Optimality and Modularity in Human Movement: From Optimal Control to Muscle Synergies

In this chapter, we review recent work related to the optimal and modular control hypotheses for human movement. Optimal control theory is often thought to imply that the brain continuously computes global optima for each motor task it faces. Modular control theory typically assumes that the brain explicitly stores genuine synergies in specific neural circuits whose combined recruitment yields task-effective motor inputs to muscles. Put this way, these two influential motor control theories are pushed to extreme positions. A more nuanced view, framed within Marr’s tri-level taxonomy of a computational theory of movement neuroscience, is discussed here. We argue that optimal control is best viewed as helping to understand “why” certain movements are preferred over others but does not say much about how the brain would practically trigger optimal strategies. We also argue that dimensionality reduction found in muscle activities may be a by-product of optimality and cannot be attributed to neurally hardwired synergies stricto sensu, in particular when the synergies are extracted from simple factorization algorithms applied to electromyographic data; their putative nature is indeed strongly dictated by the methodology itself. Hence, more modeling work is required to critically test the modularity hypothesis and assess its potential neural origins. We propose that an adequate mathematical formulation of hierarchical motor control could help to bridge the gap between optimality and modularity, thereby accounting for the most appealing aspects of the human motor controller that robotic controllers would like to mimic: rapidity, efficiency, and robustness.

[1]  A. Kuo,et al.  Mechanical Work as an Indirect Measure of Subjective Costs Influencing Human Movement , 2012, PloS one.

[2]  C. Papaxanthis,et al.  Motor planning of arm movements is direction-dependent in the gravity field , 2007, Neuroscience.

[3]  Jean-Paul Gauthier,et al.  How humans control arm movements , 2008 .

[4]  Emilio Bizzi,et al.  Representation of Muscle Synergies in the Primate Brain , 2015, The Journal of Neuroscience.

[5]  Naser Mehrabi,et al.  Predictive Simulation of Reaching Moving Targets Using Nonlinear Model Predictive Control , 2017, Front. Comput. Neurosci..

[6]  Didier Henrion,et al.  Inverse optimal control with polynomial optimization , 2014, 53rd IEEE Conference on Decision and Control.

[7]  S. Giszter,et al.  Modular Premotor Drives and Unit Bursts as Primitives for Frog Motor Behaviors , 2004, The Journal of Neuroscience.

[8]  M. A. MacIver,et al.  Neuroscience Needs Behavior: Correcting a Reductionist Bias , 2017, Neuron.

[9]  J. Krakauer,et al.  A computational neuroanatomy for motor control , 2008, Experimental Brain Research.

[10]  E. Todorov Optimality principles in sensorimotor control , 2004, Nature Neuroscience.

[11]  W. Kargo,et al.  Early Skill Learning Is Expressed through Selection and Tuning of Cortically Represented Muscle Synergies , 2003, The Journal of Neuroscience.

[12]  M. L. Chambers The Mathematical Theory of Optimal Processes , 1965 .

[13]  R. Ivry,et al.  The coordination of movement: optimal feedback control and beyond , 2010, Trends in Cognitive Sciences.

[14]  Mitsuo Kawato,et al.  Optimal control of reaching includes kinematic constraints. , 2013, Journal of neurophysiology.

[15]  Stacie A. Chvatal,et al.  Common muscle synergies for control of center of mass and force in nonstepping and stepping postural behaviors. , 2011, Journal of neurophysiology.

[16]  T. Pozzo,et al.  Visuomotor adaptation to a visual rotation is gravity dependent. , 2015, Journal of neurophysiology.

[17]  Michael S Landy,et al.  Motor control is decision-making , 2012, Current Opinion in Neurobiology.

[18]  E. Bizzi,et al.  Muscle synergies encoded within the spinal cord: evidence from focal intraspinal NMDA iontophoresis in the frog. , 2001, Journal of neurophysiology.

[19]  P. M. Hilt,et al.  Evidence for subjective values guiding posture and movement coordination in a free-endpoint whole-body reaching task , 2016, Scientific Reports.

[20]  Helen J. Huang,et al.  A Representation of Effort in Decision-Making and Motor Control , 2016, Current Biology.

[21]  Adrienne L. Fairhall,et al.  Dual Dimensionality Reduction Reveals Independent Encoding of Motor Features in a Muscle Synergy for Insect Flight Control , 2015, PLoS Comput. Biol..

[22]  Lena H Ting,et al.  A limited set of muscle synergies for force control during a postural task. , 2005, Journal of neurophysiology.

[23]  Andrea d'Avella,et al.  Matrix factorization algorithms for the identification of muscle synergies: evaluation on simulated and experimental data sets. , 2006, Journal of neurophysiology.

[24]  T. Sejnowski,et al.  Conservation law for self-paced movements , 2016, Proceedings of the National Academy of Sciences.

[25]  Reza Shadmehr,et al.  Motor Adaptation as a Process of Reoptimization , 2008, The Journal of Neuroscience.

[26]  Karl E Zelik,et al.  Preferred Barefoot Step Frequency is Influenced by Factors Beyond Minimizing Metabolic Rate , 2016, Scientific Reports.

[27]  J. Gauthier,et al.  A biomechanical inactivation principle , 2010 .

[28]  F. Lacquaniti,et al.  Motor patterns in human walking and running. , 2006, Journal of neurophysiology.

[29]  Dora E Angelaki,et al.  Direction-dependent arm kinematics reveal optimal integration of gravity cues , 2016, eLife.

[30]  Jean-Paul Gauthier,et al.  The Inactivation Principle: Mathematical Solutions Minimizing the Absolute Work and Biological Implications for the Planning of Arm Movements , 2008, PLoS Comput. Biol..

[31]  J. F. Soechting,et al.  Postural Hand Synergies for Tool Use , 1998, The Journal of Neuroscience.

[32]  Keisuke Kushiro,et al.  Direction-dependent differences in temporal kinematics for vertical prehension movements , 2013, Experimental Brain Research.

[33]  Francisco J. Valero Cuevas,et al.  Challenges and New Approaches to Proving the Existence of Muscle Synergies of Neural Origin , 2012, PLoS Comput. Biol..

[34]  Emanuel Todorov,et al.  From task parameters to motor synergies: A hierarchical framework for approximately optimal control of redundant manipulators , 2005, J. Field Robotics.

[35]  Joseph McIntyre,et al.  Multimodal reference frame for the planning of vertical arms movements , 2007, Neuroscience Letters.

[36]  Ruggero Frezza,et al.  A control theory approach to the analysis and synthesis of the experimentally observed motion primitives , 2005, Biological Cybernetics.

[37]  Emilio Bizzi,et al.  Shared and specific muscle synergies in natural motor behaviors. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[38]  R. Weale Vision. A Computational Investigation Into the Human Representation and Processing of Visual Information. David Marr , 1983 .

[39]  Aymar de Rugy,et al.  Are muscle synergies useful for neural control? , 2013, Front. Comput. Neurosci..

[40]  F. Lacquaniti,et al.  Five basic muscle activation patterns account for muscle activity during human locomotion , 2004, The Journal of physiology.

[41]  F. Lacquaniti,et al.  Coordination of Locomotion with Voluntary Movements in Humans , 2005, The Journal of Neuroscience.

[42]  Andrea d'Avella,et al.  Modularity for Sensorimotor Control: Evidence and a New Prediction , 2010, Journal of motor behavior.

[43]  C. Papaxanthis,et al.  The Temporal Structure of Vertical Arm Movements , 2011, PloS one.

[44]  Ferdinando A. Mussa-Ivaldi,et al.  Nonlinear force fields: a distributed system of control primitives for representing and learning movements , 1997, Proceedings 1997 IEEE International Symposium on Computational Intelligence in Robotics and Automation CIRA'97. 'Towards New Computational Principles for Robotics and Automation'.

[45]  Nicolas Mansard,et al.  Regularized Hierarchical Differential Dynamic Programming , 2017, IEEE Transactions on Robotics.

[46]  Naser Mehrabi,et al.  A model-based approach to predict muscle synergies using optimization: application to feedback control , 2015, Front. Comput. Neurosci..

[47]  Dario Farina,et al.  Identifying representative synergy matrices for describing muscular activation patterns during multidirectional reaching in the horizontal plane. , 2010, Journal of neurophysiology.

[48]  Jean-Paul Gauthier,et al.  How humans fly , 2013 .

[49]  E. Bizzi,et al.  Linear combinations of primitives in vertebrate motor control. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[50]  W. L. Nelson Physical principles for economies of skilled movements , 1983, Biological Cybernetics.

[51]  John M. Hollerbach,et al.  Dynamic interactions between limb segments during planar arm movement , 1982, Biological Cybernetics.

[52]  Andrea d'Avella,et al.  Differences in Adaptation Rates after Virtual Surgeries Provide Direct Evidence for Modularity , 2013, The Journal of Neuroscience.

[53]  Marco Jacono,et al.  Reach endpoint formation during the visuomotor planning of free arm pointing , 2014, The European journal of neuroscience.

[54]  Corey B. Hart,et al.  Motor primitives and synergies in the spinal cord and after injury—the current state of play , 2013, Annals of the New York Academy of Sciences.

[55]  Pietro G. Morasso,et al.  Passive Motion Paradigm: An Alternative to Optimal Control , 2011, Front. Neurorobot..

[56]  Martha Flanders,et al.  Muscular and postural synergies of the human hand. , 2004, Journal of neurophysiology.

[57]  Francesco Lacquaniti,et al.  Control of Fast-Reaching Movements by Muscle Synergy Combinations , 2006, The Journal of Neuroscience.

[58]  Surya Ganguli,et al.  On simplicity and complexity in the brave new world of large-scale neuroscience , 2015, Current Opinion in Neurobiology.

[59]  L. Ting,et al.  Muscle synergies characterizing human postural responses. , 2007, Journal of neurophysiology.

[60]  Aymar de Rugy,et al.  Muscle Coordination Is Habitual Rather than Optimal , 2012, The Journal of Neuroscience.

[61]  Richard R Neptune,et al.  Merging of healthy motor modules predicts reduced locomotor performance and muscle coordination complexity post-stroke. , 2010, Journal of neurophysiology.

[62]  E. Bizzi,et al.  The construction of movement by the spinal cord , 1999, Nature Neuroscience.

[63]  Francesco Lacquaniti,et al.  Dimensionality of joint torques and muscle patterns for reaching , 2014, Front. Comput. Neurosci..

[64]  E. Bizzi,et al.  Modules in the brain stem and spinal cord underlying motor behaviors. , 2011, Journal of neurophysiology.

[65]  Jinsook Roh,et al.  Evidence for altered upper extremity muscle synergies in chronic stroke survivors with mild and moderate impairment , 2015, Front. Hum. Neurosci..

[66]  E Burdet,et al.  Motor memory and local minimization of error and effort, not global optimization, determine motor behavior. , 2010, Journal of neurophysiology.

[67]  G. Lewis,et al.  Interactions with compliant loads alter stretch reflex gains but not intermuscular coordination. , 2008, Journal of neurophysiology.

[68]  J. McIntyre,et al.  Kinematic and dynamic processes for the control of pointing movements in humans revealed by short-term exposure to microgravity , 2005, Neuroscience.

[69]  Giulio Sandini,et al.  Visual gravity influences arm movement planning. , 2012, Journal of neurophysiology.

[70]  Francesco Nori,et al.  Evidence for Composite Cost Functions in Arm Movement Planning: An Inverse Optimal Control Approach , 2011, PLoS Comput. Biol..

[71]  Nicolas Mansard,et al.  Prioritized optimal control: A hierarchical differential dynamic programming approach , 2015, 2015 IEEE International Conference on Robotics and Automation (ICRA).

[72]  Lena H Ting,et al.  Muscle synergy organization is robust across a variety of postural perturbations. , 2006, Journal of neurophysiology.

[73]  Emmanuel Guigon,et al.  Generating human-like reaching movements with a humanoid robot: A computational approach , 2013, J. Comput. Sci..

[74]  Simon A. Overduin,et al.  Modulation of Muscle Synergy Recruitment in Primate Grasping , 2008, The Journal of Neuroscience.

[75]  Stephen H Scott,et al.  Apparent and Actual Trajectory Control Depend on the Behavioral Context in Upper Limb Motor Tasks , 2015, The Journal of Neuroscience.

[76]  Neville Hogan,et al.  Experimenting with Theoretical Motor Neuroscience , 2010, Journal of motor behavior.

[77]  Frédéric Jean,et al.  On the Duration of Human Movement: From Self-paced to Slow/Fast Reaches up to Fitts's Law , 2017, Geometric and Numerical Foundations of Movements.

[78]  宇野 洋二,et al.  Formation and control of optimal trajectory in human multijoint arm movement : minimum torque-change model , 1988 .

[79]  Paolo Mason,et al.  On Inverse Optimal Control Problems of Human Locomotion: Stability and Robustness of the Minimizers , 2013 .

[80]  Ruggero Frezza,et al.  Linear Optimal Control Problems and Quadratic Cost Functions Estimation , 2004 .

[81]  Jessica C. Selinger,et al.  Humans Can Continuously Optimize Energetic Cost during Walking , 2015, Current Biology.

[82]  A. Guével,et al.  Is interindividual variability of EMG patterns in trained cyclists related to different muscle synergies? , 2010, Journal of applied physiology.

[83]  Mark L Latash,et al.  An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension , 2010, Journal of mathematical biology.

[84]  L. Miller,et al.  Primary motor cortical neurons encode functional muscle synergies , 2002, Experimental Brain Research.

[85]  Masaya Hirashima,et al.  How does the brain solve muscle redundancy? Filling the gap between optimization and muscle synergy hypotheses , 2016, Neuroscience Research.

[86]  Matthew C. Tresch,et al.  The number and choice of muscles impact the results of muscle synergy analyses , 2013, Front. Comput. Neurosci..

[87]  T Brochier,et al.  Patterns of muscle activity underlying object-specific grasp by the macaque monkey. , 2004, Journal of neurophysiology.

[88]  S. Giszter,et al.  A Neural Basis for Motor Primitives in the Spinal Cord , 2010, The Journal of Neuroscience.

[89]  Lena H Ting,et al.  Neuromechanics of muscle synergies for posture and movement , 2007, Current Opinion in Neurobiology.

[90]  Stefano Panzeri,et al.  Quantitative evaluation of muscle synergy models: a single-trial task decoding approach , 2013, Front. Comput. Neurosci..

[91]  T. Flash,et al.  The coordination of arm movements: an experimentally confirmed mathematical model , 1985, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[92]  S. Giszter,et al.  Output Units of Motor Behavior: An Experimental and Modeling Study , 2000, Journal of Cognitive Neuroscience.

[93]  Van Hoan Vu,et al.  On the nature of motor planning variables during arm pointing movement: Compositeness and speed dependence , 2016, Neuroscience.

[94]  Stephen H Scott,et al.  Influence of the behavioral goal and environmental obstacles on rapid feedback responses. , 2012, Journal of neurophysiology.

[95]  Stefano Panzeri,et al.  Investigating reduction of dimensionality during single-joint elbow movements: a case study on muscle synergies , 2013, Front. Comput. Neurosci..

[96]  Dario Farina,et al.  Impulses of activation but not motor modules are preserved in the locomotion of subacute stroke patients. , 2011, Journal of neurophysiology.

[97]  Andrea d'Avella,et al.  Modularity in the motor system: decomposition of muscle patterns as combinations of time-varying synergies , 2001, NIPS.

[98]  Emilio Bizzi,et al.  An Optogenetic Demonstration of Motor Modularity in the Mammalian Spinal Cord , 2016, Scientific Reports.

[99]  Michael I. Jordan,et al.  Optimal feedback control as a theory of motor coordination , 2002, Nature Neuroscience.

[100]  Zachary Danziger,et al.  The Influence of Visual Motion on Motor Learning , 2012, The Journal of Neuroscience.

[101]  J. Krakauer,et al.  The basal ganglia: from motor commands to the control of vigor , 2016, Current Opinion in Neurobiology.

[102]  H. Kappen Optimal control theory and the linear bellman equation , 2011 .

[103]  Charalambos Papaxanthis,et al.  Sensorimotor adaptation of point-to-point arm movements after spaceflight: the role of internal representation of gravity force in trajectory planning. , 2011, Journal of neurophysiology.

[104]  Sascha E. Engelbrecht,et al.  Minimum Principles in Motor Control. , 2001, Journal of mathematical psychology.

[105]  S. Scott The computational and neural basis of voluntary motor control and planning , 2012, Trends in Cognitive Sciences.

[106]  A. d’Avella,et al.  Locomotor Primitives in Newborn Babies and Their Development , 2011, Science.

[107]  G. Ermentrout Dynamic patterns: The self-organization of brain and behavior , 1997 .

[108]  M. Kawato,et al.  Formation and control of optimal trajectory in human multijoint arm movement , 1989, Biological Cybernetics.

[109]  E. Bizzi,et al.  Stability of muscle synergies for voluntary actions after cortical stroke in humans , 2009, Proceedings of the National Academy of Sciences.

[110]  Konrad P. Körding,et al.  Deep networks for motor control functions , 2015, Front. Comput. Neurosci..

[111]  Francesco Nori,et al.  Manifold reaching paradigm: how do we handle target redundancy? , 2011, Journal of neurophysiology.

[112]  C. Alessandro,et al.  Identification of synergies by optimization of trajectory tracking tasks , 2012, 2012 4th IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob).

[113]  Jessica L. Allen,et al.  Neuromechanical Principles Underlying Movement Modularity and Their Implications for Rehabilitation , 2015, Neuron.

[114]  M. Flanders,et al.  Basic features of phasic activation for reaching in vertical planes , 1996, Experimental Brain Research.

[115]  Lena H. Ting,et al.  Optimization of Muscle Activity for Task-Level Goals Predicts Complex Changes in Limb Forces across Biomechanical Contexts , 2012, PLoS Comput. Biol..

[116]  Philippe Lefèvre,et al.  Optimal integration of gravity in trajectory planning of vertical pointing movements. , 2009, Journal of neurophysiology.

[117]  F. Lacquaniti,et al.  Temporal components of the motor patterns expressed by the human spinal cord reflect foot kinematics. , 2003, Journal of neurophysiology.

[118]  Matthew T. Kaufman,et al.  Neural population dynamics during reaching , 2012, Nature.

[119]  R. E. Kalman,et al.  When Is a Linear Control System Optimal , 1964 .

[120]  Sylvain Dorel,et al.  Consistency of muscle synergies during pedaling across different mechanical constraints. , 2011, Journal of neurophysiology.

[121]  Karl J. Friston What Is Optimal about Motor Control? , 2011, Neuron.

[122]  Francesco Lacquaniti,et al.  Patterned control of human locomotion , 2012, The Journal of physiology.

[123]  Stefano Panzeri,et al.  A unifying model of concurrent spatial and temporal modularity in muscle activity. , 2014, Journal of neurophysiology.

[124]  M. Tresch,et al.  The case for and against muscle synergies , 2022 .

[125]  Ning Qian,et al.  An optimization principle for determining movement duration. , 2006, Journal of neurophysiology.

[126]  A. Schwartz Movement: How the Brain Communicates with the World , 2016, Cell.

[127]  Francisco J. Valero Cuevas,et al.  Muscle Synergies Heavily Influence the Neural Control of Arm Endpoint Stiffness and Energy Consumption , 2016, PLoS Comput. Biol..

[128]  Richard R Neptune,et al.  Three-dimensional modular control of human walking. , 2012, Journal of biomechanics.

[129]  Francesco Lacquaniti,et al.  Modulation of phasic and tonic muscle synergies with reaching direction and speed. , 2008, Journal of neurophysiology.

[130]  Daniel M. Wolpert,et al.  Signal-dependent noise determines motor planning , 1998, Nature.

[131]  R. Bellman Dynamic programming. , 1957, Science.

[132]  Jesse Douglas Solution of the inverse problem of the calculus of variations , 1941 .

[133]  Lorenzo Ntogramatzidis,et al.  A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions , 2005, Autom..

[134]  Emanuel Todorov,et al.  Iterative linearization methods for approximately optimal control and estimation of non-linear stochastic system , 2007, Int. J. Control.

[135]  Emilio Bizzi,et al.  Combinations of muscle synergies in the construction of a natural motor behavior , 2003, Nature Neuroscience.

[136]  Richard R Neptune,et al.  Modular control of human walking: a simulation study. , 2009, Journal of biomechanics.

[137]  Stefano Panzeri,et al.  Task-discriminative space-by-time factorization of muscle activity , 2015, Front. Hum. Neurosci..

[138]  S. Scott Optimal feedback control and the neural basis of volitional motor control , 2004, Nature Reviews Neuroscience.

[139]  Emanuel Todorov,et al.  Optimal Control Theory , 2006 .

[140]  鍋谷 清治,et al.  The Mathematical Theory of Optimal Processes by L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko , 1964 .

[141]  Stefano Panzeri,et al.  Muscle synergies in neuroscience and robotics: from input-space to task-space perspectives , 2013, Front. Comput. Neurosci..

[142]  François Bonnetblanc,et al.  Modular Control of Pointing beyond Arm's Length , 2009, The Journal of Neuroscience.

[143]  S. Bai,et al.  Microfluidic Electrochemical Impedance Spectroscopy of Carbon Composite Nanofluids , 2017, Scientific Reports.

[144]  Robert A. Jacobs,et al.  Properties of Synergies Arising from a Theory of Optimal Motor Behavior , 2006, Neural Computation.

[145]  W. Rymer,et al.  Alterations in upper limb muscle synergy structure in chronic stroke survivors. , 2013, Journal of neurophysiology.

[146]  J. M. Hondzinski,et al.  Different damping responses explain vertical endpoint error differences between visual conditions , 2016, Experimental Brain Research.

[147]  Rajiv Ranganathan,et al.  Sensory Agreement Guides Kinetic Energy Optimization of Arm Movements during Object Manipulation , 2016, PLoS Comput. Biol..

[148]  Emanuel Todorov,et al.  Compositionality of optimal control laws , 2009, NIPS.

[149]  Ziaul Hasan,et al.  Kinematic and kinetic constraints on arm, trunk, and leg segments in target-reaching movements. , 2005, Journal of neurophysiology.

[150]  Gerald E. Loeb,et al.  Optimal isn’t good enough , 2012, Biological Cybernetics.

[151]  C. Atkeson,et al.  Kinematic features of unrestrained vertical arm movements , 1985, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[152]  R. Shadmehr Control of movements and temporal discounting of reward , 2010, Current Opinion in Neurobiology.

[153]  Emanuel Todorov,et al.  Efficient computation of optimal actions , 2009, Proceedings of the National Academy of Sciences.

[154]  M. Flanders,et al.  Muscle activation patterns for reaching: the representation of distance and time. , 1994, Journal of neurophysiology.

[155]  Shunta Togo,et al.  Control strategy of hand movement depends on target redundancy , 2017, Scientific Reports.

[156]  Emanuel Todorov,et al.  Structured variability of muscle activations supports the minimal intervention principle of motor control. , 2009, Journal of neurophysiology.

[157]  Simon A. Overduin,et al.  Microstimulation Activates a Handful of Muscle Synergies , 2012, Neuron.

[158]  L. Fadiga,et al.  Energy-related optimal control accounts for gravitational load: comparing shoulder, elbow, and wrist rotations. , 2014, Journal of neurophysiology.

[159]  Andrea d'Avella,et al.  A computational analysis of motor synergies by dynamic response decomposition , 2014, Front. Comput. Neurosci..

[160]  Stefano Panzeri,et al.  A methodology for assessing the effect of correlations among muscle synergy activations on task-discriminating information , 2013, Front. Comput. Neurosci..

[161]  F. Jean,et al.  Why Don't We Move Slower? The Value of Time in the Neural Control of Action , 2016, The Journal of Neuroscience.

[162]  Van Hoan Vu,et al.  Adaptive use of interaction torque during arm reaching movement from the optimal control viewpoint , 2016, Scientific reports.

[163]  Manu Chhabra,et al.  Flexible, Task-Dependent Use of Sensory Feedback to Control Hand Movements , 2011, The Journal of Neuroscience.

[164]  E. Bizzi,et al.  Muscle synergy patterns as physiological markers of motor cortical damage , 2012, Proceedings of the National Academy of Sciences.

[165]  Sergey Lebedev,et al.  Drawing Movements as an Outcome of the Principle of Least Action. , 2001, Journal of mathematical psychology.

[166]  Lena H Ting,et al.  Subject-specific muscle synergies in human balance control are consistent across different biomechanical contexts. , 2010, Journal of neurophysiology.

[167]  E. Bizzi,et al.  Central and Sensory Contributions to the Activation and Organization of Muscle Synergies during Natural Motor Behaviors , 2005, The Journal of Neuroscience.

[168]  William J Kargo,et al.  Individual Premotor Drive Pulses, Not Time-Varying Synergies, Are the Units of Adjustment for Limb Trajectories Constructed in Spinal Cord , 2008, The Journal of Neuroscience.

[169]  T. Pozzo,et al.  Trajectories of arm pointing movements on the sagittal plane vary with both direction and speed , 2003, Experimental Brain Research.