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“Time-Scales and Fluctuations in Motor
Learning” |
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The principle goal in the hard sciences
such as physics has always been to extract universal eternal laws
about nature that can be expressed in elegant mathematical equations.
The most famous example of this paradigm is Einstein’s E=m c2 that
reveals a deep connection between energy, mass, and the speed of
light. Attempts in the area of “soft-sciences” (specifically in the
area of motor learning) to uncover similar fundamental laws have
been less successful. One example is the widely discussed “universal power law of practice”. This “law” claims that performance of a participant who practices a given task will increase with practice time according to a mathematical function that is known as a “power law”. It is simply the practice time raised to a constant power. It is curious, that in this “universal law” it is not specified if the sign of the power constant is positive or negative. The sign, however, has fundamental consequences with applications all the way to the practicing athlete and coach. For negative constants, the performance value will never cross an asymptotic value, no matter how long the athlete continues to practice. If the constant is positive, however, the performance continues to increase beyond all limits, as long as the practice time is long enough. It is self-evident that both cases do not describe the observed reality. In a number of recent publications it could be shown that a careful re-analysis of classic motor learning data could be equally well be described by exponential functions. We could show in some classic example data (Snoddy 1928) that the assumption of two exponential functions being involved in the processes during practice and rest, the data could even be described more accurately. These results also have some deep theoretical implications for the general view of how learning and adaptation both contribute to performance increase during and even between practice sessions. The first observation is that both exponential functions (with negative exponents) as well as power laws (with constant exponent smaller than one) exhibit the common behavior of initially a fast performance increase that eventually slows down. Often this is referred to as “learning rates” that decrease after long practice times. The main difference between the two functions, however, is that exponential functions have characteristic, constant learning rates whereas for power laws the learning rate itself continuously decreases. Since every rate constant defines a time scale we can observe that exponential laws have characteristic time-scales, whereas power laws possess an infinite continuum of steadily changing time scales. Our theoretical postulate is that every learning or adaptive process is governed by a (finite) number of sub-processes that can be identified according to their characteristic time scales. This hypothesis allowed us to identify two independent exponential processes in the Snoddy data that differ by an order of magnitude. |
TIf we follow these data over several
practice days we can make a second surprising observation. The
behavior of the two processes between practice sessions appears to
be fundamentally different. Whereas the process with the slow time
scale appears to continue even after the end of the practice session,
the fast time scale process reverses direction in the sense that the
expected performance is actually decreasing. That implies that this
fast process does not describe motor learning because the behavioral
changes as reflected by the performance values are not persistent.
This is indeed observed in practice as the well known phenomenon of
“warm-up decrement”. An interesting consequence of this observation
is that our analysis of sub processes according to their
characteristic time scales allows us to discriminate between
adaptive processes and true learning processes. As mentioned above, all exponents that define the characteristic time-scales are negative, that means each process will asymptotically lead to a fixed point with a definite limit to the performance that the participant cannot exceed. Mathematically we express this structure in the context of performance landscapes in many behavioral dimensions. The elevation of a point in that landscape represents then the performance value that we would expect for the coordinated behavioral patterns given by its location in the landscape. Note, for historical reasons, high elevations correspond to lower performance. Learning and adaptation dynamics then would correspond to following a path of steepest descent and maximum performance improvement until we reach a local minimum at a learning rate, given by the characteristic time-scale associated with this minimum. In the context of our original question “Does motor learning lead to an asymptote in performance or does it allow improvement without limits?” this model would give a definite answer in favor of the “asymptotic limit” option. This is where the second ingredient of the title of this presentation comes in: namely, fluctuations. Whereas in many deterministic models noise and fluctuations are discarded as unwanted side effect that are not part of the relevant learning dynamics, our position is that fluctuations play a similarly important fundamental role in motor learning as time scales. Going back to the landscape framework we recognize, that for given initial conditions we have a finite chance that by following the fastest descent the system will approach a minimum corresponding to a process that is not associated with the best possible performance (“global minimum”). A purely deterministic system has a-priory no chance to improve beyond the elevation of the local minimum, let alone reach the global minimum. If we add fluctuations to the system, however, it now becomes possible that we escape the local minimum and continue to improve the performance. This is very different from the continuous improvement claimed by the power law with positive exponent: It predicts that we can expect to observe continued performance improvement as long as we can find a new process within the accessibility region in the landscape of a given participant and as long as the system exhibits fluctuations to induce a transition to the basin of attraction of that new process. We claim that our model thus reconciles the paradoxical situation of bounded versus unlimited performance increase in motor learning. |
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