visuomotor tracking tasks with delayed pursuit and …

Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC 2011August 28-31, 2011, Washington, DC, USA

DETC2011-47312

VISUOMOTOR TRACKING TASKS WITH DELAYED PURSUIT AND ESCAPE

John Milton∗Joint Science DepartmentThe Claremont Colleges

Claremont, California 91711Email: [email protected]

Joshua LippaiDepartment of Mathematics

Pomona CollegeClaremont, California 991711

Email: [email protected]

Rachel BellowsJoint Science Department

Pitzer CollegeClaremont, California 91711


Andrew BlombergJoint Science Department

Claremont McKenna CollegeClaremont, California 91711


Atsushi KamimuraInstitute of Industrial Science

The University of TokyoTokyo, JAPAN


Toru Ohira†

Joint Science DepartmentThe Claremont Colleges

Claremont, California 91711Email: [email protected]

ABSTRACT

Virtual stick balancing (VSB) is a manual visuomotor track-ing task that involves interplay between a human and a computerin which the movements are programmed to resemble those ofbalancing a stick at the fingertip. Since time delays and ran-dom perturbations (“noise”) are intrinsic properties of this task,we modeled VSB as a delayed pursuit-escape process: the tar-get movements are described by a simple random walk and thosemovements controlled by the computer mouse by a delayed ran-dom walk biased towards the target. As subjects become moreskilled, a stereotyped and recurring pursuit–escape pattern de-velops in which the mouse pursues the target until it overtakes it,causing the target to move in a different direction, followed, aftera lag, by the pursing mouse. The delayed pursuit-escape randomwalk model captured the qualitative nature of this tracking taskand provided insights into why this tracking task always fails atsome point in time, even for the most expert subjects.

∗Address all correspondence to this author.†Sony Computer Science Laboratories,Inc., Tokyo, Japan

INTRODUCTIONHow does the time–delayed nervous system successfully

track targets that move quickly in a complicated manner? Nu-merous experimental observations indicate that human subjectslearn to perform such tasks remarkably well; in fact they out per-form other primates [1]. Moreover they master this task evenwhen an additional time delay is added to the visual feedback[2–4]. Two explanations, not necessarily contradictory, havebeen proposed. The first assumes that the nervous system usesthe same control strategies employed by engineers to control ma-chines. Thus it is argued that there must exist strategies that an-ticipate, or compensate, for the effects of the time delay [5], suchas feedforward controllers [6], ”inverse dynamics” [7], and soon. The second approach draws attention to the inherent proper-ties of time–delayed dynamical systems. Surprisingly nonlineartime–delayed feedback controllers can successfully track chaot-ically moving targets through a mechanism known as ”anticipa-tory synchronization” [8]. Neither of these approaches takes intoconsiderations the effects of the interplay between noise and de-lay on tracking performance [9–11].

Virtual stick balancing (VSB) is a manual visuomotor track-ing task that requires a human subject to track a target that moveson the computer screen by a dot that moves under the control

1 Copyright c© 2011 by ASME

of a computer mouse [12–15]. In contrast to other trackingtasks, VSB is inherently unstable since the task is programmedto mimic the dynamics of an inverted pendulum. Consequentlythe rate of escape of the target rapidly increases with the track-ing error. Noise and time delay are important components ofthe control. Experimental [9] and theoretical [9–11] observa-tions indicate that transient control of unstable dynamical sys-tems can arise from the interplay between time–delayed feedbackand noise.

It has been argued that in real stick balancing (RSB) at thefingertip the upright position is statistically stabilized [9–11,16],i.e., the fluctuations in the vertical displacement angle resemblea random walk whose mean displacement is approximately zero.Here we consider VSB as a pursuit–escape task using a modelbased on a delayed random walk. Although pursuit–escape taskshave attracted mathematical interest since the 5th century BC(for a review see [17]), the effect of a delay on such processeshas only recently begun to attract attention [18]. The advantageof this modeling approach is that analytical insight may eventu-ally be possible. We first briefly review the properties of simpleand delayed random walks and then compare the development ofVSB skill with practice with numerical simulations of the ran-dom walk model.

BACKGROUNDThe analysis of random walks provides a convenient frame-

work for investigating the interplay between noise and time–delayed feedback [18,19]. Here we briefly describe the two typesof random walk we utilize in the delayed pursuit–escape model.

Simple random walksA simple random walk describes the movements of a walker

confined to move by taking steps of discrete length along a line.Let X(t) denote the position of the walker after the t–th step. Ifwe start at the origin, X(0) = 0, then the probability to take a stepto the right is p and the probability to take a step to the left is q,so that p+q = 1. The probability that the walker is at position xafter t steps is P(x, t) where

∞

∑x=−∞

P(x, t) = 1

The dynamics of the changes in P(x, t) are described by the mas-ter equation

P(x,0) = δx,0 (1)P(x, t) = pP(x−1, t−1)+qP(x+1, t +1)

where

δx,0 = 1 if x = 0= 0 otherwise

For the special case of a symmetric random walk, p = q = 0.5,we have for its mean and variance, respectively,

〈X(t)〉 = 0 (2)σ

2(t) = t (3)

Delayed random walksFor a delayed random walk, the transition probabilities de-

pend on the position of the walker at a time τ in the past. Agoal has been to formulate these transition probabilities so thatthe statistical properties of the random walker are equivalent tothose of the delayed Langevin equation [19, 20]

dx(t) =−kx(t− τ)dt +dW (4)

where x(t),x(t− τ) are, respectively, the values of the state vari-able at times t and t−τ , τ is the time delay, k is a constant, and Wdescribes the Weiner process. In order to do this it is necessary tointroduce the notion of stability into a random walk. This is doneby assuming that the random walk takes place within a quadratic(harmonic) potential, i.e., transition probability towards the ori-gin increases linearly with distance from the origin (of course upto a point). In particular the transition probability for the walkerto move toward the origin increases linearly at a rate of β as thedistance increases from the origin up to the position ±a beyondwhich it is constant (since the transition probability is between 0and 1). Equation (1) becomes

P(xc, t +1) = ∑xp

g(xp)P(xc−1, t;xp, t− τ)

+∑xp

f (xp)P(xc +1, t;xp, t− τ), (5)

where the position of the walker at time t is X(t), P(x, t) is theprobability for the walker to be at X(t) = x and P(xc, t;xp, t− τ)is the joint probability such that X(t) = xc and X(t−τ) = xp takesplace. f (x) and g(x) are transition probabilities for the walker totake the step to the negative (−1) and positive (+1) directions,respectively, given as

f (x) =

1+2d

2 x > a1+βn

2 −a≤ x≤ a1−2d

2 x <−a


g(x) =

1−2d

2 x > a1−βn

2 −a≤ x≤ a1+2d

2 x <−a

where a and d are positive parameters, β = 2d/a, where

f (x)+g(x) = 1 (6)

Since we have a quadratic potential it follows that f (−x) = g(x).The random walk is said to be attractive with respect to the ori-gin if f > g and repulsive if g > f . When f = g = 0.5 theseprobabilities reduce to the simple random walk.

Assume that with sufficiently large a, we can ignore theprobability that the walker is outside of the range (-a, a). In thiscase, the probability distribution function P(x, t) approximatelysatisfies the equation

P(xc, t +1) = ∑xp

12(1−βxp)P(xc−1, t;xp, t− τ) (7)

+∑xp

12(1+βxp)P(xc +1, t;xp, t− τ).

Then, the autocorrelation function with τ steps apart satisfies theinvariant relation

〈X(t + τ)X(t)〉= C(τ) =1

2β

Using this relation it is possible to determine the autocorrelationfunction, C(∆) for small β . When τ = 0 we have

C(∆)∼ 12β

e−β |∆|

and the auto–correlation decays exponentially with ∆. A curiousproperty of a delayed random walk is that for sufficiently longτ , C(∆), becomes oscillatory even though there is no change instability of the corresponding deterministic system. For example,when 0 < τ < ∆ we have

C(∆)∼C(0)cosβ∆− 12∆

sinβ∆ (8)

where

C(0) =1+ sinβτ

2β cosβτ

and hence the autocorrelation function is oscillatory. These an-alytical results agree with those obtained by a direct derivationfrom the delayed Langevin equation (4) with k = β [19–21].

METHODSSubjects

Data was collected from 23 healthy subjects between theages of 18 and 24. This study was approved by the institutionalreview board at Claremont McKenna College in accordance withthe currently applicable U. S. Public Health Service Guidelines.All participants provided informed consent for all research test-ing.

Virtual stick balancingThe subject views a target and a dot on a computer screen

(CRT monitor with a 100 Hz refresh rate): the dot reflects themovements controlled by the computer mouse (400 dpi); and themovements of the target are controlled by the computer usingrules that depend on the position of the mouse and target [13–15].The subject’s task was to move the dot to keep both the dot andtarget on the screen as long as possible. The movements of thetarget are programmed so that the target moves within a parabolicpotential that is centered at the mouse position. Specifically theequation describing the movements of the target (~xt ) is

d2~xt

dt2 = k[~xt −~xd ] (9)

where ~xd ,~xt are, respectively, the vectorial positions of the dot(mouse) and target and 4.5 < k < 7 is an adjustable constant thatcontrols the difficulty of the task. Since k > 0 the position of~xt is unstable (a saddle point as observed for an inverted pendu-lum). At t = 0, the position of the dot and target are identical; aslight movement of the mouse is sufficient to start the task. Un-der these conditions intermittent corrective movements describedpreviously [13–15] were not observed. The movements were rep-resented by calculating the length, D(x,y), of the position vectorto the target or dot from the common reference point (0,0) usingthe relation

D(x(t),y(t)) =√

x2(t)+ y2(t)

The software program was written using Vision Egg [22], a highlevel interface between Python and OpenGL.

Training protocolSubjects were required to perform stick balancing each day

for as long as it takes to accumulate 10–15 minutes of total bal-ance time (about 1–2 hours for beginners; 20–25 minutes forexperts). Fifteen subjects practiced real stick balancing (RSB)using a ∼ 55cm wooden dowels (diameter 6.35mm) and eightsubjects practiced virtual stick balancing (VSB). Financial in-centives were given to encourage the subject to increase their


FIGURE 1. MEAN STICK BALANCING TIME (〈ts〉) VERSUSDAYS OF PRACTICE: a) REAL STICK BALANCING ON THEFINGERTIP (15 SUBJECTS), b) VIRTUAL STICK BALANCING (8SUBJECTS).

skill level. Stick balancing skill was measured by estimating themean stick survival time, 〈ts〉, from 25 consecutive trials. Forboth RSB and VSB, the survival function, P(ts > t), has the formof a Weibull survival function, exp(λ t)γ where γ > 1 [10]. It wasobserved that statistically significant difference in survival curvesat the P = 0.05 level corresponded to a∼ 4−5s difference in 〈ts〉(Mann-Whitney U-test).

RESULTSFigure 1 plots 〈ts〉 as a function of consecutive days of prac-

tice for RSB and VSB. For VSB we adjusted k (typically k = 4.9)so that 〈ts〉 for subjects on day 1 were comparable to those forRSB : 9.7s (2.2s–16.7s) for 8 VSB subjects compared to 8.5(2.8–16.7)s for 15 RSB subjects; data expressed as mean(min – max)s.As anticipated skill increases with practice; however, this in-crease in skill is not always monotone. With 5 days of prac-tice 〈ts〉 increased 7–fold for RSB (64.5(12.8–337.2)s) comparedto 6-fold for VSB (44.7(31.3-60.4)s). Although this differencebetween RSB and VSB subjects was not statistically different,visual inspection of Figure 1 suggests that there may exist a sub-population of RSB subjects with increased performance.

Figure 2 compares D(x,y) for the movements of the targetand dot for a subject on the first and after five days of practice. Ineach case the shown time series corresponds to the trial for whichthe survival time was∼ 〈ts〉. When the subject is least skilled, thedot (mouse) predominately pursues the target and changes in di-rection (∗ in Figure 2a) occur infrequently. In contrast, when

FIGURE 2. COMPARISON OF POSITION OF TARGET (SOLIDLINE) AND DOT (DASHED LINE) FOR a) NOVICE SUBJECT ANDb), c), A SKILLED SUBJECT. SEE TEXT FOR DISCUSSION.

the subject becomes more skilled, changes in the direction of themovement of the target due to the dot overtaking the target occurmuch more frequently (every 0.5−0.9s in Figure 2b). This givesthe time series an oscillatory appearance consisting of recurringand stereotyped pursuit escapes: Initially the movements of thedot pursue those of the target (hence we refer to this phase aspursuit); however, at some point the dot catches up and passesthe target producing a change in direction of the movements ofthe target (hence we refer to this as a turn). After a lag, the move-ments of the dot again pursue those of the target and the cycle re-peats. However, for all skill levels the movements of the dot lagbehind those of the target by about the same amount. This is bestseen at a turn. Computing the cross–correlation between D(x,y)for dot and target gives a lag of 0.13(0.09− 0.18)s (estimatedfor 8 subjects on Day 5). This lag is consistent with the delaymeasured for RSB using the same technique [16], but is shorterthan estimates of delay for VSB∼ 0.22−0.35s determined usingother techniques [12–14].

An important question is why the pursuit–escape task alwaysfails to the point that the target escapes off the computer screen.Since a turn is associated with a increase in the distance betweentarget and dot (Figures 2 and 3), the risk of escape just after a turnis highest. This is because the larger the distance between targetand dot, the faster the target will move (see Eq. (9)). Providedthat the target’s movements are not too quick, it is possible for thedot to move fast enough to overtake the target and prevent escape.Figure 3 plots a typical escape. It can be seen that after eachturn the speed of the movements of the dot transiently increasesin order to overtake and pass the target (Figure 3b). The turn


FIGURE 3. COMPARISON OF a) POSITION AND b) SPEED OFTARGET (•) AND DOT (◦) FOR THE 7S PRECEDING AN ESCAPE.SEE TEXT FOR DISCUSSION. A.U. IS ARBITRARY UNIT.

just before an uncatchable escape occurs is characterized by alarger gap between the position of the dot and target. Presumablythis gap becomes large enough so that the dot (mouse) cannotovertake the target.

The maximum speed of the movement of the dot while thetarget was confined on the screen by a skilled subject was∼±2.5arbitrary units (a.u.) per ms in Figure 3b (3.5(2.5-5.3) a.u. perms for 6 subjects). This speed is about twice the maximum speedobserved for the same subject on day 1 (1.9(1.2-3) a.u. per msfor 6 subjects). These observations suggest that as the subjectincreases their skill level they are able to make faster correctivemovements.

ModelThe dynamics of VSB consist of transient behaviors; even

for the most skilled subject the target always eventually escapesoff the screen. In order to better understand the pursuit–escapenature of VSB we modeled the dynamics in terms of a delayedrandom walk model. There are many ways such a model can bedeveloped (see also Discussion). As a starting point we assumedthat noise enters into the pursuit–escape task at two points: 1)the perception of the position of the target by the pursuer, and2) the movement made by the pursuer on the basis of the per-ception. This model purposely does not include a specific delaycompensating mechanism. Thus departures between predictedand observed dynamics would anticipated to establish both thepresence of such mechanisms and identify the precise role thatthey play.

In view of these observations we modeled the movements of

the target as a symmetric random walk whose transition proba-bility is governed by

PT (y,0) = δy,0 (10)

PT (y, t +1) =12

PT (y−1, t)+12

PT (y+1, t)

where PT (y, t) is the probability that the target is at positionYT (t) = y at time step t. The movements of the mouse is modeledas a random walker whose transitions probabilities are biased to-ward the target with delay. Thus it is modeled as a delayed ran-dom walker, whose transition probability depends on the distancebetween the position of the target y at τ steps in the past and themouse’s current position.

PM(x, t +1) = ∑y

12{1−β (x−1− y)}P(x−1, t;y, t− τ) (11)

+∑y

12{1+β (x+1− y)}P(x+1, t;y, t− τ).

where PM(x, t) is the probability that the mouse is at positionXM(t) = x at time step t, while P(x, t;y, t− τ) is the mixed jointprobability with the target at the position y at t−τ and the mouseat the position x at t. When τ = 0 the mouse is biased to steptowards the target. In this model τ corresponds to the reactiontime and hence includes both the neural conduction delay (“deadtime”) as well as the time to complete the corrective movement.

The observation that the speed of the dot movements in-crease with skill can be incorporated into the delayed pursuit–escape model by assuming that the steepness of the walls of thequadratic potential increases with skill, i.e. by an increase in d(included in definition of f and g). This is equivalent to increas-ing the feedback gain. Figure 4 compares the dynamics of (10)–(11) for two different values of d. The oscillatory relationshipbetween target and dot appears as for the larger d (compare Fig-ures 4a with b). Thus, the model qualitatively captures the resultsof the experiment shown in Figures 2 and 3. Even further we canestimate the time scale by roughly matching the interval betweenthe large peaks between Figure 2,3 and Figure 4. With this wecan roughly get 100 steps in the model corresponds to 1s. So, thedelay of 20 steps in Figure 4 gives about 200ms delay, which is inthe correct order. The explanation for the apparent dependenceof the oscillatory relationship of d is unknown. When τ = 0 theappearance of an oscillation is usually the result of a change instability. However, when τ 6= 0 oscillations can arise when afixed–point is unstable simply from the interplay between noiseand delay [11, 23].


FIGURE 4. COMPARISON OF POSITION OF TARGET (SOLIDLINE) AND DOT (DASHED LINE) FOR A DELAYED PURSUIT–ESCAPE MODELED BY (10)–(11) AS A FUNCTION OF d: a) d =0.025 AND b) d = 2.5. OTHER PARAMETER VALUES WERE τ =20 AND a = 10.

DISCUSSIONThere are important differences between VSB and RSB.

First, whereas the vertical movements of the fingertip providea form of parametric stabilization for RSB [9, 24], these move-ments are absent in VSB. This observation likely explains whythe power law behaviors related to intermittency in RSB are notobserved for VSB (data not shown). However, it must be kept inmind that in this study the key parameter for VSB, k, was tunedto match survival times observed for RSB. Thus it is possible thatin other parameter regimes for VSB power law behaviors occurwhich are related to other mechanisms [15,25,26]. A second dif-ference is that in RSB the movements of the tips of the stick arenecessarily strongly correlated, but this is not true for the relativemovements of dot and target in VSB. Finally, mechanoreceptorsplay a role in RSB (K. Oki and J. Milton, unpublished observa-tions), but not in VSB. Consequently we believe that our VSBtask more closely resembles a delayed pursuit–escape task.

As VSB skill develops the subject makes faster movementsof the mouse. In real stick balancing the increase in skill over fiveconsecutive days of practice is similarly related to the subject’sincreasing ability to make faster movements: the Levy distribu-tion which describes the changes in speed made by the fingertipdevelops broad tails as skill increases [16, 27]. Thus when thenervous system is faced with the task of tracking a target thatmoves quickly on a parabolic potential it appears to make dowith time–delayed feedback by, for example, adjusting the gain.

An important question concerns the role played by delay

compensatory mechanisms in VSB. It is widely held that theexpert control of quick and complex voluntary movements re-quires the development of anticipatory mechanisms that com-pensate for the effects of the delay [5, 28]; however, not all in-vestigators agree [8]. In VSB and the pursuit–escape randomwalk model the movements of the mouse always lag those of thetarget. Moreover VSB dynamics at least qualitatively resemblethose of a pursuit–escape random walk model in which there isno delay compensatory mechanism. Thus if such mechanismsare present it is not obvious what aspect of the dynamics theyaccount for. However, it is possible that the reduction of a 2-Ddynamical system to a 1-D one as we have done may obscure therole for delay compensation. A limitation of the 1-D approxima-tion is that changes in pursuit strategy cannot be readily studied.Moreover, it is possible that the best strategy for keeping the tar-get on the screen as long as possible may not be the same as thatfor keeping the dot as close to the target as possible [13–15].

Obviously escape off the screen in VSB occurs because themovements controlled by the mouse are not fast enough to over-take the target and cause the target to change direction. The unan-swered question is why do escapes occur when they do? Escapesmight be related to temporary lapses in attention and/or a poordecision made at a critical point, for example, at a “turn”. Al-though we observed that the frequency of eye blinks, a qualita-tive measure of attention [29], decreases by about 50% duringVSB (from 19 blinks/min to 9 blinks/min for one subject; from11 blinks/min to 4 blinks/min in a second subject), there was noclear relationship between the timing of an eye blink and the oc-currence of a successful escape. In order to account for theseescapes it will be necessary to modify the pursuit–escape ran-dom walk model so that the speed of the movements of the targetincreases more dramatically with tracking error. One way to dothis would be to model the movements of the target as a repulsiverandom walk by turning the quadratic potential upside–down.

Pursuit–escape tasks with time delays frequently arise. Ex-amples include hunting of prey by predators [18], fixational eyemovements [30], car following [31], and military interactions be-tween opposing enemies. Thus it is surprising that this topic hasonly recently begun to attract the attention of mathematical mod-elers. The hope is that it may be possible to derive statisticalproperties that describe these dynamics, such as the auto– andcross–correlations functions, that can be applied to the study ofdelayed pursuit–escape.

ACKNOWLEDGMENTWe acknowledge useful discussions with A. Radunskaya.

Research was supported by the William R. Kenan, Jr. Foun-dation and the National Science Foundation (NSF-1028970 andNSF-0634592).


REFERENCES[1] Brooks, V. B., Reed, D. J., and Eastman, M. J., 1978.

“Learning of pursuit visuomotor tracking by monkeys”.Physiol. Behav., 21(6), pp. 887–892.

[2] Miall, R. C., Weir, D. J., and Stein, J. F., 1985. “Visuomo-tor tracking with delayed visual feedback”. Neuroscience,16(3), pp. 511–520.

[3] Beuter, A., Milton, J. G., Labrie, C., Glass, L., and Gau-thier, S., 2000. “Delayed visual feedback and movementcontrol in Parkinson’s disease”. Exp. Neurol., 110(2),pp. 228–235.

[4] Foulkes, A. J. M., and Miall, R. C., 2000. “Adaptation tovisual feedback delays in a human manual tracking task”.Exp. Brain Res., 131(1), pp. 101–110.

[5] Nijhawan, R., 2008. “Visual prediction: Psychophysics andneurophysiology of compensation for time delays”. Behav.Brain Sci., 31, pp. 179–239.

[6] Desmurget, M., and Grafton, S., 2000. “Forward model-ing allows feedback control for fast reaching movements”.Trends Cog. Sci., 4(11), pp. 423–431.

[7] Kawato, M., 1999. “Internal models for motor controland trajectory planning”. Current Opin. Neurobiol., 9(6),pp. 718–727.

[8] Stepp, N., 2009. “Anticipation in feedback–delayed manualtracking of a chaotic oscillator”. Exp. Brain Res., 198(4),pp. 521–525.

[9] Cabrera, J. L., and Milton, J. G., 2002. “On–off intermit-tency in a human balancing task”. Phys. Rev. Lett., 89(15),p. 158702.

[10] Cabrera, J. L., Luciani, C., and Milton, J., 2006. “Neuralcontrol on multiple time scales: Insights from human stickbalancing”. Condensed Matter Physics, 9(2), pp. 373–383.

[11] Milton, J. G., Cabrera, J. L., and Ohira, T., 2008. “Unsta-ble dynamical systems: Delays, noise and control”. EPL,83(4), p. 48001.

[12] Mehta, B., and Schaal, S., 2002. “Forward models in vi-suomotor control”. J. Neurophysiol., 88(4), pp. 942–953.

[13] Bormann, R., Cabrera, J. L., Milton, J. G., and Eurich,C. W., 2004. “Visuomotor tracking on a computer screen:an experimental paradigm to study the dynamics of motorcontrol”. Neurocomputing, 58–60, pp. 517–523.

[14] Cabrera, J. L., Bormann, R., Eurich, C. W., Ohira, T., andMilton, J. G., 2004. “State–dependent noise and humanbalance control”. Fluct. Noise Lett., 4(1), pp. L107–L117.

[15] Patzelt, F., Riegel, M., Ernst, U., and Pawelzik, K.,2007. “Self–organized critical noise amplification in hu-man closed loop control”. Front. Comp. Neurosci., 1, p. 4.

[16] Cabrera, J. L., and Milton, J. G., 2004. “Human stick bal-ancing: Tuning Levy flights to improve balance control”.Chaos, 14(3), pp. 691–698.

[17] Nahin, P. J., 2007. Chases and Escapes: The mathematicsof pursuit and evasion. Princeton University Press, Prince-

ton, N.J.[18] Kamimura, A., and Ohira, T., 2010. “Group chase and es-

cape”. New J. Phy., 12, p. 053013.[19] Ohira, T., and Milton, J., 2009. “Delayed random walks:

Investigating the interplay between delay and noise”. In De-lay Differential Equations: Recent advances and new direc-tions, B. Balachandran, T. Kalmar-Nagy, and D. E. Gilsinn,eds., Springer, pp. 305–335.

[20] Ohira, T., and Yamane, T., 2000. “Delayed stochastic sys-tems”. Phys. Rev. E, 61(2), pp. 1247–1257.

[21] Guillouzic, S., L’Heureux, I., and Longtin, A., 1999.“Small delay approximation of stochastic delay differentialequations”. Phys. Rev. E, 59(4), pp. 3970–3892.

[22] Straw, A. D., 2008. “Vision egg: An open-source library forrealtime visual stimulus generation”. Front. Neuroinform.,2, p. 4.doi:10.3389.neuro.11.004.2008.

[23] Pakdaman, K., Grotta-Ragazzo, C., and Malta, C. P., 1998.“Transient regime duration in continuous–time neural net-works with delay”. Phys. Rev. E, 58(3), pp. 3623–3627.

[24] Milton, J. G., Ohira, T., Cabrera, J. L., Fraiser, R. M., Gy-orffy, J. B., Ruiz, F. K., Strauss, M. A., Balch, E. C., Marin,P. J., and Alexander, J. L., 2009. “Balancing and vibra-tion: Evidence for “drift-and-act” control of human bal-ance”. PLoS ONE, 4, p. e7427.

[25] Cabrera, J. L., 2005. “Controlling instability with delayedantagonistic stochastic dynamics”. Physica A, 356, pp. 25–30.

[26] Eurich, C. W., and Pawelzik, K., 2005. “Optimal con-trol yields power law behaviors”. In Artificial Neural Net-works: Formal Models and Their Applications, W. Duch,J. Kacprzyk, E. Oja, and S. Zadrionzny, eds., Springer Lec-ture Notes in Computer Science, Volume 3679, Springer–Verlag, Berlin, pp. 365–370.

[27] Cluff, T., and Balasubramaniam, R., 2009. “Motor learningcharacterized by changing Levy distributions”. PLoS ONE,4, p. e5998.

[28] Nijhawan, R., and Wu, S., 2009. “Compensating time de-lays with neural prediction: are predictions sensory or mo-tor?”. Proc. Trans. R. Soc. A, 367, pp. 1063–1078.

[29] Fogarty, C., and Stern, J., 1989. “Eye movements andblinks: their relationship to higher cognitive processes”.Int. J. Psychophysiol., 8(1), pp. 35–42.

[30] Mergenthaler, K., and Enghert, R., 2007. “Modeling thecontrol of fixational eye movements with neurophysiologi-cal delays”. Phys. Rev. Lett., 98(13), p. 138104.

[31] Sugiyama, Y., Fukui, M., Kikuchi, M., Hosebe, K.,Nakayama, A., Nishinari, K., Tadaki, S., and Yukawa, S.,2008. “Traffic jam without bottleneck - experimental evi-dence for the physical mechanism of a jam”. New J. Phys.,10, p. 033001.


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