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This article was submitted to the journal Frontiers in Human Neuroscience.

Edited by: Jessica Phillips-Silver, Georgetown University Medical Center, USA

Reviewed by: Petri Toiviainen, University of Jyväskylä, Finland

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Entrained behavior coordinates, predicts, and modulates multi-scale rhythmic gestures with high spatio-temporal precision even as it shows flexible adaptation in response to perturbation (Clayton et al.,

This paper aims to bolster the theoretical case for the transformational potential of entrainment therapy by casting it in the framework of contemporary engineering mathematics, in particular applying the concepts of change of basis, Fourier transform, and most importantly, the growing body of work on Joint Sparse Representation (JSR) (Bruckstein et al.,

Many of the engineering marvels around us have, as a keystone of their mathematical foundations, a change of basis (Kreyszig,

This spatio-temporal Cartesian basis is our most intuitive approach to representing the world around us, but also a very poor representation for solving many engineering problems. One of the most commonly used changes of basis is the family of Fourier or frequency-domain transforms, in which a function is represented on a basis of sinusoidal

While conceptually cumbersome at first, Fourier transformation has many advantages for not only the analysis, but the storing and compression, of many kids of data. Take, for example, a sample of a single musical note, vibrating at a particular frequency, that would appear on an oscilloscope as a complex periodic waveform. In the time domain, this signal will be dense, that is, it will contain few if any zeros and most of the signal will be required for its reconstruction as specified by the the Nyquist-Shannon sampling theorem (Shannon,

Mathematically, a signal and its Fourier transform are one-to-one mappings. The frequency-domain representation of the signal is often much more efficient, however, in the sense that far more of the signal information is packed into a small subset of the vectors that span the basis. JPEG (Skodras et al.,

Frequency-domain transform is hard-wired into the anatomy of the cochlea, whose hair cells of varying stiffness resonate with stimuli of specific frequencies, triggering action potentials via auditory transduction. The inner ear thus performs a frequency transform of incoming auditory information across a small temporal window, known in its simplest form as a short-time Fourier transform (STFT), though actual observed performance resembles a somewhat more complex transform known as time-frequency reassignment (Auger et al.,

If sufficient information about the signal can be deduced from a small portion of a signal via a mathematical transform, then the benefits to the actor are obvious. Both computationally and metabolically, the organism that can reduce processing demands by such a large amount can expect to reap benefits. If frequency-domain and similar bases yield such improvements in information coding efficiency, the key question for modeling neural coding is to ask how that information might be coded to its optimum.

The optimal basis for a signal in a least-squares sense is its Singular Value Decomposition (SVD) (Strang,

However, the SVD of a single signal is not necessarily the sparsest representation of that signal in the context of a set of signals such as that encoded in neural memory. Much greater compression can be attained through the re-use of common basis vectors to transform many signals. In this approach the process of neural memory is modeled as manipulation of a set of learned basis vectors known as a “dictionary,” in which incoming signals are decomposed in the sparsest possible way using the atomic vectors or “atoms” that make up the dictionary (Rubinstein et al.,

Finally the atoms of the dictionary must adapt to the new signals in accordance with the principles of Hebbian and Bayesian learning. Efficient algorithms have been discovered for this process as well, whether the classic K-SVD (Aharon et al.,

Perhaps unsurprisingly, there is abundant experimental evidence for such sparse coding in human and animal brains (Olshausen and Field,

The sparsity argument for entrainment is then as follows: Phenomena that contain regularities are more efficiently encoded in the frequency domain. We can therefore expect that the optimal basis, such as that obtained through SVD, would be much more similar to the frequency-domain mapping of signal, by a common similarity measure such as tangent distance (Simard et al.,

Returning to the descriptions of entrainment in the literature, many of the characteristic behaviors found in entrainment can be accounted for with greater conceptual economy by applying sparsity-related concepts. Entrained movement is not necessarily more skillful than rhythmically independent movement, but rather entrained movement is more efficiently coded and less computationally demanding when projected onto a frequency-domain basis. Entrainments across multiple time scales (Large,

What experiment might validate the hypothesis that entrainment facilitates sparse coding? While we cannot observe information coding directly, we can observe behavior, and while we do not have access to the atomic dictionaries within a subject, we can determine the SVD of a subject's actions. The singular values of the SVD further provide an effective measurement tool for how sparsely the information is encoded known as

From this hypothesis for the cognitive impact of entrainment, a second hypothesis for entrainment-based therapy may be additionally derived: if the lasting result of a repeated entrainment-based intervention is a persistent shift in kinematic SVE of a behavior, even independent of the intervention, the SVE alteration is evidence of entrainment-driven neuroplastic change.

As the presence of the cochlea has long suggested to anatomists, and as neural coding theory now asserts, the brain is much more aligned to the frequency domain than our everyday, spatio-temporal accounts of the world might lead us to think. Consequently, the impact of entrainment-based instruction and therapy is likely much greater than that which can be forecasted by spatiotemporal analysis of actions. Entrainment is everywhere; entrainment is powerful; but perhaps most importantly, entrainment is sparse. A sparsity model of entrainment therapy suggests that entrainment therapy is much more than a way to scaffold the re-learning of movements: it is potentially one of the most powerful approaches to the changing of behavior in the contemporary repertoire.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author would like to gratefully acknowledge discussions with Dr. Katie Overy, University of Edinburgh.