2013, 2013(special): 643-652. doi: 10.3934/proc.2013.2013.643

Parameter dependent stability/instability in a human respiratory control system model

1. 

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, United States

2. 

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083, United States

Received  September 2012 Published  November 2013

In this paper a computational procedure is presented to study the development of stable/unstable patterns in a system of three nonlinear, parameter dependent delay differential equations with two transport delays representing a simplified model of human respiration. It is demonstrated using simulations how sequences of changes in internal and external parameter values can lead to complex dynamic behavior due to forced transitions between stable/unstable equilibrium positions determined by particular parameter combinations. Since changes in the transport delays only influence the stability/instability of an equilibrium position a stability chart is constructed in that case by finding the roots of the characteristic equation of the corresponding linear variational system. Illustrative examples are included.
Citation: Saroj P. Pradhan, Janos Turi. Parameter dependent stability/instability in a human respiratory control system model. Conference Publications, 2013, 2013 (special) : 643-652. doi: 10.3934/proc.2013.2013.643
References:
[1]

D. Breda, S. Maset and R. Vermiglio, TRACE-DDE:a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations, Topics in Time Delay Systems: Analysis, Algorithms, and Control, Lecture Notes in Control and Information Sciences, 388 (2009), 145-155.

[2]

K. L. Cooke, and J. Turi, Stability, instability in delay equations modeling human respiration, J. Math. Biol., 32 (1994), 535-543.

[3]

I. Őri, F. Hartung, and J. Turi, Numerical approximations for a class of differential equations with time- and state-dependent delays, Applied Math. Letters, 8:6 (1995), 19-24.

[4]

F. Hartung, and J. Turi, Parameter identification in a Respiratory Control System Model, to appear in "Cardiovascular and Respiratory Modeling", Lecture Notes in Mathematical Biosciences, J. Batzel and F. Kappel eds., Springer, (2012).

[5]

M. C. K. Khoo, A. Gottschalk and A. I. Pack, Sleep-induced periodic breathing and apnea:a theoritical study American Physiological Society,(1991) 2014-2024.

[6]

M. C. K. Khoo, R. E. Kronauer, K. P. Strohl, and A. S. Slutsky, Factors inducing periodic breathingin humans: a general model American Physiological Society, (1992) 644-659.

[7]

L. E. Kollar, and J. Turi, Numerical Stability Analysis in Respiratory Control System Models, Electronic Journal of Differential Equations, Conference 12,(2005) 65-78.

[8]

G. S. Longobardo, N. S. Cherniak and A. P. Fishman, Cheyne-Stokes breathing produced by a model of the human respiratory system, Journal of Applied Physiology, 21(6),(1966) 1839- 1846.

[9]

G. B Longobardo, M. D. Goldman, and N. S. Cherniack, Sleep apnea considered as a control system instability, Respiratory Physiology, 50,(1982) 311-333.

[10]

M. C. Mackey, and L. Glass, Oscillation and Chaos in Physiological Control Systems, Science, 197,(1977) 287-289.

[11]

S. P. Pradhan, The role of peripheral and central chemoreceptors in the stability of the human respiratory system, (Doctoral dissertation), University of Texas at Dallas,(2010).

[12]

J. B. West, Respiratory Physiology The Essentials, Lippincott Williams and Wilkins, (2005).

show all references

References:
[1]

D. Breda, S. Maset and R. Vermiglio, TRACE-DDE:a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations, Topics in Time Delay Systems: Analysis, Algorithms, and Control, Lecture Notes in Control and Information Sciences, 388 (2009), 145-155.

[2]

K. L. Cooke, and J. Turi, Stability, instability in delay equations modeling human respiration, J. Math. Biol., 32 (1994), 535-543.

[3]

I. Őri, F. Hartung, and J. Turi, Numerical approximations for a class of differential equations with time- and state-dependent delays, Applied Math. Letters, 8:6 (1995), 19-24.

[4]

F. Hartung, and J. Turi, Parameter identification in a Respiratory Control System Model, to appear in "Cardiovascular and Respiratory Modeling", Lecture Notes in Mathematical Biosciences, J. Batzel and F. Kappel eds., Springer, (2012).

[5]

M. C. K. Khoo, A. Gottschalk and A. I. Pack, Sleep-induced periodic breathing and apnea:a theoritical study American Physiological Society,(1991) 2014-2024.

[6]

M. C. K. Khoo, R. E. Kronauer, K. P. Strohl, and A. S. Slutsky, Factors inducing periodic breathingin humans: a general model American Physiological Society, (1992) 644-659.

[7]

L. E. Kollar, and J. Turi, Numerical Stability Analysis in Respiratory Control System Models, Electronic Journal of Differential Equations, Conference 12,(2005) 65-78.

[8]

G. S. Longobardo, N. S. Cherniak and A. P. Fishman, Cheyne-Stokes breathing produced by a model of the human respiratory system, Journal of Applied Physiology, 21(6),(1966) 1839- 1846.

[9]

G. B Longobardo, M. D. Goldman, and N. S. Cherniack, Sleep apnea considered as a control system instability, Respiratory Physiology, 50,(1982) 311-333.

[10]

M. C. Mackey, and L. Glass, Oscillation and Chaos in Physiological Control Systems, Science, 197,(1977) 287-289.

[11]

S. P. Pradhan, The role of peripheral and central chemoreceptors in the stability of the human respiratory system, (Doctoral dissertation), University of Texas at Dallas,(2010).

[12]

J. B. West, Respiratory Physiology The Essentials, Lippincott Williams and Wilkins, (2005).

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