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Fundamentals of Neuromechanics pp 135–157Cite as

The Nature and Structure of Feasible Sets

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Part of the book series: Biosystems & Biorobotics ((BIOSYSROB,volume 8))

Abstract

An engineering perspective is inherently incomplete when applied to science. However, as per the words of Galileo Galilei at the beginning of this book, science is also not complete without a mathematical foundation. Our large community applied this mathematics-based perspective for decades to understand motor control. This has resulted in a large, informative, useful, and fruitful body of work. I now comment briefly on how the neuromechanical framework of this book applies to some current tenets, theories, and debates in motor control. In particular, if we agree that the mechanical principles outlined in this book are relevant to the structure of vertebrate limbs, then the nature and structure of the feasible sets they allow are relevant to their neural control. In this chapter I present brief descriptions of how our community has approached understanding the nature and structure of the high-dimensional feasible activation sets.

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Notes

  1. 1.

    My preferred use of PCA is as the singular value decomposition of the covariance matrix of the data. In this way the PCs are the left singular vectors in matrix U, and the \(\sigma _i\) are the singular values in the diagonal matrix \(\Sigma \) [3, 4].

  2. 2.

    Sometimes the abbreviation NNMF is also used.

  3. 3.

    Recall that a basis need not have orthogonal basis vectors.

  4. 4.

    A manifold is a type of subspace of a given dimensionality that is locally Euclidean—but not necessarily linear—that can be embedded in a higher dimensional space. Some examples are lines, planes, spheres, toruses, etc. [36].

  5. 5.

    Spatial in the sense that PCA, NMF, etc. provide the low dimensional structure in ‘activation space.’

  6. 6.

    The general model has 7 DOFs, but in this analysis hip ad- ab-duction was frozen so the model is really a 6 DOF model.

  7. 7.

    That is, constraints were added to enforce that the torque elements of the output wrench be zero.

  8. 8.

    Sometimes called the inscribed ball. The term ball is used in geometry to mean a sphere in arbitrary dimensions. The circumscribed ball is the one in which the convex set fits.

  9. 9.

    Spatial in the sense of meeting the constraints in the space of neural activations, and temporal in the sense of implementing certain temporal dynamics.

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Authors and Affiliations

  1. Department of Biomedical Engineering, The University of Southern California, Los Angeles, CA, USA

    Francisco J. Valero-Cuevas

  2. Division of Biokinesiology and Physical Therapy, The University of Southern California, Los Angeles, CA, USA

    Francisco J. Valero-Cuevas

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Valero-Cuevas, F.J. (2016). The Nature and Structure of Feasible Sets. In: Fundamentals of Neuromechanics. Biosystems & Biorobotics, vol 8. Springer, London. https://doi.org/10.1007/978-1-4471-6747-1_9

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