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Multifractal Study of Parkinson’s and Huntington’s Diseases with Human Gait Data

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Book cover Multifractals and Chronic Diseases of the Central Nervous System

Abstract

In this chapter we have presented how multifractal detrended cross-correlation analysis technique can be used to study Parkinson’s disease from human gait pattern of those patients when compared to those of normal people. The chapter further emphasizes that this study is important as a new novel technique whereby data from the correlation between the two feet provides status of the degree of neurodegenerative disorder. The chapter also presents how multifractal methodologies can also be applied in Huntington’s disease.

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References

  • Ashkenazy Y, Hausdorff JM, Ivanov P, Goldberger AL, Stanley HE (2002) A stochastic model of human gait dynamics. Phys A 316:662–670

    Article  Google Scholar 

  • Baltadjieva R, Giladi N, Gruendlinger L, Peretz C, Hausdorff JM (2006) Marked alterations in the gait timing and rhythmicity of patients with de novo Parkinson’s disease. Eur J Neurosci 24:1815–1820

    Article  PubMed  Google Scholar 

  • Beauchet O, Herrmann FR, Grandjean R, Dubost V, Allali G (2008) Concurrent validity of SMTEC® footswitches system for the measurement of temporal gait parameters. Gait Posture 27:156–159

    Article  PubMed  Google Scholar 

  • Blin O, Ferrandez AM, Serratrice G (1990) Quantitative analysis of gait in Parkinson patients: increased variability of stride length. J Neurol Sci 98:91–97

    Article  CAS  PubMed  Google Scholar 

  • Bunde A, Knopp J, Schellnhuber HJ (eds) (2002) The science of disasters. Springer, Berlin

    Google Scholar 

  • Buzzi UH, Ulrich BD (2004) Dynamic stability of gait cycles as a function of speed and system constraints. Mot Control 8:241–254

    Article  Google Scholar 

  • Cavanaugh JT, Coleman KL, Gaines JM et al (2007) Using step activity monitoring to characterize ambulatory activity in community-dwelling older adults. J Am Geriatr Soc 55:120–124

    Article  PubMed  Google Scholar 

  • Collins JJ, Richmond SA (1994) Hard-wired central pattern generators for quadrupedal locomotion. Biol Cybern 71:375–385

    Article  Google Scholar 

  • Collins JJ, Stewart IN (1993) Coupled non-linear oscillators and the symmetries of animal gaits. J Nonlinear Sci 3:349–392

    Article  Google Scholar 

  • Costa M, Peng CK, Goldberger AL, Hausdorff JM (2003) Multiscale entropy analysis of human gait dynamics. Phys A 330:53–60

    Article  Google Scholar 

  • Davis R, Õunpuu S, Tyburski D, Gage JR (1991) A gait analysis data collection and reduction technique. Hum Mov Sci 10:575–587

    Article  Google Scholar 

  • Decker LM, Cignettia F, Stergiou N (2010) Complexity and human gait. Revista Andaluza de Medicina del Deporte 3:2–12

    Google Scholar 

  • Delignières D, Torre K (2009) Fractal dynamics of human gait: a reassessment of the 1996 data of Hausdorff et al. J Appl Physiol 106:1272–1279

    Article  PubMed  Google Scholar 

  • Dingwell JB, Cusumano JP (2000) Non-linear time series analysis of normal and pathological human walking. Chaos 10:848–863

    Article  PubMed  Google Scholar 

  • Dingwell JB, Kang HG (2007) Differences between local and orbital dynamic stability during human walking. J Biomech Eng 129:586–593

    Article  PubMed  Google Scholar 

  • Dutta S, Ghosh D, Chatterjee S (2013) Multifractal Detrended fluctuation analysis of human gait diseases. Front Physiol 4:274

    Article  PubMed  PubMed Central  Google Scholar 

  • Dutta S, Ghosh D, Samanta S (2016) Non linear approach to study the dynamics of neurodegenerative diseases by multifractal Detrended cross-correlation analysis—a quantitative assessment on gait disease. Phys A 448:181–195

    Article  Google Scholar 

  • Feder J (1988) Fractals. Plenum Press, New York

    Book  Google Scholar 

  • Frenkel-Toledo S, Giladi N, Peretz C, Herman T, Gruendlinger L et al (2005a) Treadmill walking as an external pacemaker to improve gait rhythm and stability in Parkinson’s disease. Mov Disord 20:1109–1114

    Article  PubMed  Google Scholar 

  • Frenkel-Toledo S, Giladi N, Peretz C, Herman T, Gruendlinger L et al (2005b) Effect of gait speed on gait rhythmicity in Parkinson's disease: variability of stride time and swing time respond differently. J Neuroeng Rehabil 2:23

    Article  PubMed  PubMed Central  Google Scholar 

  • Gabell A, Nayak US (1984) The effect of age on variability in gait. J Gerontol 39:662–666

    Article  CAS  PubMed  Google Scholar 

  • Goldberger AL, Peng CK, Lipsitz LA (2002a) What is physiologic complexity and how does it change with aging and disease? Neurobiol Aging 23:23–26

    Article  PubMed  Google Scholar 

  • Goldberger AL, Amaral LA, Hausdorff JM, Ivanov PC, Peng CK et al (2002b) Fractal dynamics in physiology: alterations with disease and aging. Proc Natl Acad Sci U S A 99:2466–2472

    Article  PubMed  PubMed Central  Google Scholar 

  • Griffin L, West DJ, West BJ (2000) Random stride intervals with memory. J Biol Phys 26:185

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Grimbergen YA, Knol MJ, Bloem BR, Kremer BP, Roos RA et al (2008) Falls and gait disturbances in Huntington's disease. Mov Disord 23:970–976

    Article  PubMed  Google Scholar 

  • Guimares RM, Isaacs B (1980) Characteristics of the gait in old people who fall. Int Rehabil Med 2:177–180

    Article  Google Scholar 

  • Harbourne RT, Stergiou N (2009) Movement variability and the use of non-linear tools: principles to guide physical therapy practice. Phys Ther 89:267–282

    Article  PubMed  PubMed Central  Google Scholar 

  • Hausdorf JM, Alexander NB (2005) Gait disorders: evaluation and management. Informa Healthcare, New York

    Book  Google Scholar 

  • Hausdorf JM, Mitchell SL, Firtion R, Peng CK, Cudkowiczet ME, al. (1997) Altered fractal dynamics of gait: reduced stride-interval correlations with aging and Huntington’s disease. J Appl Physiol 82:262–269

    Article  Google Scholar 

  • Hausdorf JM, Ashkenazy Y, Peng CK, Ivanov PC, Stanley HE et al (2001) When human walking becomes random walking: fractal analysis and modeling of gait rhythm fuctuations. Phys A 302:138–147 and references cited therein

    Article  Google Scholar 

  • Hausdorf JM, Lowenthal J, Herman T, Gruendlinger L, Peretz C et al (2007) Rhythmic auditory stimulation modulates gait variability in Parkinson's disease. Eur J Neurosci 26:2369–2375

    Article  Google Scholar 

  • Hausdorff JM (2007) Gait dynamics, fractals and falls: finding meaning in the stride-to-stride fluctuations of human walking. Hum Mov Sci 26:555–589

    Article  PubMed  PubMed Central  Google Scholar 

  • Hausdorff JM, Peng CK, Ladin Z, Wei JY, Goldberger AL (1995) Is walking a random walk? Evidence for long-range correlations in stride interval of human gait. J Appl Physiol 78:349–358

    Article  CAS  PubMed  Google Scholar 

  • Hausdorff JM, Purdon PL, Peng CK, Ladin Z, Wei JY et al (1996) Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations. J Appl Physiol 80:1448–1457

    Article  CAS  PubMed  Google Scholar 

  • Hausdorff JM, Cudkowicz ME, Firtion R, Wei JY, Goldberger AL (1998) Gait variability and basal ganglia disorders: stride-to-stride variations of gait cycle timing in Parkinson’s disease and Huntington’s disease. Mov Disord 13:428–437

    Article  CAS  PubMed  Google Scholar 

  • Hausdorff JM, Zemany L, Peng CK, Goldberger AL (1999) Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children. J Appl Physiol 86:1040–1047

    Article  CAS  PubMed  Google Scholar 

  • Hausdorff JM, Lertratanakul A, Cudkowicz ME, Peterson AL, Kaliton D, Goldberger AL (2000) Dynamic markers of altered gait rhythm in amyotrophic lateral sclerosis. J Appl Physiol 88:2045–2053

    Article  CAS  PubMed  Google Scholar 

  • He LY, Chen SP (2011) Multifractal Detrended cross-correlation analysis of agricultural futures markets. Chaos, Solitons Fractals 44:355–361

    Article  Google Scholar 

  • Herman T, Giladi N, Gurevich T, Hausdorff JM (2005) Gait instability and fractal dynamics of older adults with a cautious gait: why do certain older adults walk fearfully? Gait Posture 21:178–185

    Article  CAS  PubMed  Google Scholar 

  • Hillmana SJ, Stansfieldb BW, Richardsonc AM, Robb JE (2009) Development of temporal and distance parameters of gait in normal children. Gait Posture 29:81–85

    Article  Google Scholar 

  • Holt KG, Saltzman E, Ho CL, Ulrich BD (2007) Scaling of dynamics in the earliest stages of walking. Phys Ther 87:1458–1467

    Article  PubMed  Google Scholar 

  • Hurst HE (1951) Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng 116:770–808

    Google Scholar 

  • Inman VT, Ralston HJ, Todd F (1981) Human Walking. Williams & Wilkins, Baltimore, p 154

    Google Scholar 

  • Inman VT, Ralston HJ, Todd F (2006) In: Rose J, Gamble Lippincott JG (eds) Human walking. Williams & Wilkins, Philadelphia, pp 7–18

    Google Scholar 

  • Ivanov PC, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG et al (1999) Multifractality in human heartbeat dynamics. Nature 399:461–465

    Article  CAS  PubMed  Google Scholar 

  • Ivanov P, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG et al (2001) From 1/f noise to multifractal cascades in heartbeat dynamics. Chaos 11:641–652

    Article  PubMed  Google Scholar 

  • Ivanov PC, Ma QDY, Bartsch R, Hausdorff JM, Amaral LAN et al (2009) Levels of complexity in scale-invariant neural signals. Phys Rev E 79:041920

    Article  CAS  Google Scholar 

  • Jordan K, Challis JH, Newell KM (2006) Long range correlations in the stride interval of running. Gait Posture 24:120–125

    Article  PubMed  Google Scholar 

  • Jordan K, Challis JH, Newell KM (2007a) Walking speed influences on gait cycle variability. Gait Posture 26:128–134

    Article  PubMed  Google Scholar 

  • Jordan K, Challis JH, Newell KM (2007b) Speed influences on the scaling behavior of gait cycle fluctuations during treadmill running. Hum Mov Sci 26:87–102

    Article  PubMed  Google Scholar 

  • Kadaba MP, Ramakrishnan HK, Wootten ME, Gainey J, Gorton G et al (1989) Repeatability of kinematic, kinetic, and electromyographic data in normal adult gait. J Orthop Res 7:849–860

    Article  CAS  PubMed  Google Scholar 

  • Khandoker AH, Palaniswami R, Begg RK (2008) A comparative study on approximate entropy measure and poincaré plot indexes of minimum foot clearance variability in the elderly during walking. J Neuroeng Rehabil 5:4

    Article  PubMed  PubMed Central  Google Scholar 

  • Kuo AD (2002) The relative roles of feedforward and feedback in the control of rhythmic movements. Mot Control 6:129

    Article  Google Scholar 

  • Kurz M, Stergiou N (2003) The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait. Neurosci Lett 348:155–158

    Article  CAS  PubMed  Google Scholar 

  • Kurz M, Stergiou N (2007) Do horizontal propulsive forces influence the non-linear structure of locomotion? J Neuroeng Rehabil 1:4–30

    Google Scholar 

  • Lomax RG (2007) Statistical concepts: a second course for education and the behavioral sciences. Lawrence Erlbaum Associates, Mahwah

    Google Scholar 

  • Malamud BD, Turcotte DL (1999) Self-affine time series: i. generation and analysis. Adv Geophys 40:1–90

    Article  Google Scholar 

  • Matjaz P (2005) The dynamics of human gait. Eur J Phys 26:525

    Article  Google Scholar 

  • Miller RA, Thaut MH, McIntosh GC, Rice RR (1996) Components of EMG symmetry and variability in parkinsonian and healthy elderly gait. Electroencephalogr Clin Neurophysiol 101:1–7

    Article  CAS  PubMed  Google Scholar 

  • Miller DJ, Stergiou N, Kurz MJ (2006) An improved surrogate method for detecting the presence of chaos in gait. J Biomech 39:2873–2876

    Article  PubMed  Google Scholar 

  • Moraiti C, Stergiou N, Ristanis S, Georgoulis AD (2007) ACL deficiency affects stride-to-stride variability as measured using non-linear methodology. Knee Surg Sports Traumatol Arthrosc 15:1406–1413

    Article  PubMed  Google Scholar 

  • Morris ME, Iansek R, Matyas TA, Summers JJ (1994a) The pathogenesis of gait hypokinesia in Parkinson’s disease. Brain 117:1169–1181

    Article  PubMed  Google Scholar 

  • Morris ME, Iansek R, Matyas TA, Summers JJ (1994b) Ability to modulate walking cadence remains intact in Parkinson’s disease. J Neurol Neurosurg Psychiatry 57:1532–1534

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Morris ME, Iansek R, Matyas TA, Summers JJ (1996a) Stride length regulation in Parkinson’s disease. Normalization strategies and underlying mechanisms. Brain 119:551–568

    Article  PubMed  Google Scholar 

  • Morris ME, Matyas TA, Iansek R, Summers JJ (1996b) Temporal stability of gait in Parkinson's disease. Phys Ther 76:763–777

    Article  CAS  PubMed  Google Scholar 

  • Morris ME, McGinley J, Huxham F, Collier J, Iansek R (1999) Constraints on the kinetic and spatiotemporal parameters of gait in Parkinson’s disease. Hum Mov Sci 18:461–483

    Article  Google Scholar 

  • Morris ME, Huxham F, McGinley J, Dodd K, Iansek R (2001) The biomechanics and motor control of gait in Parkinson disease. Clin Biomech 16:459–470

    Article  CAS  Google Scholar 

  • Movahed S, Hermanis E (2008) Fractal analysis of river flow fluctuations. Phys A 387:915–932

    Article  Google Scholar 

  • Muñoz-Diosdado A (2005) A non linear analysis of human gait time series based on multifractal analysis and cross correlations. J Phys Conf Ser 23:87–95

    Article  Google Scholar 

  • Oswiecimka P, Drożdż S, Kwapień J, Górski A (2013) Effect of detrending on multifractal characteristics. Acta Phys Pol A 123:597–603

    Article  CAS  Google Scholar 

  • Oswiecimka P, Drożdż S, Forczek M, Jadach S, Kwapien J et al (2014) Detrended cross-correlation analysis consistently extended to multifractality. Phys Rev E 89:023305

    Article  CAS  Google Scholar 

  • Pailhous J, Bonnard M (1992) Steady-state fluctuations of human walking. Behav Brain Res 47:181–190

    Article  CAS  PubMed  Google Scholar 

  • Palta AE (1985) Some characteristics of EMG patterns during locomotion: implications for locomotor control processes. J Mot Behav 17:443–461

    Article  Google Scholar 

  • Peitgen HO, Jurgens H, Saupe D (1992) Chaos and fractals. Springer-Verlag, New York Appendix B

    Book  Google Scholar 

  • Podobnik B, Fu DF, Stanley HE, Ivanov PC (2007) Power-law autocorrelated stochastic processes with long-range cross-correlations. Eur Phys J B 56:47–52

    Article  CAS  Google Scholar 

  • Podobnik B, Grosse I, Horvatic D, IIic S, Ivanov PC et al (2009) Quantifying cross-correlations using local and global detrending approaches. Eur Phys J B 71:243–250

    Article  CAS  Google Scholar 

  • Robbins SM, Astephen Wilson JL, Rutherford DJ, Hubley-Kozey CL (2013) Reliability of principal components and discrete parameters of knee angle and moment gait waveforms in individuals with moderate knee osteoarthritis. Gait Posture 38:421–427

    Article  PubMed  Google Scholar 

  • Scafetta N, Griffin L, West BJ (2003) Hölder exponent spectra for human gait. Phys A 328:561–583

    Article  Google Scholar 

  • Scafetta N, Moon RE, West BJ (2007) Fractal response of physiological signals to stress conditions, environmental changes, and neurodegenerative diseases. Complexity 12:12–17

    Article  Google Scholar 

  • Scafetta N, Marchi D, West BJ (2009) Understanding the complexity of human gait dynamics. Chaos 19:026108–026110

    Article  PubMed  Google Scholar 

  • Sekine M, Tamura T, Akay M, Fujimoto T, Togawa T et al (2002) Discrimination of walking patterns using wavelet-based fractal analysis. IEEE Trans Neural Syst Rehabil Eng 10:188–196

    Article  PubMed  Google Scholar 

  • Sekine M, Akay M, Tamura T, Higashi Y (2004) Fractal dynamics of body motion in patients with Parkinson’s disease. J Neural Eng 1:8–15

    Article  PubMed  Google Scholar 

  • Singh M, Singh M, Paramjeet (2013) Neuro-degenerative disease diagnosis using human gait: a review. IJITKMI 7:16–20

    Google Scholar 

  • Smith BA, Teulier C, Sansom J, Stergiou N, Ulrich BD (2011) Approximate entropy values demonstrate impaired neuromotor control of spontaneous leg activity in infants with myelomeningocele. Pediatr Phys Ther 23(3):241

    Article  PubMed  PubMed Central  Google Scholar 

  • Stergiou N, Decker LM (2011) Human movement variability, non-linear dynamics, and pathology: is there a connection? Hum Mov Sci 30:869–888

    Article  PubMed  PubMed Central  Google Scholar 

  • Stergiou N, Buzzi UH, Kurz MJ, Heidel J (2004) Non-linear tools in human movement. In: Stergiou N (ed) Innovative analyses of human movement. Human Kinetics, Champaign, pp 63–90

    Google Scholar 

  • Stolze H, Kuhtz-Buschbeck JP, Drucke H, Johnk K, Illert M et al (2001) Comparative analysis of the gait disorder of normal pressure hydrocephalus and Parkinson’s disease. J Neurol Neurosurg Psychiatry 70:289–297

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Tochigi Y, Segal NA, Vaseenon T, Brown TD (2012) Entropy analysis of tri-axial leg acceleration signal waveforms for measurement of decrease of physiological variability in human gait. J Orthop Res 30:879–904

    Article  Google Scholar 

  • Torres BDLC, Sánchez López MD, Sarabia Cachadiña E, Naranjo Orellana J (2013) Entropy in the analysis of gait complexity: a state of the art. Br J Appl Sci Technol 3:1097–1105

    Article  Google Scholar 

  • Van Emmerik REA, Rosenstein MT, McDermott WJ, Hamill J (2004) Non-linear dynamical approaches to human movement. J Appl Biomech 20:396–420

    Article  Google Scholar 

  • Van Orden GC, Kloos H, Wallot S (2009) Living in the pink: intentionality, wellbeing, and complexity. In: Handbook of the philosophy of science volume 10: philosophy of complex systems, vol 10. Elsevier, Amsterdam, pp 639–682

    Google Scholar 

  • Wang GJ, Xie C (2012) Cross-correlations between WTI crude oil market and US stock market: a perspective from Econophysics. Acta Phys Pol B 43:2021–2036

    Article  CAS  Google Scholar 

  • Webster KE, Wittwer JE, Feller JA (2005) Validity of the GAITRite walkway system for the measurement of averaged and individual step parameters of gait. Gait Posture 22:317–321

    Article  PubMed  Google Scholar 

  • West BJ, Scafetta N (2003) Non-linear dynamical model of human gait. Phys Rev E 67:051917

    Article  CAS  Google Scholar 

  • Winter DA (1984) Kinematic and kinetic patterns in human gait: variability and compensating effects. Hum Mov Sci 3:51–76

    Article  Google Scholar 

  • Winters JM, Crago PE (2000) Biomechanics and neural control of posture and movements. Springer, New York

    Book  Google Scholar 

  • Wu Y, Krishnan S (2010) Statistical analysis of gait rhythm in patients with Parkinson’s disease. IEEE Trans Neural Syst Rehabil Eng 18:150–158

    Article  PubMed  Google Scholar 

  • Yamasaki M, Sasaki T, Tsuzki S, Torii M (1984) Stereotyped pattern of lower limb movement during level and grade walking on treadmill. Ann Physiol Anthropol 3:291–296

    Article  CAS  PubMed  Google Scholar 

  • Yamasaki M, Sasaki T, Torii M (1991) Sex difference in the pattern of lower limb movement during treadmill walking. Eur J Appl Physiol Occup Physiol 62:99–103

    Article  CAS  PubMed  Google Scholar 

  • Yogev G, Giladi N, Peretz C, Springer S, Simon ES et al (2005) Dual tasking, gait rhythmicity, and Parkinson's disease: which aspects of gait are attention demanding? Eur J Neurosci 22:1248–1256

    Article  PubMed  Google Scholar 

  • Yoshino K, Motoshige T, Araki T, Matsuoka K (2004) Effect of prolonged free-walking fatigue on gait and physiological rhythm. J Biomech 37:1271–1280

    Article  PubMed  Google Scholar 

  • Yu ZG, Wang B (2001) A time series model for CDS sequences in complete genome. Chaos, Solitons Fractals 12:519–526

    Article  CAS  Google Scholar 

  • Zhao X, Shang P, Jin Q (2011) Multifractal detrended cross-correlation analysis of Chinese stock markets based on time delay. Fractals 19:329–338

    Article  Google Scholar 

  • Zhou WX (2008) Multifractal detrended cross-correlation analysis for two nonstationary signals. Phys Rev E 77:066211

    Article  CAS  Google Scholar 

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Acknowledgment

The authors gratefully acknowledge Physica A and Elsevier Publishing Co. for providing the copyrights of Figs. 5.2a, 5.2b, 5.2c, 5.2d, 5.2e and 5.2f, 5.3a, 5.3b, 5.4a, 5.4b, 5.5a, 5.5b, 5.6a, 5.6b, 5.7a, 5.7b, 5.9a, 5.9b and Tables 5.1 and 5.2 for use in this chapter.

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Authors and Affiliations

  1. Department of Physics, Sir C V Raman Centre for Physics and Music, Jadavpur University, Kolkata, West Bengal, India

    Dipak Ghosh

  2. Department for Physics, Seacom Engineering College, Howrah, West Bengal, India

    Shukla Samanta

  3. Electrical and Electronics Engineering, ICFAI University, Agartala, Tripura, India

    Sayantan Chakraborty

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Ghosh, D., Samanta, S., Chakraborty, S. (2019). Multifractal Study of Parkinson’s and Huntington’s Diseases with Human Gait Data. In: Multifractals and Chronic Diseases of the Central Nervous System. Springer, Singapore. https://doi.org/10.1007/978-981-13-3552-5_5

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