by aresonant mechanical system.

Abstract Human physiological hand tremor has a resonant component. Proof of this is that itsfrequency can be modified by adding mass. However, adding mass also increases the load whichmust be supported. The necessary force requires muscular contraction which will change motoroutput and is likely to increase limb stiffness. The increased stiffness will partly offset the effect ofthe increased mass and this can lead to the erroneous conclusion that factors other than resonanceare involved in determining tremor frequency. Using a human centrifuge to increase head-to-footgravitational field strength, we were able to control for the increased effort by increasing forcewithout changing mass. This revealed that the peak frequency of human hand tremor is 99%predictable on the basis of a resonant mechanism. We ask what, if anything, the peak frequencyof physiological tremor can reveal about the operation of the nervous system.

(Received 3 March 2015; accepted after revision 19 June 2015; first published online 24 June 2015)Corresponding author R. F. Reynolds: School of Sport, Exercise and Rehabilitation Sciences, University of Birmingham,Edgbaston, Birmingham B15 2TT, UK. Email: r.f.reynolds@bham.ac.uk

Abbreviations EMG, electromyography; g, gravitational field strength; RoG, radius of gyration; RF, resonant frequency.

Introduction

Physiological hand tremor has a very distinct peak in itsacceleration spectrum. For 237 subjects, aged from 9 to91 years, 90% had a peak between 7 and 11 Hz (Lakie,

1994). At least part of the explanation for this is mechanical(Stiles & Randall, 1967; Raethjen et al. 2000). The massof the limb interacts with the elasticity of the musclesand tendons. The joint is less than critically dampedso it has a resonant frequency (Lakie et al. 1984, 2012;

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society DOI: 10.1113/JP270464

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2 M. Lakie and others J Physiol 00.0

Reynolds & Lakie, 2010; Vernooij et al. 2014). Drivinginput to the resonant system comes from mechanicalperturbation from active postural muscles which do notproduce particularly smooth output because they fire atfrequencies too low for complete tetanic fusion. However,this need not be the only explanation for the tremorpeak. There have been many suggestions that tremor isat least partly produced by a central or spinal neuraloscillator firing at the tremor frequency (McAuley &Marsden, 2000). In this experiment we seek to show thatit is not necessary to invoke such oscillators. Physiologicalhand tremor frequency can be adequately explained on anentirely mechanical basis. This is important because thereis a widespread (and in our view, poorly substantiated)belief that physiological tremor peak frequency providesinsights into the nervous system, for example ‘Tremor[ . . . ] could constitute a new investigative tool providinga non-invasive “window” into the rhythmic nature ofhuman motor control’ (McAuley & Marsden, 2000).

The mechanical component of tremor has generallybeen investigated by attaching masses (artificial inertialloads) to the limb (e.g. Marshall & Walsh, 1956; Stiles& Randall, 1967; Raethjen et al. 2000). The idea is thatincreasing the inertia will reduce the resonant frequency ofthe tremor in a simple and predictable way. If the tremorfrequency fails to change as expected, the conclusion isdrawn that other (neural) factors are (partly) responsiblefor the tremor. An unrecognised serious problem withthis approach is that simply adding masses to the limb notonly increases its inertia but also its weight. Consequently,an increase in active force, and thus electromyography(EMG), is necessary to maintain the loaded limb’s positionin the gravitational field. The increased muscular activityis likely to give rise to an increase in muscle stiffness,which by itself will raise the resonant frequency. As aresult, the decrease in tremor frequency that occurs dueto increased inertia may be (partly) counteracted by theincreased force. The problem is outlined in Fig. 1 Oneway to overcome this problem would be to use a balancedmass in the manner of a flywheel. Balanced masses havebeen successfully used with a passive limb, but attachingthem to a posturally active limb without compromising itsmovement is problematic (Lakie et al. 1986).

In the present experiments, which were designedto demonstrate conclusively the mechanical basis ofphysiological tremor, we used a large horizontal humancentrifuge. By spinning subjects in the centrifuge, wewere able to record naturally occurring postural handtremor at a range of levels of gravitational field strength(g). While being centrifuged, the subjects had to contendwith an increase in force alone. In a complementary partof the experiment, when the centrifuge was stationary,mass was added to the hand to generate inertial forceloadings comparable with those produced by increased g.In this latter case, the subject had to contend with both

increased force and the increased mass. It was thus possibleto compare directly the effect of force loading and inertialloading of the limb. Subsequently, the effect produced bythe increased force could be subtracted from that caused byincreased mass so that the result of purely inertial loadingwas revealed.

Methods

Ethical approval

The experiments were approved by the QinetiQ ResearchEthics Committee (QREC) and were conducted inaccordance with standard human centrifuge operatingprocedures and the Declaration of Helsinki. Informedwritten consent was taken from all volunteers.

Subjects

The experiments were performed on seven male subjects(mean age 35 years, SD 5 years) using the Farnborough(UK) human-rated centrifuge. We intentionally onlyused subjects who had considerable experience in theFarnborough centrifuge in order to obviate anxiety orarousal which might be expected to increase tremor size.All subjects had previously experienced up to 9 g on theFarnborough machine and regarded the maximum g levelthat was used (2.5 g) as trivial.

Apparatus and procedure

The SI abbreviation for local gravitational field strengthis g and for mass is kg, which prevents unintentionalconfusion. Figure 2 depicts an instrumented subject seatedin the centrifuge. A lightweight moulded splint (thermo-plastic, 0.1 kg), which held the digits in a slightly adductedand fully extended posture, was attached to the sub-ject’s hand. A 3-axis accelerometer (ADXL335; AnalogDevices, Norwood, MA, USA) was firmly attached to thedorsum of the splint to record hand tremor. Throughoutthe experiment, subjects sat in the gondola at the endof the centrifuge arm (approximately 10 m from theaxis of rotation, and were harnessed into an aircraftpilot’s seat (Mk 16; Martin Baker Aircraft Company Ltd,Higher Denham, Middlesex, UK) with the pronated leftforearm supported by an armrest and foam splintingat approximately chest level. The subject was asked tohold his hand in a horizontal position. A retroreflectivelaser rangefinder (Model YP11MGV80; Wenglor sensoricGmbH, Tettnang, Germany) was focused on the dorsumof the hand in order to monitor hand position and thissignal was recorded and displayed to the subject on amonitor screen at eye level at �1 m distance. The gondolawas hinged so that the effective acceleration vector always

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

J Physiol 00.0 Effects of increased gravity upon tremor 3

passed precisely in the head-to-foot (typically referred toas +Gz) direction (Latham, 1955). In the first part of theexperiment, the centrifuge was rotated at angular velocitiesfrom zero to a maximum of approximately 15 r.p.m. togenerate forces of 1, 1.5, 2 and 2.5 g. The onset and offsetrate of change of acceleration was 1 g s−1 and the sub-jects were spun at the designated velocity for 45 s; the firstand the last 1 s of the steady state record were discarded

from analysis. Each subject performed three repeats of eachdesignated velocity. To establish a ‘noise floor’ and controlfor minor accelerations produced by small changes inangular velocity of the centrifuge or vibration, accelerationwas recorded while the accelerometer was attached to aninanimate object in the capsule. This broadband noisewas two orders of magnitude less than the tremor signalat the highest (noisiest) rotation velocity. In addition to

Sag

F F2F

mg

2 mg

3 mg

mg

2 mg

3 mg

3F 2F 3F

mm m m

mm

mmm

m mm

Sag Sag No Sag No Sag No Sag

Figure 1. The dual effect of added massA mass is supported by a spring. Resonant frequency (RF, in Hz) is dictated by spring stiffness (k) and by mass(m). The equation is RF = 1/(2π )�(k/m). Left panel: increasing mass produces increased force (F) which causesextension (sag) in the supporting spring if k does not change. The decrease in RF with added mass is indicatedin the amplitude frequency spectra (top row). The reduction in resonant frequency reflects only the added mass.Right panel: this symbolises the human postural system when mass is added to a limb. In order to support theload without sag (to maintain posture), increased muscular effort is required to generate increased upward force.This muscular activity increases stiffness, and the anticipated reduction in resonant frequency is partly offset. Thespectra in the top row show the spectra with sag (copied from the left panel) with the spectra with no sagsuperimposed (spectra are displaced to the right, indicated by arrows).

Figure 2. Experimental setupTo the left of the picture is the centrifugewith one of its two pivoted gondolas at theend of its �10 m radius arm. The insetshows a subject strapped into the seat. Athermoplastic splint ensured the hand andfingers moved as one. A laser rangefindersituated above the hand was used tomonitor its vertical position, which wasdisplayed to the subject on a screen just outof shot on the right of the picture. Theaccelerometer is located on top of the handsplint but is too small to be seen distinctly.The EMG electrodes lie underneath theprotective bandaging on the forearm.

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

4 M. Lakie and others J Physiol 00.0

Table 1. Loads applied in two conditions (altered gravitationalfield (g) and added mass (kg))

Load Gravitational field(g) Added mass(kg)

1 1 02 1.5 0.2503 2 0.5004 2.5 0.750

hand tremor and position we recorded surface EMG fromthe extensor digitorum communis and flexor digitorumsuperficialis muscles using a two channel Delsys Bagnolisystem (Delsys Inc., Boston, MA, USA). As part ofroutine centrifuge subject monitoring, limb lead ECGand blood pressure (Portapres, Finapres Medical Systems,Amsterdam, The Netherlands) were also recorded.

In the second part of the experiment, tremor wasrecorded with masses attached to the hand while thecentrifuge remained stationary. These were made fromlead sheet and had masses of 0.250 kg, 0.500 kg and0.750 kg. Each subject performed three repeats with eachmass. Because the average mass of the instrumentedhand is approximately 0.500 kg and the weights werepositioned as closely as possible above the centre of massof the hand, the subject had to exert an upward forceapproximately equivalent to that experienced at 1.5 g,2 g and 2.5 g. The hand mass for each subject wascalculated using a simple anthropometric model and ascanned and dimensioned image of the hand (for detailssee Appendix). To enable direct comparison between thetwo conditions (increased gravity and increased mass),we present data recorded at four equivalent levels, calledsimply ‘load’ (Table 1). All data were sampled at 1 kHz. Thehand would more correctly be considered as a torsionaloscillator with the wrist having an angular stiffness andthe hand having a moment of inertia (Lakie et al. 1986).For simplicity, and following common practice, we useda linear approximation where the mass of the hand issuspended by muscles and tendons with an elastic stiffness.This approximation does not introduce much inaccuracyand we explain our reasoning in the Appendix.

Analysis

Analysis was performed by custom-written programs inMATLAB software (Matlab 2011a; MathWorks, USA).EMG was band-pass filtered between 30 and 200 Hzand rectified. For each trial, the amplitude spectra of thehand acceleration and EMG were obtained by NeuroSpecsoftware (version 2.0, 2008, http://www.neurospec.org/)and the maximum amplitude and associated frequencywere detected. For all subjects, the resulting data(maximum amplitude and associated frequency) werethen averaged, including the three trials for each condition.

Repeated-measures ANOVA was used to test for significanteffects of load. P < 0.05 was considered significant.

Results

Tremor acceleration and frequency

The results from a typical subject are shown in Fig. 3 Withincreasing g (left panel) tremor acceleration increasedprogressively from its baseline level until it becameapproximately four times bigger. The peak frequency alsoincreased progressively, in this case from �6.5 Hz to�7.5 Hz. The effects of increasing mass (right panel) wereopposite: tremor size decreased and the frequency fell,in this case from �6.5 Hz to �5 Hz. These results weretypical of all subjects and Fig. 3 shows the mean (±SEM)frequency for both conditions.

In this study we are mainly interested in the peakfrequency of tremor. However, with added mass, thefrequency spectrum in this representative subject alsoshows typically variable sub-peaks at higher frequencies(10–20 Hz) which are close to those seen in the EMG(Fig. 6).

Figure 4 shows the progressive decrease in tremorfrequency as inertia is increased and the progressiveincrease in tremor frequency as g is increased (F3,18 � 15.9;P < 0.001). The decrease in frequency with added mass isthe expected effect of adding mass to a spring mass system.The rise in frequency with added g is simply explainedas the result of increased limb stiffness as a result ofincreased muscle activity required to support the hand.Increasing g has no direct effect on the resonance of thespring and mass. Accordingly, it now becomes possibleto correct for the inevitable increased stiffness in the massloading condition. In Fig. 4 this has been done by piecewisecorrection at each load condition. For example, the effectof mass at load 4 has been to lower the resonant frequency

0.1 0.03

0.02

0.01

00

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load 2load 3load 4

load 1

Acc

eler

atio

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/s2 )

0.06

0.08

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00 5 10 15 5 10 15

Figure 3. Representative subject dataTremor acceleration spectra when load is increased in the centrifuge(left panel) or by adding mass (right panel). With added gravitationalfield strength the tremor size increases and the frequency risessomewhat. With added mass, the tremor size decreases slightly andthe frequency falls.

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

J Physiol 00.0 Effects of increased gravity upon tremor 5

by approximately 1.4 Hz. However, this mass loading willhave also produced an increase in resonant frequencyof approximately 0.9 Hz because of necessary muscularstiffening and this will have partly offset the reductioncaused by the added mass. Correction reveals the truereduction (approximately 2.3 Hz) that would be producedby solely the mass. We can use piecewise correction for eachcondition, since the added masses were chosen to matcheach g level.

There are two lines of evidence that suggest that wecan directly compare the two conditions. First, we canuse the data from Fig. 4 to calculate the mass of thehand and compare this to our estimation of hand massbased on hand size (see Appendix). Addition of mass toa resonant system will produce a linear increase in theperiod of oscillation (‘Period’) squared (Lakie et al. 1986).Backwards extrapolation of this relationship will reveal thenotional mass that has to be removed to make the periodzero – that is, the original mass of the limb (plus splint).This relationship is shown in Fig. 5 and further describedin the Appendix.

In Fig. 5 the frequency values obtained from thecondition where mass is added have been replotted interms of Period2. Using uncorrected frequency valuesfrom Fig. 4 (dark grey traces) implies a mass of 1.09 kgfor the hand plus splint, whereas the use of correctedvalues (light grey traces) implies a mass of 0.493 kg. Thesevalues may be compared with the retrospectively estimatedmass of our subjects’ hands (see Appendix) which was0.460 kg. The agreement when the corrected values areused is close. The second line of evidence comes fromthe EMG, because we can compare whether the amountof EMG activity required to support different loads wassimilar in the two conditions. We used a standard added

8

Gravitational fieldAdded massAdded mass corrected

7

6

5

4

1 2Load

Freq

uenc

y (H

z)

3 4

Figure 4. Group mean tremor frequencyThe tremor peak frequency (mean and SEM) in both conditions;increased g (black) and increased mass (dark grey) for each load. Thecorrected values for each load (decrease caused by mass minusincrease caused by g) are also plotted (light grey).

mass which assumed that the mass of the hand and splintwas �0.500 kg. Therefore the effect of adding a mass of0.500 kg should double the effort, and thus EMG, requiredto support the hand. Increasing the gravitational fieldstrength to 2 g should have the same effect. The closecorrespondence between the two conditions is shown inFig. 6 Addition of mass or g both similarly increased EMG(F3,18 � 11.3; P < 0.001).

Figure 6 shows that there is a very good agreementbetween the sizes of EMG at each load condition. Thisfigure therefore also strongly supports the assumptionthat the mass of the hand plus splint is close to0.500 kg.

The increase in tremor frequency with increased g wasshown in Fig. 4 and we have attributed it to increasedmuscular stiffness contingent on the effort. Are there other

−1.000 −0.500 0Added mass

0.03

0.06

r2= 0.9963y = 0.0479x + 0.0245Corrected:

Per

iod2

1.0000.500

r2 = 0.9949y = 0.0224x + 0.0245Non-corrected:

Figure 5. Estimated hand massThe corrected values and non-corrected values of the added masscondition from Fig. 4 are plotted as oscillation period (Period)squared vs. added mass. Both lines provide a very good fit to thedata points. Extrapolation to the point where the regression line cutsthe x-axis yields the amount of mass that would have to be removedto reduce the Period to zero – that is, the mass of the hand andsplint. The uncorrected values predict a mass of �1.1 kg, whereasthe corrected values predict a mass of �0.5 kg.

3.5

3.0

2.5

2.01 32

Load

Mea

n am

plitu

de (

mV

)

4

4.0Gravitational field

Added mass

x10−3

Figure 6. Mean filtered rectified extensor EMG at each load(±SEM)

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

6 M. Lakie and others J Physiol 00.0

possible reasons why tremor frequency might increaseunder conditions of increased gravitational field strength?A possible candidate would be that it is due to a progressive,gravity dependent change in EMG frequency. This isexamined in Fig. 7 which shows the mean rectified extensorEMG spectra in each condition.

Three features are clear in Fig. 7 First, the general shapesof the spectra are always similar, with a broad peak between8 and 14 Hz in each load condition which does not changesystematically with load. Second, the sizes of the spectraare well matched between added g and added mass foreach load condition, further supporting the statement thatgravitational and mass loads are equivalent. Third, thereis a small sub-peak in EMG activity at the approximatefrequency of the tremor in the increased g conditions. Itis noteworthy that these are the conditions in which thetremor acceleration is particularly large.

In addition to the effect on frequency, increasing theload on the hand also affected the size of tremor. Thetremor acceleration (size) was shown for an individualsubject in Fig. 3 and is shown for all subjects in Fig. 8.

Figure 8 shows that increasing gravitational fieldproduces a large increase in tremor acceleration(F3,18 = 11.7; P < 0.001). Acceleration is related to force,which rises progressively as g increases. The effect is

5

Gravitational field Added mass

load 1load 2load 3load 4

x10−3 x10−3

4

3

2

1

Am

plitu

de (

mV

) 5

4

3

2

10 5 10 15 20 25

Frequency (Hz) Frequency (Hz)30 0 5 10 15 20 25 30

Figure 7. Mean extensor EMG spectraLeft panel shows the effect of increasing g. Right panel shows theeffect of increasing mass. At each load the overall level of EMG issimilar and the spectra all have a similar shape. It is clear that in thehigher g conditions, but not in any of the other conditions, there isan emergent small peak close to the frequency of the tremor.

0.12

0.08

0.04

1 2 3 4Load

Gravitational field

Mea

n am

plitu

de (

m/s

2 )

Added mass

Figure 8. Mean amplitude of peak tremor acceleration at eachload (±SEM)

progressive but not quite linear. Conversely, increasingthe mass produced no significant change (F3,18 = 0.53;P = 0.67).

Discussion

We discuss our results under five headings.

(1) The effect of increased mass

In our experiments, the addition of mass systematicallydecreased the frequency of tremor (Fig. 4). There is somedisagreement in the literature about the effect of addedmass on tremor frequency. Marshall & Walsh (1956) foundthat frequency did not change. Hamoen (1962) was thefirst to show that human tremor frequency was decreasedby added mass. This was subsequently confirmed by Stiles& Randall (1967), and confirmed in a reduced animalmodel (Rietz & Stiles, 1974). However, in a subsequentstudy (Stiles, 1980) the author found that the effect ofadded mass was variable and depended on the posture ofthe hand.

When a decrease in tremor frequency occurred, it hasalways been attributed to a reduction in frequency of amechanically resonant limb. It is difficult to think of anyalternative explanation. A load sensitive neural oscillatoris one possibility, but it would have to possess the featurethat its frequency of operation decreases as limb massincreases. Prima facie this seems unlikely. There is nopeak in the EMG at the tremor frequency (Fig. 7 rightpanel). Additionally, this figure shows that, if anything, asmass is added, there is a slight increase in the frequencywhere the rectified EMG is largest. Neural drive that shiftsto a lower frequency with added mass thus seems highlyunlikely. Furthermore, we show in these experiments thatincreasing the load, and thus EMG, by increasing g isassociated with a clear rise in tremor frequency. Therefore,the most likely cause of the reduction in tremor frequencywith added mass is a reduction in the resonant frequencyof the limb. Furthermore, our experiments provide anatural explanation for the apparently capricious responseto increased mass described by other authors, above. Asmass produces an inevitable increase in stiffness its effectwill depend on whether the increased inertia (loweringfrequency), or the increased stiffness (raising frequency),prevails. In turn, this will depend on the precise detail ofhow the procedure is performed. Because load is a linearfunction of the moment arm but inertia depends on thesquare of the radius of gyration, much will depend onwhere the additional mass is positioned (see Appendix).

(2) The effect of increased gravitational field strength

Increasing g increased the frequency of tremor (Fig. 4).This change has not been previously described and seems

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

J Physiol 00.0 Effects of increased gravity upon tremor 7

to have only two possible explanations. It is conceivablethat there is an increase in the frequency of a putativetremor oscillator as gravitational load increases. Note thatit has to be gravitational load because increasing load bythe addition of mass is associated with a reduction oftremor frequency as described above. Such a specificallygravity sensitive oscillator seems highly unlikely. It isundeniable that in Fig. 7 there is some prominent EMGactivity close to the tremor frequency as g is increased.While this might possibly support the notion of a tremoroscillator which is uniquely sensitive to gravitational loadit seems much more likely that it actually reflects somereflexive modulation of EMG activity by limb motionunder conditions where tremor is particularly large. FromFig. 8 it can be seen that tremor is approximately fourtimes larger at 2.5 g compared to 1 g. Modulation ofEMG in enhanced tremor has been previously described(Hagbarth & Young, 1979) and is probably an inevitableconsequence of reflex modulation of motor output drivenby muscle spindles or other afferents. There is no reasonto suppose that increased depth of modulation as tremorsize increases will automatically produce an associatedincrease in frequency. However, a natural explanationfor the elevated tremor frequency in increased g is thatincreased muscular effort leads to inevitable stiffeningof the muscles. Many studies have shown an activitydependent increase in skeletal muscle stiffness (Joyce &Rack, 1969; Cannon & Zahalak, 1982; Kearney & Hunter,1990). Although this stiffening will be mitigated to somedegree by the compliance of the series tendons, it willbe sufficient to increase the resonant frequency andtremor frequency will rise. Resonant frequency squaredis proportional to stiffness. Figure 4 suggests that in goingfrom 1 g to 2.5 g the stiffness increases by approximately25%. As we show below, increased wrist stiffness providesan almost perfect explanation for our results. We studiedthe effect of increased g. However, our results can also beused to predict the effect of reduced g and we describe thisrelationship in the Appendix.

(3) The effect of the combination of increased massand increased stiffness

As described in section (1) the addition of mass decreasesresonant frequency of a limb. However, as describedin section (2) the increased muscular effort as mass isadded will cause a degree of increased limb stiffeningand therefore an increase in resonant frequency. Becausethese two changes will occur simultaneously, the increasedstiffness will partly offset the reduction in frequencycaused by the mass. The extent of this offset is revealedin Figs 4 and 5. In these figures we have plotted thereduction in resonant frequency caused by added masswhen the increased stiffness is both ignored and included.In Fig. 5 we have plotted this in terms of Period2 plotted

Table 2. Subject hand parameters

Hand Hand Mass Hand Centre of massSubject area(cm2) volume(l) (kg) length(cm) (cm from axis)

1 109.7 0.302 0.332 18.4 8.242 91.4 0.226 0.249 17.3 7.733 130.1 0.397 0.437 20.5 9.164 131.4 0.403 0.443 20.0 8.965 106.1 0.287 0.316 18.5 8.296 127.5 0.384 0.422 20.7 9.257 107.6 0.293 0.322 17.7 7.91Mean 114.8 0.327 0.360 18.7 8.64

Table 3. Prediction of human hand tremor frequency in non-terrestrial gravitational environments

Environment g RF

Space station �0 5.830952Moon 0.16 5.93579Mars 0.38 6.076989Earth 1 6.458328Jupiter 2.5 7.298973

g, gravitational field strength; RF, resonant frequency.

against added mass. By doing this, it is possible to deducethe mass of the hand and the stiffness of the wrist (seeAppendix). The use of uncorrected values leads to anestimate of > 1.0 kg for hand mass and > 1700 N m−1

for wrist stiffness. When the values are corrected bypiecewise correction of the increased frequency producedby muscular stiffening, these estimates have much morerealistic values (0.493 kg and 793 N m−1). The estimatedmass includes 0.1 kg for the mass of the hand splint so ourpredicted hand mass is 0.393 kg. This value is very closeto the retrospectively measured mean hand mass for oursubjects: (0.360 kg) (Appendix).

There are several published estimates for the mass ofthe human hand. The classic cadaveric study by Clauseret al. (1969) gives values of 0.380 kg, 0.446 kg and 0.548 kgfor small, medium and large male subjects. Drillis et al.(1964) tabulate hand mass measurements as a percentageof body mass found by other investigators. From theirTable 6, estimated hand mass for a 70 kg man wouldbe 0.420–0.588 kg. Our subjects may have been slightlysmaller than average, but unfortunately we do not havedata for their height or weight. However, the main pointis clear. Hand mass, even for the largest subjects, will notexceed 600 g. Only corrected values predict a hand massof anything close to a realistic estimate. It is absolutelynecessary to include the increased stiffening.

Furthermore, the gradients of the regression linesin Fig. 5 reflect stiffness (details in Appendix). The

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

8 M. Lakie and others J Physiol 00.0

use of uncorrected values in our experiment predicts astiffness of 1761 N m−1 whereas corrected values pre-dict a stiffness of 793 N m−1. There are several publishedestimates of human wrist stiffness for flexion/extension.A complication is that these values are often expressed inangular units, that is, as N m rad−1. In order to convert tolinear units as used in our study it is necessary to know themoment arm of the wrist for flexion/extension. A sensibleestimate for this is 1.3 cm (see Figs 3 and 4 in Gonzalezet al. 1997). When published values are adjusted in this waya definitive study on 10 subjects by Halaki et al. (2006)gives a mean value for wrist stiffness of 534.8 N m−1

(range 320.7–1015.4 N m−1; their Table 3). Clearly, ouruncorrected stiffness values are impossibly high whereasthe corrected values are in the appropriate range. There isalso the fact that had we used a possibly more appropriatetorsional model of the hand the equivalent masses wouldhave been applied at the radius of gyration rather than thecentre of mass. Had we done so there would have been agreater reduction in resonant frequency and our stiffness

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100

00 50 100

Hand area (cm2)

Han

d vo

lum

e (I

) y = 0.1703×1.5927

r2 = 0.9695

150 200

Figure 9. The relationship between area of the hand andvolume for 14 subjectsThe equation of the regression line was used subsequently toestimate the hand volume of the 7 different subjects that we used inthe centrifuge experiments.

0.15

Corrected

Non-corrected

y = 0.0497× + 0.0245

y = 0.0224× + 0.0245r2 = 0.9949

k = 400k = 800k = 1600

r2 = 0.9963

0.10

0.05

−1.00 −1.50 0 0.50

Added mass (kg)

Per

iod2

1.00

Figure 10. The relationship between oscillation periodsquared and added mass for a resonant system

estimate would reduce further bringing them even closerto published values (details in Appendix).

These results imply that, to generate realistic estimatesof hand mass and wrist stiffness, it is essential to includethe increased stiffness that is a consequence of increasedload. When this is done, tremor frequency provides veryrobust estimates of limb mass and stiffness and this is verystrong evidence that the hand tremor frequency resultsfrom resonance only. The coefficients of determination ofthe regression lines of Fig. 5 show that more than 99%of the change in frequency can be explained by a simplespring/mass oscillator. It is common to describe physio-logical tremor by the frequency of its spectral peak, forexample, 8–12 Hz tremor or �10 Hz tremor. What isclear from our experiments is that the peak frequencyof tremor says a lot about the stiffness and mass of thehand and wrist, but little about the neural input. TheEMG spectrum is a clear reflection of the output of theCNS but the acceleration spectrum of tremor is mainlydetermined by mechanical factors. The peak frequency ofthe physiological tremor we recorded was greatly alteredby changed loads whereas the EMG was not. Thus tremorfrequency revealed much more about the load than aboutneural oscillators, central or spinal. From these resultsit is difficult to believe that the study of physiologicalpeak tremor frequency can show anything more than themechanical features of the musculoskeletal system. It doesnot provide an insight into neural rhythmicity (McAuley& Marsden, 2000). However, because tremor frequencydoes reflect the stiffness of the muscles it can provide auseful insight into their state (Vernooij, 2014).

(4) Comparison of EMG levels for each load condition

Central to our approach was the belief that the mass ofthe instrumented hand was close to 0.500 kg. This is

60

50

40

30

200 0.5 1

Gravitational field

Res

onan

t fre

quen

cy2

1.5 2.0

y = 7.7135x + 34.081r2 = 0.8723

2.5 3

Figure 11. The relationship between resonant frequencysquared and g

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

J Physiol 00.0 Effects of increased gravity upon tremor 9

supported by our own and other estimates (see Appendixand section (3) above). However, this assumption alsounderpinned our belief in the equality of the mass andgravitational loading. The equality is confirmed by Figs 6and 7, which show that very similar levels of EMG wereassociated in both conditions with the corresponding loadso that, for example, the effect of the addition of 0.500 kgwas the same as doubling the gravitational field strength.The shape of the EMG spectrum always had a very broadpeak of activity around 8–14 Hz, which was not close tothe tremor frequency. This feature of the EMG has beenfrequently described (Timmer et al. 1998; Halliday et al.1999; Raethjen et al. 2000). EMG spectra are similar inboth conditions at each load level and the only observabledifference is a small peak close to the tremor frequencywhen the tremor is particularly large (discussed in section(2) above).

(5) Tremor size increased considerably with increasedg and decreased slightly with added mass

Tremor acceleration represents force fluctuation whichis approximately proportional to background force(signal dependent noise; Schmidt et al. 1979; Slifkin &Newell, 1999; Enoka et al. 1999; Laidlaw et al. 2000).Because of Newton’s second law (A = F/M), as forcefluctuations increase, tremor acceleration becomes larger.Consequently, an increase in tremor size with g is anti-cipated. Figure 8 shows that the relationship is not quitelinear. A likely explanation is that limb damping alsoincreases with activity and this acts to attenuate thetremor as g is increased. With mass loading, becauseA = F/M, as mass is increased acceleration is decreased.However, this is largely offset by the associated increasein force fluctuation as load increases. It is interesting thatFig. 8 shows that tremor acceleration tends to decrease alittle as mass is added although the reduction does notreach statistical significance. The implication is that theeffect of added mass may outweigh the increased forcefluctuations.

Conclusion

The effects on physiological tremor of adding inertia to alimb can only be explained if the concomitant changesin stiffness are taken into consideration. Peak tremorfrequency reflects the inertia and stiffness of the limb andis not the result of a central oscillator. These results arerestricted to physiological tremors. In most pathologicaltremors the tremor rhythm is probably dominated byneural drive at a specific frequency; however, even inthat situation, mechanical factors will play an inevitablerole.

Appendix

Calculation of hand mass

Ideally, we would have measured the volume of eachsubject’s hand at the time of making the tremormeasurements, but this was not possible. We were ableto obtain high quality 2-D scans of each subject’s handand the surface area of this scan was used as the inputto an equation which we subsequently derived from aseparate group of 14 subjects with a broad range of handsizes. This equation defined the relationship between handsurface area and hand volume and was calculated in thefollowing way. Subjects placed their left hand, palmar sidedown, on the surface of a scanner. The resultant imagewas processed using an edge detection algorithm (definehand edges: Adobe Photoshop; convert edges to binaryx–y coordinates: ImageJ; Schneider et al. 2012) whichoutlined the hand. The hand surface was defined as thearea bounded by this margin and the line of the radiocarpaljoint. Hand volume was measured by water displacementwhen the hand was immersed up to the line of the radio-carpal joint. The relationship between measured volumeand hand surface area was y = 0.1703x1.5927 (r2 = 0.9695;see Fig. 9). Note that this proportionality is very similar tothe surface–volume relationship of a regular rectangularobject; i.e. x1.5. This equation was then used to estimatethe hand volume of the subjects who participated in ourstudy (see Table 2). These volumes were finally convertedto mass using a value for density of 1.1, which reflectsthe preponderance of bone in the hand. This value liesapproximately midway between two previously reportedvalues for hand density. (Harless, 1860, cited in Drilliset al. 1964) gave a value of 1.1128 and, more recently,Clarys & Marfell-Jones (1986) gave a value of 1.0823 (ourcalculation of overall hand density from their Table 3).

There are, unsurprisingly, few relevant studies directlymeasuring hand mass. Clauser et al. (1969) give valuesof 0.380 kg, 0.446 kg and 0.548 kg for the cadaverichands of small, medium and large male subjects. Clarys &Marfell-Jones (1986) found a mean hand mass of 0.345 kgfor three male and three female subjects. The estimatedmeasured volumes of our 14 subjects (Fig. 9) rangedfrom approximately 0.200–0.500 l implying a mass of0.220–0.550 kg, which is similar to the above values. Themass of the splint plus accelerometer (0.100 kg) must beadded to the calculated hand masses of the subjects whotook part in the experiment.

The weights that we used were designed so that sub-jects when loaded would have to exert an upwards forcenominally equivalent to that generated when they werecentrifuged at 1.5, 2 and 2.5 g. These weights had to be pre-pared in advance of the experiment. We used an estimatedhand mass for our subjects of 0.400 kg. The estimate wasappropriate; the subsequently calculated mean hand mass

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

10 M. Lakie and others J Physiol 00.0

of the subjects that we used in the centrifuge was 0.360 kg(Table 2). For both conditions, i.e. force and inertia, themass of the splint plus accelerometer (0.100 kg) must beadded to the calculated hand masses. This meant that thetotal mass became 0.460 kg, which was very close to thevalue which we had assumed would produce equal massand gravitational loading (0.500 kg).

The relationship between the hand mass and the addedmoment of inertia we wanted to apply with the weightsis quite complex because it depends on the shape aswell as the size of the hand. Using data from de Leva(1996) the centre of mass was individually located at44.8% of the hand length, defined as the distance betweenthe radiocarpal joint and dactylion (see Table 2). Whenmass was added to the hand it was applied as closely aspossible above this point. In doing this we reproducedwhat is usually done in ‘added inertia’ tremor experimentsbecause we used a linear approximation of a torsionalsystem. Technically, we should have applied loads whichwere multiples of hand mass at the radius of gyration(RoG). We did not do this for four reasons. (a) There arevery few estimates of moment of inertia of the hand andthis is required to calculate the RoG (RoG2 = Moment ofInertia/Mass). (b) We followed common practice in ‘addedload’ tremor experiments. (c) The difference is probablyquite small. Winter (2009) gives 51% of hand length forthe centre of mass (CoM) and 59% for RoG. (d) Thesplint that we used also adds some inertia. We mentionthe possible size of the resultant error below.

The relationship between added inertia and resonantfrequency

The relationship (when frequency is measured in Hz)between resonant frequency (RF), mass (m) and stiffness(k) is

RF = 1

2π

√k

m

When mass is added to a resonant system its resonantfrequency is decreased. The relationship was describedby Lakie et al. (1986). The relationship between addedmass and oscillation period (Period) squared is linear, andthe x-intercept indicates the notional mass that must beremoved to reduce the mass to zero: that is, the originalmass of the system. When resonant frequency is plotted interms of Period2 against added mass the result is a straightline with the equation:

y = (2π)2

kx + (2π)2

km

for y = 0

0 = (2π)2

kx + (2π)2

km

Consequently, when y = 0, m = −x. Stiffness (k) is thereciprocal of ‘gradient’/(2π)2. This relationship is exploredin Fig. 10.

In Fig. 10, as described above, the x-intercept representsthe original mass and the gradient reflects the stiffnessof the system. The experimental results are fitted bycontinuous regression lines. For corrected results (blackcircles) this line (r2 = 0.996) indicates an original massof 0.493 kg and a stiffness of 793 N m−1. To indicate thesensitivity of the system to stiffness, the figure also showsmodelled results for the mass of 0.493 kg stabilised bythree different values of stiffness (squares, 1600 N m−1;crosses, 800 N m−1; and circles, 400 N m−1, all grey andfitted by dashed regression lines). Clearly, the stiffness thatsatisfies our corrected experimental results cannot be veryfar away from 800 N m−1. When the uncorrected resultsare plotted (black diamonds) they are also well fitted by alinear regression line (r2 = 0.995), but this line predicts anoriginal mass of 1.09 kg and a stiffness of 1760.9 N m−1,both unfeasibly large values.

If the hand is considered as a torsional oscillator, todouble the moment of inertia a mass equivalent to thehand should have been placed at the RoG. Because weplaced the equivalent mass at the CoM, which is moreproximal (45% rather than 59% of hand length) (Winter,2009), we have possibly added less inertia than we claimand not produced as big a reduction in frequency as weshould have. With a greater reduction in frequency therewould have been a bigger increase in Period2 in Fig. 10.Correcting for this difference gives a �28% lower valueof stiffness which would move our value to 571 N m−1,even closer to those calculated by Halaki et al. (2006).However, this is partly offset by the additional moment ofinertia added by our splint. With the uncertainties aboutthe precise location of RoG, load positioning, splint inertiaand the conversion factor used to convert angular intolinear wrist stiffness we do not wish to overstate this claim,but it is much more likely that we have overestimated,rather than underestimated, stiffness.

The effect of reduced g

The relationship between gravitational field strengthand tremor frequency is restated in Fig. 11. Because,as explained, the effect of increased gravity is to alterstiffness there is a linear relationship between g and tremorfrequency squared.

This suggests the potentially testable hypothesis thathuman hand tremor frequency will be different innon-terrestrial gravitational environments. The equationis

RF = √7.7g + 34

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

J Physiol 00.0 Effects of increased gravity upon tremor 11

where g is gravitational field strength relative to Earth andresonant frequency (RF) is in Hz.

This predicts the values in Table 3.

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Additional information

Competing interests

The authors have no competing interests to declare.

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12 M. Lakie and others J Physiol 00.0

Author contributions

Authors are presented in alphabetical order. M.L., R.F.R., C.A.V.:conception of the experiments. M.L., C.J.O., R.F.R., A.T.S.,J.P.R.S., C.A.V.: experimental design and collection and assemblyof data. M.L., R.F.R., C.A.V.: analysis and interpretation of data.M.L., R.F.R., C.A.V.: drafting the article and revising it criticallyfor important intellectual content. All authors approved the finalversion of the manuscript, all persons designated as authorsqualify for authorship, and all those who qualify for authorshipare listed. The experiments were performed at the humancentrifuge facility, QinetiQ, Farnborough. UK.

Funding

This work was funded by a BBSRC Industry Interchange Awardto J.P.R.S. and R.F.R. C.J.O. was funded by BBSRC grantBB/I00579X/1. C.A.V. was funded by A∗Midex (Aix-MarseilleInitiative of Excellence).

Acknowledgements

We thank the technical support staff at QinetiQ centrifugefacility.

Translational perspective

Physiological tremor has been studied for over a century but the mechanisms remain contentious.Two explanations are generally proffered. Firstly, that oscillations within the central nervous systemmanifest as movement, and secondly, that tremor is simply the consequence of the mechanicalproperties of the limb. Here we attempt to distinguish between these explanations by changing themechanical properties of the hand and observing the consequences upon tremor. In one case, weadded mass to the hand to increase its inertia. This reduced tremor frequency, but not quite asmuch as predicted by a purely resonant mechanical system. However, adding mass also increased therequired muscular effort to hold the hand against gravity, potentially increasing joint stiffness andthus resonant frequency. To account for this effect, we devised a situation whereby increased musculareffort was necessary, but without any change in inertia. This was achieved by using a human centrifugeto increase gravitational acceleration (g). We observed that, unlike added mass, higher g did indeedcause an increase in tremor frequency. By matching the g level to the amount of added mass in theprevious condition, we were able to effectively subtract the effect of muscular effort. When we didthis, the change in tremor frequency was entirely explicable on the basis of a mechanical resonantsystem. This suggests that physiological tremor primarily reflects the physical properties of the limbin question, rather than its neural input.

C© 2015 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society