580.431/631 Computational
Motor Control
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
Press,
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.
Muscle
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
Stability
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
Limb
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.
Encoding
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.
Encoding
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.
Computing
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.
Coding
of movement direction and force in the primary motor cortex Updated
10/16/03
Contents: Directional tuning of premotor and motor cortex cells; population
coding; dependence of tuning on arm configuration.
Realignment of
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.
Neural
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.
Generalization
Updated 10/30/03
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.
Remapping Updated
10/30/03
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.
Planning trajectories
Updated 11/04/03
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.
Next-state
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
Signal
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.
Dynamics as
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
Learning dynamics
Updated 12/23/03
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