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Distinct timing mechanisms are implicated in distinct circle drawing tasks
Neuroscience Letters 472 (2010) 24–28
Contents lists available at ScienceDirect
Neuroscience Letters journal homepage: www.elsevier.com/locate/neulet
Distinct timing mechanisms are implicated in distinct circle drawing tasks Raoul Huys a,∗ , Breanna E. Studenka b , Howard N. Zelaznik c , Viktor K. Jirsa a,d a
Theoretical Neuroscience Group, UMR 6152 Institut des Sciences du Mouvement, CNRS & Université de la Méditerranée, 163 av. de Luminy, CP910, F-13288 Marseille, France Department of Kinesiology, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S4K1 c Health and Kinesiology, Purdue University, 800 W. Stadium Blvd., West Lafayette, IN 47907, USA d Center for Complex Systems & Brain Sciences, Physics Department, Florida Atlantic University, Boca Raton, FL 33431, USA b
a r t i c l e
i n f o
Article history: Received 7 October 2009 Received in revised form 22 January 2010 Accepted 22 January 2010 Keywords: Timing Dynamics Circle drawing Saddle-node on invariant circle bifurcation
a b s t r a c t That individual timing variability is correlated across some tasks but not others has instigated the notion of distinct timing processes, referred to as ‘event’ timing and ‘emergent’ timing for tasks with and without salient events, respectively. The delineation of the event-emergent framework owes much to the circle drawing task as it can be performed with or without such events, all other factors being equal. We investigated continuous and intermittent circle drawing from a principled perspective allowing for the classification of timing mechanisms based on mathematical theorems. We propose a one-dimensional dynamical model to portray the essential dynamics of circle drawing that exhibits either fixed point or oscillator dynamics, each associated with distinct timing mechanisms. Eight participants drew circles under three different instruction conditions (no specific instructions, to draw as a fast or as smoothly as possible) at seven frequencies ranging from 0.5 Hz to 3.5 Hz. We computed the angle between the circle drawing’s horizontal and vertical component and reconstructed the corresponding vector fields prescribing the temporal evolution. Fixed point dynamics were present only in the ‘fast’ condition at low movement frequencies (i.e., in intermittent circle drawing representative of event timing). All other conditions were realized via oscillator dynamics representative of emergent timing. The transition between both dynamical mechanisms involved a saddle-node on invariant circle bifurcation, which was accompanied by increased trajectory variability, which is a key signature of phase transitions. These findings demonstrate the involvement of distinct timing mechanisms in different circle drawing tasks. © 2010 Elsevier Ireland Ltd. All rights reserved.
Whether timing draws upon a ubiquitous task-independent mechanism has become a central question in the neurosciences [2,6,14]. Recently, Zelaznik and colleagues [1,21] had participants perform several motor timing tasks—finger tapping, continuous line and circle drawing, and intermittent circle drawing. They reasoned that if tasks share the same timing mechanism, then individual differences in within-trial timing variability should be correlated across tasks. Correlation analysis dissociated timing in tasks involving salient ‘discrete’ events (tapping and intermittent circle drawing) and ‘continuous’ tasks (continuous circle and line drawing), referred to as ‘event’ and ‘emergent’ timing, respectively [9]. Importantly, the physical space and musculature involved in continuous and intermittent circle drawing are identical, ruling out alternative interpretations along those lines. Further evidence for this framework was provided by a study using participants with cerebellar lesions, which were associated with increased timing variability in ‘event’ timing tasks but not in ‘emergent’ timing tasks [17].
∗ Corresponding author. Tel.: +33 49 11 72 201; fax: +33 49 11 72 252. E-mail address: [email protected] (R. Huys). 0304-3940/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.neulet.2010.01.047
In contrast to relying on empirical findings to posit different types of timing, dynamical systems theory offers an analytical basis to a priori postulate distinct movement classes associated with or without timing mechanisms based on the concept of phase flows [6,10]. A phase space is the space spanned by a system’s state variables.1 The flow therein unambiguously describes a system’s evolution, and its topology allows for unambiguous system classification [19]. In two-dimensional systems, the types of topological structures are limited to fixed points, separatrices (structures that locally divides the phase space in regions with opposing flow), and limit cycles. Stable fixed points and limit cycles are associated with discrete and (continuous) rhythmic movements, respectively. We recently examined the implication of motor control mechanisms associated with fixed point and limit cycle dynamics in an experiment where participants executed repetitive index finger flexion/extension movements (tapping without surface contact) at paces from 0.5 Hz to 3.5 Hz [6]. The participants were either
1 In line with the literature, we here adopt the assumption that movements along a one-dimensional physical direction can be sufficiently represented by a two-dimensional phase space, which in that case is spanned by position and velocity.
R. Huys et al. / Neuroscience Letters 472 (2010) 24–28
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Fig. 1. 3D projections of 4D circle drawing. Single participant data in the ‘fast’ condition at 0.5 Hz (left panel) and 3.5 Hz (right panel), respectively. Circular black trajectories represent the projections of the 4D trajectory into the 3D subspace spanned by x, y, and dx/dt; grey trajectories represent their projection on the xy-plane. The straight black lines in the xy-plane schematically represent the angle between the x- and y-components (see text).
instructed to produce a maximal flexion coincident with the metronome tone (the ‘natural’ condition), or to execute the movements as smoothly as possible (the ‘smooth’ condition) or as fast as possible (the ‘fast’ condition), and in all cases coincide maximum flexion with the metronome. We identified the phase flow via vector field reconstruction and found clear evidence for fixed point dynamics at low frequencies (up to approximately 2.0 Hz) for all participants in the ‘fast’ condition and for some in the ‘natural’ condition. At higher frequencies, the participants, irrespective of the instructions, utilized limit cycle dynamics. In other words, we found evidence in favour of the utilization of two distinct motor control and timing mechanisms (see also below) whose implementation is dictated by movement frequency and to some extent instruction. Through topological phase space analysis we examined the implication of different timing mechanisms in intermittent and continuous circle drawing, tasks that have played a crucial role in developing the notions of ‘event’ and ‘emergent’ timing. We predict the implication of fixed point dynamics in intermittent (slow pace) circle drawing but not at higher paces in which the drawing becomes continuous. Eight participants (mean age = 27.9 years) took part in the experiment (seven right-handed, one left-handed). The Purdue University Committee on the Usage of Human Research Participants approved the protocol, which was in agreement with the Declaration of Helsinki. All participants provided informed consent. Data were collected using a Polhemus Liberty-8 receiver (23 mm × 13 mm × 11 mm, 27 g), which was attached to the participant’s index finger with adhesive tape. The receiver was controlled by Matlab using an AuSIM-AuTrakMatlab USB driver and collection interface via library C++ calls. Displacement data were collected at 240 Hz. Participants were seated at a 79-cm high table. A white sheet of paper with a printed circle (diameter of 7.5 cm) was placed in front of them. At the ‘top’ of the circumference (12 o’clock; the point farthest away from the participant) a 1 cm diameter target circle was printed. The participants, moving counter-clockwise, attempted to produce one circle per beat of the metronome, with fixed wrist, moving with the elbow and shoulder only, and to coincide the passing of their finger tip over the “target” circle on the metronome beat. Timing accuracy, rather than spatial accuracy, was emphasized in circumscribing the circle. The participants were instructed to perform the circle drawing task without touching the table surface in order to avoid feedback from friction. The circle drawing was performed under three instruction conditions (see also [6]). In the ‘natural’ condition, the participants were instructed to pass over the target cycle at the moment of the metronome tone. In the ‘smooth’ condition, they were furthermore
instructed to move as smooth as possible. In the ‘fast’ condition, they were instructed to execute the movement in between metronome tones as fast as possible (i.e., staccato-like) while maintaining the requirement of arriving with their finger at the target circle at the moment the metronome tone sounded, which at the lower movement frequencies typically resulted in a short pause over the target circle. Thus, the ‘smooth’ and ‘fast’ conditions were constructed to force the participants to adopt a particular ‘mode of operation’ (associated with fixed point or limit cycle dynamics; see below). The participants performed 10 trials in each of the three instruction conditions on three different days. In five of each set of 10 trials, the metronome pace increased from 0.5 Hz to 3.5 Hz in steps of 0.5 Hz; on the other five trials, the pace decreased from 3.5 Hz to 0.5 Hz with equal steps. Fifteen metronome tones were presented at each frequency plateau. The order of the increasing (decreasing) set of trials was performed in a blocked design. All participants performed the ‘natural’ condition on day 1. The order of the ‘smooth’ and ‘fast’ conditions was balanced for all subjects. Movements along a single spatial dimension, such as tapping, are commonly portrayed as a two-dimensional dynamical system with position and velocity as state variables [11]. In circle drawing, two spatial dimensions exist, rendering its dynamical description four-dimensional. Rather than investigating the four-dimensional system we developed a reduced description of circle drawing that more easily allowed us to focus on its underlying phase space topology. Thereto, we concentrated on the dynamics of the angle between the (constant) vector pointing from the origin to the drawing’s maximal excursion and the vector pointing from origin to the drawing’s evolving horizontal (x) and vertical (y) component (i.e., [x(t), y(t)]). These vectors are defined on the drawing’s projection onto the xy-plane as illustrated in Fig. 1. The dynamics of is given by ˙ = f (), where the dot notation signifies the time derivative. The minimal model for which ˙ may exhibit either zero, one or two fixed points is ˙ = ω − cos(), where ω equals angular frequency. The amplitude of the cos() term may well scale with ω; we here set this amplitude scaling to 1/ω in view of the general tendency of movements to become more harmonic as their frequency increases. (This scaling does not impact the system’s qualitative behaviour as long as the corresponding parameter does not equal 0.) Fig. 2 portrays the system’s (saddle-node) bifurcation diagram (see also [19]). As the saddle-node bifurcation occurs on an attractive invariant circle, it represents a saddlenode-on-invariant-circle bifurcation (SNIC). Notice that for ω < 1, will settle at the stable fixed point, and a movement will be executed only if an external input I brings the system across the unstable fixed point (which acts as a separatrix). The full system,
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R. Huys et al. / Neuroscience Letters 472 (2010) 24–28
subsequently calculated based on P( ,t + t|, t) according to D
Fig. 2. Saddle-node on invariant circle bifurcation diagram for ˙ = ω − (1/ω)cos(). Each black line represents ˙ as a function of and ω. Fixed points are identified ˙ through the crossings of -curves with the (grey) ˙ = 0 plane. Stable and unstable fixed points are indicated in the − ω-plane as a black and grey line, respectively. For ˙ ω < 1, the -curve crosses the ˙ = 0 plane twice. As frequency ω increases, stable and unstable fixed points collide at ω = 1; the remaining fixed point vanishes for ω > 1.
˙ = ω − (1/ω)cos() + I, thus belongs to the class of excitable systems [15]. We reconstructed the circle drawing’s vector field on a line by extracting the deterministic component ˙ (i.e., the angle ’s time derivative). Human motion is intrinsically stochastic; it contains a deterministic and a stochastic (i.e., noise) dynamical component [5] that can be disentangled by computing the motion’s conditional probability distribution [4,20]. Importantly, extracting the deterministic component allows one to reconstruct vector fields, which allows us to establish the link between behavioural data and concepts from dynamical systems theory. The latter is not possible by ˙ We here focus on the deterministic dynamics. Thereto analysing . all trials were filtered with a low-pass 4th order Butterworth filter (cut-off frequency: 20 Hz). For each trial, the drawing movements’ horizontal (x) and vertical component (y) were mean subtracted and divided by their maximal value. Next, the continuous angle () between both components was computed as = arctangent (y/x). Each trial’s first three drawn circles, at each frequency, were omitted from analysis. The remaining twelve circles were divided into segments from − to . For each segment, the two-dimensional conditional probability matrix P( ,t + t|,t), which denotes the probability to find the system at state at a time t + t given its state at an earlier time step t, was computed using a bin size of 2/71. For each participant and each condition, the average conditional probability matrix across all segments of all trials was computed. Each condition’s so-called drift coefficients were
(n)
1 (, t) = lim t→0 t
n
( − ) P( , t + t|, t) d n!
Presently, the first drift coefficients D1 reveal the system’s deterministic dynamics, which we use to recover the vector field on the line in − ˙ space. We examined ’s trajectory variability of model simulations and human data (as a function of ω and movement frequency, respectively) using principal component analysis (PCA [3,7]). Trials and simulations were segmented as above (from − to ), resampled to the median length of all segments (N = 380), and for each participant, instruction and frequency conditions (or value of ω for simulations) were organized into a state vector q(t) that was subjected to PCA. Fig. 1 illustrates that in the fast condition at 0.5 Hz each drawn circle is followed by a period in which slowly converges to an approximately stationary state or moves around it with little amplitude. Thus intermittent circle drawing shows up as the area with locally reversing and intersecting trajectories, suggesting the existence of a fixed point. This suggestion was corroborated by the extracted drift coefficients (D1 ) shown in Fig. 3 for all three conditions as a function of movement frequency (compare with Fig. 2). In the fast condition, the line in − D1 space reveals a marked local minimum that almost crosses the D1 = 0 line at low frequencies (0.5 Hz and 1.0 Hz). For all participants, conditions, and trials, we examined whether the absence of line crossing could have been due to averaging and small drift in the fixed point’s position. This appeared to be so: for the fast condition at 0.5 Hz, the trial average minimal D1 -value was smaller than zero for each participant, and for 10 out of 16 cases (two ramps per participant) for 1.0 Hz, zero-crossings were observed several times per trial for each participant. The depth of the local minimum decreased and vanished with increasing frequency. For the natural and smooth conditions, the local minimum was hardly present and the line was never crossed (except for one participant in the natural condition at 0.5 Hz). In all conditions, the D1 -line became approximately straight at about 2.0 Hz, indicating harmonic circle drawing. The model predicts lower system stability the closer it is to the bifurcation [19], which should reveal itself in increased trajectory variability. PCA of the human data confirmed this prediction partly; the first eigenvalue (1 ) increased (i.e., trajectory variability decreased) monotonically with increasing movement frequency in the limit cycle regime (Fig. 4), and revealed a local minimum at or around the bifurcation point. These results were statistically significant: an ANOVA of the first eigenvalue revealed a main effect for condition (F(2,14) = 18.320, P < 0.001) and an Instruction × Movement frequency interaction (F(12,84) = 10.627,
Fig. 3. Extracted drift coefficients D1 as a function of instruction and movement frequency. The left, middle, and right panels display the D1 coefficients for the natural, fast, and smooth conditions, respectively, as a function of movement frequency ω and angle (averaged across trials, ramps [increasing and decreasing], and participants). Each line represents one movement frequency condition. On average, D1 (almost) intersected the D1 = 0 line in the fast condition at low frequencies. In all conditions, D1 increased with increasing movement frequency and the D1 line became approximately straight.
R. Huys et al. / Neuroscience Letters 472 (2010) 24–28
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Fig. 4. First principal component’s eigenvalue 1 for human data (left panel) and simulations (right panel) as a function of movement frequency and ω, respectively. The model was simulated at various ω’s and three different noise levels (increasing from the light grey to black line; right panel). 1 only depends on movement frequency in the fast condition; the local minimum around the bifurcation point is mimicked in the simulations.
P < 0.001). Newman–Keuls post hoc tests (˛ = 0.05) did not identify significant differences for the natural and smooth condition between frequencies whereas the fast condition’s local minimum differed significantly from its neighbouring points. Trajectory variability of the model simulations (Fig. 4) mimicked that of the human data (fast condition). The (human data’s) deviation from an approximately symmetrical pattern around the bifurcation point is likely largely due to the longer duration spent at the fixed point for slower movements. This interpretation is supported by a PCA on model simulations under constant ω = 0.7 but varying between stimuli (I; see the model above) durations showed that 1 increased with increasing inter-stimulus duration (Spearman correlation r = .97, p < .01). In the present study, participants drew circles at different frequencies and under distinct instructions. We constructed the corresponding vector fields, which allows for the linkage between behavioural data and mathematical concepts. We show that qualitatively distinct dynamics are associated with circle drawing depending on instruction and movement frequency: fixed point dynamics were observed under instructions leading to intermittent circle drawing (i.e., the ‘fast’, staccato-like condition at low frequencies), while limit cycle dynamics were observed under instructions not specifying strategy or demanding smooth performance. At higher frequencies there was no evidence for fixed point dynamics regardless the instruction, and the approximately straight line of the drift coefficients in − D1 space indicates that the circles were drawn almost harmonically. That is, we found evidence that qualitatively distinct dynamics are associated with intermittent and continuous circle drawing that have previously been associated with ‘event’ and ‘emergent’ timing [9,17,21]. The transition from fixed point dynamics to oscillator dynamics occurred via a SNIC bifurcation, in which the fixed points coalesce and a limit cycle is formed. A characteristic thereof is that in a small parameter regime the fixed point, while no longer existing, is still felt. In the presence of such a so-called ‘ghost’ (see Fig. 3, middle panel), dynamical systems theory predicts that trajectories locally slow down [20]. The existence of the ghost further testifies to the notion that a single model, ˙ = ω − (1/ω)cos() + I, governs intermittent as well as continuous circle drawing even though the corresponding topologies and flow patterns are frequency dependent. Moreover, the increased trajectory variability around the bifurcation point, indicative of phase transitions in nonlinear systems [5], suggests that the motor timing mechanisms here reported belong to the realm of self-organized systems.
While ‘event’ timing and ‘emergent’ timing are associated with motor tasks involving or lacking salient events, classes of tasks evoking different behaviours, it is important to show that distinct dynamics underlie this distinction. The current framework takes up this challenge, and identifies task constraint-dependent distinct dynamic mechanisms under given task constraints. Movements on a limit cycle are, mathematically speaking, autonomous and evolve in a self-sustained manner. In contrast, the presence of a (stable) fixed point implies non-autonomous dynamics. A system settled at a fixed point will maintain its position unless an external impact forces it away, in which case a movement is executed. In the context of motor control, such an impact has been associated with a neural (network) timing structure functioning as a ‘clock’ mechanism [6]. It may be more appropriate, however, to dissociate timing and movement initiation mechanisms—even though they likely operate in coordination. Both mechanisms will be implicated in intermittent circle drawing but not in the other circle drawing realizations. We previously reported the implication of distinct timing mechanisms in (two-dimensional) finger tapping [6]. Circle drawing arguably is a more complex task whose dynamics are four-dimensional. Our present results not only consolidate the implication of distinct mechanisms—their utilization appears to be not task specific, but also indicates that the framework applied can be extended to higher dimensional phase spaces. In intermittent circle drawing, in contrast to its continuous counterpart, timing cannot be attributed solely to the movement dynamics. The brain areas associated with rhythmic movements (i.e., mainly primary motor areas) are, by and large, a subset of those involved in discrete movements (with additional activity in several non-primary motor areas and stronger, bilateral activity in the cerebellum and cerebrum [16]). Each ‘mode of operation’ is thus likely supported by different neural networks – maybe generating distinct neural dynamics – rather than by distinct neural dynamics arising from the same network. It is less clear, however, which areas are specific to the timing per se (if any). Candidate structures are the cerebellum [8,9,17], in particular the superior cerebellar vermis [16,18], and basal ganglia [2,13], although their identification as timers has also been refuted [8,13]. The lack of agreement as to what structure(s) underlies timing may be partly due to the general failure to distinguish timing and movement initiation mechanisms. Our result contrasts the existence of a ubiquitous taskindependent timing mechanism. Several dichotomies underlying distinct timing mechanisms have been proposed, notably in terms of explicit versus implicit timing, in which the (explicit) temporal
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nature of the task (or not) is the distinguishing feature [2], and automatic versus cognitively controlled timing, distinguished by task duration and predictability and the presence (or absence) of movement [12]. While our present distinction in terms of autonomous versus non-autonomous dynamics (practically) overlaps to some degree with both dichotomies above, it stands apart through its mathematical motivation, which, in our opinion provides a fruitful basis to the study of timing, and other issues pertinent to the neurosciences more in general. References [1] J. Biberstine, H.N. Zelaznik, L. Kennedy, E. Whetter, Timing precision in circle drawing does not depend on spatial precision of the timing target, J. Motor Behav. 37 (2005) 447–453. [2] J.T. Coull, A.C. Nobre, Dissociating explicit timing from temporal expectation with fMRI, Curr. Opin. Neurobiol. 18 (2008) 1–8. [3] A. Daffertshofer, C.J. Lamoth, O.G. Meijer, P.J. Beek, PCA in studying coordination and variability: a tutorial, Clin. Biomech. 19 (2004) 415–428. [4] R. Friedrich, J. Peinke, Statistical properties of a turbulent cascade, Physica D 102 (1997) 147–155. [5] H. Haken, Synergetics, An Introduction: Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry and Biology, 3rd rev. enl., SpringerVerlag, New York, 1983. [6] R. Huys, B.E. Studenka, N. Rheame, H.N. Zelaznik, V.K. Jirsa, Distinct timing mechanisms produce discrete and continuous movements, PLoS Comp. Biol. 4 (2008) e1000061. [7] R. Huys, L. Fernandez, R.J. Bootsma, V.K. Jirsa, Fitts’ law is not continuous in reciprocal aiming, Proc. R. Soc., Lond. B: Biol. Sci. (2010).
[8] R.B. Ivry, R.M.C. Spencer, The neural representation of time, Curr. Opin. Neurobiol. 14 (2004) 225–232. [9] R.B. Ivry, R.M.C. Spencer, H.N. Zelaznik, J. Diedrichsen, The cerebellum and event timing, Ann. N. Y. Acad. Sci. 978 (2002) 302–317. [10] V.K. Jirsa, J.A.S. Kelso, The Excitator as a minimal model for the coordination dynamics of discrete and rhythmic movement generation, J. Motor. Behav. 37 (2005) 35–51. [11] J.A.S. Kelso, Dynamic Patterns: The Self-organization of Brain and Behaviour, MIT Press, Cambridge, MA, 1995. [12] P.A. Lewis, R.C. Miall, Distinct systems for automatic and cognitively controlled time measurement: evidence from neuroimaging, Curr. Opin. Biol. 13 (2003) 250–255. [13] M.D. Mauk, D.V. Buonomano, The neural basis of temporal processing, Annu. Rev. Neurosci. 27 (2004) 307–340. [14] R.C. Miall, R. Ivry, Moving to a different beat, Nat. Neurosci. 7 (2004) 1025– 1026. [15] J.D. Murray, Mathematical Biology, Springer, New York, 1993. [16] S. Schaal, D. Sternad, R. Osu, M. Kawato, Rhythmic arm movement is not discrete, Nat. Neurosci. 7 (2004) 1136–1143. [17] R.M.C. Spencer, H.N. Zelaznik, J. Diedrichsen, R.B. Ivry, Disrupted timing of discontinuous but not continuous movements by cerebellar lesions, Science 300 (2003) 1437–1439. [18] R.M.C. Spencer, T. Verstynen, M. Brett, R.B. Ivry, Cerebellar activation during discrete and not continuous timed movements: an fMRI study, NeuroImage 36 (2007) 378–387. [19] S.H. Strogatz, Nonlinear Dynamics and Chaos. With Applications to Physics, Biology, Chemistry, and Engineering, Perseus, Cambridge, MA, 1994. [20] A.M. van Mourik, A. Daffertshofer, P.J. Beek, Estimating Kramers–Moyal coefficients in short and non-stationary data sets, Phys. Lett. A 351 (2006) 13–17. [21] H.N. Zelaznik, R.M.C. Spencer, R.B. Ivry, Dissociation between explicit and implicit timing in repetitive tapping and drawing movements, J. Exp. Psychol. Hum. Percept. Perform. 28 (2002) 575–588.
Contents lists available at ScienceDirect
Neuroscience Letters journal homepage: www.elsevier.com/locate/neulet
Distinct timing mechanisms are implicated in distinct circle drawing tasks Raoul Huys a,∗ , Breanna E. Studenka b , Howard N. Zelaznik c , Viktor K. Jirsa a,d a
Theoretical Neuroscience Group, UMR 6152 Institut des Sciences du Mouvement, CNRS & Université de la Méditerranée, 163 av. de Luminy, CP910, F-13288 Marseille, France Department of Kinesiology, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S4K1 c Health and Kinesiology, Purdue University, 800 W. Stadium Blvd., West Lafayette, IN 47907, USA d Center for Complex Systems & Brain Sciences, Physics Department, Florida Atlantic University, Boca Raton, FL 33431, USA b
a r t i c l e
i n f o
Article history: Received 7 October 2009 Received in revised form 22 January 2010 Accepted 22 January 2010 Keywords: Timing Dynamics Circle drawing Saddle-node on invariant circle bifurcation
a b s t r a c t That individual timing variability is correlated across some tasks but not others has instigated the notion of distinct timing processes, referred to as ‘event’ timing and ‘emergent’ timing for tasks with and without salient events, respectively. The delineation of the event-emergent framework owes much to the circle drawing task as it can be performed with or without such events, all other factors being equal. We investigated continuous and intermittent circle drawing from a principled perspective allowing for the classification of timing mechanisms based on mathematical theorems. We propose a one-dimensional dynamical model to portray the essential dynamics of circle drawing that exhibits either fixed point or oscillator dynamics, each associated with distinct timing mechanisms. Eight participants drew circles under three different instruction conditions (no specific instructions, to draw as a fast or as smoothly as possible) at seven frequencies ranging from 0.5 Hz to 3.5 Hz. We computed the angle between the circle drawing’s horizontal and vertical component and reconstructed the corresponding vector fields prescribing the temporal evolution. Fixed point dynamics were present only in the ‘fast’ condition at low movement frequencies (i.e., in intermittent circle drawing representative of event timing). All other conditions were realized via oscillator dynamics representative of emergent timing. The transition between both dynamical mechanisms involved a saddle-node on invariant circle bifurcation, which was accompanied by increased trajectory variability, which is a key signature of phase transitions. These findings demonstrate the involvement of distinct timing mechanisms in different circle drawing tasks. © 2010 Elsevier Ireland Ltd. All rights reserved.
Whether timing draws upon a ubiquitous task-independent mechanism has become a central question in the neurosciences [2,6,14]. Recently, Zelaznik and colleagues [1,21] had participants perform several motor timing tasks—finger tapping, continuous line and circle drawing, and intermittent circle drawing. They reasoned that if tasks share the same timing mechanism, then individual differences in within-trial timing variability should be correlated across tasks. Correlation analysis dissociated timing in tasks involving salient ‘discrete’ events (tapping and intermittent circle drawing) and ‘continuous’ tasks (continuous circle and line drawing), referred to as ‘event’ and ‘emergent’ timing, respectively [9]. Importantly, the physical space and musculature involved in continuous and intermittent circle drawing are identical, ruling out alternative interpretations along those lines. Further evidence for this framework was provided by a study using participants with cerebellar lesions, which were associated with increased timing variability in ‘event’ timing tasks but not in ‘emergent’ timing tasks [17].
∗ Corresponding author. Tel.: +33 49 11 72 201; fax: +33 49 11 72 252. E-mail address: [email protected] (R. Huys). 0304-3940/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.neulet.2010.01.047
In contrast to relying on empirical findings to posit different types of timing, dynamical systems theory offers an analytical basis to a priori postulate distinct movement classes associated with or without timing mechanisms based on the concept of phase flows [6,10]. A phase space is the space spanned by a system’s state variables.1 The flow therein unambiguously describes a system’s evolution, and its topology allows for unambiguous system classification [19]. In two-dimensional systems, the types of topological structures are limited to fixed points, separatrices (structures that locally divides the phase space in regions with opposing flow), and limit cycles. Stable fixed points and limit cycles are associated with discrete and (continuous) rhythmic movements, respectively. We recently examined the implication of motor control mechanisms associated with fixed point and limit cycle dynamics in an experiment where participants executed repetitive index finger flexion/extension movements (tapping without surface contact) at paces from 0.5 Hz to 3.5 Hz [6]. The participants were either
1 In line with the literature, we here adopt the assumption that movements along a one-dimensional physical direction can be sufficiently represented by a two-dimensional phase space, which in that case is spanned by position and velocity.
R. Huys et al. / Neuroscience Letters 472 (2010) 24–28
25
Fig. 1. 3D projections of 4D circle drawing. Single participant data in the ‘fast’ condition at 0.5 Hz (left panel) and 3.5 Hz (right panel), respectively. Circular black trajectories represent the projections of the 4D trajectory into the 3D subspace spanned by x, y, and dx/dt; grey trajectories represent their projection on the xy-plane. The straight black lines in the xy-plane schematically represent the angle between the x- and y-components (see text).
instructed to produce a maximal flexion coincident with the metronome tone (the ‘natural’ condition), or to execute the movements as smoothly as possible (the ‘smooth’ condition) or as fast as possible (the ‘fast’ condition), and in all cases coincide maximum flexion with the metronome. We identified the phase flow via vector field reconstruction and found clear evidence for fixed point dynamics at low frequencies (up to approximately 2.0 Hz) for all participants in the ‘fast’ condition and for some in the ‘natural’ condition. At higher frequencies, the participants, irrespective of the instructions, utilized limit cycle dynamics. In other words, we found evidence in favour of the utilization of two distinct motor control and timing mechanisms (see also below) whose implementation is dictated by movement frequency and to some extent instruction. Through topological phase space analysis we examined the implication of different timing mechanisms in intermittent and continuous circle drawing, tasks that have played a crucial role in developing the notions of ‘event’ and ‘emergent’ timing. We predict the implication of fixed point dynamics in intermittent (slow pace) circle drawing but not at higher paces in which the drawing becomes continuous. Eight participants (mean age = 27.9 years) took part in the experiment (seven right-handed, one left-handed). The Purdue University Committee on the Usage of Human Research Participants approved the protocol, which was in agreement with the Declaration of Helsinki. All participants provided informed consent. Data were collected using a Polhemus Liberty-8 receiver (23 mm × 13 mm × 11 mm, 27 g), which was attached to the participant’s index finger with adhesive tape. The receiver was controlled by Matlab using an AuSIM-AuTrakMatlab USB driver and collection interface via library C++ calls. Displacement data were collected at 240 Hz. Participants were seated at a 79-cm high table. A white sheet of paper with a printed circle (diameter of 7.5 cm) was placed in front of them. At the ‘top’ of the circumference (12 o’clock; the point farthest away from the participant) a 1 cm diameter target circle was printed. The participants, moving counter-clockwise, attempted to produce one circle per beat of the metronome, with fixed wrist, moving with the elbow and shoulder only, and to coincide the passing of their finger tip over the “target” circle on the metronome beat. Timing accuracy, rather than spatial accuracy, was emphasized in circumscribing the circle. The participants were instructed to perform the circle drawing task without touching the table surface in order to avoid feedback from friction. The circle drawing was performed under three instruction conditions (see also [6]). In the ‘natural’ condition, the participants were instructed to pass over the target cycle at the moment of the metronome tone. In the ‘smooth’ condition, they were furthermore
instructed to move as smooth as possible. In the ‘fast’ condition, they were instructed to execute the movement in between metronome tones as fast as possible (i.e., staccato-like) while maintaining the requirement of arriving with their finger at the target circle at the moment the metronome tone sounded, which at the lower movement frequencies typically resulted in a short pause over the target circle. Thus, the ‘smooth’ and ‘fast’ conditions were constructed to force the participants to adopt a particular ‘mode of operation’ (associated with fixed point or limit cycle dynamics; see below). The participants performed 10 trials in each of the three instruction conditions on three different days. In five of each set of 10 trials, the metronome pace increased from 0.5 Hz to 3.5 Hz in steps of 0.5 Hz; on the other five trials, the pace decreased from 3.5 Hz to 0.5 Hz with equal steps. Fifteen metronome tones were presented at each frequency plateau. The order of the increasing (decreasing) set of trials was performed in a blocked design. All participants performed the ‘natural’ condition on day 1. The order of the ‘smooth’ and ‘fast’ conditions was balanced for all subjects. Movements along a single spatial dimension, such as tapping, are commonly portrayed as a two-dimensional dynamical system with position and velocity as state variables [11]. In circle drawing, two spatial dimensions exist, rendering its dynamical description four-dimensional. Rather than investigating the four-dimensional system we developed a reduced description of circle drawing that more easily allowed us to focus on its underlying phase space topology. Thereto, we concentrated on the dynamics of the angle between the (constant) vector pointing from the origin to the drawing’s maximal excursion and the vector pointing from origin to the drawing’s evolving horizontal (x) and vertical (y) component (i.e., [x(t), y(t)]). These vectors are defined on the drawing’s projection onto the xy-plane as illustrated in Fig. 1. The dynamics of is given by ˙ = f (), where the dot notation signifies the time derivative. The minimal model for which ˙ may exhibit either zero, one or two fixed points is ˙ = ω − cos(), where ω equals angular frequency. The amplitude of the cos() term may well scale with ω; we here set this amplitude scaling to 1/ω in view of the general tendency of movements to become more harmonic as their frequency increases. (This scaling does not impact the system’s qualitative behaviour as long as the corresponding parameter does not equal 0.) Fig. 2 portrays the system’s (saddle-node) bifurcation diagram (see also [19]). As the saddle-node bifurcation occurs on an attractive invariant circle, it represents a saddlenode-on-invariant-circle bifurcation (SNIC). Notice that for ω < 1, will settle at the stable fixed point, and a movement will be executed only if an external input I brings the system across the unstable fixed point (which acts as a separatrix). The full system,
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subsequently calculated based on P( ,t + t|, t) according to D
Fig. 2. Saddle-node on invariant circle bifurcation diagram for ˙ = ω − (1/ω)cos(). Each black line represents ˙ as a function of and ω. Fixed points are identified ˙ through the crossings of -curves with the (grey) ˙ = 0 plane. Stable and unstable fixed points are indicated in the − ω-plane as a black and grey line, respectively. For ˙ ω < 1, the -curve crosses the ˙ = 0 plane twice. As frequency ω increases, stable and unstable fixed points collide at ω = 1; the remaining fixed point vanishes for ω > 1.
˙ = ω − (1/ω)cos() + I, thus belongs to the class of excitable systems [15]. We reconstructed the circle drawing’s vector field on a line by extracting the deterministic component ˙ (i.e., the angle ’s time derivative). Human motion is intrinsically stochastic; it contains a deterministic and a stochastic (i.e., noise) dynamical component [5] that can be disentangled by computing the motion’s conditional probability distribution [4,20]. Importantly, extracting the deterministic component allows one to reconstruct vector fields, which allows us to establish the link between behavioural data and concepts from dynamical systems theory. The latter is not possible by ˙ We here focus on the deterministic dynamics. Thereto analysing . all trials were filtered with a low-pass 4th order Butterworth filter (cut-off frequency: 20 Hz). For each trial, the drawing movements’ horizontal (x) and vertical component (y) were mean subtracted and divided by their maximal value. Next, the continuous angle () between both components was computed as = arctangent (y/x). Each trial’s first three drawn circles, at each frequency, were omitted from analysis. The remaining twelve circles were divided into segments from − to . For each segment, the two-dimensional conditional probability matrix P( ,t + t|,t), which denotes the probability to find the system at state at a time t + t given its state at an earlier time step t, was computed using a bin size of 2/71. For each participant and each condition, the average conditional probability matrix across all segments of all trials was computed. Each condition’s so-called drift coefficients were
(n)
1 (, t) = lim t→0 t
n
( − ) P( , t + t|, t) d n!
Presently, the first drift coefficients D1 reveal the system’s deterministic dynamics, which we use to recover the vector field on the line in − ˙ space. We examined ’s trajectory variability of model simulations and human data (as a function of ω and movement frequency, respectively) using principal component analysis (PCA [3,7]). Trials and simulations were segmented as above (from − to ), resampled to the median length of all segments (N = 380), and for each participant, instruction and frequency conditions (or value of ω for simulations) were organized into a state vector q(t) that was subjected to PCA. Fig. 1 illustrates that in the fast condition at 0.5 Hz each drawn circle is followed by a period in which slowly converges to an approximately stationary state or moves around it with little amplitude. Thus intermittent circle drawing shows up as the area with locally reversing and intersecting trajectories, suggesting the existence of a fixed point. This suggestion was corroborated by the extracted drift coefficients (D1 ) shown in Fig. 3 for all three conditions as a function of movement frequency (compare with Fig. 2). In the fast condition, the line in − D1 space reveals a marked local minimum that almost crosses the D1 = 0 line at low frequencies (0.5 Hz and 1.0 Hz). For all participants, conditions, and trials, we examined whether the absence of line crossing could have been due to averaging and small drift in the fixed point’s position. This appeared to be so: for the fast condition at 0.5 Hz, the trial average minimal D1 -value was smaller than zero for each participant, and for 10 out of 16 cases (two ramps per participant) for 1.0 Hz, zero-crossings were observed several times per trial for each participant. The depth of the local minimum decreased and vanished with increasing frequency. For the natural and smooth conditions, the local minimum was hardly present and the line was never crossed (except for one participant in the natural condition at 0.5 Hz). In all conditions, the D1 -line became approximately straight at about 2.0 Hz, indicating harmonic circle drawing. The model predicts lower system stability the closer it is to the bifurcation [19], which should reveal itself in increased trajectory variability. PCA of the human data confirmed this prediction partly; the first eigenvalue (1 ) increased (i.e., trajectory variability decreased) monotonically with increasing movement frequency in the limit cycle regime (Fig. 4), and revealed a local minimum at or around the bifurcation point. These results were statistically significant: an ANOVA of the first eigenvalue revealed a main effect for condition (F(2,14) = 18.320, P < 0.001) and an Instruction × Movement frequency interaction (F(12,84) = 10.627,
Fig. 3. Extracted drift coefficients D1 as a function of instruction and movement frequency. The left, middle, and right panels display the D1 coefficients for the natural, fast, and smooth conditions, respectively, as a function of movement frequency ω and angle (averaged across trials, ramps [increasing and decreasing], and participants). Each line represents one movement frequency condition. On average, D1 (almost) intersected the D1 = 0 line in the fast condition at low frequencies. In all conditions, D1 increased with increasing movement frequency and the D1 line became approximately straight.
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Fig. 4. First principal component’s eigenvalue 1 for human data (left panel) and simulations (right panel) as a function of movement frequency and ω, respectively. The model was simulated at various ω’s and three different noise levels (increasing from the light grey to black line; right panel). 1 only depends on movement frequency in the fast condition; the local minimum around the bifurcation point is mimicked in the simulations.
P < 0.001). Newman–Keuls post hoc tests (˛ = 0.05) did not identify significant differences for the natural and smooth condition between frequencies whereas the fast condition’s local minimum differed significantly from its neighbouring points. Trajectory variability of the model simulations (Fig. 4) mimicked that of the human data (fast condition). The (human data’s) deviation from an approximately symmetrical pattern around the bifurcation point is likely largely due to the longer duration spent at the fixed point for slower movements. This interpretation is supported by a PCA on model simulations under constant ω = 0.7 but varying between stimuli (I; see the model above) durations showed that 1 increased with increasing inter-stimulus duration (Spearman correlation r = .97, p < .01). In the present study, participants drew circles at different frequencies and under distinct instructions. We constructed the corresponding vector fields, which allows for the linkage between behavioural data and mathematical concepts. We show that qualitatively distinct dynamics are associated with circle drawing depending on instruction and movement frequency: fixed point dynamics were observed under instructions leading to intermittent circle drawing (i.e., the ‘fast’, staccato-like condition at low frequencies), while limit cycle dynamics were observed under instructions not specifying strategy or demanding smooth performance. At higher frequencies there was no evidence for fixed point dynamics regardless the instruction, and the approximately straight line of the drift coefficients in − D1 space indicates that the circles were drawn almost harmonically. That is, we found evidence that qualitatively distinct dynamics are associated with intermittent and continuous circle drawing that have previously been associated with ‘event’ and ‘emergent’ timing [9,17,21]. The transition from fixed point dynamics to oscillator dynamics occurred via a SNIC bifurcation, in which the fixed points coalesce and a limit cycle is formed. A characteristic thereof is that in a small parameter regime the fixed point, while no longer existing, is still felt. In the presence of such a so-called ‘ghost’ (see Fig. 3, middle panel), dynamical systems theory predicts that trajectories locally slow down [20]. The existence of the ghost further testifies to the notion that a single model, ˙ = ω − (1/ω)cos() + I, governs intermittent as well as continuous circle drawing even though the corresponding topologies and flow patterns are frequency dependent. Moreover, the increased trajectory variability around the bifurcation point, indicative of phase transitions in nonlinear systems [5], suggests that the motor timing mechanisms here reported belong to the realm of self-organized systems.
While ‘event’ timing and ‘emergent’ timing are associated with motor tasks involving or lacking salient events, classes of tasks evoking different behaviours, it is important to show that distinct dynamics underlie this distinction. The current framework takes up this challenge, and identifies task constraint-dependent distinct dynamic mechanisms under given task constraints. Movements on a limit cycle are, mathematically speaking, autonomous and evolve in a self-sustained manner. In contrast, the presence of a (stable) fixed point implies non-autonomous dynamics. A system settled at a fixed point will maintain its position unless an external impact forces it away, in which case a movement is executed. In the context of motor control, such an impact has been associated with a neural (network) timing structure functioning as a ‘clock’ mechanism [6]. It may be more appropriate, however, to dissociate timing and movement initiation mechanisms—even though they likely operate in coordination. Both mechanisms will be implicated in intermittent circle drawing but not in the other circle drawing realizations. We previously reported the implication of distinct timing mechanisms in (two-dimensional) finger tapping [6]. Circle drawing arguably is a more complex task whose dynamics are four-dimensional. Our present results not only consolidate the implication of distinct mechanisms—their utilization appears to be not task specific, but also indicates that the framework applied can be extended to higher dimensional phase spaces. In intermittent circle drawing, in contrast to its continuous counterpart, timing cannot be attributed solely to the movement dynamics. The brain areas associated with rhythmic movements (i.e., mainly primary motor areas) are, by and large, a subset of those involved in discrete movements (with additional activity in several non-primary motor areas and stronger, bilateral activity in the cerebellum and cerebrum [16]). Each ‘mode of operation’ is thus likely supported by different neural networks – maybe generating distinct neural dynamics – rather than by distinct neural dynamics arising from the same network. It is less clear, however, which areas are specific to the timing per se (if any). Candidate structures are the cerebellum [8,9,17], in particular the superior cerebellar vermis [16,18], and basal ganglia [2,13], although their identification as timers has also been refuted [8,13]. The lack of agreement as to what structure(s) underlies timing may be partly due to the general failure to distinguish timing and movement initiation mechanisms. Our result contrasts the existence of a ubiquitous taskindependent timing mechanism. Several dichotomies underlying distinct timing mechanisms have been proposed, notably in terms of explicit versus implicit timing, in which the (explicit) temporal
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nature of the task (or not) is the distinguishing feature [2], and automatic versus cognitively controlled timing, distinguished by task duration and predictability and the presence (or absence) of movement [12]. While our present distinction in terms of autonomous versus non-autonomous dynamics (practically) overlaps to some degree with both dichotomies above, it stands apart through its mathematical motivation, which, in our opinion provides a fruitful basis to the study of timing, and other issues pertinent to the neurosciences more in general. References [1] J. Biberstine, H.N. Zelaznik, L. Kennedy, E. Whetter, Timing precision in circle drawing does not depend on spatial precision of the timing target, J. Motor Behav. 37 (2005) 447–453. [2] J.T. Coull, A.C. Nobre, Dissociating explicit timing from temporal expectation with fMRI, Curr. Opin. Neurobiol. 18 (2008) 1–8. [3] A. Daffertshofer, C.J. Lamoth, O.G. Meijer, P.J. Beek, PCA in studying coordination and variability: a tutorial, Clin. Biomech. 19 (2004) 415–428. [4] R. Friedrich, J. Peinke, Statistical properties of a turbulent cascade, Physica D 102 (1997) 147–155. [5] H. Haken, Synergetics, An Introduction: Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry and Biology, 3rd rev. enl., SpringerVerlag, New York, 1983. [6] R. Huys, B.E. Studenka, N. Rheame, H.N. Zelaznik, V.K. Jirsa, Distinct timing mechanisms produce discrete and continuous movements, PLoS Comp. Biol. 4 (2008) e1000061. [7] R. Huys, L. Fernandez, R.J. Bootsma, V.K. Jirsa, Fitts’ law is not continuous in reciprocal aiming, Proc. R. Soc., Lond. B: Biol. Sci. (2010).
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