Judging Time Intervals Using a Modelof Perceptuo-Motor Control

Madeleine A. Grealy1, Cathy M. Craig2, Christophe Bourdin2,and Simon G. Coleman3

Abstract

& Estimating a time interval and temporally coordinatingmovements in space are fundamental skills, but the relation-ships between these different forms of timing, and the neuralprocesses that they incur, are not well understood. Whiledifferent theories have been proposed to account for timeperception, time estimation, and the temporal patterns ofcoordination, there are no general mechanisms which unifythese various timing skills. This study considers whether amodel of perceptuo-motor timing, the tGUIDE, can alsodescribe how certain judgements of elapsed time are made.

To evaluate this, an equation for determining interval estimateswas derived from the tGUIDE model and tested in a task whereparticipants had to throw a ball and estimate when it would hitthe floor. The results showed that in accordance with themodel, very accurate judgements could be made withoutvision (mean timing error �19.24 msec), and the model was agood predictor of skilled participants’ estimate timing. It wasconcluded that since the tGUIDE principle provides temporalinformation in a generic form, it could be a unitary process thatlinks different forms of timing. &

INTRODUCTION

Sensing time is an essential survival skill. Without itmaking sense of the world would be impossible andinteracting with ever-changing environments hazardous.Timing is not a single entity though, and because timingabilities take many forms, it is not surprising that anumber of different theoretical accounts have beenproposed. For example, there are substantial bodies ofwork on time perception, time estimation, time discrim-ination, conditioning, circadian rhythms, and the tem-poral patterns of motor control. The theories andmodels that have emerged from this work differ notonly in the tasks they explain but also in scale, rangingfrom milliseconds to days. Each has implications for thetype of neural structures and mechanisms involved andthere are instances within the literature where views onthese matters differ radically, or even conflict.

Assessing the extent to which there might be com-monality in the neural basis for various timing abilities isnot an easy task because many of the models have beendeveloped and tested using specific paradigms. Thismakes it difficult to create experimental tests that coulddemonstrate generalization to other timing skills and,consequently, the question of whether there is a com-mon neural mechanism which operates across differenttime frames or a unitary process that accounts for

multiple skills, such as time estimation and the temporalcontrol of movement, has not been fully answered.However, there is one recent model that appears tohave the potential to apply to, and be tested on, a rangeof timing skills. It is called the tGUIDE (Lee, 1998, 2004),and the author’s claim for this model is that it provides acomplete description of the temporal control of allactions, and suggests a neural process that is bothfundamental to and common to all motor tasks. If thisis the case, then it would be interesting to see whetherthis model can be extended beyond the motor domainto other forms of timing. As an initial test, the currentstudy sets out to evaluate whether the tGUIDE model ofaction timing could provide an explanation of how thepassage of relatively short intervals are estimated. To dothis, a mathematical model, which describes how tem-poral information relating to an action could be used tomake an interval judgement, was derived from theoriginal tGUIDE equation. This gave rise to the predictionthat when asked to throw a ball in the air without theuse of vision, participants would be able to make anaccurate estimation of when the ball would hit theground. This was duly tested along with the extent towhich the model predicted performance.

In terms of the neural basis of timing, one of thecontroversies within the literature is the extent to whichneural ‘‘clocks’’ are under the control of central ordistributed processes. The idea of a central clock,proposed notably by Treisman (1963), is that there isa neural pacemaker which generates pulses that are

1University of Strathclyde, 2Universite de la Mediterranee,3University of Edinburgh

D 2004 Massachusetts Institute of Technology Journal of Cognitive Neuroscience 16:7, pp. 1185–1195

used for all timing tasks. This central timer has beenseen as independent of, and dissociable from, thetiming of a motor response (Wing & Kristofferson,1973), and implies the existence of neurons that areresponsible for generating pulses and a counter mech-anism which keeps track of the number of pulsesemitted. In contrast, more recent research on distribut-ed mechanisms sees timing as an emergent propertyarising from nonlinear oscillatory processes in both thebrain and the motor system. At a neural level, it hasbeen proposed that this could be achieved by differentgroups of neurons timing different intervals (Ivry &Richardson, 2002), or networks of neurons in whichtemporal processing is a property of a population asopposed to any single neuron (Buonomano & Karmar-kar, 2002).

Lesion and imaging studies point to various corticalnetworks distributed throughout the brain that areinvolved in perceptual or motor timing (or both), andwhile links between neural processes and overt behav-iors are increasingly being made (e.g., Meck & Benson,2002; Harrington & Haaland, 1999; Quintana & Fuster,1999), there are few instances where a close correspon-dence between behavioral models and neural mecha-nisms has been shown. For instance, Ivry and Richardson(2002) proposed a multiple timer model which de-scribes a mechanism that could be used to perceivetime intervals and regulate the onset of a movementfrom which the temporal properties of the action mightemerge, but they recognize that evidence at a structurallevel (such as chronotopic maps in the cerebellum orthe coordination of networks of temporally tuned neu-rons) is currently lacking.

In contrast to these approaches, the generalizedtGUIDE model suggests a very different kind of temporalmechanism for coordinating actions. It has parallels withthe theory of population vector coding for directionalinformation (Georgopoulos, Schwartz, & Kettner, 1986)in that just as the weighted directional sum of a neuralpopulation determines the direction of movement, thet pattern of firing rates determines the timing of theaction (Lee, in press). If this temporal information has ageneric form, regardless of the specifics of the action,then the merging of these two basic processes providesa way in which spatial and temporal activity in differentbrain regions can be multiplexed (Georgopoulos, 2002).Lee and colleagues have provided substantial evidenceto show that the tGUIDE model can account for thetemporal control of action at both the behavioral andneural levels (see Lee, in press for a review). The time-to-closure of a physical gap at its current closurerate (referred to as tX) forms the basic temporal mea-sure, and the model predicts that the neural code formovement production (the tGUIDE) corresponds to time-varying changes in neuronal firing patterns that acceler-ate from a rest level at a constant rate. Studying theactivity of a population of neurons in the motor and

parietal cortices, Lee and colleagues have shown a highlevel of correspondence between the model, the ‘‘neuralpower’’ (defined as the rate of firing at rest divided bythe change in firing rate), and the ensuing reachingaction. The model also provides a robust description ofthe temporal evolution of a large range of goal-directedactions, such as striking (Craig, Delay, Grealy, & Lee,2000), reaching (Lee, Craig, & Grealy, 1999; Lee, Geor-gopoulos, Clark, Craig, & Port, 2001), and sucking (Craig& Lee, 1999). However, if the tGUIDE model of motortiming is to account for how judgments of intervals oftime are made, then the temporal patterns that themodel generates should not only describe the kinemat-ics and strategy behind the action, but should alsocharacterize the temporal representation of other eventswhere there is motion of an object and/or an actor.Perceiving these temporal structures could provide away of predicting and synchronizing events.

The generalized tGUIDE model is based on Newtonianequations of motion and assumes a constant gravita-tional force, although not necessarily set at 1 g. Theequation describing the neural activity associated withan action is

tGUIDE ¼ ½ðtX ;0 þ kTÞt2� kT2t�tX ;0T

2�=½2tðtX ;0 þ kTÞ � kT2� ð1Þ

(see Appendix for derivation), where t is time runningfrom zero when the tGUIDE starts, tX,0 is the time-to-closure of a gap (X) when the tGUIDE is started (e.g.,the t of the gap between the ball and the ground atthe point of release), k is a constant, and T is theduration of the event that is guided. Using this modelto predict the total time taken for an event to occur,such as judging when a thrown ball would hit theground, would require the use of various sources ofinformation to form a temporal representation (T ) forthat interval estimate.

To evaluate this, an equation expressing T was de-rived from Equation 1 by noting that the velocity ofclosure of the gap between the ball and the floor was

�

X ¼ X0ð1=kÞfð1 þ ðkT=tX ;0ÞÞð2t=T2Þ � ðkT=tX ;0Þð1=TÞgfF½1 þ ðkT=tX ;0Þ�ðt=TÞ2 � ðkT=tX ;0Þðt=TÞ � 1�ð1=kÞ�1g

ð2Þ

and that this becomes zero when the ball reaches itszenith, that is

t ¼ T=f2½1 þ ðtX ;0=kTÞ�g ð3Þ

Rearranging Equation 3, the temporal estimate (T) canbe expressed as

T ¼ t Fffiffið

pt2 þ 2tðtX ;0=kÞÞ ð4Þ

Therefore, when parameters associated with the re-lease point of a vertically launched ball are fed into this

1186 Journal of Cognitive Neuroscience Volume 16, Number 7

equation, it should accurately describe the ball’s tem-poral trajectory, where k = 1 under normal gravitationalconditions.

To estimate the time interval T, the parameters thatthe thrower would need to perceive are the temporalgap between the ball and the floor (tball to floor at release)when the ball is launched, and the time taken for theball to reach its zenith. tball to floor at release could besensed directly by monitoring how the gap betweenthe ball (in the hand) and the feet (on the ground)changes during the throw, or derived by dividingthe distance between the ball–ground surface by itsinstantaneous velocity. The time-to-zenith would alsohave to be estimated but because this equals the veloc-ity of the hand and ball at release divided by g,sensing the constituent components of tX,0 (the ball’srelease velocity and the hand/ball to floor gap), andhaving an intuitive knowledge of gravity, should be suffi-cient information to estimate the flight duration T.

Figure 1 shows simulated flight trajectories derivedusing this model for different values of tX,0. If thismodel is correct, then sensing the hand’s positionrelative to the feet and the velocity of the arm move-ment at the point of ball release should give sufficientinformation to estimate the ball’s flight time. Becauseneither of these parameters are dependent on vision,one prediction of this model is that it should bepossible to estimate flight duration accurately withoutsight. To test this, participants were asked to throw aball with or without vision, and estimate when it wouldhit the ground. Data from the release point were usedto model the ball’s flight duration (T ) and these

modeled intervals were compared with the participants’estimates and actual ball flight times.

RESULTS

The tGUIDE model predicts that accurate estimates offlight duration should have been made without vision,as the information required for this task could be gainedproprioceptively. Figure 2 shows that participants weretemporally very accurate in both visual conditions andthere were no significant differences in mean errors[t(9) = 0.07, p = .47] or absolute errors [t(9) = 0.68,p = .26].

To see whether accuracy levels were consistent acrossthe range of flight times, correlation and regressionanalyses were carried out. A slight tendency for theparticipants to conform to Vierordt’s Law was found.This states that shorter events are overestimated andlonger events are underestimated. There were significantcorrelations between timing error and flight duration forboth the vision (r = �.16 p = .02) and the no-visionconditions (r = �.28 p = .01), but the regressionanalyses showed that these relationships only accountedfor a very small percentage of variance within the data(with vision r2 = .03, without vision r2 = .08). Furtherinspection of the timing errors showed that theyspanned from �102 to 72 msec (a range of 174 msec)and because ball flight times were in the range of331 msec, it would appear that the participants werenot performing stereotypical throws but were using astrategy that allowed them to predict flight times.

Figure 3 shows the relationships between the esti-mated, modeled and actual flight times for all partic-ipants combined. Without vision the model was foundto be a good predictor of estimated flight times in thatit accounted for 80% of the variance in the data(Figure 3B). Estimated errors in this prediction were±46.5 msec (with a 95% confidence level). Not surpris-ingly, given the high level of accuracy shown, the ball’sactual flight time was also a good predictor of theparticipants’ estimation of flight time (Figure 3D; r2 =.79, estimated error ±52.5 msec). The relationshipbetween the model and the actual ball flight times wassimilar, accounting for 79% of the variance, with errorsin prediction being estimated at ±48.0 msec (Figure 3F).

A comparable picture emerged when the participantscompleted the task with vision. The model predicted86% of the variance in estimated flight times, with theestimated error in this prediction being ±39.6 msec(Figure 3A). The relationship between the actual andestimated flight times explained 84% of the variance,with the estimated error being ±47.2 msec (Figure 3C).Finally, the relationship between the modeled andactual flight times was marginally weaker, accountingfor 79% of the variance in the data, with errors inprediction being estimated at ±52.7 msec (Figure 3E).Further analysis of the heteroskedastic nature of these

Figure 1. Modeled ball trajectories for different values of tX,0.

Where the amplitude of the gap (X ) = X0{±[[1 + (kT/tX,0)](t/T )2 �(kT/tX,0)(t/T ) � 1]}(1/k) and the initial gap size and overall duration(T ) are set to 1.

Grealy et al. 1187

relationships (Figure 4) showed no systematic bias.However, other factors, such as sequential effects ofhow a prior judgment might influence a current re-sponse or fatigue effects, were not considered.

Taking the velocity of the hand and the gap betweenthe hand and the floor at the point of release, the tGUIDE

model was able to predict the participants’ estimate offlight duration to within ±46.5 msec and the actualflight duration to within ±52.5 msec. This level ofaccuracy was similar in magnitude to that shown bythe participants whose overall absolute mean timingerror was 27.7 msec when performing without vision.This suggests that the precision level of the modelmatched that of the humans.

Although the participants were not using a stereo-typical action during the experiment, it should be notedthat they were well practiced at the task. It could bethat by only looking at highly skilled performances, thedegree to which the tGUIDE model truly reflects a basictemporal neural process is exaggerated. To assess this,a second experiment was carried out where a differentgroup of participants were asked to perform the sametask without prior practice or feedback being givenduring the trials. Additionally, to evaluate whether thefindings were specific to the use of a particular objector limb, participants were asked to throw two balls ofdiffering weights and use their preferred and nonpre-ferred hand (see Methods for details).

Performance without Practice or Feedback

While increases in timing errors were noted under theconditions of no practice or feedback, the participantsstill showed great temporal accuracy in both visual con-ditions. With vision the overall mean timing error in-creased from �19.47 msec with practice to �78.29 msecwithout, and from �19.24 to �40.32 msec in the no-vision condition. Surprisingly, when practice and feed-back were denied, the overall mean error withoutvision (�40.32 msec) was significantly smaller thanthat with vision (�78.29 msec) [t(6) = 2.71, p = .03,ES = 0.71].

To test for differences in temporal accuracy betweenthe visual conditions, the two ball weights and the handused to throw the ball, a three-way repeated-measuresANOVA was conducted on mean timing errors. Thisrevealed no significant differences in accuracy for per-formances with or without vision [(F(1,6) = 0.40,p = .55], or when participants used their preferred ornonpreferred hand [F(1,6) = 1.67, p = .24]. There wasa trend towards significance with the two ball weights[F(1,6) = 4.90, p = .07], with more accurate responsesbeing given when the heavier ball was used.

Regression analyses were carried out for each partic-ipant separately to assess the relationships betweenthe modeled, estimated, and actual flight times. The data(Table 1) clearly indicate that the model fitted the

Figure 2. Top panels show

distribution of errors when

throwing (A) with vision and

(B) without vision. Lowerpanels show (C) mean timing

errors and (D) absolute timing

errors for all participants under

the two visual conditions. Errorbars show standard deviations.

1188 Journal of Cognitive Neuroscience Volume 16, Number 7

behavior of some participants better than others, withthe percentage of variance in the estimation data themodel accounted for ranging from 2% to 74%. However,an analysis of the relationship between timing errors andthe predictive power of the model (Figure 5) revealedthat those participants who were poor at making esti-mate judgments under these conditions also showedthe model to be a poor predictor of their estimation.

DISCUSSION

Georgopoulos (2002) argues that the cognitive controlof action requires the merging of separate processes

that share common general principles for extractinginformation in space and executing tasks in time. ThetGUIDE model is offered as one such general principle.Until now this has been seen as a temporal measure thatdefines the evolving time course of an action at bothneural and behavioral levels. The results of this studysuggest that the same process could be used for judgingshort time intervals associated with the consequences ofan action.

With practiced performers, the regression equationsdescribing the relationship between the tGUIDE modeland estimates of flight duration accounted for 80% (novision) and 86% (vision) of the variance in the data.

Figure 3. Left panels show

the relationship between

(A) estimated and modeled

estimates, (C) estimated andactual flight durations, and

(E) modeled and actual flight

durations for performances

with vision. Right panelsshow performances without

vision. Data are from all

participants combined.

Grealy et al. 1189

The model also predicted that participants would beable to make very accurate estimates of flight timewithout the use of visual information. The overall meantiming error of �19.24 msec confirmed this. However,without the benefits of practice or feedback, the mod-el’s ability to predict estimated flight times revealedconsiderable variations between individuals, but these

differences were mirrored in the participants’ ability tomake accurate responses. Those whose temporal accu-racy was acute for making interval judgments showed agood correspondence between the model, their tem-poral estimates and actual flight times, and for thosewho were less accurate the strength of these relation-ships declined. However, consistent patterns in ac-

Figure 4. Plots of

standardized residuals againstthe standardized model for the

relationship between the

estimated versus modeleddurations (A and B), estimated

and actual durations (C and D),

and modeled and actual flight

times (E and F). Left panelsshow performances with vision

and the right panels without

vision. The lack of a linear trend

in residuals showsno heteroskedasticity.

1190 Journal of Cognitive Neuroscience Volume 16, Number 7

curacy were seen when using the dominant and non-dominant hand and the two different ball weights,suggesting that the results reflect a general ability forperformance of this skill. The fact that some of theparticipants were better than others at this task, whichis not one which is performed routinely in everydaylife, could reflect the sensitivity of this task to revealdifferences in the ability to use perceptual informationin making timing judgments. This was not confined tothe use of proprioceptive information since when theywere allowed to see the upper part of the ball’s flight(including when it reached its zenith) the participants’accuracy did not improve—in fact the overall meantiming error increased.

An alternative explanation for the diversity of resultscould come from differences in our intuitive knowledgeof gravitational forces. An assumption of the model isthat we can accurately sense and take into accountgravity. Although this seems reasonable given that gravityis the fundamental characteristic that the earth’s envi-ronment affords to all organisms, it might not be thecase. The effect that changing gravity has on the control

of movements has been well documented (Fisk, Lackner,& Dizio, 1993; Bock, Howard, Money, & Arnold, 1992),and considerable adaptation of the neuromuscular sys-tem has been shown which suggests that we do have anintuitive knowledge of gravitational forces (Avela, Santos,Kyrolainen, & Komi, 1994). The ability to visually per-ceive gravitational accelerations has also been demon-strated by Stappers and Waller (1993) and Watson,Banks, von Hofsten, and Royden (1992), who arguedthat gravity could be used as a monocular cue for theperception of absolute distance or size. However, oursensitivity to this has been questioned by Hecht, Kaiser,and Banks (1996), who showed that observers weremore sensitive to the average velocity than to the grav-itational acceleration pattern of a simulated ball throw.

McIntyre, Zago, Berthoz, and Lacquaniti (2001) havealso suggested that a knowledge of gravity is used tomake temporal judgments. They studied behaviors inaltered gravitational states and were able to show howperceptual estimates of distance, velocity, and the accel-eration of an object, along with a knowledge of gravity,could be used in an internal model to make temporal

Table 1. Summaries of Individual Regression Analyses for the Relationships between Modeled and Estimated Flight Durations

No Vision Vision Dominant Hand Nondominant Hand

Participant r2 SEE r2 SEE r2 SEE r2 SEE

2 .74 55 .46 37 .62 77 .68 75

1 .69 55 .47 42 .72 72 .57 73

5 .41 32 .36 51 .47 72 .44 73

4 .34 59 .33 40 .36 77 .39 74

3 .25 43 .32 72 .38 73 .39 74

7 .22 69 .26 47 .22 83 .32 80

6 .02 44 .11 51 .03 75 .23 80

Figure 5. Relationships

between the percentage of

variance accounted for by the

regression equation relatingmodeled and estimated

flight times as a function of

timing accuracy.

Grealy et al. 1191

judgments for the time of arrival of a dropped ball.However, if the ball is initially thrown upwards and vi-sual information is not available, as in the present study,then using this model to make temporal estimateswould be quite demanding. It would first require makingan approximation of the distance the ball travels fromrelease to its zenith (which in turn requires an estimateof the time taken for the ball to get there), and thenusing this to calculate the overall flight time. An internaltiming system for judging when this interval has elapsedwould also be required to allow for an appropriateresponse to be made. Additionally, while this modelcan account for how some temporal estimates are made,it cannot account for the temporal control of actions.

In contrast to the work which suggests we have anintuitive knowledge of gravity there are reports that ourconceptual and perceptual knowledge of projectile mo-tion is sometimes quite poor. Hecht and Bertamini(2000) reported poor judgments of the maximal velocityof a thrown ball. Half of their adult sample (n = 176)erroneously believed that the ball would continue toaccelerate after it had left the hand, and 31% thought,again erroneously, that the ball would speed up thenslow down. Furthermore, Krist, Fieberg, and Wilkening(1993) showed a dissociation between perceptuo-motorand judgmental abilities in children aged 5–6 years.While their actions showed an understanding of gravita-tional forces, their ratings of speed were poor comparedwith older children and adults. Therefore, it is possiblethat a conceptual misunderstanding of gravity couldexplain why some of the participants performed poorlywhen they were not given the opportunity to learn howto perform the task. However, in the visual condition ofthe second experiment, they could track the ball for atleast half of its trajectory and this did not improveperformance, which suggests that a more likely expla-nation is that some of the participants simply had a poorsense of timing.

Apart from an understanding of gravity, a furtherrequirement of the tGUIDE model is its use of intervalsof time, notably the parameter t in Equation 1. Lee andcolleagues (see Lee, in press) have shown how thismechanism could be represented by the ‘‘neural power’’of neuronal firing rates, a solution which negates theneed for a centralized clock or specialized interval units.They argue that if the pattern of neural firing rates take atGUIDE form, this in itself provides a neural chronometerfor each action or imagined event. Modulations in firingrates that could occur through priming, or changingsynaptic thresholds, suggest a means by which ‘‘internaltiming’’ (the parameter t) could be modified. This high-lights the disparity between this model and models suchas Wing and Kristoffersen’s in that the tGUIDE does notrely on a central pacemaker that provides regular beatsnor is it dissociable from the timing of a motor response,rather it links neural and behavioral events through acommon process.

Nevertheless, many questions about the tGUIDE modelremain. For example, there is the issue of how simplicityat one level can be integrated with complexity atanother. Kerr and Worringham (2002) discuss the com-plexity of perceiving the hand’s velocity from proprio-ceptors that are distributed within muscles, tendons,joints, and the skin, and it is not clear how this infor-mation could be amalgamated and used in a fashionsimilar to that described by Equation 4. Considerationalso needs to be given as to how, within this framework,information from different sensory systems is temporallyintegrated and how actions are synchronized to eventsso as to give a perception of simultaneity (Aschersleben& Prinz, 1995; Mates, Muller, Radil, & Poppel, 1994;Poppel, Schill, & von Steinbuchel, 1990). Additionally, ifthis model is an accurate descriptor of an estimatetiming process, then two other interesting findings thatemerged from this study also need to be considered.The first is that the participants’ estimates did not im-prove with vision, that is, having a visual cue as to whento stop ‘‘timing’’ did not improve performance. Thiswould imply that the use of proprioceptive informationgained at the point of release allowed participants tomake extremely accurate judgments about the conse-quences of the throwing action. However, the extent towhich the model’s utility is dependent on propriocep-tive information or a sense of agency needs to beaddressed.

The second is that the degree of accuracy remainedfairly constant over the range of time intervals studied(see Figure 4C and D), whereas it might be expectedthat the variance in estimates would increase as thejudgement length increased. Although it could be ar-gued that the range of intervals studied was relativelysmall, if this is found to be the case over a larger rangethen this would suggest that the process associated withmaking these judgments is dissimilar to the pacemakeror accumulator mechanisms that have previously beenproposed (e.g., Gibbon, Malapani, Dale, & Gallistel,1997). If a lack of scalar variance is found, then thereare two ways in which this model could operate. Partic-ipants could either use information gained at the pointof release to define the initial parameters for flight andthen mentally track the trajectory visualizing the ball’stemporal path following a tGUIDE like form, or theycould calculate the flight duration using equations ofmotion or Equation 4. From these data, it is not possibleto say which of these methods might have to be used.One speculation though is that being able to imaginethe motion of an object during the interval to beestimated, that is, having an internal representation ofthe ball’s temporal trajectory, could reduce the likeli-hood of systematic errors such as scalar variance. How-ever, although further work is needed to address theseissues, it would seem that in trying to answer thequestion of whether there is a single mechanism whichforms a unitary process for different timing abilities,

1192 Journal of Cognitive Neuroscience Volume 16, Number 7

the tGUIDE model (or a variant of it) could help providethe solution.

METHODS

Experiment 1

Participants

Nine women and one man (mean age 26.2 years, SD =4.5 years) participated. All gave informed consent andlocal ethical approval was granted in line with theDeclaration of Helsinki.

Procedure

Participants stood in front of two pressure-sensitivemats and were instructed to throw a lacrosse ballupwards with their dominant hand so that it landedon one mat. Their throwing arm was splinted so thatthere was no flexion at the wrist and a cup with apressure sensitive switch was attached to the palm oftheir hand. The ball was placed in this cup and whenit was thrown the switch was released starting twomillisecond timers. One timer was stopped when theball hit the mat, while the other was stopped by theparticipant tapping the other mat with a stick whichthey held in their nondominant hand. Participantswere asked to use the stick to indicate when theythought the ball would hit the ground. They were toldto hold it in a stationary position above the mat duringthe throwing action and to make a single downwardmovement so that it struck the mat at the same timeas when the ball hit the other mat. Participants wereallowed to practice until they felt they were makingaccurate judgments and then performed 12 trials withvision and 12 trials wearing a blindfold, presented in arandom order.

A Flock of Birds motion analysis system (AscensionTechnology, North America) was used to record thethrowing action and the movement of the stick. Onthe throwing arm sensors were placed on the shoulderand elbow joints and on the underside of the handbelow the switch. A fourth sensor was placed on thetop of the stick that was held in the nondominanthand. Trials were recorded at 120 Hz, and digitallyfiltered using a gaussian filter with a cutoff of 8 Hz.

The accuracy of the participant’s estimate was calcu-lated by noting the difference between the two milli-second timers. Mean (signed) and absolute (nonsigned)errors were calculated for each condition. The param-eter t( ball to floor at release), which was required to modelthe flight time T (Equation 4), was gained from kine-matic parameters at the time of ball release. Using theball’s flight time as recorded by the millisecond timer,and noting the difference between the actual andestimated times, the time of release was calculated.

The distance between the hand and the floor, and thevelocity of the arm (calculated using finite differences)were then noted and used to model T.

Experiment 2

Participants

Five men and two women (mean age 26 years) who hadnot taken part in Experiment 1 participated. All gaveinformed consent and local ethical approval was grantedin line with the Declaration of Helsinki.

Procedure

Participants were instructed to throw a ball in the airand predict as accurately as possible when it would hita foam platform on the floor (1 � 0.5 m) beside them.They did this with vision and without vision, with theirpreferred and nonpreferred hand and with a heavy(400 g) and a light (150 g) ball. In the ‘‘with vision’’condition, a horizontal screen was placed around theparticipant’s neck so that they could see the ballarriving at its zenith and starting its descent to thelanding pit. In the ‘‘no vision’’ condition, participantswore a blindfold.

The 15 trials in each condition were randomized,giving a total of 120 per participant. To eliminatelearning effects from auditory feedback, participantswore a headset through which white noise was played.The ball’s flight time and the kinematics of the armmovement were measured using an Elite passive mo-tion capture system (BTS, North America) sampling at100 Hz. Markers were placed on the elbow and wristjoints and both balls were completely covered inreflective tape. Participants’ predictions about the ball’sarrival time were made using a switch held in theirnonthrowing hand, and this was sampled at 500 Hz viathe Elite system. The gap between the ball in the handand the floor, and the velocity of this gap derivedusing finite differences, were used to model the flightduration T (Equation 4).

APPENDIX: DERIVATION OF THE FORMULAOF THE tGUIDE

At any point during its closure a physical a gap (X) canbe described as

X ¼ ðutþ 0:5at2Þ � ðuT þ 0:5aT2Þ ðA1Þ

where u is the initial velocity, a is acceleration, t is timestarting from zero, and T is the total duration for theclosure. Because the velocity of closure at time t

�

X ¼ uþ at ðA2Þ

Grealy et al. 1193

the time to closure, or tGUIDE is

tGUIDE ¼ ½ðutþ 0:5at2Þ � ðuT þ 0:5aT2Þ�=½uþ at� ðA3Þ

Dividing numerator and denominator by u, andsimplifying:

tGUIDE ¼ ½ðt� TÞ þ 0:5ða=uÞðt2 � T2Þ�=½1 þ ða=uÞt�ðA4Þ

Thus when t = 0,

tGUIDE;0 ¼ ½�T � 0:5ða=uÞT2� ðA5Þ

Because tGUIDE,0 = (1/k)tX,0 (see Lee, 1998 for details),substituting (1/k)tX,0 for tGUIDE,0 in Equation A5 andsolving for a/u

a=u ¼ �2½tX ;0 þ kT�=½kT2� ðA6Þ

Substituting for a/u from Equation A6 in Equation A4,

tGUIDE ¼ ½kðt� TÞ þ 0:5f�2½tX ;0 þ kT�=T2gðt2 � T2Þ�=½kþ f�2½tX ;0 þ kT�=T2gt�

Simplifying

tGUIDE ¼ ½ðtX ;0 þ kTÞt2� kT2t�tX ;0T

2�=½2tðtX ;0 þ kTÞ � kT2� ðA8Þ

Acknowledgments

We thank Professor David Lee for his help and guidance.

Reprint requests should be sent to Madeleine A. Grealy,Department of Psychology, University of Strathclyde, GrahamHills Building, 40 George Street, Glasgow G1 1QE, Scotland,UK, or via e-mail: m.grealy@strath.ac.uk.

REFERENCES

Aschersleben, G., & Prinz, W. (1995). Synchronizing actionswith events—The role of sensory information. Perceptionand Psychophysics, 57, 305–317.

Avela, J., Santos, P. M., Kyrolainen, H., & Komi,P. V. (1994). Effects of different simulated gravityconditions on neuromuscular control in drop exercises.Aviation Space and Environmental Medicine, 65,301–308.

Bock, O., Howard, I. P., Money, K. E., & Arnold, K. E.(1992). Accuracy of aimed arm movements in changedgravity. Aviation Space and Environmental Medicine,63, 994–998.

Buonomano, D. V., & Karmarkar, U. R. (2002). How do wetell time? Neuroscientist, 8, 42–51.

Craig, C. M., Delay, D., Grealy, M. A., & Lee, D. N. (2000).Guiding the swing in golf putting. Nature, 405,295–296.

Craig, C. M., & Lee, D. N. (1999). Neonatal control of nutritive

sucking pressure: Evidence for an intrinsic tau-guide.Experimental Brain Research, 124, 371–382.

Fisk, J., Lackner, J. R., & Dizio, P. (1993). Gravitationalforce level influences arm movement. Journal ofNeurophysiology, 69, 504–511.

Georgopoulos, A. P. (2002). Cognitive motor control: Spatialand temporal aspects. Current Opinion in Neurobiology,12, 678–683.

Georgopoulos, A. P., Schwartz, A. B., & Kettner, R. E. (1986).Neuronal population coding of movement direction.Science, 233, 1416–1419.

Gibbon, J., Malapani, C., Dale, C. L., & Gallistel, C. R. (1997).Toward a neurobiology of temporal cognition: Advancesand challenges. Current Opinion in Neurobiology, 7,170–184.

Harrington, D. L., & Haaland, K. Y. (1999). Neuralunderpinnings of temporal processing: A review of focallesion, pharmacological, and functional imaging research.Reviews in the Neurosciences, 10, 91–116.

Hecht, H., & Bertamini, M. (2000). Understanding projectileacceleration. Journal of Experimental Psychology: HumanPerception and Performance, 26, 730–746.

Hecht, H., Kaiser, M. K., & Banks, M. S. (1996).Gravitational acceleration as a cue for absolute size anddistance. Perception and Psychophysics, 58,1066–1075.

Ivry, R. B., & Richardson, T. C. (2002). Temporal controland coordination: The multiple timer model. Brain andCognition, 48, 117–132.

Kerr, G. K., & Worringham, C. J. (2002). Velocity perceptionand proprioception. Sensorimotor Control of Movementand Posture, 508, 79–86.

Krist, H., Fieberg, E. L., & Wilkening, F. (1993). Intuitivephysics in action and judgment: The development ofknowledge about projectile motion. Journal ofExperimental Psychology: Learning, Memory andCognition, 19, 952–966.

Lee, D. N. (1998). Guiding movement by coupling ts.Ecological Psychology, 10, 221–250.

Lee, D. N. (in press). Tau in action in development. In J. J.Rieser, J. J. Lockman, & C. A. Nelson (Eds.), The roleof action in learning and development. Hillsdale, NJ:Erlbaum.

Lee, D. N., Craig, C. M., & Grealy, M. A. (1999). Sensoryand intrinsic coordination of movement. Proceedings ofthe Royal Society of London Series B—Biological Sciences,266, 2029–2035.

Lee, D. N., Georgopoulos, A. P., Clark, M. J. O., Craig, C. M.,& Port, N. L. (2001). Guiding contact by coupling thetaus of gaps. Experimental Brain Research, 139,151–159.

Mates, J., Muller, U., Radil, T., & Poppel, E. (1994). Temporalintegration in sensorimotor synchronization. Journal ofCognitive Neuroscience, 6, 332–340.

McIntyre, J., Zago, M., Berthoz, A., & Lacquaniti, F. (2001).Does the brain model Newton’s laws? Nature Neuroscience,4, 693–694.

Meck, W. H., & Benson, A. M. (2002). Dissecting thebrain’s internal clock: How frontal–striatal circuitry keepstime and shifts attention. Brain and Cognition, 48,195–211.

Poppel, E., Schill, K., & von Steinbuchel, N. (1990).Sensory integration within temporally neutral systemsstates: A hypothesis. Naturwissenschaften, 77, 89–91.

Quintana, J., & Fuster, J. M. (1999). From perception toaction: Temporal integrative functions of prefrontaland parietal neurons. Cerebral Cortex, 9,213–221.

ðA7Þ

1194 Journal of Cognitive Neuroscience Volume 16, Number 7

Stappers, P. J., & Waller, P. E. (1993). Using free fall of objectsunder gravity for visual depth estimation. Bulletin of thePsychonomic Society, 31, 125–127.

Treisman, M. (1963). Temporal discrimination and theindifference interval: Implications for a model of theinternal clock. Psychological Monographs, 77, 13, WholeNo. 576.

Watson, J. S., Banks, M. S., von Hofsten, C., & Royden, C. S.(1992). Gravity as a monocular cue for perception ofabsolute distance and/or absolute size. Perception, 21,69–76.

Wing, A. M., & Kristofferson, A. B. (1973). Response delays andthe timing of discrete motor responses. Perception andPsychophysics, 14, 5–12.

Grealy et al. 1195