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Punching unpredictability, diversity and efficiency ratio dependencies
Pearson product - moment correlation coefficients between the scaled distance dependent unpredictability as measured by entropy (H), diversity as measured by symmetry (S) and efficiency ratio (E) are presented in Table 5.
These three measures showed significant positive relationships. Change of group mean punching unpredictability and diversity as a function of the performer - target scaled distance is presented in Figure 2D. In similar vein, the change of the group mean punching unpredictability and efficiency ratio is presented in Figure 3. It is readily visible that these three measures reached their maximal values for performer - target scaled distances around D = 0.6. In other words, for the distance region where the punching efficiency ratio becomes maximal, the unpredictability and the diversity of punching action modes become maximal as well. This relationship is probably connected to the following causes: The efficiency ratio measure E increases when the sum of perceived efficiencies of selected punches increases and their difference decreases. The obtained efficiency ratio scaled distance dependence shows that around D = 0.6 this ratio is maximal.
The maximum of perceived efficiency ratio does not imply that all of the action patterns attained maximal perceived efficiency values. For example hooks and uppercuts possessed larger perceived efficiency values for smaller scaled boxer - target distances than D = 0.6. Jabs attained maximal perceived efficiency values for somewhat larger scaled distance values. The maximum of the perceived efficiency ratio signified that there is an optimal proportion between the perceived efficiencies of the actions and their differences. The maximum of this function is attained when there are relatively high perceived efficiencies of the whole set of actions and small inter-efficiency differences.
In other words, in this region the sum of the perceived punching efficiencies is relatively high and the perceived punching efficiency differences are minimal. This proportion maximizes the efficiency ratio function. On the other hand the unpredictability and diversity measures increase when the probabilities of all action patterns attain closer values. From the previously discussed results it was clear that in this specific region all types of punching action patterns were active and the probabilities of their occurrence possessed relatively similar values. This characteristic maximizes the unpredictability and diversity measures as well.
Now starting from D = 0.6 and moving toward the larger or lower D values (see Figures 2 D and 3), these criteria are satisfied to a lesser extent due to the paucity of the action repertoire of the performers available at these distances and the larger differences between the perceived efficiencies, which leads to the overall decrement of these three functions.
Overall, the relations between handstriking unpredictability (H), diversity (S) and efficiency ratio (E) show that the performers were able to discover and exploit the scaled distances D where the perceived efficiency differences of their action patterns were minimal and yet their absolute values were relatively high making the efficiency ratio (E) maximal. The observed relationships between the perceived efficiency ratio and the action unpredictability and diversity is yet another corroboration of the sensitivity of human movement systems (represented here by the novice boxers) to the interacting environmental and individual information flows which constrain their actions toward a target.
Movement selection meta-stability and diversity
The meta-stability (dynamics at the edge of instability) evident in the boxers' punching data is portrayed in Figure 2C. The probabilistic nature of the activation and temporal coupling of particular punching modes suggests this fundamentall meta - stable nature of the dynamics.
For fixed values of scaled distance to the target there are coexisting (simultaneously available) states of actions implying that the dynamics of boxers' movement systems over time is dominated by transitions between different regions of the perceptual- motor landscape. To enable the flexible switching between the punching modes the boxers' movement system has to be poised at the edge of instability from where each punching mode can be spontaneously activated under the task and the perceived environmental constraints.
It is important to note first that through a systematic gradual change of the scaled boxer - target distance, the number of available movement solutions changed abruptly. The probability of these abrupt changes (i.e. bifurcations) of specific punching action modes increased in well defined D regions, i.e. when the perceptual - motor system of the boxers became maximally susceptible to the small changes of the scaled distances, which together showed that these changes were most probably due to nonlinear dynamical causes. As presented on Figure 2C, at scaled boxer - target distance D = 1.2 the inactivity state splits (i.e bifurcates) into two punching solutions (right and left jabs). Around D = 1.05 two more motor solutions to the punching task emerged (hook punches) and eventually at D = 0.9 another two motor solutions emerged, this time the uppercut punches. In the interval 0.9 > D > 0.45 the whole set of punching modes was simultaneously available. Around D = 0.45 jab punching modes ceased to be used and at D = 0 the uppercuts ceased to exist as well. Also, as shown above (Figure 2 B), with each abrupt change in the motor solution to the task, the whole probability landscape changed making some of the already active striking patterns less or more probable than before. This finding signified that with each newly formed or annihilated movement solution, not only local, but a global restructuring of the action landscape arose allowing a greater degree of flexibility in the decision - making space of boxers. The newly formed movement solutions reduced the probability of occurrence of the other previously active punching modes and effectively created a more equilibrated medium in the decision-making space of the boxers increasing the competition between the different action modes.
The meta-stability (weak instability) of punch selection is a key characteristic of the action flexibility of boxers because it avoids maladaptive mode-locked (more rigid) forms of behaviour that are too stable (and thus predictable) in the dynamic context of the martial arts.
The meta-stability (weak instability) of punch selection is a key characteristic of the action flexibility of boxers because it avoids maladaptive mode-locked (more rigid) forms of behaviour that are too stable (and thus predictable) in the dynamic context of the martial arts. This adaptive behavior enables boxers to increase the uncertainty of their actions with a maximum occurring in the region around D = 0.6 that corresponds to the maximum values of the unpredictability diversity and efficiency ratio measures (see Figure 2D and 3). These data highlighted that one of the most important constraints on the dynamics of the movement-information couplings of boxers is the scaled relative distance to a target. This constraint shapes decision-making from moment to moment by altering the dynamic range of available action solutions via the distance-dependent perception of punching efficiency in individual boxers.
Emergence of performer - performer coordinative states
Strategic positions of boxers may be observed as performer - performer coordination dynamical states similar to those observed during sub-phases of sports such as dribbling in basketball and sailing regattas (see Araújo et al., 2005; Davids et al., 2005). These states are formed spontaneously which means that there is no specific rule imposed from an external source to which boxers should adhere in forming the strategic positions. In other words, 1 v 1 coordination states in boxing emerge (i.e. they are not imposed by external influences) in a self - organized fashion from the interactions between the perceptual, cognitive, intentional and motor subsystems of the boxers. Anecdotal observations from boxing matches suggest that boxers spend a lot of time in well-defined relative distances from each other, which are the regions around the edge of jab reachability D = 1.2. In Figure 3 it can be noted that these regions are actually regions of minimum unpredictability (H), diversity (not presented on Figure 3, but see Figure 2D) and perceived efficiency ratio (E) of an opponent's actions.
Under the task constraints of boxing, on the one hand opponents are attracted towards one another due to the main goal of the task (to strike each other), whereas on the other, the region of striking unpredictability, peaking at D = 0.6, acts as a repeller, driving opponents towards its minima D = 1.2 and D = 0. Through competition and cooperation of these contrary tendencies relatively stable regions are formed around the unpredictability and efficiency ratio minima. Data from this study supported anecdotal evidence that those minima are located at the edge of jab reachability and fluctuate so that the jabs could emerge. The incursions of the attacker (boxer A in Figure 3) on shorter scaled distances especially those around D = 0.6 enables him to increase and maximize the unpredictability, diversity and the efficiency ratio of his actions, however simultaneously he is also being exposed to increased action possibilities by the defending opponent (boxer B in Figure 3). So after the incursions of the attacker, typically, previously stable coordination states are quickly re-established either by withdrawal of the attacker or by the evasion of the defender or both. Actually, the culmination of sudden incursions of the attacker is usually twofold: either the previous "at the edge of jab reachability" (the first unpredictability and efficiency ratio minimum) dynamics are spontaneously re-established around D = 1.2, or the boxer - boxer coordination dynamics transits to the D = 0 zone where boxers stick together and lean on each other, so that only sporadic hooks could emerge (clench position). These system dynamics are characterized by the second unpredictability and efficiency ratio minimum, at D = 0, which was also observed in this investigation. One should note, however, that this minimum possesses larger values of unpredictability and efficiency ratio measures than the one at D = 1.2, meaning that it is only a local minimum. On the other hand the global minimum is the one located around D =1.2. This means that the area around the global minimum is more stable (i.e. more often visited) strategic position than the one located at D = 0. The reason why boxers do not constantly spend much time at distances close to D = 0.6, is that those distances represent repellent areas of coordination dynamics landscape (i.e. from this position it is easy for boxers to move to other specific parts of the perceptual-motor landscape). Certainly, other constraints like the defensive position of the arms of the opponent may regulate the attacker's intentions in specific ways which requires further investigation.
CONCLUSIONS
The data from this study have shown how a constraints-led framework on movement behaviour emphasizes the importance of the environmental, task and individual constraints on human movement systems. The results of this investigation on decision-making in boxers permit a number of practical conclusions to be drawn, as outlined below:
1. In training tasks involving punching static (i.e. non - moving) heavy bags, novice boxers are sensitive to two types of interacting constraints of high ecological validity: (i) The scaled performer - target distance; and (ii), the perceived efficiency of punching actions. The first constraint is instrumental in creating new and annihilating previously active striking patterns. The second influential constraint is used to fine tune the action landscape of individual boxers by suppressing less efficient, and facilitating use of more efficient boxing action patterns.
2. The typical training task constraints studied in this investigation showed that novice boxers are subject to a kind of discovery learning by exploring the efficiency of their motor repertoire when changing the performer - target distances as evident by the strong efficiency - action probability relationships observed in the data. With regard to these findings it is important that the attention of the novice boxers should be directed by coaches to the major informational constraints identified in this paper (i.e. the reachability specific to each kind of punching actions and the associated perceived punching efficiencies) so that stable movement-information couplings are formed during practice. Particularly important is the relationship, between 3 parameters in the reachability condition (see the Methods section) i.e. the physical performer - target distance (X), the forward body lean length (Llean) and the effective arm length (Leff), which itself depends on the elbow joint angle at the moment of hand - target collision. By allowing novice boxers during the basic training sessions, when the heavy bag practice is mostly used, to explore the whole spectrum of constraints enabled by each combination of parameters, they would learn how to adjust emergent motor solutions to the hitting task which are specific to their individual organismic constraints. Once these efficient coordination patterns have been established with the heavy bag, learners could move to the task of hitting moving opponents during light sparring;
3. Novice boxers are able to discover and exploit the scaled performer - target distance region that affords maximization of the unpredictability (H), diversity (S) and the efficiency ratio (E) of their punching actions. It is located around D = 0.6; The regions around this distance are those which maximize the meta-stability (i.e. dynamics at the edge of instability) of movement selection and consequently the flexibility of creating punching actions in novices. This region is an optimal area for practicing all possible temporal couplings of boxing action patterns which is not the case for other regions where some of the action patterns are decoupled or are much less likely to occur;
4. Spontaneous emergence of boxer - boxer coordinative states and strategic positioning as a consequence of boxers' perception of essential interacting constraints points to the possibility that practice should be less loaded with verbal instructions from the coach to impose decisions. Rather, practice could be directed towards creating a variety of learning situations (by manipulating the dynamics' constraints) in which trainees would themselves explore, discover and thus adapt to the information - movement demands of diverse 1 v 1 interpersonal coordinations (i.e.sparring situations).
The results obtained in this study demonstrated that dynamical systems theory could provide a useful and relevant theoretical framework for investigating decision making behaviour in sports such as boxing, as exemplified by the task of punching a heavybag. Interesting questions are raised over the differences and similarities of dynamical properties amongst a range of combat sports such as boxing, karate and kickboxing. One should expect that a variety of dynamical differences exist due to the differences in the number of limbs involved, the inter- and intra-limb coordination states that are enabled by rules and permissible actions and differences in limb lengths of participants. It is expected that differences in all of the measures and their interdependencies used in this investigation exist which would also change the properties of performer - performer coordination. These issues should be addressed in further investigations of combat sports motivated by dynamical systems theory.
KEY POINTS
During the practicing with static (i.e. non moving) heavy bags novice boxers' perceptual-action system is sensitive to the scaled distance and efficiency informational constraints. These interdependent constraints shape the action behaviour of the novice boxers;
During heavy bag practice novice boxers are subject to a kind of discovery learning by exploring the efficiency of their motor repertoire when changing the performer - target distances;
The region close to D = 0.6 maximizes the flexibility of switching among different types of punching actions and is optimal for practicing the temporal couplings between all types of boxing actions;
Strategic positioning among boxers spontaneously emerges under the pressure of the key constraints revealed in these research. Applying practice strategy that emphasizes learning in which trainees would themselves explore, discover and thus adapt to the information - movement demands of diverse sparring situations is thus of great importance.
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