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Research Papers

Clinical Facts Along With a Feedback Control Perspective Suggest That Increased Response Time Might Be the Cause of Parkinsonian Rest Tremor

[+] Author and Article Information
Vrutangkumar V. Shah

SysIDEA Lab,
Mechanical Engineering,
Indian Institute of Technology Gandhinagar,
Gandhinagar 382355, India
e-mail: shah_vrutangkumar@iitgn.ac.in

Sachin Goyal

Assistant Professor
Department of Mechanical Engineering,
University of California,
Merced, CA 95343
e-mail: sachin.goyal@ucmerced.edu

Harish J. Palanthandalam-Madapusi

Assistant Professor
SysIDEA Lab,
Mechanical Engineering,
Indian Institute of Technology Gandhinagar,
Gandhinagar 382355, India
e-mail: harish@iitgn.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 25, 2015; final manuscript received June 24, 2016; published online September 1, 2016. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 12(1), 011007 (Sep 01, 2016) (8 pages) Paper No: CND-15-1396; doi: 10.1115/1.4034050 History: Received November 25, 2015; Revised June 24, 2016

Parkinson's disease (PD) is a neurodegenerative disorder characterized by increased response times leading to a variety of biomechanical symptoms, such as tremors, stooping, and gait instability. Although the deterioration in biomechanical control can intuitively be related to sluggish response times, how the delay leads to such biomechanical symptoms as tremor is not yet understood. Only recently has it been explained from the perspective of feedback control theory that delay beyond a threshold can be the cause of Parkinsonian tremor (Palanthandalam-Madapusi and Goyal, 2011, “Is Parkinsonian Tremor a Limit Cycle?” J. Mech. Med. Biol., 11(5), pp. 1017–1023). The present paper correlates several observations from this perspective to clinical facts and reinforces them with simple numerical and experimental examples. Thus, the present work provides a framework toward developing a deeper conceptual understanding of the mechanism behind PD symptoms. Furthermore, it lays a foundation for developing tools for diagnosis and progress tracking of the disease by identifying some key trends.

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Figures

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Fig. 1

The contributions of this work

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Fig. 2

The closed-loop feedback system representing motor control of human body

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Fig. 3

Angular velocity for various time delays with zero intended velocity, initial angular position of 0.1 rad, and saturation limits as −100 to 100 N·m (pendulum numerical example)

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Fig. 4

Angular velocity for various time delays with zero intended velocity, various initial angular velocities (−4.11 rad/s for delay = 0.02 s, −3.52 rad/s for delay = 0.03 s, −5.35 rad/s for delay = 0.04 s, and −5.27 rad/s for delay = 0.05 s), and saturation limits as −1 to 1 N·m (servomotor experimental example)

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Fig. 5

Trajectories starting from various initial conditions converging to a limit cycle in phase space for delay 0.3 s and saturation limits −100 to 100 N·m (pendulum numerical example)

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Fig. 6

Trajectories starting from various initial conditions converging to a limit cycle in phase space for delay 0.05 s and saturation limits −1 to 1 N·m (servomotor experimental example)

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Fig. 7

Trajectories starting from various initial conditions converging to a limit cycle in phase space (rotary-inverted pendulum experimental example)

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Fig. 8

Velocity plots for a sinusoidal intended velocity of frequency 0.6 Hz, saturation limits −100 to 100 N·m, delay 0.2 s, and amplitudes 0, 25, and 60 rad/s (pendulum numerical example)

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Fig. 9

Velocity plots for a sinusoidal intended velocity of frequency 0.5 Hz, saturation limits −1 to 1 N·m, delay 0.05 s, and amplitudes 0, 10, and 20 rad/s (servomotor experimental example)

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Fig. 10

Frequency content of velocity plots for a sinusoidal intended velocity of frequency 0.6 Hz, saturation limits −100 to 100 N·m, delay 0.2 s, and amplitudes 0, 25, and 60 rad/s (pendulum numerical example). Here, x = 1.709 Hz is the frequency of rest tremor, and the amplitude of this rest tremor is subsequently reduced (second and third subplots) as we increased the input amplitude. Note that in the third subplot, due to saturation nonlinearity, we observed the peak (at x = 1.2 Hz) other than intended velocity frequency 0.6 Hz, which is the second harmonic of the intended velocity frequency 0.6 Hz.

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Fig. 11

Delay versus frequency and amplitude of oscillation with saturation limits −4 to 4 N·m (pendulum numerical example)

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Fig. 12

Delay versus frequency and amplitude of oscillation with saturation limits as −10 to 10 N·m (servomotor experimental example)

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Fig. 13

(a) Amplitude of the tremor as a function of saturation and delay. (b) Frequency of the tremor as a function of saturation and delay. (c) Area of the limit cycle as a function of saturation and delay. (d) Aspect ratio of the limit cycle as a function of saturation and delay (pendulum numerical example).

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Fig. 14

Effect of delay (left) and saturation (right) on the limit cycles

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Fig. 15

A candidate algorithm for the diagnosis of PD

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Fig. 16

A proof-of-concept prototype for a pocket device, which can further evolve into a convenient and compact wireless device

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Fig. 17

Prototype for a smartphone application [20]

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