View source for Degrees of freedom problem

{{Short description|Multiple ways for multi-joint objects to realize a movement}}

In [[neuroscience]] and [[motor control]], the '''degrees of freedom problem''' or '''motor equivalence problem''' states that there are multiple ways for humans or animals to perform a [[Motion (physics)|movement]] in order to achieve the same goal. In other words, under normal circumstances, no simple one-to-one correspondence exists between a motor problem (or task) and a motor solution to the problem. The motor equivalence problem was first formulated by the Russian [[neurophysiology|neurophysiologist]] [[Nikolai Bernstein]]: "It is clear that the basic difficulties for co-ordination consist precisely in the extreme abundance of degrees of freedom, with which the [nervous] centre is not at first in a position to deal."<ref name="bernstein">{{cite book|last=Bernstein|first=Nikolai|title=The Coordination and Regulation of Movements|year=1967|publisher=Pergamon Press|location=Oxford}}</ref>

Although the question of how the [[nervous system]] selects which particular [[Degree of freedom (mechanics)|degrees of freedom]] (DOFs) to use in a movement may be a problem to scientists, the abundance of DOFs is almost certainly an advantage to the [[mammal]]ian and the [[invertebrate]] [[nervous system]]s.<ref name=flash2005motor>{{cite journal|author1=[[Tamar Flash|Flash, Tamar]] |author2=Hochner, Binyamin |title=Motor primitives in vertebrates and invertebrates|year=2005|journal=Current Opinion in Neurobiology|volume = 15|issue=6| pages=660–666|doi=10.1016/j.conb.2005.10.011|pmid=16275056|s2cid=5815716 }}</ref> The human body has redundant [[Anatomy|anatomical]] DOFs (at [[muscle]]s and [[joint]]s), redundant [[kinematics|kinematic]] DOFs (movements can have different trajectories, velocities, and accelerations and yet achieve the same goal), and redundant [[neurophysiology|neurophysiological]] DOFs (multiple [[motor neuron|motoneurons]] [[neuromuscular junction|synapsing]] on the same muscle, and vice versa).<ref name=bernstein/><ref name=scott>{{cite journal|last=Scott|first=Stephen|title=Optimal feedback control and the neural basis of volitional motor control|year=2004|journal=Nature Reviews Neuroscience|volume = 5| issue = 7|pages=534–546|doi = 10.1038/nrn1427|pmid=15208695|s2cid=708532 }}</ref> How the nervous system "chooses" a subset of these near-infinite DOFs is an overarching difficulty in understanding [[motor control]] and [[motor learning]].

==History==

The study of [[motor control]] historically breaks down into two broad areas: "Western" [[neurophysiology|neurophysiological]] studies, and "Bernsteinian" functional analysis of movement. The latter has become predominant in motor control, as Bernstein's theories have held up well and are considered founding principles of the field as it exists today.

===Pre-Bernstein===

In the latter 19th and early 20th centuries, many scientists believed that all motor control came from the [[spinal cord]], as experiments with [[electrophysiology|stimulation]] in frogs displayed patterned movement ("motor primitives"), and spinalized cats were shown to be able to [[Spinal locomotion|walk]]. This tradition was closely tied with the strict nervous system [[functional specialization (brain)|localizationism]] advocated during that period; since stimulation of the frog spinal cord in different places produced different movements, it was thought that all motor impulses were localized in the spinal cord. However, fixed structure and localizationism were slowly broken down as the central dogma of [[neuroscience]]. It is now known that the primary [[motor cortex]] and [[premotor cortex]] at the highest level are responsible for most voluntary movements. Animal models, though, remain relevant in [[motor control]] and spinal cord [[Stretch reflex|reflexes]] and [[central pattern generator]]s are still a topic of study.<ref name=hart>{{cite journal
| last=Hart
| first=C.B.
| title=A Neural Basis for Motor Primitives in the Spinal Cord
| journal=Journal of Neuroscience
| year=2010
| volume = 30
| issue = 4
|pages=1322–1336
|pmc=6633785| doi = 10.1523/jneurosci.5894-08.2010
| pmid=20107059
| doi-access=free
}}</ref><ref name=mccrea>{{cite journal|last=McCrea|first=David|title=Organization of mammalian locomotor rhythm and pattern generation|year=2008|journal=Brain Research Reviews|volume = 30| issue = 1|pages=134–146|doi = 10.1016/j.brainresrev.2007.08.006|pmid=17936363|pmc=2214837}}</ref>

===Bernstein===

{{Main|Nikolai Bernstein}}

Although [[Karl Lashley|Lashley]] (1933) first formulated the motor equivalence problem,<ref name=lashley>{{cite journal|last=Lashley|first=KS|title=Integrative function of the cerebral cortex|year=1933|journal=Physiological Reviews|volume = 13| issue = 1|pages=1–42|doi=10.1152/physrev.1933.13.1.1}}</ref> it was Bernstein who articulated the DOF problem in its current form. In Bernstein's formulation, the problem results from infinite redundancy, yet flexibility between movements; thus, the nervous system apparently must choose a particular motor solution every time it acts.  In Bernstein's formulation, a single muscle never acts in isolation. Rather, large numbers of "nervous centres" cooperate in order to make a whole movement possible. Nervous impulses from different parts of the [[Central nervous system|CNS]] may converge on the [[Peripheral nervous system|periphery]] in combination to produce a movement; however, there is great difficulty for scientists in understanding and coordinating the facts linking impulses to a movement. Bernstein's rational understanding of movement and prediction of [[motor learning]] via what we now call "plasticity" was revolutionary for his time.<ref name=bernstein/>

In Bernstein's view, movements must always reflect what is contained in the "central impulse", in one way or another.  However, he recognized that effectors ([[Feed forward (control)|feed-forward]]) were not the only important component to movement; [[feedback]] was also necessary. Thus, Bernstein was one of the first to understand movement as a closed circle of interaction between the nervous system and the [[Sense|sensory environment]], rather than a simple arc toward a goal. He defined motor coordination as a means for overcoming indeterminacy due to redundant peripheral DOFs. With increasing DOFs, it is increasingly necessary for the nervous system to have a more complex, delicate organizational control.<ref name=bernstein/>

Because humans are adapted to survive, the "most important" movements tend to be [[reflex]]es -- [[pain withdrawal reflex|pain]] or defensive reflexes needed to be carried out in very short time scales in order for ancient humans to survive their harsh environment. Most of our movements, though, are voluntary; voluntary control had historically been under-emphasized or even disregarded altogether. Bernstein saw voluntary movements as structured around a "motor problem" where the nervous system needed two factors to act: a full and complete perception of reality, as accomplished by [[multisensory integration]], and objectivity of perception through constant and correct recognition of [[Action potential|signals]] by the nervous system. Only with both may the nervous system choose an appropriate motor solution.<ref name=bernstein/>

==Difficulties==

The DOF problem is still a topic of study because of the complexity of the neuromuscular system of the human body. Not only is the problem itself exceedingly difficult to tackle, but the vastness of the field of study makes synthesis of theories a challenge.

===Counting degrees of freedom===

[[File:CatHindlinbModel.tif|thumb|Cat hindlimb musculoskeletal model with redundant degrees of freedom at muscles (red lines) and joints]]

One of the largest difficulties in motor control is quantifying the exact number of DOFs in the complex neuromuscular system of the human body. In addition to having redundant [[muscle]]s and [[joint]]s, muscles may span multiple joints, further complicating the system. Properties of muscle change as the muscle length itself changes, making [[Mathematical model|mechanical models]] difficult to create and understand. Individual muscles are [[Nerve|innervated]] by multiple nerve fibers ([[motor unit]]s), and the manner in which these units are recruited is similarly complex. While each joint is commonly understood as having an [[Agonist (muscle)|agonist]]-[[Antagonist (muscle)|antagonist]] pair, not all joint movement is controlled locally. Finally, movement kinematics are not identical even when performing the same motion repeatedly; natural variation in [[Position (vector)|position]], [[velocity]], and [[acceleration]] of the [[Limb (amatomy)|limb]] occur even during seemingly identical movements.<ref name = bernstein/><ref name = scott/><ref name=guigon/>

===Types of studies===

Another difficulty in motor control is unifying the different ways to study movements. Three distinct areas in studying motor control have emerged: limb mechanics, neurophysiology, and motor behavior.<ref name=scott/>

====Limb mechanics====

Studies of limb mechanics focus on the peripheral motor system as a [[Filter (signal processing)|filter]] which converts patterns of muscle activation into purposeful movement. In this paradigm, the building block is a [[motor unit]] (a [[Motor neuron|neuron]] and all the [[Skeletal striated muscle|muscle fibers]] it innervates) and complex models are built to understand the multitude of biological factors influencing motion. These models become increasingly complicated when multiple joints or environmental factors such as [[ground reaction force]]s are introduced.<ref name=scott/>

====Neurophysiology====

In [[Neurophysiology|neurophysiological]] studies, the motor system is modeled as a distributed, often hierarchical system with the [[spinal cord]] controlling the "most automatic" of movements such as stretch reflexes, and the [[Motor cortex|cortex]] controlling the "most voluntary" actions such as reaching for an object, with the [[brainstem]] performing a function somewhere in between the two. Such studies seek to investigate how the primary motor cortex (M1) controls planning and execution of motor tasks. Traditionally, neurophysiological studies have used animal models with [[Electrophysiology|electrophysiological]] recordings and stimulation to better understand human motor control.<ref name=scott/>

====Motor behavior====

Studies of motor behavior focus on the [[Adaptation|adaptive]] and [[feedback]] properties of the nervous system in motor control. The motor system has been shown to adapt to changes in its [[Mechanics|mechanical]] environment on relatively short timescales while simultaneously producing smooth movements; these studies investigate how this remarkable feedback takes place. Such studies investigate which [[variable (research)|variables]] the nervous system controls, which variables are less tightly controlled, and how this control is implemented. Common paradigms of study include voluntary reaching tasks and perturbations of standing balance in humans.<ref name=scott/>

===Abundance or redundancy===

Finally, the very nature of the DOF problem poses questions. For example, does the nervous system really have difficulty in choosing from DOFs, or is the abundance of DOFs necessary for [[evolution]]ary survival? In very extreme movements, humans may exhaust the limits of their DOFs—in these cases, the nervous system only has one choice. Therefore, DOFs are not always infinite. Bernstein has suggested that our vast number of DOFs allows [[motor learning]] to take place, wherein the nervous system "explores" the set of possible motor solutions before settling on an [[Optimum|optimal]] solution (learning to walk and ride a bike, for example). Finally, additional DOFs allow patients with brain or [[spinal cord injury]] to often retain movement while relying on a reduced set of [[Biomechanics|biomechanical]] DOFs.  Therefore, the "degrees of freedom problem" may be a misnomer and is better understood as the "motor equivalence problem" with redundant DOFs offering an evolutionary solution to this problem.<ref name=gelfand>{{cite journal|last=Gelfand|first=I.M.|title=On the problem of adequate language in motor control|year=1998|journal=Motor Control|volume = 2| issue = 4|pages=306–13|pmid=9758883|doi=10.1123/mcj.2.4.306}}</ref>

==Hypotheses and proposed solutions==

{{See also|Motor coordination#Theories}}

There have been many attempts to offer solutions or conceptual models that explain the DOF problem. One of the first hypotheses was [[Fitts's Law|Fitts' Law]], which states that a trade-off must occur between movement speed and movement [[Accuracy and precision|accuracy]] in a reaching task. Since then, many other theories have been offered.

===Optimal control hypothesis===

A general paradigm for understanding motor control, optimal control has been defined as "optimizing motor control for a given aspect of task performance," or as a way to minimize a certain "cost" associated with a movement.<ref name=scott/><ref name="iaroslav_02"/> This "cost function" may be different depending on the task-goal; for example, minimum [[Energy balance (biology)|energy expenditure]] might be a task-variable associated with [[Terrestrial locomotion|locomotion]], while precise trajectory and positional control could be a task-variable associated with reaching for an object.<ref name=todorov>{{cite journal|last=Todorov|first=Emmanuel|title=Optimality principles in sensorimotor control|year=2004|journal=Nature Neuroscience|volume = 7| issue = 9|pages=907–915|doi = 10.1038/nn1309|pmid=15332089|pmc=1488877}}</ref> Furthermore, the cost function may be quite complex (for instance, it may be a functional instead of function) and be also related to the representations in the [[Internal model (motor control)|internal space]]. For example, the speech produced by biomechanical tongue models, controlled by the internal model which minimizes the length of the path traveled in the internal space under the constraints related to the executed task (e.g., quality of speech, stiffness of tongue), was found to be quite realistic.<ref name="iaroslav_02"/> In essence, the goal of optimal control is to "reduce degrees of freedom in a principled way."<ref name=guigon>{{cite journal|last=Guigon|first=Emmanuel|title=Computational motor control: Redundancy and invariance|year=2007|journal=Journal of Neurophysiology|volume = 97|issue = 1|pages=331–347|doi = 10.1152/jn.00290.2006|pmid=17005621|citeseerx=10.1.1.127.3437}}</ref> Two key components of all optimal control systems are: a "state estimator" which tells the nervous system about what it is doing, including afferent sensory feedback and an [[Efference copy|efferent copy]] of the motor command; and adjustable [[Loop gain|feedback gains]] based on task goals.<ref name=shadmehr>{{cite journal|author=Reza, Shadmehr|author2=John W. Krakauer |title=A computational neuroanatomy for motor control|year=2008|journal=[[Experimental Brain Research]]|volume = 185 | issue = 3|pages=359–381| doi = 10.1007/s00221-008-1280-5|pmid=18251019 |pmc=2553854}}</ref> A component of these adjustable gains might be a "minimum intervention principle" where the nervous system only performs selective error correction rather than heavily modulating the entirety of a movement.<ref name=todorov/>

====Open and closed-loop models====

Both open-loop and closed-loop models of optimal control have been studied; the former generally ignores the role of sensory feedback, while the latter attempts to incorporate sensory [[feedback]], which includes delays and uncertainty associated with the sensory systems involved in movement.<ref>{{cite journal|last1=Mehrabi|first1=Naser|last2=Sharif Razavian|first2=Reza|last3=McPhee|first3=John|title=Steering disturbance rejection using a physics-based neuromusculoskeletal driver model|journal=Vehicle System Dynamics|date=17 June 2015|pages=1393–1415|doi=10.1080/00423114.2015.1050403|volume=53|issue=10|bibcode=2015VSD....53.1393M |hdl=10012/13424|s2cid=54173111 |url=https://uwspace.uwaterloo.ca/bitstream/10012/13424/1/Steering%20Disturbance%20Rejection%20using%20a%20Physics-based%20Neuromusculoskeletal%20Driver%20Model.pdf|hdl-access=free}}</ref> Open-loop models are simpler but have severe limitations—they model a movement as prerecorded in the nervous system, ignoring sensory feedback, and also fail to model variability between movements with the same task-goal. In both models, the primary difficulty is identifying the [[cost]] associated with a movement. A mix of cost variables such as minimum energy expenditure and a "smoothness" function is the most likely choice for a common performance criterion.<ref name=todorov/>

====Learning and optimal control====

Bernstein suggested that as humans learn a movement, we first reduce our DOFs by stiffening the musculature in order to have tight control, then gradually "loosen up" and explore the available DOFs as the task becomes more comfortable, and from there find an [[Optimization problem|optimal]] solution.<ref name=bernstein/> In terms of optimal control, it has been postulated that the nervous system can learn to find task-specific variables through an optimal control search strategy. It has been shown that adaptation in a visuomotor reaching task becomes optimally tuned so that the cost of movement trajectories decreases over trials. These results suggest that the nervous system is capable of both nonadaptive and adaptive processes of optimal control.  Furthermore, these and other results suggest that rather than being a control variable, consistent movement trajectories and velocity profiles are the natural outcome of an adaptive optimal control process.<ref name=braun>{{cite journal|author=Daniel Braun |author2=Ad Aertsen |author3=Daniel Wolpert |author4=Carsten Mehring|title=Learning Optimal Adaptation strategies in unpredictable motor tasks|year=2009|journal=The Journal of Neuroscience|volume = 29| issue = 20|pages=6472–6478|doi = 10.1523/jneurosci.3075-08.2009 |pmid=19458218 |pmc=2692080}}</ref>

====Limits of optimal control====

Optimal control is a way of understanding motor control and the motor equivalence problem, but as with most [[mathematics|mathematical]] theories about the nervous system, it has limitations. The theory must have certain information provided before it can make a behavioral prediction: what the costs and rewards of a movement are, what the constraints on the task are, and how [[state estimation]] takes place. In essence, the difficulty with optimal control lies in understanding how the nervous system precisely executes a control strategy.<ref name="iaroslav_02"/> Multiple operational time-scales complicate the process, including sensory delays, [[muscle fatigue]], changing of the external environment, and cost-learning.<ref name=todorov/><ref name=shadmehr/>

===Muscle synergy hypothesis===

In order to reduce the number of [[Human musculoskeletal system|musculoskeletal]] DOFs upon which the nervous system must operate, it has been proposed that the nervous system controls muscle synergies, or groups of co-activated muscles, rather than individual muscles. Specifically, a muscle synergy has been defined as "a vector specifying a pattern of relative muscle activation; absolute activation of each synergy is thought to be modulated by a single neural command signal."<ref name=ting>{{cite journal|author=Lena Ting|author2=J. Lucas McKay |title=Neuromechanics of muscle synergies for posture and movement|year=2007|journal=Current Opinion in Neurobiology|volume = 17| issue = 6|pages=622–628|doi = 10.1016/j.conb.2008.01.002|pmid=18304801|pmc=4350235}}</ref> Multiple muscles are contained within each synergy at fixed ratios of co-activation, and multiple synergies can contain the same muscle. It has been proposed that muscle synergies emerge from an interaction between constraints and properties of the nervous and musculoskeletal systems. This organization may require less computational effort for the nervous system than individual muscle control because fewer synergies are needed to explain a behavior than individual muscles. Furthermore, it has been proposed that synergies themselves may change as behaviors are learned and/or optimized. However, synergies may also be [[Intrinsic and extrinsic properties|innate]] to some degree, as suggested by postural responses of humans at very young ages.<ref name=ting/>

A key point of the muscle synergy hypothesis is that synergies are low-[[dimension]]al and thus just a few synergies may account for a complex movement.<ref name=davella>{{cite journal|author=Andrea d'Avella |author2=Philippe Saltiel |author3=Emilio Bizzi|title=Combinations of muscle synergies in the construction of a natural motor behavior|year=2003|journal=Nature Neuroscience|volume = 6| issue = 3|pages=622–628| doi = 10.1038/nn1010|pmid=12563264|s2cid=2437859 }}</ref> Evidence for this structure comes from [[electromyography|electromyographical]] (EMG) data in frogs, cats, and humans, where various mathematical methods such as [[Principal component analysis|principal components analysis]] and non-negative matrix factorization are used to "extract" synergies from muscle activation patterns.<ref name=ting/><ref name=davella/><ref name=tresch>{{cite journal|last=Tresch|first=MC|title=Matrix factorization algorithms for the identification of muscle synergies: Evaluation on simulated and experimental data sets|year=2006|journal=Journal of Neurophysiology|volume = 95| issue = 4|pages=2199–2212|doi = 10.1152/jn.00222.2005|pmid=16394079|citeseerx=10.1.1.136.4466}}</ref> Similarities have been observed in synergy structure even across different tasks such as kicking, jumping, swimming and walking in frogs.<ref name=davella/> Further evidence comes from [[stroke]] patients, who have been observed to use fewer synergies in certain tasks; some stroke patients used a comparable number of synergies as healthy subjects, but with reduced motor performance. These data suggest that a synergy formulation is robust and may lie at the lowest level of a hierarchical neural controller.<ref name=ting/>

===Equilibrium point hypothesis and threshold control===

In the Equilibrium Point hypothesis, all movements are generated by the nervous system through a gradual transition of [[equilibrium point]]s along a desired trajectory. "Equilibrium point" in this sense is taken to mean a state where a field has zero force, meaning opposing muscles are in a state of balance with each other, like two rubber bands pulling the joint to a stable position. Equilibrium point control is also called "threshold control" because signals sent from the CNS to the periphery are thought to modulate the threshold length of each muscle. In this theory, [[motor neuron]]s send commands to muscles, which changes the [[Muscle contraction|force–length relation]] within a muscle, resulting in a shift of the system's equilibrium point. The nervous system would not need to directly estimate limb [[Analytical dynamics|dynamics]], but rather muscles and spinal reflexes would provide all the necessary information about the system's state.<ref name=asatryan1965functional>{{cite journal|author1=Asatryan, David G |author2=Feldman, Anatol G |title=Functional tuning of the nervous system with control of movement or maintenance of a steady posture: I. Mechanographic analysis of the work of the joint or execution of a postural task|year=1965|journal=Biophysics|volume = 10| pages=925–934}}</ref> The equilibrium-point hypothesis is also reported to be well suited for the design of biomechanical robots controlled by appropriated internal models.<ref name="iaroslav_02">[https://www.researchgate.net/profile/Iaroslav_Blagouchine/publication/224080014_Control_of_a_Speech_Robot_via_an_Optimum_Neural-Network-Based_Internal_Model_With_Constraints Iaroslav Blagouchine and Eric Moreau. ''Control of a Speech Robot via an Optimum Neural-Network-Based Internal Model with Constraints.'' IEEE Transactions on Robotics, vol. 26, no. 1, pp. 142—159, February 2010.]</ref>

===Force control and internal models===

{{Main|Internal model (motor control)}}

The force control hypothesis states that the nervous system uses calculation and direct specification of [[force]]s to determine movement trajectories and reduce DOFs. In this theory, the nervous system must form [[internal model (motor control)|internal models]]—a representation of the body's dynamics in terms of the surrounding environment.<ref name="iaroslav_02"/> A nervous system which controls force must generate torques based on predicted kinematics, a process called [[inverse dynamics]]. Both [[Feed forward (control)|feed-forward]] (predictive) and [[feedback]] models of motion in the nervous system may play a role in this process.<ref name=ostry>{{cite journal|last=Ostry|first=David|title=A critical evaluation of the force control hypothesis in motor control|year=2003|journal=Experimental Brain Research|volume = 153| issue = 3|pages=275–288| doi = 10.1007/s00221-003-1624-0|pmid=14610628|s2cid=8601920 }}</ref>

===Uncontrolled manifold (UCM) hypothesis===

It has been noted that the nervous system controls particular variables relevant to performance of a task, while leaving other variables free to vary; this is called the uncontrolled manifold hypothesis (UCM).<ref name=scholz>{{cite journal|author=John P. Scholz|author2=Gregor Schöner|title=The uncontrolled manifold concept: identifying control variables for a functional task|year=1999|journal=Experimental Brain Research|volume = 126| issue = 3|pages=289–306| doi = 10.1007/s002210050738|pmid=10382616|s2cid=206924808 }}</ref> The uncontrolled manifold is defined as the set of variables not affecting task performance; variables [[perpendicular]] to this set in Jacobian space are considered controlled variables (CM). For example, during a sit-to-stand task, head and center-of-mass position in the horizontal plane are more tightly controlled than other variables such as hand motion. Another study indicates that the quality of tongue's movements produced by bio-robots, which are controlled by a specially designed internal model, is practically uncorrelated with the stiffness of the tongue; in other words, during the speech production the relevant parameter is the quality of speech, while the stiffness is rather irrelevant.<ref name="iaroslav_02"/> At the same time, the strict prescription of the stiffness' level to the tongue's body affects the speech production and creates some variability, which is however, not significant for the quality of speech (at least, in the reasonable range of stiffness' levels).<ref name="iaroslav_02"/> UCM theory makes sense in terms of Bernstein's original theory because it constrains the nervous system to only controlling variables relevant to task performance, rather than controlling individual muscles or joints.<ref name=scholz/>

==Unifying theories==

Not all theories about the selection of movement are mutually exclusive. Necessarily, they all involve reduction or elimination of redundant DOFs. Optimal feedback control is related to UCM theory in the sense that the optimal control law may not act along certain [[dimension]]s (the UCM) of lesser importance to the nervous system. Furthermore, this lack of control in certain directions implies that controlled variables will be more tightly correlated; this correlation is seen in the low-dimensionality of muscle synergies. Furthermore, most of these theories incorporate some sort of [[feedback]] and [[Feed forward (control)|feed-forward]] models that the nervous system must utilize. Most of these theories also incorporate some sort of [[Hierarchy|hierarchical]] neural control scheme, usually with cortical areas at the top and peripheral outputs at the lowest level.<ref name=todorov/> However, none of the theories is perfect; the DOF problem will continue to be relevant as long as the nervous system is imperfectly understood.

==See also==
*[[Motor control]]
*[[Motor coordination]]
*[[Motor learning]]
*[[Nikolai Bernstein]]
*[[Six degrees of freedom]]

==References==
{{reflist|2}}

[[Category:Motor control]]