Course Instructor: Reza Shadmehr
Overview: This course blends robotics, control theory, and neuroscience to understand in some depth the human motor system. Our approach is to use mathematical models to explore functions of the structures that are involved in control of movement. We will begin by studying muscles, muscle sensory organs, spinal control structures, and inertial dynamics of a multi-joint limb. This will give us a sense of the machinery that the brain must control in order to make simple movements. We will then use mathematical models to consider parts of the brain that control our movements. Our focus is on reaching movements. A central problem involves computing limb and target position by integrating feedback from various sensory modalities like proprioception and vision. We will consider how the brain updates estimates of these variables during a movement using efference copy and how prisms and other visual distortions require realignment of these maps. We will consider how a reach may be represented as a feedback control policy, and then consider the process by which planned movements are transformed into motor commands. A central theme is how the brain learns to control movements of the limb. All along the way, the theoretical control problem is compared to what we know about the neuronal properties of the motor system in the brain and how diseases of the motor system affect the ability of the individual to control movements.
The course material and associated homework will require the student to use either Mathematica or Matlab to simulate control of biomechanical systems. It is highly recommended that students who plan to take this course familiarize themselves with one of these languages
Text book: Shadmehr R, Wise SP (2005) Computational Neurobiology of Reaching and Pointing: A
Foundation for Motor Learning, MIT
Web resources for the text book.
Introduction to Computational Motor Control Updated 9/3/03
Muscles and muscle
models Updated 9/03/03
Contents: Development of a viscoelastic model of passive properties of muscle and active force generation.
afferent system; from muscle force to joint torques Updated 9/03/03
Contents: Virtual work and development of a Jacobian and its relation to moment arms; development of a model of muscle spindles; development of a model of the gamma motor neuron system and its mechanism of control of length sensitivity of spindles. Supplementary materials for muscle models
and equilibrium of multi-muscle systems. Updated 9/17/03
Contents: Stimulation of antagonist muscles produces an equilibrium position for the limb; co-activation changes how the limb responds to a perturbation; Rapid movement through sequential activation of muscles; origin of the 3-burst EMG pattern; essential tremor; deafferentation
stiffness and time delays in feedback control. Updated 9/17/03
Contents: measurement of limb stiffness; gradient descent and estimation of hand stiffness and joint stiffness; time delays in spinal and supra-spinal feedback pathways. Supplementary materials for stiffness calculation
Estimating position of the
hand: alignment of vision with proprioception Updated 9/25/03
Contents: Variation in the proprioceptive input changes the brain’s estimate of hand position, even if other sensory modalities do not signal a change in hand position. The brain’s estimate of hand position is an alignment between current inputs from various sensory cues. This alignment may be influenced by the likelihood of the sensory cues. Recurrent networks provide one method for computationally representing this alignment. Mathematica code for simulating alignment between two noisy sensory maps
Encoding limb position in the
primate cortex Updated 9/30/03
Contents: Cortical fields in the posterior parietal cortex (PPC) and the frontal motor areas; a linear encoding of static limb position in the spinocerebellar tract, primary somatosensory cortex, and motor cortex; non-uniform distribution of preferred displacement; sensitivity to both proprioception and vision in PPC; effect of lesion in PPC on sense of limb position and estimating location of visible targets.
target position in the posterior parietal cortex Updated 10/02/03
Contents: Representing target position in fixation centered coordinates through a multiplicative combination of eye and head position signals with signals from the retina; Computational models of how a group of neurons can multiplicatively encode different sensory variables; Updating estimate of target position due to an intervening eye movement.
a difference vector Updated 10/07/03
Contents: Computing a hand-to-target vector (a difference vector) by subtracting an estimate of hand position from target position; errors in computing a difference vector accumulate for sequential movements; shoulder-centered vs. fixation centered coordinates; Area 5d as an intermediate layer in the coding of the difference vector; sensitivity of neuronal discharge to changes in positions of target and hand in PPC; model for computing a difference vector.
a movement plan Updated 10/09/03
Contents: Directional tuning and delay period activity in PPC and frontal cortex; maintaining a movement plan after target disappears; planning in terms of kinematics, not dynamics; planning saccades vs. reaching movements; planning a movement when no spatial information is available from the stimulus; covertly planning a movement but not performing it; planning multiple movements in a sequence.
Representing a difference
vector in the premotor cortex Updated 10/14/03
Contents: The premotor cortex appears to participate in a mapping that aligns end effector displacements (in fixation centered coordinates) to joint rotations. These cells have directional tuning that is generally not affected by direction of gaze or arm orientation. In the primary motor cortex, however, directional tuning is more affected by arm orientation. However, sensory cues that instruct a movement also affect discharge of cells in PM cortex.
of movement direction and force in the primary motor cortex Updated
Contents: Directional tuning of premotor and motor cortex cells; population coding; dependence of tuning on arm configuration.
proprioception with vision: prism adaptation Updated 10/21/03
Contents: Adapting reaching movements to prisms involves changes in two maps: a realignment of proprioceptive sense of position of the arm with vision of the hand, and a realignment of visually observed displacement of the hand with proprioceptively sensed motion of the arm. Short-term training often results in realignment of existing maps. Long-term training results in formation of new maps that can be instantiated based on context.
systems involved in prism adaptation Updated 10/23/03
Contents: Posterior parietal regions of cortex receive visual information primarily from the visual region contralateral to the fixation point. Damage in the right PPC may result in neglect of the left visual field. Adapting to a left-shifting prism depends on the left premotor cortex, as well as the cerebellum. Prism adaptation improves neglect. This is perhaps because it engages the cortical regions on the same hemisphere that has been damaged.
Contents: Prism adaptation and other visual perturbations produce a change in the maps that align vision with proprioception. The patterns of generalization indicate a broad tuning of arm position in proprioceptive space, and narrow tuning of direction of motion.
Contents: Reaching and pointing movements involve continuous monitoring of target- and end-effector location in fixation-centered coordinates with the goal of reducing the difference vector to zero. The CNS re-computes the kinematic maps that estimate target- and end-effector location as the eyes, targets and end effector move. Because this remapping depends on a copy of motor commands to the eyes, the head, and the arm, the CNS can update these estimates predictively. Systems that predict consequences of motor commands in sensory coordinates are called forward models. Forward models may also underlie your ability to imagine movements.
Contents: Reaching movements can entail an infinite number of trajectories for the hand. For most reaches, however, your CNS plans the movement so that the hand (or the end-effector) moves straight to the target in visual coordinates. One way to describe these trajectories is with a function that maximizes smoothness of hand position.
planners and control policies Updated 12/23/03
Contents: During reaching, our goal is to get the hand to the target. If something perturbs the hand, or the target, we want to be robust to the perturbations. A control policy describes a feedback system that evaluates the state of the limb at any given time, and estimates the motor commands that are sufficient to bring the hand to the target. The selection of the policy depends on the choice of what is being optimized. This lecture introduces the work of Bruce Hoff and Michael Arbib in deriving a feedback controller that has a minimum jerk control policy. It also introduces the approach used by
dependent noise and redundancy Updated 12/23/03
Contents: Why should movements be smooth? This lecture introduces the signal dependent noise theory and experiments of Harris and Wolpert. It also introduces the issue of redundancy and the theory of Todorov and Jordan for selecting actions to take advantage of redundancy. The lecture concludes by reviewing data on feedback control of reaching in patients with Huntington’s disease. The data suggests that the next-state planner may be affected in these patients.
minimization of an energy cost Updated 12/23/03
Contents: Motion of a system is one that minimizes an energy cost, the difference between the kinetic and potential energy of the system. We show how optimization of this cost produces
Contents: How do we learn internal models of dynamics? The brain appears to rely on the cerebellum and perhaps the motor cortex to form a model of the dynamics of reaching and the tools that we may hold in our hand. The basis functions with which this model is computed produces generalization patterns that one can observe in behavior of people