Motor Control & Sensory Integration

Testing of Rhythmic Pattern Production 149 The system shows periodic behaviour when the winding number is a rational fraction p/q. Quasiperiodic behaviour occurs for irrational winding numbers. Chaos does not occur provided the function g is a diffeomorphism--that is, it is smooth and invertible (Jensen, Bak & Bohr, 1984). Where this condition fails to obtain, chaotic behavior is found. For the sine circle map this occurs when K>I. The regime diagram for the sine circle map (Figure 1) shows its iterative behaviour as a function of the two parameters K (coupling strength) and if2 (uncoupled frequency ratio). The Arnol'd tongues show the regions in which a given winding number is stable. They are attractors. Treffner & Turvey (1993) make the point that the width of the Arnol'd tongue is an index of its stability. 1.50 i ~,,,~';..'..,. ~~z:,: .;:. t,:'.:,;.:.:7 ,~.,;J.x',,.,i~.:.:. :~' ~'~'':\"\"it't~ I 1.25 :\"' ' '::': ' ' :; - 91 I Iz 1.0 v 0.75 O.50 0.25 0.0 0.0 0.2 0.4 0.6 0.8 tO Figure 1. Regime diagram for the sine circlemap. (Reprintedby permission from Jensen, Bak & Bohr, 1984,p.1965). Part of the attraction of the use of the circle map is that its global qualitative properties (and some quantitative ones) are not dependent on the particular form of the function g. This is critical, because there is no a priori reason to suspect that the sine function will fit every or indeed any experimental situation. The logic in attributing universality to the circle map has been given by Bohr, Bak, and Jensen (1984):

150 J. Pressing Physical system Differential equation 2D discrete return map 1D circle map (universal behaviour) And indeed it is found that the general structure of the regime diagram looks similar for different functions g. Exponents characterizing various properties like fractal dimension of the measure of quasi-periodic orbits and other properties of these systems are found to be independent of the details of the g function. This strengthens the case for applying general nonlinear methods to the problem at hand. As mentioned by Treffner & Turvey (1993) the circle map has an interesting relation to the Farey tree (see Figure 2), an ordering of all rational numbers with important implications for mode-locking behaviour (Allen, 1983). Positions of maximal adjacency on this tree are definable as standing in the relation of unimodularity, defined, for two ratios qmlp/q & m/n, as [pn - = 1. These tum out to be isomorphic to the adjacent Arnol'd tongue positions. Treffner & Turvey (1993) propose and test a unimodularity shift hypothesis, which states that transitions between different mode-locked states that stand in unimodular relation (mod 1 shifts) will be more likely than those between non-unimodular states (non-mod 1 shifts). For example, this would suggest that transitions from 3:5 to 2:3 (unimodular, since 13\"3 - 5\"21 = 1) should be more common than transitions from 3:5 to 1:3 (non-unimodular, since 13\"3 - 5\"11 = 4). This prediction is a plausible one in view of the structure of the regime diagram for the circle map, and Treffner & Turvey (1993) fred that the hypothesis is strongly supported by the results of Experiments 1 & 2. This finding is consistent with work on other physical systems, as reviewed by Treffner & Turvey. Treffner & Turvey (1993) go on to enunciate a second hypothesis, the continued fraction substructure hypothesis, which states that more mod-1 shifts will occur for Fibonacci ratio states than from non-Fibonacci ratio states. This hypothesis rests on the assumption that the decomposition of a rational fraction into continued fraction format is a satisfactory way of assessing the convergence properties of the corresponding mode- locked ratio. In one sense such a decomposition measures the complexity of the ratio. The hypothesis is certainly interesting and testable, and Treffner & Turvey (1993) found that the hypothesis was supported for non-musicians but not supported for musicians.

Testing of Rhythmic Pattern Production 151 0I Level 0 I I Level I eve,, 1 2 --- 3 Level 3 12 33 45 / \\--- Level 4 57 87 78 54 --- --- 75 Figure 2. The Farey Tree Now, two objections can be raised to all this, beyond the ambiguous fundamental status of the circle map for this experimental set-up. First, these hypotheses are not very strong in their effect, being simply binary preference rules. Can the theory make more powerful and specific suggestions? Second, how do these predictions and findings of significance compare with those of cognitive models? But before addressing these questions, the cognitive side of the story. 6. CENTRAL COGNITIVE HYPOTHESES Here the central ideas are those of information processing and the effects of limited memory and attentional capacity on performance. These are both directly affected by the complexity of the message being processed. The most natural assumption seems to be that stability of an informational state (especially in the absence of special previous learning) should be negatively (perhaps inversely) correlated with its complexity. Transitions between states should occur based on a different parameter, similarity. The more similar two states are, the easier transitions between them should be. Similarity of states can be measured by distance between them, using one or more variables to represent the locations of the states in state space. The use of spatial proximity to represent similarity, or equivalently, distance between states to indicate dissimilarity, is a standard feature of psychological similiarity theory (e.g. Gregson, 1975; Shepard, 1962).

152 J. Pressing 7. ANALYTICAL F R A M E W O R K We can now return to the two questions posed at the end of Section 5. The second question we address first, as it is easy to answer. Cognitive models also provide a statistically significant fit to the data of Treffner & Turvey (1993). They make sensible predictions of preferred binary choice in pattern transitions that are well-supported by Treffner & Turvey's experimental results. However, I do not give an analysis based on binary preference hypotheses because it is subsumed by the more elaborate analysis to follow. Tt{e rationale for extending the existing theories is that we need a better test to discriminate between the two, and this we obtain by addressing the first question. To compare the two approaches I have assumed that there are two types of phenomena reflected in the data of Treffner & Turvey (1993): stability and interpattem transitions. This assumption is quite general in any study of a system which possesses distinct states and can make transitions between them. The relation between these two variables can be readily expressed formally, as in the so-called master equation (Gardiner, 1983; van Kampen, 1976) of statistical mechanics: 3Wn (5) 3t - Z [P( m ~ n)W m - 19(n --~ m)W n ] m where Wn is the probability ( = stability) of a certain state and p(m ~ n) is the transition rate or transition probability per unit time of changing from state m to n. The equation is simply a probability conservation law operating over a statistical ensemble of systems but it is useful in formalizing the arguments presented here. If we prepare the ensemble to be in a certain (mode-locked) state n then the fraction of the entire ensemble found in another state m after a certain elapsed time (as in the experiments here) is attributable to three processes: direct transitions from n to m, possible transitions via intermediary states from n to m such as n ~ k(---~...) --, m, and decay out of state m before the elapsed time of measurement by trajectories that previously entered it. Simple integration of the equation and rearrangment of terms formally yields the fraction of the time that an ensemble of systems initially purely in state n will be found in state m after time t:

Testing of Rhythmic Pattern Production 153 t tt (6) Wm (t)= p(n --->m)~Wn('C)d'c + ~, p(k ~ m)~Wk('C)d'c - Y . p ( m --->k)~Wm(X)d'c 0 k~n 0 k 0 which shows the three processes as three respective terms in the equation, where I have assumed, as is normally done in the absence of time-varying external \"forces,\" that the probabilities of transition during the equilibration period in which a new resonance is found are independent of time (the same form is generated if probabilities vary with time but can be approximated by a time average ~). In this equation \"c is a dummy variable of integration. Treffner & Turvey's (1993) results do not provide the many explicit parameters required to solve these equations exactly. In fact, accurate measurements of these would be very time-consuming. However, Treffner & Turvey (1993) did make choices of experimental boundary conditions which allow some simplifying assumptions. Treffner & Turvey (1993) discounted unparseable runs, and runs from subjects who could not stabilize properly or readily. They also found that there seemed to be little wandering between different resonances in the equilibration period and thereafter, by detailed examination of the individual time series. The effect of this can be considered to be, for m r n, to exclude or at least minimize terms 2 & 3 on the fight-hand side of equation (6), since term 2 corresponds to transitions proceeding via intermediaries and term 3 corresponds to runs that do not stabilize, as measured by frequency ratios deviant from target rational fractions or t~lure to mode lock. For the case of m = n, Treffner & Turvey's selected experimental conditions mean that the third term in the equation should predominate, since the first term is zero by default (a state does not make a transition to itself), and the second term corresponds to a mid-experiment transition that would be excluded as unstable or yielding a non-selected frequency ratio. Under these assumptions the equations simplify to: t for all m~n (7) Win(t) = p(n ---~m)fWn ('r (8) 0 t W n (t) = 1 - ~ p(n ~ m)fW n ('r m~n 0 with solutions

154 J. Pressing (9) W i n ( t ) _ P( n Q---~m ) ( 1 - e - Q t ) f ~ all m~: n ' a n d (10) W n (t) - e -Qt where Q- ~p(n---~ k) (11) k,n This solution, though only approximate, shows that stability of the initial state is characterized by exponential decay governed by a time constant 1/Q while the growth of other states is (at least initially) similarly exponential in time but also approximately linearly related to their individual transition probabilities. The linearity is only approximate due to the presence of a p(n--)m) term within the sum of probabilities Q. This nonlinearity is not expected to be a large effect. Furthermore, we can predict that linearity will be most closely followed when p(n--)m) is small, since this specific probability will then make minimal contribution to the sum making up Q that causes the nonlinearity. The stability S of the state n is clearly related to Q. The form of this relation is unknown, and so we use two types of assumed relations for correlation testing. The first type looks for linear relations between experiment measurements of Wn and stability indicators (si) which reflect S. The second type considers that stability might be linear in Q, as for example S = 1 - Q. Then the expectation is, from equation (10), for linearity between log Wn and the si reflecting S. Hence a log-linear relation is also tested. 8. STABILITY INDICATORS AND TRANSITION INDICATORS The next step is to identify indicators (predictors) of stability and transition probability that can be examined for their correlation with experiment. I assume that a properly chosen indicator will clearly show the main effects of stability and relative transition probabilities, despite their possibly containing some effects of averaging over time and over trajectories. Explicitly, the assumptions become the following: 1. The amount of the time that the subjects remain at the W-expected (that is, they make no shift of frequency ratio) is directly related to (positively correlated with) the stability of the given resonance.

Testing of Rhythmic Pattern Production 155 2. The number of transitions from W-expected to W-actual is inversely related to (positively correlated with the inverse of or negatively correlated with) the \"distance\" between the two states. These hypotheses seem plausible, and they are widely found in statistical mechanics, but they require interpretation within each theoretical domain here. I therefore propose a number of indicators of stability and transition probability for each of the two theoretical perspectives. The correlations between these indicators and experimental observations are used to shed light on which perspective gives a better fit to the data. It should be emphasized that such con'elation coefficients test only for linear correlation effects. The stability indicators examined are as follows, for an n:m polyrhythm: Stability indicators Dynamical indicators Cognitive indicators 1/Farey level I/total # of pattern elements = 1/(n+m) Arnol'd tongue width 1/product of #s of elements = 1~(n'm) 1/# of continued fraction levels 1/min(n,m) 1/max(n,m) 1/\"chunks\" (Deutsch) The dynamical indicators seem natural extensions of the theory of Treffner & Turvey (1993). Farey level is as indicated in Figure 2. The Arnol'd tongue widths are those used by Treffner & Turvey (1993) derived from the sine circle map with K = 1. The number of continued fraction levels is computed from Figure 4 of Treffner & Turvey (1993). For this purpose I use the continued fraction representation and compute (# of continued fraction levels) by the length of the continued fraction. For example, if p/q = 3/8, so that p/q = 1 (12) 2+1 1+1 ,, 1+1 which can be written p/q = (2,1,1), the number of levels (below the top 1) is 3. The cognitive indicators used are based on different estimators of cognitive complexity, which according to the cognitive hypotheses above, should be negatively correlated with stability. The exact functional nature of this relationship is, of course, unknown. Here we

156 J. Pressing look for an inverse relation, S oc I/C. 1/(n+m ) is the inverse of the total number of cognitive events in the cycle, and seems an obvious choice based upon simple counting. 1/Max(n,m) appears to correspond to one indicator of cognitive complexity used by Deutsch (1983) for polyrhythms, although it is difficult to be sure because the article only defines this quantity by example and not by explicit rule. Another complexity measure used by Deutsch in this same article is based on the number of memory chunks required to code the pattern. It is not obvious why this indicator, based on a short-term memory concept, should apply to well-learned patterns. The parameter is also only defined by example rather than by rule set. However, I have included this measure, using the following values for Deutsch chunk complexity: 1:2 (2), 1:3 (3), 1:4 (4), 1:5 (5), 2:3 (2), 2:5 (4), 3:5 (5), 3:4 (4), 4:5 (6), 4:7 (8), & 5:8 (10). Stabilities are the inverses of these numbers. The use of lln*m has no obvious precedent, and was included to see if nonlinear effects were present. For transition from an n:m rhythm to a p:q rhythm, the indicators examined are Transition probability indicators Dynamical indicators Cognitive_indicators A (Arnol'd tongue width) 1/J analog shift[ = 1/([n/m - p/q[) l/A(Farey level) I/modular distance = 1/( [nq - mp [) 1/distance between Arnol'd tongues Change in Arnord tongue width as a transition indicator has been at least alluded to by at least one previous research team (Peper, Beek, & van Wieringen, 1991). It is not directly suggested by Treffner and Turvey (1993) but it seems a natural extension of their discussion of page 1226 that compares the hypothesis of Mod 1 resonance change to percentage of the K=I line covered by the relevant Arnord tongues. In my view this discussion is somewhat surprising because it is not the size of Arnord tongues which one would a priori expect to primarily govern relative transition rates, but distances between them. Although mechanisms of motion in regime diagrams are rarely made explicit, if they are noise-based, as seems likely, then a random walk or diffusion model might be more sensible, or at least a model incorporating the idea of proximity in phase space (see section 10 for further discussion).

Testing of Rhythmic Pattern Production 157 Inverse of change in Farey level is a natural extension to the discussion of Treffner & Turvey (1993). Inverse of modular distance seems to be a natural extension to the theory of Treffner & Turvey (1993) that should allow a more discriminating test of the Farey structure than the binary hypothesis they used. The \"cognitive\" indicator 1/[ analog shift I presumes that the system moves on the basis of simple perceived decimal frequency ratio similarity, as a continuously controllable programmed mechanical system might. Note that these last two are related: modular distance = mq Ianalog shift I. Note that I explicitly do not consider differences in pattern complexity to determine transitions. The Arnord intertongue distance may require some motivation. This was computed by measuring the distance (in arbitrary units) in the sine circle map diagram from the center of the n:m mode lock region to the edge of the p:q region for K = 1. These distances were obtained from the detailed regime diagram of the sine circle map given by Jensen, Bak & Bohr (1984). These were measurable with an accuracy commensurate with that of other indicators used, and the relative distances were calibrated for consistency by comparison of Arnol'd tongue width measurements from Jensen et. al. with those of Treffner & Turvey (1993). The agreement was excellent (e.g. correlation of tongue widths for Experiment 1 from the two sources: .994). The Arnord tongue distance measurement might have been made from the K = 0 line, but in this case the indicator becomes identical with the analog shift indicator. Furthermore this seems inconsistent with Treffner & Turvey's preference for the K=I line. Instead, the process has been conceived of as a motion from one tongue (with the centre as the most likely starting point) to the boundary of the other, using a random walk process. Since each tongue is an attractor, once the boundary is passed (somehow passing over other tongues) mode-locking is highly likely ( noise within the system makes it less than certain). In practice, this indicator is highly correlated with the analog shift indicator. Overall, transition indicators have been defined to look for strong positive correlations with 1/distance. An alternative route, not detailed here, of looking for strong negative correlations with distance, showed the same general effects but with less overall significance. The relevant data from Treffner & Turvey (1993) Experiment 1, along with the values of data used for the proposed indicators, are given in Table 1.

158 J. Pressing Table 1 Results from Treffner & Turvey (1993) Experiment 1, with dynamical and cognitive stability and transition indicators W-expected Experimental measure 1:2 1:3 2.3 2.5 3:5 3:4 65.00 47.00 57.00 51.00 64.00 57.00 number at any resonance remaining at W-expected 48.00 22.00 17.00 12.00 3.00 7.00 remaining at W/total parseable 0.74 0.47 0.30 0.24 0.05 0.12 shifts to 1:1 1.00 0.00 2.00 0.00 3.00 11.00 shifts to 1:2 2.00 15.00 12.00 36.00 6.00 unparseable ~ 53.00 43.00 49.00 36.00 43.00 35.00 Cognitive stability indicators ll(n+m) 1/3 1/4 1/5 1/7 1/8 1/7 lln*m 1/2 1/3 1/6 1/10 1/15 1/12 llmin(n,m) 1 1 1/2 1/2 1/3 1/3 llmax(n,m) 1/2 1/3 1/3 1/5 1/5 1/4 l/\"chunks\" (Deutsch) 1/2 1/3 1/2 1/4 1/5 1/4 Dynamic stability indicators 1 1/2 1/2 1/3 1/3 1/3 0.0792 0.0344 0.0344 0.0152 0.0152 0.0152 1/Farey level 1 1 1/2 1/2 1/3 1/2 Arnord tongue width (Treffner & Turvey (1993)) I/continued fraction levels Cognitive transition indicators 2.000 1.500 3.000 1.667 2.500 4.000 1/absolute analog shift to 1:1 6.000 6.000 10.000 10.000 4.000 1/absolute analog shift to 1:2 Dynamic transition indicators .0636 .108 .108 .128 .128 .128 A(Arnord tongue width) to 1:1 0.00 .045 .045 .064 .064 .064 A(Arnol'd tongue width) to 1:2 1 1/2 1/2 1/3 1/3 1/3 1/A(Farey level) 1:1 m 1 1 112 1/2 1/2 l/A(Farey level) 1:2 1 112 1 113 112 1 1/modular distance to 1:1 m 1 1 1 1 112 1/modular distance to 1:2 0.0655 0.0453 0.1149 0.0512 0.0895 0.1779 l/Amord tongue dist. to 1:1 ~ 0.1960 0.1980 0.3922 0.3937 0.1234 l/Arnord tongue dist. to 1:2

Testing of Rhythmic Pattern Production 159 Similar tables were prepared for the data of Experiments 2 & 3. The results for the various stability indicators, correlating W-remaining and stability indicators directly, are given in Table 2. For each column, the best-fitting indicator is marked by **, and second best-fitting indicator by *. Indicator correlations that achieve significance at o~ = .05 (two-tailed) are underscored for each experiment and the composite Z-score. Table 2 Correlations of different stability indicators with measured stability (% of trials where subjects remained at the initial W-value) Correlation type Expt 1 Expt 2 Expt 3 Comp- Composite osite Z-score r(% unchanged, 1/(n+m)) 0.970 0.936 0.867 0.936 5.641 r(% unchanged, 1~n'm) 0.963 **0.983 0.747 0.944 *6.352 r(% unchanged, 1/min(n,m)) 0.849 0.953 0.436 0.831 4.477 r(% unchanged, llmax(n,m)) 0.954 0.893 *'0.917 0.926 5.231 .:L~.~ ~.~:.Y. .~..c~u.n~s). ~............................ 9:760.................. 0.:74. ] . ............................... 0:7.5.9 ....................... 3:2.5...5. ....... r(% unchanged, 1/Fareylevel) *0.977 \"0.981 \"0.901 0.965 **6.893 r(% unchanged, A-tonguewidth) **0.979 0.976 0.892 0.962 \"6.701 r(% unchanged, 1/cont.frac.level) 0.837 0.888 0.421 0.772 3.693 Correlations were also performed between all the indicators and the fraction remaining/total parseable. The differences in relative rankings of the different indicators were only very minor and hence these are not separately reported. Nearly all indicators correlate well. In Experiments 2 & 3 the best indicator was cognitive, while in Experiment 1 the best indicator was dynamical. However, the differences between best cognitive and best dynamical indicators were too small to be significant. This rather obvious fact was verified by converting the correlations (e.g. for Arnol'd tongue width and 1/(n+m) in Experiment 1) to Fisher z-distribution scores, since the resulting values are known to be, to good approximation, normally distributed with -I/2 standard deviation Cz = (N - 3) (e.g. Rosenthal and Rosnow, 1991; van der Waerden, 1969). (Note: small z is used for the Fisher transform, large Z for normal Z-distribution.) The Fisher z-transform is an inverse hyperbolic tangent, l+rj (~3) zj -X21n l_r, I

160 J. Pressing The resultant difference in normalized Fisher z-distribution values, (zl u / | ,,h distributed as Z (normally), termed here d, was nonsignificant Z2) ~(J~ + (J~, when various pairings of best cognitive and dynamic indicators were chosen on an experiment-by-experiment basis. It is possible to combine the correlations from the three different experiments using statistical meta-analysis even though the experimental designs were different, if we suppose that they are basically measuring the same phenomenon. This seems a plausible assumption here, but we must interpret the results carefully in light of its possible limitation. In the composite column, a composite correlation value is given based on Fisher z-transform of sample correlations, discounting slight differences in number of cases for the three experiments. This might be used to determine significance directly. But we follow a more travelled route by combining the Z-scores of the three experiments according to the formula z _ Z z, (14) 42d , where the different experiments are Weighted by their degrees of freedom (df's), and where Zj = (zj - gz) / r z (Rosenthal and Rosnow, 1991). As indicated above, the composite Z must be viewed with a little caution, but the clear trend here is for highly significant correlation for all indicators, with Deutsch chunks the poorest cognitive indicator. The best indicators are, in order, 1/Farey level, Amord tongue width, lln*m, ll(n+m), and llmax(n,m). However, on the basis of these data this ordering is arbitrary as there are no significant differences at the .05 level within this group. The correlations of stability indicators with log W-remaining are given in Table 3. Best-fitting indicator for each experiment is marked by **, and second best-fitting parameter by *. Indicator correlations that achieve significance at a = .05 (two-tailed) are underscored for each experiment and for the composite Z-score. Similar trends are evident here, though the correlations here were generally poorer. However, most indicators reached significance for most experiments and all were significant overall. Overall the same 5 indicators showed the highest significance levels as in linear correlations with the stability indicators (as found in Table 2). There were no

Testing of Rhythmic Pattern Production 161 significant differences overall between these indicators. As before, the best dynamical and cognitive indicators were equally powerful predictors. Table 3 Correlations of different stability indicators with logarithm of measured stability (In(% of trials where subjects remained at the initial W-value)) Correlation type Expt 1 Expt 2 Expt 3 Comp- Comp. osite Z-~ r(ln(% unchanged),ll(n+m)) **0.908 0.898 0.786 0.873 *4.529 r(ln(% unchanged),lln*m) *0.879 *0.925 0.624 0.846 4.333 r(ln(% unchanged),llmin(n,m)) 0.855 *0.925 0.327 0.793 3.884 r(ln(% unchanged),llmax(n,m)) 0.869 0.855 **0.874 0.866 4.345 ..~.(.!.n..(..~..u.n~.h.~.~.).,JL,..S.h.u~.). ............................. 0:.8..!..7. ................... 0:.7.9...4...................... 0:.7~.! ............. .O.7...S...9. ..................... ~.~.~.. r(ln(% unchanged),l/Farey level) 0.819 0,924 *0.869 0.878 \"4.67! r(ln(% unchanged),A-tonguewidth) 0.840 **0.929 0.850 0.880 *'4.718 r(ln(%unchanged),l/cont.frac, level) 0.850 0.852 0.305 0.737 3.313 Table 4 shows the results for transition indicators. The results of Experiment 3 were not useable because of the paucity of transitions to 1\"1 and 1:2. Best-fitting indicator for each experiment is marked by **, and second best-fitting parameter by *. Indicator correlations that achieve significance at r = .05 (two-tailed) are underscored for each experiment and for the composite Z-score. Table 4 Correlations of different transition indicators with shift incidence (% of trials where subjects movedfrom the starting ratio to either 1:1 or 1:2) Correlations with 1:1 Expt 1 Expt 2 Comp- Comp. osite Z-score r(shifts to 1:1, l/lanalog shift to 1:11) \"0.914 *0.954 0.937 4.978 r(shifts to 1:1, A(Arnord tongue width)) -0.362 -0.344 r(shifts to 1:1, 1/A(Fareylevel)) -0.341 -0.340 -0.353 -1.032 r(shifts to 1:1, 1/modulardistance to 1:1) 0.498 0.520 -0.341 -1.000 r(shifts to 1:1, 1/tonguedistance to 1:1) *'0.941 **0.977 0.509 1.597 0.963 5.821 Correlations with 1:2 .T.(..s...h~..m..~ ;..2.,..y.!~o.~...s.~i.ft...t?...! ;..2.!.)............................. .*..0:.~.9. ................. .*..9:..5..3..6................ 0:_~7 ................... .1.:..8.6.8.. ............. r(shifts to 1:2, A(Amord tongue width)) -0.392 0.221 -0.094 0.062 r(shifts to 1:2, 1/A(Fareylevel)) -0.394 0.218 -0.097 0.054 r(shifts to 1:2, 1/mod.distance to 1:2)) 0.347 **0.566 0.464 1.553 r(shifts to 1:2, 1/tonguedistance to 1:2) *'0.681 0.510 0.602 1.820

162 J. Pressing No significant differences were found between the best cognitive and dynamical indicators on individual experiments or overall. These best indicators, and only these (l/intertongue distance, 1/analog shift), achieved very high significance in the composite analysis for the 1:1 transition. Although there was a clear trend for significance of these same indicators in the 1:2 case, these failed to achieve significance individually or overall at the .05 level. In attempting to understand this lack of significance in the second case, I have considered the following proposition: transition probability in the regime diagram depends not only on distance between states, but on direction of motion. In the simple random walk idea of movement, this just means that the chance of taking a step to the fight is not the same as the chance of taking a step to the left. A tittle consideration of the nature of the experiments suggests that this is likelyto hold: the end-of-interval attractor in right- directed motion in the regime is 1:1, with both hands moving. The end-of-interval attractor for left-directed motion is 0:1 which corresponds to one of the subject's hands stopping, which violates the experimental conditions. Hence it is possible to consider that the instructions given to the subjects required them to impose cognitive constraints on movement within the regime diagram that produced an asymmetry of drift. Can we test for this? Yes, in a crude way. We might do a full regression treatment of the data, but in the spirit of simple correlation we can simply perform a separate correlation analysis for transitions to the fight and to the left. When we examine the data for Experiments 1 & 2 we see that most of the transitions are in fact to the left. There are in fact too few examples of fight transitions to make a meaningful test. However, it is not too implausible to combine data for left transitions from Experiments 1 & 2, averaging shift data where both experiments have values to get more reliable data points. We are testing then, transitions from 2:3, 3:5, 3:4, 4:7, 5:8, and 4:5, to 1:2. The results are as given in Table 5. Results for transitions to 1:1 are given to show that this method is compatible with the procedure used for that case. The pattern of significance for the 1:1 case is maintained. In fact, correlations for the two important indicators improve there, although this may well be fortuitous. Palpably improved correlations for the 1:2 case suggest that the interpretation given here of different fight- and left-directed transitions in regime diagrams here may have some validity. The best cognitive indicator achieves significance at .05 (two-tailed) and the best dynamical indicator only fails by a minute amount to do so (cut-off = .811). The main result, that the best cognitive and dynamical indicators are equally effective, is maintained.

Testing of Rhythmic Pattern Production 163 Table 5 Correlations of different transition indicators with shift incidence (% of trials where subjects movedfrom the starting ratio to either 1:1 or 1:2)for unidirected transitions only Correlations with 1:1 Experiments right-directed l&2 transitions r(shifts to 1:1, l/lanalogshift to 1:11) *0,970 r(shifts to 1:1,A(Amordtonguewidth)) -0.392 r(shifts to 1:1, l/A(Fareylevel)) -0.386 r(shifts to 1:1, 1/modulardistanceto 1:1) 0.469 r(shifts to 1:1, 1/tonguedistanceto 1:1) **0.985 Correlations with 1:2 .~.(s.h..~..t..o....[!.g..~!!~o.~..~..h..!.ft....~..~..~!).. .......................... *.?...9.:.8..~.7......................................... le,-direaed r(shifts to 1:2,A(Amol'dtonguewidth)) 0.004 transitions r(shifts to 1:2, I/A (Fareylevel)) 0.012 r(shifts to 1:2, l/rood, distanceto 1:2)) 0.787 r(shifts to 1:2, 1/tonguedistanceto 1:2) *0.804 9. DISCUSSION A significant correlation was found between measured stability of polyrhythms and a number of cognitive and dynamical indicators of stability in the individual experiments of Treffner & Turvey (1993). When the experiments are considered as a whole, all indicators were significantly correlated. To attempt to refine the comideradon of whether cognitive or dynamical indicators were better predictors, two kinds of functional variation were explored between stability indicators and the measured variable (% unchanged). In tests for linear relation between the stability indicators and either % unchanged or logarithm(% unchanged), correlations were strongest for dynamic indicators 1/Farey level, Arnord tongue width, and cognitive indicators l l n * m , 1/(n+m), and llmax(n,m). Differences between the indicators were not significant. Overall, correlations were better for the direct linear relationship. The amount of variation accounted for is the square of the correlations and for the best indicators with a linear relation between stability and experimental measure this was about 90%. Evidently, dynamical and cognitive indicators are equally effective in interpreting the results of these experiments, if the procedures for evaluation here are accepted. The high correlations found for the best indicators appears to argue for the applicability of the modeling procedures proposed here. With regard to transitions, the 2 best indicators were unequivocally 1/analog shift and l/intertongue distance for 1:1 and 1:2 transitions. Correlations here for the 1:1 case were

164 J. Pressing highly significant, but not in the first instance significant at .05 level (two-tailed) for 1:2 case. However, these correlations were improved to probable significance by considering that transitions to the left in the regime diagram might be characterized by a different linear relation than those to the right. The experimental plausibility of this relied on an argument which suggested that cognitive factors might be introducing asymmetry into regime transition processes. The residual somewhat lower correlations for the 1:2 case may also be interpreted. In the master equation derivation above the prediction was made that linear correlation should be most clear for transitions to weakly populated states. In fact it has been found that the (linear) correlations for transitions to the weakly populated 1:1 state are stronger than to the much more common 1:2 state. Given this, it is natural to suggest that the residual correlation unaccounted for in this case may well be due to the neglected terms in the master equation given earlier involving multiple transitions between states. However, further evidence is needed. Little support was found for principles of transition based on dynamic indicators such as Arnol'd tongue width and Farey level as indicators, or a new proposed indicator, modular distance. However, another new dynamical measure proposed here, intertongue distance, does compare well with the best cognitive indicator, analog shift. In fact, the two indicators are necessarily highly correlated, and it will not be easy to find an experimental design that can distinguish between them. Without this new dynamical indicator, we would have been forced to conclude that cognitive principles were more successful than dynamical ones in interpreting some basic features of dynamics-oriented experiments. 10. COMMENTARY ON THE UTILITY OF CIRCLE MAPS AND R E G I M E DIAGRAMS There is a somewhat covert problem with the use of circle maps and Arnord tongue measurements, be they widths or intertongue distances, to which I now turn. First, relative Arnord tongue distances and widths vary as a function of K, even for the sine circle map. Treffner & Turvey (1993) are not correct when they claim the contrary: \"the relative widths of the Arnord tongues are maintained over the range K < 1.\" (p. 1226). This statement is not supported by Treffner & Turvey's (1993) own sine circle map regime diagram or that given by Jensen, Bak & Bohr (1984) or derivations of asymptotic forms of tongue widths given by Jensen, Bak & Bohr (1984). Arnol'd (1965), for example, found

Testing of Rhythmic Pattern Production 165 that the relative width of the 1:2 and 1:3 tongues for a cosine map is given by (for K not too small) 6 (15) 1:2 / ~ 1 : 3 -- Kq7 Second, although as stated earlier, some properties of circle maps are invariant with respect to driving function g, relative Arnold tongue widths and distances are not. For example, Jensen, Bak & Bohr (1984) investigated maps of the form 0+ 1-0 + [2-(K / 2rt)[sin(2rt0)+ asin3(2rt0)] (16) to address questions of universality, finding that relative Arnol'd tongue widths differed from that of the simple sine circle map. This means that the choice of sine circle map K=I used in some of the above indicators is fairly arbitrary. This objection is the more compelling, in that other workers searching for motor Farey structure have found supercriticality (K>I) (e.g., Peper, Beek, & van Wieringen, in press). Furthermore, measurement of K is not simple, and there may be significant individual differences based presumably on skill, differential learning, and multiplicity of possible strategies (Peper, Beek, & van Wieringen, in press; Summers & Pressing, 1994). Yet the K=I sine circle map values basically worked very well, although admittedly no better than cognitive indicators. Why? This is not completely clear. Experimental work in general reveals patterns that often agree extremely well and sometimes differ from the predictions of the simple sine circle map (e.g. Glazier & Libchaber, 1988). The issue is apparently whether the system happens to have a relative phase-like variable and operate in the regime where attractors are one-dimensional. It is an empirical question whether the circle map predicts well this particular aspect of motor behaviour, or whether higher dimensional maps like the Standard map or the Henon map are required. Since the applicability of collapsing (unstated) more general equations has not been assessed by a systematic procedure, the use of the sine circle map does not rest on a fh'rn theoretical foundation. Nevertheless, it is clearly highly useful and successful as a computational metaphor, and theoretically germane due to universality arguments. However, it would be useful to see the theory develop to the point where it can tackle cognitive experimental protocols on their home ground. At the moment the modeling of temporal covariance

166 J. Pressing structure remains the province of motor-programming/cognitive methodologies. These methods, of course, also do not proceed from first principles. The general notion of transitions within a regime diagram is evidently governed by some sort of Fokker-Planck equation, which can describe the diffusion of probability over an ensemble of systems prepared initially with identical boundary conditions (Risken, 1984), as in the experiments of Treffner & Turvey (1993). However, general discussion of this will be deferred until another occasion (Pressing, in preparation). 11. CONCLUSIONS This paper has found a direct way to compare dynamical and cognitive approaches to the modeling of motor behaviour, albeit initially within the limited domain of polyrhythms or polyrhythmlike behaviour. This has been achieved by extending both modeling procedures into a dynamical-oriented experimental situation, using as a basis the master equation of statistical mechanics and several simple assumptions. These assumptions yield predictions that fit the experimental data very well, which argues for their utility and plausibility. This close fit allowed us to conclude that polyrhythm state stability is equally well predicted by dynamical and cognitive approaches. Likewise, interstate transitions are equally well predicted by dynamical and cognitive models, if the extensions to dynamical principles proposed here are accepted. Without these extensions, cognitive indicators perform clearly better. Since this experimental design was designed to exclude cognition as completely as possible, this falls considerably short of unambiguous support for the dynamic approach. By Occam's razor alone the cognitive approach would seem to have attractions, since it is certainly simpler than deriving circle maps and regime diagrams and justifying their universality by proof. On the other hand, it can be criticized as ad hoc, although this may be the nature of the diversity that adaptive cognition presents us with. However, it is would be very useful to have elaborations of dynamical theory that allow cognitive-type predictions in experimental designs ostensibly favouring cognitive control of motor movement. Further comparative work seems essential. ACKNOWLEDGMENTS I am very pleased to be able to thank Jeffery Summers, Pip Pattison and Philip Smith for useful discussions in the preparation of this article.

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Motor Control and Sensory Motor Integration: Issues and Directions 171 D.J. Glencross and J.P. Piek (Editors) 9 1995 Elsevier Science B.V. All rights reserved. Chapter 7 TEMPORAL COORDINATION OF HUMAN GAIT Bruce Abernethy, Robin Burgess-Limerick, Craig Engstrom, Alastair Hanna and Robert J. Neal Department of Human Movement Studies, The University of Queensland, Australia Locomotion is undoubtedly one of the most natural and functional of all human actions yet examination of the control of walking and running has traditionally been the domain of the biomechanist rather than the motor control theorist. This chapter focuses upon the use of human locomotion as a vehicle for the examination of two pivotal contemporary notions about motor control First, studies are described in which kinematic and electromyographic data from the walking and running gaits are used to examine the contentious issue of whether temporal proporn'onality (or relative timing) is an immutable, velocity-independent property of human movement. Second, datafrom a current project inspired by a dynamical systems perspective are presented which examine the characteristics and determinants of pattern transitions in human gait. Some comments are advanced from these examples on the respective utility of programming and dynamical systems approaches in providing insight into the control of human gait. 1. INTRODUCTION Effective and efficient locomotion is central to the life-long survival of individual animals and the generation-to-generation development and evolution of whole animal species. Perhaps because of this primacy, gait has been extensively studied in a diverse range of animal species, including humans. As a consequence of the focus biomechanists have traditionally placed on the study of human gait, much is now known about the kinematics and kinetics of a number of different human gait forms [Vaughan, Murphy, & du Toit's bibliography of studies on the biomechanics of human gait, for example, lists nearly 3,000 studies in this area at the time of its compilation in 1987]. Much less is known, however, about the control of human gait, especially the temporal co- ordination of human gait. Many of the favoured views of human gait control are based on inference either from studies of lower species (this is especially true of the notion of 'central pattern generators' for instance; e.g., Delcomyn, 1975) or from studies, in humans, of syrmnetrical

172 B. Abernethy et al. paired actions other than gait (this is especially true of the notions of self-organisation derived recently from studies of paired finger movements; e.g., Haken, Kelso, & Bunz, 1985; Kelso, 1984; SchSner & Kelso, 1988 and more recently four-limb co-ordination; e.g., Jeka, Kelso, & Kiemel, 1993; SchSner, Jiang, & Kelso, 1990). It is noteworthy, in the studies of symmetrical paired actions, to see frequent reference to walking-running transitions in human gait as a classical example of emergent, self-organised control (e.g., Kelso, 1992; Kugler, Kelso, & Turvey, 1980; Turvey, Carello, & Kim, 1990) yet little empirical evidence to verify such claims. This chapter briefly summarises a number of completed and ongoing studies within our laboratory which use human gait as a vehicle for examining some contemporary motor control notions -- notions, in most cases, derived from the study of movements much simpler in form and ~ss ecologically-relevant in function than gait. Two key control issues are examined, namely:. (i) whether temporal proportionality (or relative timing) is an immutable, velocity- independent property of human gait; and (ii) whether walking-nmning transitions in human gait show the same self-organising characteristics that typify complex pattern formation and transitions in other systems, both biological and non-biologicaL The former issue is one inspired primarily, but not exclusively, by traditional programming perspectives on motor control (e.g., Schmidt, 1980, 1985); the latter issue emanates directly from contemporary dynamical systems perspectives on motor control (e.g., Kelso, 1986; Kugler & Turvey, 1987; Sch~Sner& Kelso, 1988; Turvey, 1990). 2. IS TEMPORAL PROPORTIONALITY (RELATIVE TIMING) AN IN ARIANT FEATURE OF HUMAN GAIT?. 2.1 Theoretical Foundations Both traditional programming and dynamical theorists of motor control seek to find invariant features of movement because such features potentially provide insight into the underlying control of movement (Abernethy, 1993). The search for invariant or essential features is a central ~ h issue r e g ~ of whether the search is driven by a theoretical model that hypoflw,sises control to be in the structured form of a prescriptive program (or a general plan) or is driven rather by a perspective that views movement features as an emergent signature of the underlying dynamics of the system or organism. One feature, proposed by both programming (Schmidt,

Temporal Coordinationof Human Gait 173 1985, 1988) and dynamical flr~orists (Kelso, 1981; Warren & Kelso, 1985; Zanone & Kelso, 1991) as a prime candidate for invafiance, is relative timing. Relative timing is the proportion of total movement duration spent in any one phase of the movement. The temporal proportionality model of movement control (Gentner, 1987) posits that the relative time spent in any one phase of movement remains unchanged regardless of the total movement duration. In examinations of this notion movement duration is classically manipulated by imposing different speed constraints on the movement. If relative timing is indeed an invariant feature of movement then all identifiable phases, components or segments within a given movement should speed up or slow down in direct proportion to changes in the overall movement duration. Preservation of relative timing across changes in movement duration could be achieved if time was accurately metered out through the central nervous system by some structure, such as the cerebellum, acting in a manner analogous to a 'real time clock' (Keele, Pokomy, Corcos, & Ivry, 1985), (the programming explanation). Alternatively, temporal proportionality could be achieved without reliance on either the representation of time or the presence of an intrinsic time piece, through the preservation of critical phase angles between co-acting effectors (Kelso & Tuller, 1985; Tuller, Kelso, & Harris, 1982), (the dynamicalexplanation). 2.2 Empirical Foundations Substantial support for the concept of relative timing invadance has accrued over the past two decades. Data from studies on fine motor activities such as speaking (Gracco, 1986; Tuller, Kelso, & Harris, 1983), handwriting (Merton, 1972), typing (Terzuolo & Viviani, 1980) and serial key pressing (Summers, 1975); as well as gross activities such as jumping (Clark, 1986), hopping (Roberton, 1986; Roberton & Halverson, 1988) and throwing (Roth, 1988); have all at one time or another, been cited as support for the preservation of relative timing and the primacy of relative timing within movement control Several current motor control texts (e.g., MagilL 1993; Schmidt, 1988) also posit the view that relative timing is an invariant feature of human walking and running. The empirical basis for this view is generally the data of Shapiro, Zernicke, Gregor, and Diestal (1981). Shapiro et al. (1981) used the Philippson (1905) step cycle conventions to calculate the relative times that five male subjects spent in each of four step cycle phases as they walked and ran on a motor driven treadmill at stepped velocities of 1 km.hr-1 for 3-6 krn.hr-1 (for walking) and 8-12 km.hr~ (for running). The proportions of total cycle time spent in each phase at each speed of locomotion

174 B. Abernethy et al. were then pooled and averaged across subjects and an analysis conducted to ascertain if the reladve time spent in any of the phases of movement changed significandy as a function of speed of locomotion. While different relative times for the movement phases were apparent between the walking and running gaits (attributed to walking and running being controlled by different programs) there were no s t a t i s ~ y significant speed of locomotion effects on any of the gait phases within either the walking or running gaits. This was taken by Shapiro et al. (and many other subsequendy) as evidence that relative timing is preserved invariant within the walking gait and within the running gait of humans. Following the lead provided by Gentner (1987), we argue both here and elsewhere (Abernethy, Neal, & Burgess-Limerick, 1995; Burgess-Limerick, Neal, & Abemethy, 1992) that there are sound reasons to examine this conclusion about relative timing invafiance in human gait more thoroughly. In particular, there are three major ~ of concern with the acceptance of the Shapiro et aL conclusions (and indeed, as Gentner, 1987 demonstrates, the interpretation of much of the other data purporting to support the temporal proportionality model). F'wst,the notion of relative timing being preserved invariant across all phases of the step cycle is inconsistent with a wealth of animal locomotion data (e.g., Arshavskii, Kots, Orlovskii, Rodionov, & Shik, 1965; Goslow, Reinking, & Stuart 1973; Macmillan, 1975; Pearson, 1976), and some other human locomotion data (e.g., Gdllner, Halbertsma, Nilsson, & Thorstensson, 1979; Henmn, Wirta, Bampton, & Finley, 1976; Nilsson, Thorstensson, & Halbertsma, 1985; Winter, 1983), which indicate that as speed of locomotion is incwased the time spent by any one limb in contact with the ground (the stance phase) d ~ rapidly and systematically whereas the time spent with the limb in the air (the sw/ng phase) reduces only minimally. Second, the Shapiro et al. analysis, like many others conducted on relative timing, uses group data pooled across a small number of subjects. Data averaged in this way can appear to support relative timing even though, as Gentner (1987) elegantly demonstrates, none of the individual subject data may actually be consistent with the notion of preserved temporal proportionality. The third concern is with the use of null data (in this case the absence of statistically significant main effects for speed) to support the concept of invariance (cf. Corcos, Agarwal & Gottlieb, 1985). Conditions of high intra-condition variability (inflated by the use of grouped data and small numbers of subjects and observations) can result in failure to reject the null hypothesis even when the mean values of relative times per segment are quite variant. Lack of statistically significant differences is not, by itself, sutYacientgrounds to support a concept of relative timing invariance.

Temporal Coordination of Human Gait 175 The constant proportion test devised by Gentner (1987) is one possible approach to more stringent statistical testing of temporal proportionality. In this procedure the proportion of time spent in any given movement phase is regressed linearly against total movement cycle duration and the slope of the line of best fit tested against zero (zero being the hypothetical value if relative timing is preserved invariant). The analysis is conducted independently for each individual subject and the notion of relative timing invafiance rejected if more than 10% of subjects display slopes with gradients differing significantly from zero. ~ alpha level is raised beyond the conventional 5% level as an adjustment for the lack of independence between the variables within the regression equation and from which the estimate of slope is derived). 2.3 Recent Examinations of Relative Timing in Gait Kinematics Given our concerns with the widely cited Shapiro et al. data and the emergence of some more stringent analytical methods for testing the relative timing notion we (Abemethy et al., submitted) mx~entlyre-examined the question of temporal proportionality in human gait by applying Gentner's (1987) constant proportionality test to a considerably larger sample size and number of step cycle observations than had been used in the previous studies. Our intent was to ascertain if relative timing invariance indeed emerges in human gait once conservative (and, in our view, more appropriate) analytic techniques are applied. The gait kinematics of a total of 20 healthy, young subjects (10 male and 10 female) with no known gait abnormalities were recorded using high speed video as the subjects locomoted on a motor-driven treadmill for approximately three minutes at each of two walking (4 & 6 km.hr-1) and four running (8, 10, 12 & 14 krn.hr-1) velocities. The timing ofheel str~e and toe-offin each step cycle was determined from footswitches connected to the front and rear of both shoes while the continuous displacemem-time arrays for the hip, knee, and ankle joints were derived from the videotape records of the motion, in the sagittal plane, of retromflective markers placed on the left greater trochanter, lateral epicondyle of the femur, lateral malleolus of the fibula and head of the fifth metatarsal recorded and digitised at 100Hz. Times spent in the stance phase (defined from heel str~e to toe-off for the same limb), in the flexion component of the swing phase (defined from toe-off to maximum knee flexion) and in the extension component of the swing phase (defined from maximum knee flexion to heel stn~e) were determined for each of 10 s u ~ i v e step cycles and then the proportional time spent in each of these three components of the step

176 B. Abernethy et al. cycle was regressed against step cycle duration. Gentner's (1987) constant proportion test was then applied separately to the data derived for the walking and running gaits. The analyses revealed significant departures for both the walking and running data from the predictions of the invadant temporal proportionality model For the walking data deviations from the expected zero slope were relatively minor in magnitude (mean gradients ranged from -0.034 to +0.023 across the various phases) but nevertheless sut~ient to show significant slope deviations from zero for 15 (75%) of the subjects on the stance phase, 9 (45%) of the subjects on the extension component of the swing phase and 12 (60%) of the subjects on the flexion component of the swing phase. In all cases this was much more than the maximum of 10% of subjects showing non-zero slopes that Genmer suggested be the upper limit permissible as support for relative timing invariance. The departut~ from reladve timing invariance were more pronounced for the nmning gait than for walking. Gradients in running ranged from -0.284 to +0.256, with 18 (95%), 15 (79%) and 18 (95%) of the 19 subjects for whom full data were available showing significant non-zero slopes for the stance, swing (flexion) and swing (extension) phases of the gait cycle respectively. As had been observed previously (primarily in the animal literature but also in the human literature) increased speed of both walking and especially running was achieved primadly through a reduction in stance time rather than a proportionate reduction of all phases of the gait cycle. Our analyses of gait in this setting, indicating that relative timing is not an invariant feature of human gait kinematics, were consistent with our earlier analyses of stair climbing (Burgess-Limerick et al., 1992) but clearly inconsistent with the conclusions of Shapiro et ak (1981). 2.4 Possible Contaminants of the Kinematic Observation of Relative Timing Invariance The absence of preserved temporal proportionality at the level of kinematic observation does not, and cannot, as Heuer (1988) correctly points out, preclude the possibility of temporal proportionality being preserved at other, more central levek of observation. It remains possible, for example, that efferent commands are issued centrally with strictly preserved temporal proportionality but that a number of intervening effects may prevent this temporal invariance from being expressed at the more peripheral level of kinematic observation. There are a number of possible s o ~ of peripheral obscurity of relative timing invariance. Foremost amongst these are: (i) d i f f e r e ~ in efferent neural transmission times between proximal and distal muscle groups involved in the same action (this has the effect of adding a constant rather than a scaled

Temporal Coordination of Human Gait 177 delay to the temporal expression of the central commands); (ii) muscle-to-muscle (and even motor unit-to-motor unit) differences in electromechanical delay;, and ('hi) variations in the inertia needed to be overcome by different muscles before observable movement can take place. The latter may be especially important in gait where the duration of the swing phase of the limb may be limited largely by inertial constraints (McMahon, 1984). Given that the first of these possible sources of peripheral blurring is likely to induce only relatively trivial deviations in temporal co-ordination, we have focussed our empirical examinations on the impact of electromechanical delay and inertial resistance on relative timing. We have tested the prediction that relative timing is preserved centrally but not expressed peripherally due to the effect of electromechanical delay and inertial resistance by comparing the relative proximity to temporal proportionality:. (i) of relative times calculated from kinematic data and relative times calculated from electromyographic (EMG) records of muscle onset-offsets and (ii) of relative times derived from the kinematics of actions produced with and without supra- normal levels of lower limb inertia. If electromechanical delay and inertial resistance are source~ of peripheral blurring which prevent the expression of a central pattern of preserved temporal proportionality, then a closer proximity to temporal proportionality should be expected for relative times detennire,d: (i) electromyographically rather than kinematically; and (ii) under unloaded as opposed to loaded limb conditions. A closer proximity to temporal proportionality could be expressed in such comparisons either: (i) in terms of a ~ slope significantly closer to zero; or (ii) in terms of a greater proportion of subjects displaying slopes (from the constant proportion test) which do not differ significantly from zero. The comparison of temporal proportionality between relative times detemained kinematicaUy and relative times determined from EMG records was undertaken by simultaneously recording lower limb kinematics and surface EMG from the right vastus medialis, biceps femoris and medial gastrocnemius muscles of 20 young, healthy subjects (10 males and 10 females) as they ran at velocities of 9, 12, 14, 16 and 18 km.hr-1 on a motor driven treadmill The kinematic analysis of temporal proportionality for these subjects resulted in constant proportion test slopes and rates of zero slope rejections comparable with those obtained for the subjects used in the earlier exclusively kinematic study. The assessment of temporal proportionality at the EMG level was undertaken by determining the duration of burst activity in each muscle, expressing this duration as a proportion of cycle duration and then regressing this proportion against cycle duration. Previous studies on upper limb positioning have been equivocal with respect to whether temporal

178 B. Abernethy et al. proportionality is amplified or dimims\"hed at the EMG level compared to the kinematic level (Carter & Shapiro, 1984; Shapiro & Walter, 1986). The mean constant proportion test slope values were found to be lower for the EMG-derived measures than for the previous observations on kinematic data although the percentage of individual subjects with regression gradient significantly different from zero was still such to warrant rejection of the notion of invariant relative timing using Gentner's (1987) criterion. Moving to the surface EMG level of observation therefore appears to remove some of the relative timing variance evident at the kinematic level but nevertheless instdfcient to support the notion that the commands being sent to the periphery contain strictly preserved temporal proportionality. This finding is consistent with some existing data on the EMG changes in walking with increased cadence which also conclude against a simple change of the EMG pattern by a multiplicative constant (Yang & Winter, 1985). The companion test on the putative role of inertial resistance in possible peripheral blurring of relative timing in gait was performed by having five male subjects run on the treadmill at velocities of 9, 12, 15 and 18 km.hr1 under conditions where they were either unloaded (i.e., normal running) or loaded with masses of either 1.0 kg or 1.5 kg attached bilaterally to the lower leg at a position just above each ankle. If modulations in inertia are indeed a cause of kinematic blurring of relative timing invariance it was expected that the incremental addition of lower limb inertia though the loading conditions would induce a systematic regression away from the zero slope sought with the constant proportion test. Preliminary analyses of these data have indicated that despite the additional energetic cost of running with leg weights (see also Miller & Stamford, 1987; Soule & Goldman, 1969) no changes attributable to the added limb inertia were apparent either in overall gait kinematics or in the proportional stance and swing time components of the gait cycle. At this stage fle,se data are unable to support the notion that relative timing is preserved centrally but merely prevented from expression in the gait kinematics by lower limb inertia. 2.5 Some Conclusions and Unresolved Issues Despite assertions to the contrary in some of the existing literature on human motor control, the findings from our series of studies are clearly inconsistent with the conclusion that relative timing is an invadant feature of the kinematics of human gait. Moreover, the more recent electromyographic and inertial loading studies have been able to provide only very modest

Temporal Coordination of Human Gait 179 support for the contention that relative timing is preserved invariant at a central level but that this invariance is not exhibited in the gait kinematics because of a variety of central-to-peripheral distortions and non-linearities. While the available empirical data, when considered collectively, argue quite strongly against relative timing being an invariant and fundamental property of human gait, interest in the more general notion of temporal proportionality as an invariant feature generic to a vast array of motor skills persists unabated (e.g., see Fagard & Wolff, 1991). A number of fundamental issues with respect to relative timing invariance remain unresolved which are directly relevant to the current focus on gait control. One major issue relates to the rigorous use of Gentner's (1987) constant proportion test as a means of statistically testing temporal proportionality. A number of authors (e.g., Viviani & Laissard, 1991) have expressed concern that the constant proportion test may be too strict (and conservative) to demonstrate, as statistically significant, temporal propordonafity invariances which may be biologically significant. Some altemative tests of bio-equivalence are available (e.g., Hauk & Anderson, 1986) which commence a priori with an accepted error band for equivalence (or invariance) and then test statistically whether two or more mean values fall within these a priori bounds. While such methods have been employed effectively in comparative drug treatment trials, for instance, where the issue of biological significance and equivalence are paramount, these methods have not, as yet, been applied to questions of motor system invariance. The major difficulty, of course, is determining, in a non-arbitrary way, the a priori bounds for ~ p t i n g biological equivalence and avoiding the inference that the error bounds have been set post hoc to support the ~ r e t i c a l bias of the researcher. A second issue relates to the duration of temporal proportionality and the question of whether temporal proportionality ought to be expected to be preserved over the total movement duration or whether it might reasonably be expected to be applied selectively to sub-parts of movements (Heuer, 1991). In the case of gait, for instance, is the total cycle duration the appropriate unit over which to seek relative timing invariance or might gait rather be viewed as two essentially independent movements (nominally a stance movement and a swing movement), each with its own separate invariant relative timing? While the latter proposal may provide a better explanation of the empirical data, such an approach is clearly not parsimonious because it doubles the number of elements which need to be independently controlled. This problem of increasing control elements (or 'programs' in the traditional sense) is further exacerbated if one entertains the

180 B. Abernethy et al. otherwise plausible notion that temporal proportionality may not generalise to any particular action but may be both task- and/or context-specific (Heuer, 1991). Finally, the suggestion by Heuer (1991) that relative timing might be best considered not as a categorical variable which can assume only a limited, discrete set of values (e.g., one value for a walking gait, a second value for a running gait etc.) but rather as a continuous variable that can be adjusted by parameters, warrants serious consideration. Such a perspective raises the issue of how optimal states of relative timing are achieved and whether in actions as highly learnt as gait there am preferred 'natural' values of relative timing. Further, ff there am preferred relative timing states, the question of how one might determine such states experimentally provides a 'stepping off\" point from conventional, cognitive approaches to movement control to more dynamical, natural-physical methods of analysis. 3. THE STUDY OF PATrERN FORMATION AND TRANSITIONS IN HUMAN GAIT 3.1 Synergetics as a Framework One way of attempting to understand movement control is to examine single movement patterns of relative stability and seek to identify invafiances by changing various aspects of the conditions within the boundary conditions for that movement pattern (Abemethy, 1993). This is precisely the approach taken in the studies described thus far in this chapter where single co- ordination modes such as walking and nmning have been examined separately with velocity being changed within the comfort bounds of each movement pattern to ascertain if relative timing remains immutable across these changes. An alternative approach is to examine the transitions between different patterns of co-ordination (such as between walking and nmning) with the view that the parameters exposed as key ones for order and control at the transition point also reveal themselves as valuable for control under conditions of pattern stability (Jeka & Kelso, 1989). This alternative approach is derived from synergetics (Haken, 1983), which is the science concerned with the understanding of pattern formation in complex systems. The attractions of applying a synergetic approach to complex movement control are, among other things, that the approach: (i) offers increased parsimony, in the sense of seeking a generic explanation for multiple rather than single patterns of movement; (ii) is less arbitrary than many existing cognitive models of movement control with respect to the a priori assumptions it need (or more importantly need not) make regarding the type of representation and computation which is performed; and (iii)

Temporal Coordination of Human Gait 181 provides the potential to seek paraUels between pattern formation in the motor system and pattern formation in other systems (both biological and non-biological), thus providing a tlw.am of assessing the extent to which dynamical models of movement control, grounded in natural law, may be feasible. The next section of the chapter describes our rationale and preliminary attempts in applying the synergetics approach to examination of walking-nmning phase transitions in human gait. There are a number of precedents in the use of dynamical approaches, but not necessarily synergetic one& to the study of animal gait (e.g., Collins & Stewart, 1993; Taga, Yamaguchi, & Shimizu, 1991). In animals, particularly quadrupeds, it has been clearly demonstrated that a relatively small range of velocities of locomotion are naturally favoured within each gait mode (nominally walking, trotting and galloping in quadrupeds). It has been traditionally accepted that animals naturally select speeds of locomotion in each gait form where efficiency (measured in terms of metabolic costs per distance moved) is optimised (e.g., Alexander, 1989; Heglund & Taylor, 1988; Hoyt & Taylor, 1981) although some alternative evidence (e.g., Hreljac, 1993) suggests that the trigger for transition from one gait mode to another may not be inefficiency but some other factor such as the attainment of a critical level of skeletal force during limb-ground impact (Farley & Taylor, 1991). The possibility of a physical trigger for the initiation of changes in gait mode is clearly counter to the traditional human motor control perspective that significant changes in movement pattern are accomplished by switching (consciously or otherwise) from one motor program to another but gains some additional support from the biological scaling data of Kugler and Turvey (1987) and Turvey, Schmidt, Rosenblum, and Kugler (1988). Within the constraints of scaling methods the biological scaling data suggest, that for a wide range of quadrupeds, them is a relatively linear relationship between simple mass and leg length ratios and the preferred natural frequency for each of the walking, trotting and cantering gait modes. Given that walking-nmning transitions may be driven primarily by physical rather than cognitive variables it appears sensible to seek to explain more fully the control of walking-running patterns and their stability through the use of physical rather than cognitive ~ r i e s . Further, given that synergetics provides a principled insight into pattern self-organisation and re-organisation in non- biological systems, synergetics would appear to offer a useful set of tools and premises to commence the search for a natural-physical explanation of walking-running transitions. Phase transitions in self-organising systems (biological or non-biological) are characterised by spontaneous changes in the form of spatial organization (the order parameter) as the value of

182 B. Abernethy et al. another parameter (the control parameter) is altered beyond critical levels. Water, for example, undergoes discrete and spontaneous changes in its molecular spatial organisation (from solid to liquid to gas) as the control parameter (temperature) is continuously altered through a range of (a) H20 gas- Order liquid-- Parameter (structure) solid I (b) Finger Movements 0 100 in phase-- Control Parameter (Temp. ~ Order C> Parameter (relative phase) out of phase iI (c) Human Gait Slow Fast Control Parameter (Cadence) Order Parameter (????) walk II Slow Fast Control Parameter (Locomotion Speed) Figure 1. Typicalorderparameter-controlparameterrelationshipsfor known patterntransitionsin (a) non-biological systems; Oa)pairedfinger movements and (c) human gait. values from below 0~to above 100~ (Figure la). Paired finger movements may be characterised in a similar manner with phase transitions within the order parameter of relative phase occurring at critical values of the control parameter of cycling frequency (Sch6ner & Kelso, 1988). Movements commencing in anti-phase switch spontaneously to in-phase as cycling frequency is continuously incremented but remain in the more stable in-phase state when cycling frequency is

Temporal Coordination of Human Gait 183 decreased below the initial critical frequency (Figure l b). Given that water and other non- biological systems lacking nervous tissue are able to self-organise their component elements into orderly, stable patterns in the absence of any form of higher control or mental representations of the required pattern it is clearly tempting to question the necessity and/or role of any such higher control and representation for control of (at least superficially) comparable patterns of self- organisation in human movement. The parallels between phase transitions in other systems (both biological and non-biological) and phase transitions in some human actions certainly seems sutticient to warrant closer examination. Turvey (1990), drawing primarily on the work of Haken et al. (1985), has outlined a number of properties known to characterise phase transitions in a broad range of different physical systems. These properties are: 9 modality - the order parameter has two or more distinct values (in the case of water these are its solid, liquid and gas states; in the case of paired finger movements the in-phase and anti-phase states); 9 i n a c c e s s i b i l i t y - spatial organisations other than the preferred one cannot be reliably maintained (e.g., ice cannot be reliably maintained when the control temperature e x ~ 0~ just as anti-phase paired finger movements cannot be reliably maintained when the critical cycling frequency is exceeded); 9 sudden jumps - small changes in the control parameter can induce rapid changes in the order parameter (e.g., small changes in temperature around 0~ or 100~ or small changes in cycling frequency around the transition point bring about complete changes in macroscopic spatial organisation whereas comparable increments during periods of pattern stability typically induce no perceivable change in organisation); 9 hysteresis - the magnitude of the control parameter at the point of transition in the order parameter from one form of spatial organisation to another may vary dependent on the direction in which the control parameter is changed (e.g., in paired finger movements reliable transitions from anti-phase to in-phase are apparent when cycling frequency is incremented but comparable transitions back to anti-phase do not occur when the control parameter is decreased back through the ascending transition value); 9 critical f l u c t u a t i o n s - variance in the order parameter increases as the transition point is approached (e.g., variance in relative phase is markedly higher around the critical cycling frequency for anti-phase to in-phase transitions of paired finger movements than it is at

184 B. Abernethy et al. other cycling frequencies where pattern stability is greater; Kelso, Scholz, & SchSner, 1986); critical slowing down - the dme taken for an order parameter to return to equilibrium after a perturbation increases as the control parameter value approaches the transition point (e.g., critical slowing down is a demonstrated property within paired finger movements; Scholz, Kelso, & Sch6ner, 1987). The crucial question in the current context therefore is do these characteristics also hold true for phase transitions in human gait? A number of theorists (e.g., Kelso, 1992; Kelso et al., 1986; Kugler et al., 1980; Turvey et al., 1990) have, at various times, drawn attention to the parallels between the phase transitions which occur as a nmam of self-organising purely physical systems and the spontaneous shifts in movement organisation which occur from walking to running in the locomoting human but the extent of such parallels has not been subjected to systematic empirical analysis. The concern, of course, is to ascertain whether such similarities are real or merely illusory and whether, in turn, if the parallel proves to be a reasonable one, it provides any additional explanatory power with respect to gait control beyond that afforded by traditional explanations. 3.2 Preliminary Empirical Observations While it is self-evident that there are a number of identifiable, distinct human gait modes (of which walking and running are the most common), thus satisfying the moda/ity characteristics of self-organising systems, it is far ~ apparent how best to succinctly describe ~ modes through the use of a quantifiable order parameter (see Figure lc). The selection of an appropriate order parameter to describe human gait modes becomes most pronounced in attempting to satisfy the sudden jumps characteristic. The logical starting point for a synergetic explanation of phase transitions in human gait is therefore with the search for an order parameter which is continuously ~ u r a b l e , demonstrates sudden rather than monotonic jumps in value as the transition between walking and nmning is enacted, and yet remains essentially stable during periods of comfortable walking and running. What then might be the gait equivalent to relative phase in paired finger movements?

Temporal Coordinationof Human Gait 185 600 - Running Cycle Walking Cycle 400 - 200 - Transition 0 \"Cycle\" > 0- < -200- -400 - -6O0 IIIIl 1 -120 -100 -80 -60 -40 -20 0 Angular Displacement (wrthorizontal) Figure2. A typicalphaseprofileofthe shankduringprogressivelyincrementedgaiton a treadmill. Characteristic~ h i c changesbetweenthewalkingandrunninggaitarereadilyapparent. A useful starting point for discovering potential order parameters for human gait is to examine the many excellent perceptual studies of gait (Cutting, 1981; Cutting, Proffitt, & Kozlowski, 1978; Hoenkamp, 1978; Todd, 1983)that have sought to identify the perceptual parameters that human observers use to discriminate, among other things, between walking and nmning gaits. So fight is the reciprocal coupling between perception and action (Turvey & Carello, 1986) and so sensitive is the human perceptual system to biologically relevant perceptual information (Runeson, 1977) that it would not surprise if the perceptual parameters used by human observers to visually determine if someone else is walking or running were also the same order parameter (or a close derivative thereof) which fully describes the spatial organisational differences between the two modes of co-ordination. Todd's (1983) analyses suggest that the motion of the lower leg (i.e., the tibia) provides the pivotal perceptual information for the discrimination of walking and running gaits and indeed phase profiles (angular velocity vs angular displacement) plotted for the shank with respect to the horizontal for running and walking data collected in our laboratory appear to provide clear qualitative differentiation of the two gait modes (Figure 2). Ongoing work in our laboratory is concerned with reducing the continuous

186 B. Abernethy et al. data contained within the phase profile to an order parameter with discrete, unitary values. The centroid of each cycle appears at this time as a prime candidate for fiarther scrutiny as an order parameter for describing human gait transitions in terms consistent with the synergetic notion of sudden jumps. 8.1 - (a) 7.8- 7.5- I-I Ascending Transition K~ Decending Transition .~ 7.2 ID r~ 6.9 ,..-, 6.6 [.-, 6 . 3 - ! ! i, 1 2 6 Trial Day (b) 8.1 7.8 IK-\"~ .t:: \\-,4 7.5 7.2 INNI \\\\ IXNI \\\\ o') 6.9 IXNI \\\\ .=_ \\\\ ! \\\\ 6.6 \\\\ 1 tD x.\\ \"x\\ 6.3 x\\ 6 i! 234 5 Trial Series 8.1 - ,~ 7 . 8 - 7.5- ~ 7.2- r~ ,_.., 6 . 9 - ~ 6.6- [\"' 6.3- I 1i i I 6 1 2 34 5 Trial Series Figure 3. Velocity at gait transition for 4 subjects under conditions where treathnill speed is progressively either increased or decreased in a continuous (ramped) fashion. Data are plotted as group mean values averaged across 2 days (a) as well as separately for days l(b) and 2 (c).

Temporal Coordination of Human Gait 187 (a) 8.1 -- [\"l Ascending Transition [] Decending Transition ---\" 7.8- |I..~ 7.5- \\xj \\NI \\'q \\NI _..,,.... ~ 7.2- \\\\1 \\NI %N rm 6.9- \\xJ \\NI \\NI N\\ E 6.6- -,,',4 \\xa \\NI \\NI N x. ~ 6.3- \\%1 \\NI \\',J X.\\ %M \\\"J \\\\1 \",,',4 \\NI N\\ \\NJ \\NI ,,.',l \\\\1 %\\ \\x.J \\NI \\',q \\NI %% \\\\J \\',4 \\NI \\\\ \\NJ \\xa \\NI \\x.j \\xa \\NI \"-.. N \\,q \\NI %% \\NJ \\ -,4 \\NI \\\\ \\\\1 \\-4 \\NI %'x. \\x.j %% 234 \\x. \\\\1 \"xx, 1 \\\\ 5 Trial Series (b) 8.1 -- -~\" 7.8- 7.5- \\NI \\'xl \\\\ \\Nl W'k\"! \"~ 7.2- \\NI %% \\Nl \\\\ \\NI \\NI \\-,, \\Nl 6.9 \\Ni ',.%! \\\\ \\',4 ~ 6.6- %'xJ \\\\ \\Nl [... 6.3- \\%J \\\\i \\\\ \\N] \\\\l \\\\J \\'M \\\\ \\\\1 \\\\ \\Ni \\\\ \\\",I \\Nl \\\\ \\Nl \\\\ \\Nl \\\\ \\Ni \\\\ \\\\I \\N! \\\\ \\Nl \\\\ \\'M \\\\ \\\\ \\\\1 \\Nl \\\\ \\Nl \\\\ \\\\l \\\\ \\Ni \\\\ 1 \\Nl \\\\ \\Ni \\\\ \\\\l \"\\\\ \\Nl \\\\ \\'M \\\\ \\Ni \"x\\ \\'M \\Nl \\'M 3 \\ \\1 5 \\NI 4 \\ \\1 2 Trial Series Figure 4. Velocity at gait transition for 15 subjectsunder conditions where treadmill speed is progressivelyeither increasedor decreasedin 0.3 km. hr~ steps. Datashown are groupmean values for day 1 (a) and day 2 (b). Some preliminm3, data are also available at this stage to examine whether or not hysteresis and critical fluctuations are characteristics of human gait transitions, although such examinations are n~sarily preliminary until the issue of the most appropriate order parameter is established with greater certainty. Examinations to date using both continuously ramped and discretely stepped ascending and descending series of manipulated treadmill velocities have failed as yet to reveal any systematic hysteresis effect (Figures 3 & 4). While mean treadmill speed at the transitions from walking to nmning for the ramped protocol have been, on average, higher than the treadmill speed at which transitions from running to walking occurred for each of 2 days of

(a) 8 0 0 - ~, 600- o 400- U 200- >1 ) 0- ~>o-200 - < 8 -400- :~ -600 - -800 I 1 i' 1 i l '1 40 60 80 100 120 140 160 180 Knee Angular Displacement (~ (c) 800- ,~ 600- o 400- U o 200- 0- ~-2oo- < -400 - %1 -600 - -800 I I \"i ..... w I w I 40 60 80 100 120 140 160 180 Knee Angular Displacement (~ Figure 5. Phase profiles of the knee joint in running at (a) 8, (b) 10, (c) 12 and (d) 14 k usual walking transition point.

(b) 800 - I#l 6 0 0 - o 400 - oU 2 0 0 - >O 0- r- o -400 - t~ -600 - -800 L_ 1 I I I I I I 40 60 80 100 120 140 160 180 Knee Angular Displacement (~ (d) 8 0 0 - -l~#t 6 0 0 - o 400 - o,,.d 200- II > 0- \"~o-200 - o -400 - O i~ -600 - -800 ,l I l I I I 40 60 80 100 120 140 160 180 Knee Angular Displacement (~ km.hra showing greater variability for the running velocity (8 km.hr-1) closest to the

Temporal Coordination of Human Gait 189 testing (Figure 3a), perhaps indicating a preference for subjects to maintain existing modes rather than change modes, the data when examined more closely on a series-by-series basis are clearly variable and non-systematic (Figures 3b & 3c). Data generated using the stepped protocol (treadmill velocity steps of 0.3 km.hr1) are much more consistent however, and again are in the direction of indicating higher treadmill velocities for the walking-running transition (M=7.77 krn.hr1) than for the nmning-walking transition (M=7.39 krn.hr~) (Figure 4). This observation is consistent with those made by Thorstensson and Roberthson (1987) and Beuter and Lalonde (1989). However, while on first approximation our data are suggestive of hysteresis, such a conclusion is not ~ a r i l y warranted as the directionally-specific ~ differences in the control parameter value (treadmill speed) at the transition point fall within a single (0.3 km.hr~) step increment (the 7.4 - 7.7 km.hr1 step). If hysteresis effects indeed exist they must be within this step interval and necessitate a finer graded step protocol than has currently been employed to uncover them. With respect to critical fluctuations, some existing phase portraits of the knee joint of a subject nmning at 8, 10, 12 and 14 krn.hr1 (Figure 5) show qualitatively the expected greater variability of the phase portrait at velocities close to the transition point (the 8 krn.hr~ condition) than velocities more distant from the transition point (the higher velocities) but these differences have not as yet been quantified nor assessed directly using the putative order parameter. The remaining synergetic characteristics of inaccessibility and critical slowing down have not yet been examined but, in some sense, will represent the more critical tests with respect to the value or otherwise of assessing human gait transitions in synergetic terms. The issue of the inaccessibility characteristic is especially pertinent in the case of human gait as it is readily apparent, from events such as competitive race walking (van Wieringen, 1989), that it is possible for humans to alter the normal walking-running transition point voluntarily and sustain non- preferred modes of co-ordination apparently to a much greater extent than is possible in other favoured phase transition tasks such as those involving paired finger movements. The key issue is whether the overriding of the natural (preferred) system dynamics is achieved cognitively and, if so, at what attentional cost? Work we are currently undertaking involves the use of dual-task methods to determine the attentional requirements of preferred gait (walking at a treadmill speed consistent with walking and nmning at a treadmill speed consistent with running) and non- preferred gait (walking at speeds normally consistent with running and running at speeds normally consistent with walking). If sustaining non-preferred gait requires increased allocation of

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Motor Control and Sensory Motor Integration: Issues and Directions 199 D.J. Glencross and J.P. Piek (Editors) 9 1995 Elsevier Science B.V. All rights reserved. Chapter 8 THE CONTRIBUTION OF SPONTANEOUS MOVEMENTS IN THE ACQUISITION OF MOTOR COORDINATION IN INFANTS Jan P. Piek School of Psychology, Curtin University of Technology A growing number of researchers in the area of motor development are acknowledging the role that spontaneous movements play in the acquisition of motor coordination in infants. The nature and characteristics of spontaneous movements have been determined through both qualitative and quantitative methods. Many of the qualitative studies have described spontaneous movements as representing part of a maturational process, dependent on the development of the appropriate neural pathways. Several recent quantitative studies investigating spontaneous leg kicking in infants have described these types of movements in terms of Bernstein's (1967) coordinative structures, suggesting a dynamical systems approach to the understanding of motor development and control. In the present study, it is proposed that infant spontaneous movements freeze or reduce the degrees of freedom through rigid co-contractions in order to facilitate the learning of temporal and spatial parameters that are essential for the development of voluntary motor control Spontaneous movements then disappear as they are replaced by voluntary-controlled coordinative structures different in nature to the earlier rigid couplings imposed during spontaneous movements. 1. INTRODUCTION Infants appear motorically uncoordinated at birth, producing movements which suggest a lack of organisation or intent. Primitive reflexes such as sucking and the protective withdrawal provide the basic responses needed for the neonate to survive. Reflexes have been studied in great detail over this century and have formed the basis for many of the original theories of infant motor development. In a recent review, McDonnell and Corkum (1991) described the possible role of reflexes in the development of motor coordination. The notion that reflexes form the \"ouilding blocks, for later motor control was termed the 'motor- continuity' theory (McDonnell & Corkum, 1991). An alternative view was that primitive reflexes interfered with normal motor development and needed to be inhibited before motor coordination could develop. For example, the continuation of the asymmetrical tonic neck reflex beyond 2 - 3 months can prevent the infant from rolling over, resulting in a delay in motor development. McDonnell, Corkum and Wilson (1989) have suggested that primitive