degrees of freedom

Learning Objectives

  • Recognize the challenges and opportunities presented by the abundance of Degrees of Freedom (DoF) in perception and action. This refers to the vast number of ways the human body can move to achieve a single goal.

  • Understand how the brain addresses these challenges by employing strategies that result in computationally efficient solutions, such as motor synergies and coupling.

Quiz Question (MCQ)

  • Assessing results from the foundational study by Woodworth (1899) regarding movement speed and visual feedback.

  • Options:

    • A. At highest velocity, no difference in accuracy between eyes closed and eyes open when aiming at a target.

    • B. At lowest velocity, no difference in accuracy between eyes closed and eyes open when aiming at a target.

    • C. At highest velocity, significant difference in accuracy between eyes closed and eyes open when aiming at a target.

  • Correct Answer Insight: When movements are very fast, there is no time for the visual system to process feedback and adjust the trajectory, making eyes-open and eyes-closed performance comparable.

The Degrees of Freedom Problem

  • Introduction: The DoF problem addresses how the central nervous system (CNS) selects one specific movement pattern out of an infinite number of possibilities.

  • Coupling: This represents the reduction of independent variables by linking joints together (e.g., the wrist and shoulder moving as a single unit).

  • Mechanics: Physical considerations, such as gravity and friction, influence how the brain plans movement to minimize energy expenditure.

  • Soft Constraints: These are flexible rules applied during action planning, such as maintaining balance or avoiding obstacles while moving.

Understanding Degrees of Freedom in Statistics

  • Definition: In statistics, DoF is the maximum number of logically independent values that can vary in a data sample without violating constraints.

  • Statistical Data Example:

    • A study comparing energy levels with sugar (M=4.2, SD=1.3) vs. no sugar (M=2.2, SD=0.84).

    • Results: t(8)=2.89, p=0.020. The value in the parenthesis represents the degrees of freedom.

Motor Control

  • Definition: The number of parameters (joints, muscles, neurons) that can vary independently to produce a specific action.

  • Joint Parameters: To move a hand to a point in space, we must coordinate:

    • Shoulder pitch, shoulder roll, shoulder yaw.

    • Elbow pitch.

    • Wrist yaw and wrist pitch.

Historical Context of Degrees of Freedom

  • Quote by Nikolai Bernstein (1896-1966):

    • ‐‐ “It is clear that the basic difficulties for coordination consist precisely in the extreme abundance of degrees of freedom…” (1967).

    • Bernstein argued that the primary task of motor control is mastering these redundant degrees of freedom.

Action Selection

  • Theme: Selecting one action from an infinite set of alternatives. For example, there are infinite ways to reach for a coffee cup, yet we consistently choose a trajectory that is efficient and comfortable.

Redundancy and Motor Control

  • Importance: Redundancy is often seen as a problem, but it provides flexibility if one part of the system fails.

  • Technique Used: ‐‐ Freezing degrees of freedom: Beginners often stiffen their joints to reduce the number of variables they need to control. In marksmanship, a novice may freeze their wrist and elbow to match the shoulder movement, ensuring the gun stays on target.

Practical Exercise

  • Task: Touch your nose. Explore how different elbow and shoulder positions can achieve the same goal. This demonstrates that there is no unique joint configuration for a single spatial target.

Motor Equivalence

  • Concept: The ability of the motor system to achieve the same functional outcome using different muscle groups or limbs.

  • Reference (Raibert, 1977): Demonstrated that a person's handwriting maintains its unique characteristics (style) whether they use their:

    • A. Right hand.

    • B. Right hand constrained (wrist fixed).

    • C. Left hand.

    • D. Teeth.

    • E. Foot.

Case Study: Abigail and Brittany

  • Background: Conjoined twins who share a single body but have separate heads and brains.

  • Coordination Complexity: Each twin controls one side of the body (Abigail the right, Brittany the left). They must achieve a high level of neural coupling to perform complex actions like walking, running, or driving.

  • Shared Systems: Despite shared internal organs, their motor signals must be perfectly synchronized at the midline.

Reduction Through Coupling

  • Mechanism: The brain reduces the complexity of a task by coupling variables. If four joints (4 \text{ DOF}) are paired to move together in a fixed ratio, the control problem is simplified to only 3 \text{ DOF} or fewer.

Biological and Mechanical Coupling

  • Example Illustrations:

    • Fish fins: When a dorsal fin moves, it creates mechanical undulations that influence the oscillation of the side fins.

    • Human arms: When oscillating both arms at different frequencies, they tend to drift into a common rhythm due to neural and mechanical coupling.

Context-Dependent Movements

  • Neural Mechanism: Reflexes are context-sensitive. The same sensory stimulus (e.g., a tap on the leg) can produce different motor responses depending on whether the individual is standing still or in the middle of a walking stride.

Historical Insights on Physical Coupling

  • Christiaan Huygens: Observed that two pendulum clocks hanging on the same wall would eventually synchronize their swings; this physical principle applies to biological systems as well.

  • Michael Turvey: Used oscilloscopes to study how humans maintain rhythm, showing that motor systems naturally gravitate toward stable, coupled states.

Walking vs. Running

  • R. McNeill Alexander (1984): Defined the transition based on the Froude number.

  • Transition speed: Humans typically switch from walking to running at approximately 3 \text{ m/s}. At this point, the centrifugal force of the leg's swing exceeds the pull of gravity, making walking physically impossible.

Energy Efficiency in Gait

  • Finding: Our bodies naturally choose the gait that minimizes metabolic cost. The transition to running occurs exactly when the energy required to walk at that speed becomes greater than the energy required to run.

Technical Challenges in Motor Control

  • Studying mechanics is difficult because of non-linearities, sensory delays (roughly 100-200 \text{ ms}), and the changing environment.

Preflex Mechanisms

  • Definition: Immediate, zero-delay responses caused by the inherent mechanical properties (stiffness and damping) of muscles and tendons. These allow for stability even before the nervous system can react.

Passive Robotics Exploration

  • Focus: Passive bipedal walking robots use only gravity and their own physical structure to walk down a slight incline.

  • Example: The Compass-Gait Robot (Dr. Aaron Ames) demonstrates that walking can be a mechanical property rather than a complex computational task.

The Complexity of Motor Actions

  • Even when we appear still, the motor system is active. Posture is a dynamic process involving constant micro-adjustments to counteract gravity.

Muscle Coordination Example

  • Agonist and Antagonist: Normal movement requires the coordination of muscles that pull in opposite directions (e.g., bicep and tricep). If both contract equally, the joint is ‐‐frozen.‐‐

Frogs Study

  • Emilio Bizzi: Stimulated the spinal cords of frogs to map "force fields."

  • Equifinality: Regardless of where the frog's leg started, stimulation of a specific spinal site always drove the limb toward the same final equilibrium position.

Equifinality

  • Concept: The property of a system where the same final state is reached from different initial conditions and through different pathways. This simplifies control because the brain only needs to specify the target endpoint.

Neuronal Activity Vector Summation

  • A principle where the direction of a movement is determined by the weighted sum of activity across a population of neurons, rather than by a single cell.

Posture Neurons Study

  • Graziano et al. (2002): Microstimulation of the motor cortex in monkeys resulted in the limb moving to a specific complex posture, suggesting the brain encodes meaningful end-states rather than just individual muscle contractions.

Grasp Height Effect

  • Data (Cohen & Rosenbaum, 2004): When people pick up an object to move it, they adjust their initial grasp height based on the final destination.

    • If the target is high, they grasp the object low, and vice versa. This ensures a comfortable arm position at the end of the movement.

Second Order Grasping

  • First-order Planning: Thinking only about the immediate act of picking up an object.

  • Second-order Planning: Planning the initial action so that the subsequent action is efficient. This is known as the End-State Comfort Effect.

Precrastination Study

  • Concept: The tendency to complete a sub-goal as soon as possible, even at the cost of extra physical effort.

  • Fournier et al. (2018): Participants would pick up a heavy bucket early in a walk rather than later, just to "check off" the task in their mental to-do list.

  • Findings: Cognitive load significantly increases the likelihood of precrastination.

Summary of Interaction Among Perception, Action, and Cognition

  • These are not three separate systems but one integrated cycle. Perception informs action, action changes perception, and cognition governs the goals of both.

  • It is a burgeoning field in neural science that uses simple behavioral tasks to uncover deep principles of brain organization.

Conclusion and Reading Recommendations

  • Next Week: Full recap focusing on Learning, Feedback (reacting to errors), and Feedforward (predicting outcomes).

  • Recommended Reading: Latash (2012). "The bliss (not the problem) of motor abundance." Experimental Brain Research. This article argues that having many degrees of freedom is an advantage for flexibility, not just a problem to be solved.

Agonist and antagonist muscles are functional pairs that work in opposition to control joint movement and manage the body's vast Degrees of Freedom.

1. Functional Roles

  • Agonist (Prime Mover): This is the muscle primarily responsible for generating a specific movement. For example, during elbow flexion (curling the arm), the biceps act as the agonist.

  • Antagonist: This muscle opposes the agonist's action. To allow for smooth movement, the antagonist must relax and lengthen while the agonist contracts. In the elbow flexion example, the triceps serve as the antagonist.

2. Motor Control Strategies

  • Reciprocal Inhibition: This is a neural mechanism where the activation of an agonist muscle automatically triggers a signal to relax the antagonist, preventing internal resistance and ensuring movement efficiency.

  • Co-contraction (Co-activation): This occurs when both the agonist and antagonist contract simultaneously. While this is less energy-efficient, it serves critical purposes:

    • Freezing Degrees of Freedom: As noted by Bernstein, beginners often co-contract muscles to "freeze" joints. By activating both the biceps and triceps, a novice can simplify a task by turning a multi-joint limb into a single rigid unit, reducing the number of independent variables (DOF) the brain must manage.

    • Stability and Preflexes: Co-contraction increases joint stiffness, which enhances "preflex" responses—the immediate, mechanical resistance to external perturbations that occurs before neural feedback can kick in.

3. The Equilibrium Point (Equifinality)

The interaction between these opposing muscles is central to the concept of Equifinality. As demonstrated in Emilio Bizzi's research, the brain can specify a target position by adjusting the balance of tension between agonist and antagonist pairs. This creates a "force field" that pulls the limb toward a specific equilibrium point, regardless of where the movement started.