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A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners
1.  Department of Mathematics & Computer Science, Clarkson University, Potsdam, NY 13676, United States, United States, United States 
2.  Applied Physiology Laboratory, Eastern Michigan University, Ypsilanti, MI 48197, United States, United States 
References:
[1] 
W. Aziz and M. Arif, Complexity analysis of stride interval time series by threshold dependent symbolic entropy, Eur. J. Appl. Physiol., 98 (2006), 3040. doi: 10.1007/s0042100602265. 
[2] 
O. Beauchet, V. Dubost, F. R. Herrmann and R. W. Kressig, Stridetostride variability while backward counting among healthy young adults, J. Neuroeng. Rehabil., 2 (2005), 26. doi: 10.1186/17430003226. 
[3] 
B. R. Bloem, V. V. Valkenburg, M. Slabbekoorn and M. D. Willemsen, The multiple tasks test: Development and normal strategies, Gait Posture, 14 (2001), 191202. 
[4] 
E. M. Bollt, J. D. Skufca and S. J. McGregor, Control entropy: A complexity measure for nonstationary signals, Mathematical Biosciences and Engineering, 6 (2009), 125. doi: 10.3934/mbe.2009.6.1. 
[5] 
U. H. Buzzi and B. D. Ulrich, Dynamic stability of gait cycles as a function of speed and system constraints, Motor Control, 8 (2004), 241254. 
[6] 
P. Cavanagh, The mechanics of distance running: A historical perspective, in "Biomechanics of Distance Running" (ed. P. Cavanagh), Human Kinetics, 1990. 
[7] 
T. M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley Series in Telecommunications, A WileyInterscience Publication, John Wiley & Sons, Inc., New York, 1991. doi: 10.1002/0471200611. 
[8] 
K. Davids, S. Bennett and K. M. Newell, "Movement System Variability," Human Kinetics, Champaign, IL, 2006. 
[9] 
D. P. Ferris, G. S. Sawicki and M. A. Daley, A physiologist's perspective on robotic exoskeletons for human locomotion, Int. J. HR, 4 (2007), 507528. doi: 10.1142/S0219843607001138. 
[10] 
A. D. Georgoulis, C. Moraiti, S. Ristanis and N. Stergiou, A novel approach to measure variability in the anterior cruciate ligament deficient knee during walking: The use of the approximate entropy in orthopaedics, J. Clin. Monit. Comput., 20 (2006), 1118. doi: 10.1007/s1087700610327. 
[11] 
P. S. Glazier and K. Davids, Constraints on the complete optimization of human motion, Sports Med., 39 (2009), 1528. doi: 10.2165/0000725620093901000002. 
[12] 
G. H. Golub and C. F. Van Loan, "Matrix Computations," The Johns Hopkins University Press, 1996. 
[13] 
P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, 28 (1983), 25912593. doi: 10.1103/PhysRevA.28.2591. 
[14] 
J. M. Hausdorff, Gait dynamics, fractals and falls: Finding meaning in the stridetostride fluctuations of human walking, Hum. Mov. Sci., 26 (2007), 555589. 
[15] 
H. Kantz and T. Schreiber, "Nonlinear Time Series Analysis," Second edition, Cambridge University Press, Cambridge, 2004. 
[16] 
C. K. Karmakar, A. H. Khandoker, R. K. Begg, M. Palaniswami and S. Taylor, Understanding ageing effects by approximate entropy analysis of gait variability, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2007), 19651968. doi: 10.1109/IEMBS.2007.4352703. 
[17] 
A. H. Khandoker, M. Palaniswami and R. K. Begg, A comparative study on approximate entropy measure and poincaire plot indexes of minimum foot clearance variability in the elderly during walking, J. Neuroeng. Rehabil., 5 (2008), 4. doi: 10.1186/1743000354. 
[18] 
M. J. Kurz and N. Stergiou, The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait, Neurosci. Lett., 348 (2003), 155158. doi: 10.1016/S03043940(03)007365. 
[19] 
F. Liao, J. Wang and P. He, Multiresolution entropy analysis of gait symmetry in neurological degenerative diseases and amyotrophic lateral sclerosis, Med. Eng. Phys., 30 (2008), 299310. doi: 10.1016/j.medengphy.2007.04.014. 
[20] 
K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis," Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press [Harcourt Brace Jovanovich, Publishers], LondonNew YorkToronto, Ont., 1979. 
[21] 
S. J. McGregor, M. A. Busa, J. D. Skufca, J. A. Yaggie and E. M. Bollt, Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners, Chaos, 19 (2009), 026109, 13 pp. 
[22] 
S. J. McGregor, M. A. Busa, J. A. Yaggie and E. M. Bollt, High resolution MEMS accelerometers to estimate VO2 and compare running mechanics between highly trained intercollegiate and untrained runners, PLoS One, 4 (2009), e7355. doi: 10.1371/journal.pone.0007355. 
[23] 
S. P. Messier, C. Legault, C. R. Schoenlank, J. J. Newman, D. F. Martin and P. Devita, Risk factors and mechanisms of knee injury in runners, Med. Sci. Sports Exerc., 40 (2008), 18731879. doi: 10.1249/MSS.0b013e31817ed272. 
[24] 
D. J. Miller, N. Stergiou and M. J. Kurz, An improved surrogate method for detecting the presence of chaos in gait, J. Biomech., 39 (2006), 28732876. doi: 10.1016/j.jbiomech.2005.10.019. 
[25] 
R. MoeNilssen, A new method for evaluating motor control in gait under reallife environmental conditions. Part 2: Gait analysis, Clin. Biomech. (Bristol, Avon), 13 (1998), 328335. doi: 10.1016/S02680033(98)000904. 
[26] 
K. M. Newell, Constraints on the development of coordination, in "Motor Development in Children: Aspects of Coordination and Control" (eds. M. G. Wade and W. H. Dordect), Nihjoff, (1986), 341360. 
[27] 
K. M. Newell and D. E. Vaillancourt, Dimensional change in motor learning, Hum. Mov. Sci., 20 (2001), 695715. doi: 10.1016/S01679457(01)000732. 
[28] 
S. M. Pincus, Approximate entropy as a measure of system complexity, Proceedings of the National Academy of Sciences of the United States of America, 88 (1991), 22972301. 
[29] 
S. M. Pincus, Assessing serial irregularity and its implications for health, Annals of the New York Academy of Sciences, 954 (2001), 245. doi: 10.1111/j.17496632.2001.tb02755.x. 
[30] 
A. Renyi, On measures of entropy and information, Proceedings of the 4th Berkeley Sympo sium on Mathematical Statistics and Probability, 1 (1961), 547561. 
[31] 
J. S. Richman and J. R. Moorman, Physiological timeseries analysis using approximate en tropy and sample entropy, American Journal of Physiology Heart and Circulatory Physiology, 278 (2000), 20392049. 
[32] 
C. Robinson, "Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE and the Theory of Global Attractors," 2^{nd} edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. 
[33] 
L. A. Schrodt, V. S. Mercer, C. A. Giuliani and M. Hartman, Characteristics of stepping over an obstacle in community dwelling older adults under dualtask conditions, Gait Posture, 19 (2004), 279287. doi: 10.1016/S09666362(03)000675. 
[34] 
C. E. Shannon and W. Weaver, "The Mathematical Theory of Information," Uni versity of Illinois Press, 97, Urbana, 1949. 
[35] 
J. S. Slawinski and V. L. Billat, Difference in mechanical and energy cost between highly, well, and nontrained runners, Med. Sci. Sports Exerc., 36 (2004), 14401446. doi: 10.1249/01.MSS.0000135785.68760.96. 
[36] 
G. YogevSeligmann, J. M. Hausdorff and N. Giladi, The role of executive function and attention in gait, Mov. Disord., 23 (2008), 329342. doi: 10.1002/mds.21720. 
[37] 
M. Joyner and E. Coyle, Endurance exercise performance: The physiology of champions, The Journal of Physiology, 586 (2008), 3544. 
[38] 
J. Lin, E. Keogh, S. Lonardi and B. Chiu, A symbolic representation of time series, with implications for streaming algorithms, Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, (2003), 211. 
[39] 
Y. Nakayama, K. Kudo and T. Ohtsuki, Variability and fluctuation in running gait cycle of trained runners and nonrunners, Gait Posture, 31 (2009), 331333. doi: 10.1016/j.gaitpost.2009.12.003. 
[40] 
K. Jordan and K. M. Newell, The structure of variability in human walking and running is speeddependent, Exerc. Sport Sci. Rev., 36 (2008), 200204. doi: 10.1097/JES.0b013e3181877d71. 
show all references
References:
[1] 
W. Aziz and M. Arif, Complexity analysis of stride interval time series by threshold dependent symbolic entropy, Eur. J. Appl. Physiol., 98 (2006), 3040. doi: 10.1007/s0042100602265. 
[2] 
O. Beauchet, V. Dubost, F. R. Herrmann and R. W. Kressig, Stridetostride variability while backward counting among healthy young adults, J. Neuroeng. Rehabil., 2 (2005), 26. doi: 10.1186/17430003226. 
[3] 
B. R. Bloem, V. V. Valkenburg, M. Slabbekoorn and M. D. Willemsen, The multiple tasks test: Development and normal strategies, Gait Posture, 14 (2001), 191202. 
[4] 
E. M. Bollt, J. D. Skufca and S. J. McGregor, Control entropy: A complexity measure for nonstationary signals, Mathematical Biosciences and Engineering, 6 (2009), 125. doi: 10.3934/mbe.2009.6.1. 
[5] 
U. H. Buzzi and B. D. Ulrich, Dynamic stability of gait cycles as a function of speed and system constraints, Motor Control, 8 (2004), 241254. 
[6] 
P. Cavanagh, The mechanics of distance running: A historical perspective, in "Biomechanics of Distance Running" (ed. P. Cavanagh), Human Kinetics, 1990. 
[7] 
T. M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley Series in Telecommunications, A WileyInterscience Publication, John Wiley & Sons, Inc., New York, 1991. doi: 10.1002/0471200611. 
[8] 
K. Davids, S. Bennett and K. M. Newell, "Movement System Variability," Human Kinetics, Champaign, IL, 2006. 
[9] 
D. P. Ferris, G. S. Sawicki and M. A. Daley, A physiologist's perspective on robotic exoskeletons for human locomotion, Int. J. HR, 4 (2007), 507528. doi: 10.1142/S0219843607001138. 
[10] 
A. D. Georgoulis, C. Moraiti, S. Ristanis and N. Stergiou, A novel approach to measure variability in the anterior cruciate ligament deficient knee during walking: The use of the approximate entropy in orthopaedics, J. Clin. Monit. Comput., 20 (2006), 1118. doi: 10.1007/s1087700610327. 
[11] 
P. S. Glazier and K. Davids, Constraints on the complete optimization of human motion, Sports Med., 39 (2009), 1528. doi: 10.2165/0000725620093901000002. 
[12] 
G. H. Golub and C. F. Van Loan, "Matrix Computations," The Johns Hopkins University Press, 1996. 
[13] 
P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, 28 (1983), 25912593. doi: 10.1103/PhysRevA.28.2591. 
[14] 
J. M. Hausdorff, Gait dynamics, fractals and falls: Finding meaning in the stridetostride fluctuations of human walking, Hum. Mov. Sci., 26 (2007), 555589. 
[15] 
H. Kantz and T. Schreiber, "Nonlinear Time Series Analysis," Second edition, Cambridge University Press, Cambridge, 2004. 
[16] 
C. K. Karmakar, A. H. Khandoker, R. K. Begg, M. Palaniswami and S. Taylor, Understanding ageing effects by approximate entropy analysis of gait variability, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2007), 19651968. doi: 10.1109/IEMBS.2007.4352703. 
[17] 
A. H. Khandoker, M. Palaniswami and R. K. Begg, A comparative study on approximate entropy measure and poincaire plot indexes of minimum foot clearance variability in the elderly during walking, J. Neuroeng. Rehabil., 5 (2008), 4. doi: 10.1186/1743000354. 
[18] 
M. J. Kurz and N. Stergiou, The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait, Neurosci. Lett., 348 (2003), 155158. doi: 10.1016/S03043940(03)007365. 
[19] 
F. Liao, J. Wang and P. He, Multiresolution entropy analysis of gait symmetry in neurological degenerative diseases and amyotrophic lateral sclerosis, Med. Eng. Phys., 30 (2008), 299310. doi: 10.1016/j.medengphy.2007.04.014. 
[20] 
K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis," Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press [Harcourt Brace Jovanovich, Publishers], LondonNew YorkToronto, Ont., 1979. 
[21] 
S. J. McGregor, M. A. Busa, J. D. Skufca, J. A. Yaggie and E. M. Bollt, Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners, Chaos, 19 (2009), 026109, 13 pp. 
[22] 
S. J. McGregor, M. A. Busa, J. A. Yaggie and E. M. Bollt, High resolution MEMS accelerometers to estimate VO2 and compare running mechanics between highly trained intercollegiate and untrained runners, PLoS One, 4 (2009), e7355. doi: 10.1371/journal.pone.0007355. 
[23] 
S. P. Messier, C. Legault, C. R. Schoenlank, J. J. Newman, D. F. Martin and P. Devita, Risk factors and mechanisms of knee injury in runners, Med. Sci. Sports Exerc., 40 (2008), 18731879. doi: 10.1249/MSS.0b013e31817ed272. 
[24] 
D. J. Miller, N. Stergiou and M. J. Kurz, An improved surrogate method for detecting the presence of chaos in gait, J. Biomech., 39 (2006), 28732876. doi: 10.1016/j.jbiomech.2005.10.019. 
[25] 
R. MoeNilssen, A new method for evaluating motor control in gait under reallife environmental conditions. Part 2: Gait analysis, Clin. Biomech. (Bristol, Avon), 13 (1998), 328335. doi: 10.1016/S02680033(98)000904. 
[26] 
K. M. Newell, Constraints on the development of coordination, in "Motor Development in Children: Aspects of Coordination and Control" (eds. M. G. Wade and W. H. Dordect), Nihjoff, (1986), 341360. 
[27] 
K. M. Newell and D. E. Vaillancourt, Dimensional change in motor learning, Hum. Mov. Sci., 20 (2001), 695715. doi: 10.1016/S01679457(01)000732. 
[28] 
S. M. Pincus, Approximate entropy as a measure of system complexity, Proceedings of the National Academy of Sciences of the United States of America, 88 (1991), 22972301. 
[29] 
S. M. Pincus, Assessing serial irregularity and its implications for health, Annals of the New York Academy of Sciences, 954 (2001), 245. doi: 10.1111/j.17496632.2001.tb02755.x. 
[30] 
A. Renyi, On measures of entropy and information, Proceedings of the 4th Berkeley Sympo sium on Mathematical Statistics and Probability, 1 (1961), 547561. 
[31] 
J. S. Richman and J. R. Moorman, Physiological timeseries analysis using approximate en tropy and sample entropy, American Journal of Physiology Heart and Circulatory Physiology, 278 (2000), 20392049. 
[32] 
C. Robinson, "Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE and the Theory of Global Attractors," 2^{nd} edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. 
[33] 
L. A. Schrodt, V. S. Mercer, C. A. Giuliani and M. Hartman, Characteristics of stepping over an obstacle in community dwelling older adults under dualtask conditions, Gait Posture, 19 (2004), 279287. doi: 10.1016/S09666362(03)000675. 
[34] 
C. E. Shannon and W. Weaver, "The Mathematical Theory of Information," Uni versity of Illinois Press, 97, Urbana, 1949. 
[35] 
J. S. Slawinski and V. L. Billat, Difference in mechanical and energy cost between highly, well, and nontrained runners, Med. Sci. Sports Exerc., 36 (2004), 14401446. doi: 10.1249/01.MSS.0000135785.68760.96. 
[36] 
G. YogevSeligmann, J. M. Hausdorff and N. Giladi, The role of executive function and attention in gait, Mov. Disord., 23 (2008), 329342. doi: 10.1002/mds.21720. 
[37] 
M. Joyner and E. Coyle, Endurance exercise performance: The physiology of champions, The Journal of Physiology, 586 (2008), 3544. 
[38] 
J. Lin, E. Keogh, S. Lonardi and B. Chiu, A symbolic representation of time series, with implications for streaming algorithms, Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, (2003), 211. 
[39] 
Y. Nakayama, K. Kudo and T. Ohtsuki, Variability and fluctuation in running gait cycle of trained runners and nonrunners, Gait Posture, 31 (2009), 331333. doi: 10.1016/j.gaitpost.2009.12.003. 
[40] 
K. Jordan and K. M. Newell, The structure of variability in human walking and running is speeddependent, Exerc. Sport Sci. Rev., 36 (2008), 200204. doi: 10.1097/JES.0b013e3181877d71. 
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