Doug Weber

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System Identification / Systems Analysis in
neuromechanical control systems

• Doug Weber
• January 22, 2009

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Topics

• What is a system (static vs. dynamic)?
• What is system ID?
• Why/when is it useful or necessary?
• How is it done?
• Examples from the neuroscience/neural engineering
  literature
• Terzuolo CA, and Poppele RE. Myotatic reflex Its
  input-output relation. Science 159 743-745,
  1968.
• Humphrey DR. Relating motor cortex spike trains
  to measures of motor performance. Brain Res 40
  7-18, 1972.
• Supplemental reading Intro to Dynamic Systems by
  Dr. Sanjeev Shroff (sections A-D, pp. 1-4)

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What is a system?

• A system is a a bounded collection of
  interacting elements giving rise to some
  collective behavior of interest
• Examples
• Muscles and bones
• Neurons and muscles
• Neurons and other neurons
• Channels and neurons

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Static and Dynamic systems

• Static systems output (y) at time t depends only
  on input (u) at time t

• Dynamic systems a system with "memory
• The time evolution of system behavior depends on
  the history of that behavior. 
• In other words, one cannot predict the output of
  dynamic system based simply on the current input
  only one needs to know the past, including the
  initial conditions.

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Static and Dynamic systems (contd)

• All physical systems are dynamic
• In equilibrium, dynamic systems become static,
  but only as long as they reside in the
  equilibrium state

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What is system ID?

• Analytical methodology for modeling dynamic
  systems based on empirical data
• Different model types involve different methods
  for ID
• Mechanistic models (structural or parametric
  models) require a priori knowledge of the
  elements of the system and how they interact.
• If you can fully describe all of properties of
  the elements, you have a white box model and do
  not require system ID (e.g. model of mass-spring
  system with known mass and stiffness)
• If you need to identify those properties, you
  have a gray box model and need to make
  measurements to determine them. (e.g. model of
  mass-spring system with unknown mass and
  stiffness).
• Descriptive (black-box models non-parametric)
  describe only the input-output behaviors of the
  system you cant see inside the box, but you can
  measure the responses over a range of inputs and
  time.

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Why is system ID used?

• Develop controller for driving the system
• Learn something about the behavior of the system
• study how behavior changes under different
  conditions
• example pendulum test for spasticity

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Why is system ID used?

• Develop controller for driving the system
• Learn something about the behavior of the system
• study how behavior changes under different
  conditions
• example pendulum test for spasticity

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Methods of system ID

• Parametric methods generally involve optimization
  model parameters tuned to minimize prediction
  error
• Non-parametric methods probe system with various
  inputs and characterize input/output relationship
  by
• Time domain methods
• Impulse response (e.g. Humphrey, 1972)
• Step response
• Correlation analysis / time
• Frequency domain methods
• Sine-wave testing (e.g. Poppele Terzuolo, 1968)
• Correlation analysis / Frequency
• Fourier-analysis
• Spectral analysis

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System ID methods in neurophysiology

• Modeling input-output properties of neurons,
  particularly sensory neurons
• Many labels
• Receptive field estimation
• Spike-triggered analysis
• Peri-stimulus time histograms
• Reverse correlation
• White noise analysis

Stimulus
Response
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Myotatic Reflex its input/output relation
(Terzuolo and Poppele, 1968)
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Myotatic Reflex its input/output relation
(Terzuolo and Poppele, 1968)
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Monosynaptic stretch reflex shortest feedback
control system
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Monosynaptic stretch reflex shortest feedback
control system
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Motor Unit
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Motor Unit Dynamics
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Muscle spindle afferents
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Muscle spindle structure and function
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Muscle spindle structure and function
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Muscle spindle output in response to stretch
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Myotatic Reflex its input/output relation
(Terzuolo and Poppele, 1968)

• Goal use system ID to characterize dynamics of
  monsynaptic stretch reflex circuit
• Experiment measure input/output
  characteristicsof each element and the complete
  circuit
• Prep decerebrate or anesthetized cat
  deefferented
• Assumption Linear, time-invariant (LTI) system

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Experimental prep

• Inputs Sinusoidal stretches applied to muscle
  (a) or joint (b) frequency and amplitude varied
• Probed outputs
• Ia firing rate
• Transmembrane potential of motor-neuron
• Muscle (Force, EMG)

1
2
3
2
3
1
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Frequency Response(Gain Phase) lationships(0
- 6 Hz)

• Input 1 Hz sinusoidal stretches, amplitudes lt
  2.2 mm
• Outputs
• Muscle
• EMG (filled triangle) produced during sinusoidal
  stretching
• Muscle tension produced during sinusoidal
  stretching (open triangles)
• Muscle tension () produced during muscle nerve
  stimulation
• Ia firing rate (filled circles)
• MN membrane potential (open circles)

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Gain and phase of isolated muscle (c)
Sinusoidal Muscle stim (20-35 pps)
Muscle
Force ()
Force-producing machinery in muscle introduces
phase lag and decreasing gain with frequency
muscle acts like a low-pass filter.
25

• Gain and phase of reflex loop (a,b)

Transmembrane Potential (O)
EMG (?)
Force (?)
Sinusoidal Stretches
Ia firing rates (?)
Spindle dynamics compensate for low-pass filter
properties of muscle contractile machinery
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Summary of Frequency Response Characteristics
Ia
a-MN
Muscle

• Ia response provides proportional-derivative (PD)
  control of muscle length
• Cascade of elements in reflex loop results in
  flat input-output response

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Amplitude Response

• Input 1 Hz sine stretches, amplitudes lt 2.2 mm
• Output
• Ia firing rate (filled circles)
• MN membrane potential (open circles)
• EMG (triangles, 2 cats)

MN output has wider dynamic range than spindles
likely the result of convergent afferent inputs
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Relating motor cortex spike trains to measures of
motor performance (Humphrey, 1972).

• Goal determine transfer function from motor
  cortex neurons to motor action variables
• Experiment measure motor cortex activity and
  wrist joint position and torque during
  alternating flexion/extension of wrist
• Prep awake, behaving monkey with chronic
  recording chamber over motor cortex

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PTN-musculoskeleton as a LTI system

• Impulse response (h) used to characterize
  system dynamics (i.e. transfer function)
• Least-squares estimation procedure used to
  identify h
• Model accurately predicts torque output for a
  given spike train
• dynamic model works better than static model

Note t0 50 ms
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3 types of PTNs
Tonic cells
Phasic-tonic cells

• Tonic cells co-vary with torque
• Phasic cells co-vary with rate of torque change

Phasic cells
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Assembling the complete neural control system
Motorcortex
Torque
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Muscle spindles and tendon organs provide
compensation for muscle dynamics s.t. GMHS and
GMHT are constant where K0 and K1
represent loop gains of GTO and Ia pathways
Conclusion Motor cortex output has direct
(unfiltered) access to muscle, because dynamics
of lower levels can be approximated by static
sub-systems having constant gain/phase
relationships
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Summary Cortico-muscular control system
The cortico-muscular control system behaves like
a linear, 2nd order system (i.e. dynamics of
lower levels do not increase the order of the
total system. Output from motor cortex is
appropriate to compensate for dynamics of muscle
force generation 3 distinct types of signals
from motor cortex provide drive to muscles - a
proportional component (µ), controls
stead-state force - derivative components (a,ß)
compensate for dynamics of muscle and loads
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Summary

• Systems ID/analysis tools can provide
  quantitative descriptions of neuro-musculo-skeleta
  l control systems
• Interactions among dynamic sub-systems are
  crucial determinants of overall system behavior
• Analysis of motor behavior that ignores dynamics
  of underlying systems provides limited (if any)
  insight into the underlying neural control
  mechanisms