Thanks for this wonderful question! As it happens, I was re-reading the earlier work of this French-Scots sports science team yesterday, so your question is both very interesting and timely.
The short answer is that sports scientists use implicit assumptions about how the brain processes information in the perception-movement "action nexus" without really knowing what they are assuming; AND because of their implicit acceptance of conventional (and erroneous) notions about brain processes, they MISS THE BIG PICTURE entirely. Unfortunately, this doesn't in the least deter them from forging "concepts" to describe the brain processes they purport to study, and once forged, these "concepts" take on a career-related life of their own. "Tau" and "Tau Guidance" are similar concepts.
The paper referenced by you comes from DN Lee (Psychology, Univ. of Edinburgh, Scotland), MA Grealy (Psychology, Strathclyde Univ., Glasgow), Delphine Delay (UFRAPS, Université de Joseph Fourier, Grenoble France), and CM Craig (UMR Mouvement and Perception, Université de la Méditerranée, Marseilles France). These team members have been coalescing together for papers on this same topic since the mid-1990s. The basic concept -- rather pervasive in sports science academia (just try pursuing a movement-related PhD or Masters degree without saluting this "Tau" flag!) -- is (in paraphrase): "the brain is assumed to be coordinating and regulating" the time to impact (tau) by some mechanism or process." "Tau" is the time remaining to "closure of the gap" given the current state of the closing rate.
From this, various teams concerning various movements in various sports investigate different aspects and make interpretations of their study data to make "theoretical predictions" about "how golfers regulate the spatial and temporal components of the forward swing in order to transmit the appropriate amount of kinetic energy at ball impact." The above paper is TYPICAL sports science data-gathering in academia: ten low-handicap (under 5) golfers putting across different distances of 1m, 2m, 3m, and 4m. These would be locally available volunteer students at the colleges, hereinafter referred to exclusively as "expert" golfers. The conclusion in the paper above is that, given the authors' starting concept of "tau", the golfers are "regulating" kinetic force and "guiding movements by coupling them onto intrinsic tau guides and spatially scaling the movements."
What does this last mouthful of jargon really mean? It means the authors ASSUME there is some "representation" in the brain that serves as an internal "guide" to what SHOULD BE happening with "tau" in external reality and the two (external tau reality and internal tau-guide brain representation) are "coupled". Because they "find" from their data that "tau" is kept in constant ratio with the supposed tau-guide, they congratulate themselves on confirming with data that the two indeed are "coupled". Then the curious logic approaches this final implicit self-confirming position: having "proved" with data that tau and tau-guidance are "coupled" via size-force variance to keep stroke time constant, the implicit conclusion not quite dared to be uttered is: and THEREFORE WE HAVE PROVED THE FACTUAL REALITY OF TAU-GUIDANCE IN THE BRAIN AS THE CONTROLLING PROCESS. Yeah, right!
Here's the ASSUMPTION hiding in the core of the above paper:
"The theory argues that gap closure is controlled by using perceptual information about a particular measure of a gap, denoted as tau, which is equal to the time to gap closure at the existing closure rate. The rate of gap closure can be regulated by keeping the tau value of the motion gap coupled onto (in constant ratio with) the tau value of a guiding gap. In self-paced actions like putting, this tau guide is ASSUMED TO BE GENRATED IN THE NERVOUS SYSTEM (for example, by modulating energy levels). Neural mechanisms are implicated in the intrinsic timing of movements and in their spatial specification, and there are similarities in brain activity when the performance of an action is being imagined and when it is subsequently executed."
Please note the utterly WEAK attempt to investigate or flesh out the details of neuroscience about how "neural mechanisms are implicated in the intrinsic timing of movements and in their spatial specification". All the authors do is refer blithely to a few footnotes. That really is not very serious.
How does the golfer "couple" tau and the tau-guide? Uh, plainly stated, by adujusting the SIZE of the stroke to the required impact velocity. Ever heard that before? How about: ALL pendular motion (free-fall in gravity of a rod beneath a fixed pivot) adjusts impact velocity only by adjusting downstroke / backstroke size. If you can penetrate thru the self-puffing jargon that generously laces these academic "set pieces", there is a great big fat "DUH!" waiting at the end.
Does the brain in fact "coordinate and regulate" tau with an internal tau-guide, and if so, HOW? Uh, let's plow thru the jargon on that one and see what these good academics claim about their assumptions.
Short answer: all these folks ever really say is that humans scale distance force control by adjusting the size of the stroke within the context of pendular isochrony (all strokes take the same time, no matter how large or small -- within limits at either end of too-small and too-large). Ever heard that before? DUH!
For example, the earlier paper, D Delay, V Nougier, J-P Orliaguet, and Y Coello, Movement control in putting, Human Movement Science 16 (1997), 597-619, traces the same arc thru the familiar "tau" science so pervasive in academia and concludes:
"Regarding the temporal characteristics of the movement, results showed that DS (downstroke) velocity was proportional to its amplitude so as to keep DS movement time until percussion and total DS movement time approximately constant with increasing distance of the target. IN OTHER WORDS, SUBJECTS INCREASED THE FORCE APPLIED TO THE BALL BY INCREASING THE AMPLITUDE OF THE IMPULSE RATHER THAN INCREASING ITS DURATION. [my emphasis] This isochrony principle, originally studied in unidimensional movements, characterizes various motor skills such as manual pointing, drawing or handwriting, head rotations, graspings, of kicking of a ball by children. In golf putting, the tendency towards an isochrony of the movement and the fact that peak velocity occurred approximately in the middle of the DS amplitude seem to be intrinsic characteristics of the subject-club system. Furthermore, these components are maintained constant even when modifying, through learning and experience, some components of the movement. ... In golf putting, the isochrony principle would facilitate programming of the movement, whatever be the subjects' level of expertise." (Id., page 613).
Uh, whadded-'ey-say? Clearly, the authors don't seem to "get" the idea that humans USE isochrony because the World IS that way for all pendular motions in gravity. The world constantly exhibits this fact of pendular patterning to the brain, and it is the BRAIN'S job at the instinctive movement level to "get this right or die". It's not that these aspects of movement (same duration in stroke, with impact velocity varied solely by stroke size) "seem to be intrinsic characteristics of the subject-club system" -- these aspects ARE in physics and reality intrinsic characteristics of all pendular motion on ANY PLANET in the universe. All force control is instinctively the coordination of movement size with impact or movement force in the context of the reality of pendular isochrony. Pendular isochrony has a planet-dependent TEMPO but a planet-independent rhythm or proprtionality in the backstroke-downstroke timing (2:1). The human brain uses this pattern because it simply and reliably fits the movement planning, evaluation, and execution of the animal to the requirements of the external world for success.
What is puzzling is that these academics never seem too interested in why humans do that, for a great variety of movements. They make assumptions about what happens inside the brain but don't ask why that might have come about that way. They approach the whole subject back-ass-wards. From developing a "math and physics related concept" about movement as the heart of what MUST be going on in movement "control", the authors look for a similar "math and physics" process inside the brain that "controls" the external reality: this approach is sterile and stands the academics in a tangle of confused assumptions that don't bear great scrutiny in the light of developing neuroscience about actual brain processes.
The fruitful approach is opposite to this: the right question is what is the purpose of the brain in movement and how does it accomplish this? This approach leads to common-sense answers that are immediately translatable and communicable to normal humans about how things work and how to enhance this process.
The brain is emphatically NOT making physics / math calculations and does not have an internal representation of a "tau-guide" and some small "Homer Simpson" hired to sit in the brain's control room and watch the dials to keep the guide and the external tau "coupled".
Neuroscience teaches this exact point very loudly these days. What the brain does is learn the pattern of the pendulum FROM THE WORLD and then learns that all amplitude strokes take the same time to swing for the same pendulum system (length) and then learns the point-to-point, "fact A-Size goes with fact A-velocity" "coupling" that stroke size S1.237 always goes only with stroke-velocity-at-impact V1.237. This KNOWLEDGE does not in the least require any sort of "processing" or computer time or calculating at all, and certainly does not require any Homer Simpsons inside the head at the control panels.
I once tried to explain this to a "graduate student" in sports science in England and he appeared to be turning physically ill at the very suggestion that "tau" was a dead-end approach to human movement "control". He simply wouldn't listen. He was the sports science member for the Zen Oracle teaching of putting and golf swing at Sheffield-Hallam University.
So academic sports scientists tend to be career-captives of their own jargon-dependent concepts, which take on a wholly undeserving life of their own, the constant feeding and nurturing of which generates almost impenetrably dense jargon-enshrouded nonsense that just keeps getting "curiouser and curiouser" paper after peer-reviewed paper without advancing understanding and also while actively screening investigators away from more fruitful approaches.
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