Skip to main content
SpringerLink
Log in

Dynamical structure of center-of-pressure trajectories in patients recovering from stroke

  • Research Article
  • Published:
Experimental Brain Research Aims and scope Submit manuscript

Abstract

In a recent study, De Haart et al. (Arch Phys Med Rehabil 85:886–895, 2004) investigated the recovery of balance in stroke patients using traditional analyses of center-of-pressure (COP) trajectories to assess the effects of health status, rehabilitation, and task conditions like standing with eyes open or closed and standing while performing a cognitive dual task. To unravel the underlying control processes, we reanalyzed these data in terms of stochastic dynamics using more advanced analyses. Dimensionality, local stability, regularity, and scaling behavior of COP trajectories were determined and compared with shuffled and phase-randomized surrogate data. The presence of long-range correlations discarded the possibility that the COP trajectories were purely random. Compared to the healthy controls, the COP trajectories of the stroke patients were characterized by increased dimensionality and instability, but greater regularity in the frontal plane. These findings were taken to imply that the stroke patients actively (i.e., cognitively) coped with the stroke-induced impairment of posture, as reflected in the increased regularity and decreased local stability, by recruiting additional control processes (i.e., more degrees of freedom) and/or by tightening the present control structure while releasing non-essential degrees of freedom from postural control. In the course of rehabilitation, dimensionality stayed fairly constant, whereas local stability increased and regularity decreased. The progressively less regular COP trajectories were interpreted to indicate a reduction of cognitive involvement in postural control as recovery from stroke progressed. Consistent with this interpretation, the dual task condition resulted in less regular COP trajectories of greater dimensionality, reflecting a task-related decrease of active, cognitive contributions to postural control. In comparison with conventional posturography, our results show a clear surplus value of dynamical measures in studying postural control.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. In the original study of De Haart et al. (2004), each balance assessment consisted of two consecutive test series, incorporating four quiet-standing tasks and one weight-shifting task, presented in a fixed sequence. This sequence was repeated in reverse order to control for time effects. Between the DT and EC condition, participants conducted a trial while looking at a vertical black bar, which served as a visual midline reference. The weight-shifting task was performed twice after (and preceding) the first (second) EC condition. A 1-min rest was given after each balance test, whereas a longer pause was allowed between the two test series. The arithmetic task in the DT condition consisted of a (varying) verbal sequence of eight single-digit additions (e.g., 7+4=11 or 3+5=7) equally timed over the 30-s period. The participants were instructed to verbally indicate the correctness of each summation by good or fault response.

  2. For our data, the first minimum of the mutual information occurred at 11 (mean 11.14, SE 0.15) and 10 (mean 9.91, SE 0.20) data samples for ML and AP sway components of the stroke patients, respectively. These minima did not change with rehabilitation or condition (P>0.05). For the healthy controls, the first minimum of the mutual information occurred at 11 data samples for both ML (mean 11.10, SE 0.27) and AP (mean 10.54, SE 0.21) sway components. Again, no change with condition was found (P>0.05). Note that the choice of the time delay τ was not based on these group averages but was determined independently for each trial.

  3. Correlations between consecutively sampled points can produce spurious indications of low-dimensional structure. With the introduction of the cut-off parameter W>1, it is possible to minimize these correlations (Grassberger 1986; Theiler 1986). Therefore, all pairs of points that are closer together in time than some cut-off W were excluded. W=1 returns the standard Grassberger and Procaccia (1983) formula.

  4. The dimension is often calculated by looking at the slope of the most linear segments of C m (r), requiring a means of evaluating a score for each plausible linear segment (i.e., based on the length of the segment or the goodness of fit to a line). The ‘optimal’ linear segment is chosen. In this way, these techniques emphasize the possible existence of strange attractors. A drawback of such methods is that the length scale chosen can depend discontinuously on the underlying signal, because a small change in the signal can change the relative ranks of the candidate linear segments and thereby change the calculated dimension substantially (Kaplan et al. 1991). Because the applied dimension analysis in this study did not involve examination of the linear scaling of C m (r), it would be incorrect to interpret the estimated dimension D 2 as the dimension of the attractor. Similarly, it would be incorrect to infer from this analysis that an attractor must exist.

  5. In agreement with, e.g., Lake et al. (2002) and Richman and Moorman (2000), time series were normalized to unit variance. Sample entropy software was obtained from PhysioNet (Goldberger et al. 2000).

  6. Notice that a multivariate extension of the detrended fluctuation analysis algorithm yields identical results when applied to the embedded time series (see above) since we assumed stationarity.

  7. To avoid false or spurious conclusions, Hurst exponents were also determined by means of a rescaled range analysis (Hurst 1965; Rangarajan and Ding 2000; Delignières et al. 2003; cf. Wing et al. 2004 for a related power spectral approach), yielding slightly higher estimates of the diffusion process than the DFA. To compare these two methods, the pair-wise two-tailed Pearson correlation coefficient between the scaling exponent based on the rescaled range analysis (H→H R/S) and the detrended fluctuation analysis (H→H DFA) was determined for all the trials of the stroke patients (N=990). For both the AP and ML scaling estimates, the correlation analysis showed a good agreement between H R/S and H DFA (r=0.918, P<0.01 and r=0.895, P<0.01, respectively).

  8. The significant results reported in Table 3 were all preserved when the averaged post-stroke values were replaced by the earliest post-stroke values.

References

  • Abarbanel HDI (1996) Analysis of observed chaotic data. Springer, New York

    Google Scholar 

  • Baratto L, Morasso PG, Re C, Spada G (2002) A new look at posturographic analysis in the clinical context: sway-density versus other parameterization techniques. Motor Control 6:246–270

    PubMed  Google Scholar 

  • Bohannon RW (1995) Standing balance, lower extremity muscle strength, and walking performance of patients referred for physical therapy. Percept Mot Skills 80:379–385

    PubMed  CAS  Google Scholar 

  • Brown LA, Sleik RJ, Winder TR (2002) Attentional demands for static postural control after stroke. Arch Phys Med Rehabil 83:1732–1735

    Article  PubMed  Google Scholar 

  • Brunnstrom S (1966) Motor testing procedures in hemiplegia: based on sequential recovery stages. Phys Ther 46:357–375

    PubMed  CAS  Google Scholar 

  • Buzzi UH, Stergiou N, Kurz MJ, Hageman PA, Heidel J (2003) Nonlinear dynamics indicates aging affects variability during gait. Clin Biomech 18:435–443

    Article  Google Scholar 

  • Cabrera JL, Bormann R, Eurich C, Ohira T, Milton J (2004) State-dependent noise and human balance control. Fluct Noise Lett 4:L107–L118

    Article  Google Scholar 

  • Cohen J (1988) Statistical power analysis for the behavioral sciences. Erlbaum, Hillsdale, NJ

    Google Scholar 

  • Collen FM, Wade DT, Bradshaw CM (1990) Mobility after stroke: reliability of measures of impairment and disability. Int Disabil Stud 12:6–9

    PubMed  CAS  Google Scholar 

  • Collins JJ (1999) Fishing for function in noise. Nature 402:241–242

    Article  PubMed  CAS  Google Scholar 

  • Collins JJ, De Luca CJ (1993) Open-loop and closed-loop control of posture: a random-walk analysis of center-of-pressure trajectories. Exp Brain Res 95:308–318

    Article  PubMed  CAS  Google Scholar 

  • Collins JJ, De Luca CJ (1995) Upright, correlated random walks: a statistical-biomechanics approach to the human postural control system. Chaos 5:57–63

    Article  PubMed  Google Scholar 

  • Collins JJ, Priplata AA, Gravelle DC, Niemi J Harry J, Lipsitz LA (2003) Noise-enhanced human sensorimotor function. IEEE Eng Med Biol Mag 22:76–83

    Article  PubMed  Google Scholar 

  • De Haart M, Geurts AC, Huidekoper SC, Fasotti L, van Limbeek J (2004) Recovery of standing balance in postacute stroke patients: a rehabilitation cohort study. Arch Phys Med Rehabil 85:886–895

    Article  PubMed  Google Scholar 

  • Delignières D, Deschamps T, Legros A, Caillou N (2003) A methodological note on nonlinear time series analysis: is the open- and closed-loop model of Collins and De Luca (1993) a statistical artifact? J Mot Behav 35:86–96

    PubMed  Google Scholar 

  • Frank TD, Daffertshofer A, Beek PJ (2001) Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coefficients: application to postural sway. Phys Rev E 63:0011905/1–16

    Google Scholar 

  • Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140

    Article  PubMed  Google Scholar 

  • Fugl-Meyer AR, Jaasko L, Leyman I, Olsson S, Steglind S (1975) The post-stroke hemiplegic patient 1. A method for evaluation of physical performance. Scand J Rehabil Med 7:13–31

    PubMed  CAS  Google Scholar 

  • Geurts ACH, de Haart M, van Nes IJW, Duysens J (2005) A review of standing balance recovery from stroke. Gait Posture 22:267–281

    Article  PubMed  Google Scholar 

  • Glass L (2001) Synchronization and rhythmic processes in physiology. Nature 410:277–284

    Article  PubMed  CAS  Google Scholar 

  • Goldberger AL (1996) Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. Lancet 347:1312–1314

    Article  PubMed  CAS  Google Scholar 

  • Goldberger AL (1997) Fractal variability versus pathological periodicity: complexity loss and stereotypy in disease. Perspect Biol Med 40:543–561

    PubMed  CAS  Google Scholar 

  • Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE (2000) PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101:e215–e220

    PubMed  CAS  Google Scholar 

  • Goldberger AL, Amaral LAN, Hausdorff JM, Ivanov PC, Peng C-K, Stanley HE (2002) Fractal dynamics in physiology: alterations with disease and aging. Proc Natl Acad Sci USA 99:2466–2472

    Article  PubMed  Google Scholar 

  • Grassberger P (1986) Do climate attractors exist? Nature 323:609–612

    Article  Google Scholar 

  • Grassberger P, Procaccia I (1983) Characterization of strange attractors. Phys Rev Lett 50:346–349

    Article  Google Scholar 

  • Harbourne RT, Stergiou N (2003) Nonlinear analysis of the development of sitting postural control. Dev Psychobiol 42:368–377

    Article  PubMed  Google Scholar 

  • Hurst HE (1965) Long-term storage: an experimental study. Constable, London

    Google Scholar 

  • Huys R, Beek PJ (2002) The coupling between point-of-gaze and ball movements in three-ball cascade juggling: the effects of expertise, pattern and tempo. J Sports Sci 20:171–186

    Article  PubMed  Google Scholar 

  • Huys R, Daffertshofer A, Beek PJ (2003) Learning to juggle: on the assembly of functional subsystems into a task-specific dynamical organization. Biol Cybern 88:302–318

    Article  PubMed  CAS  Google Scholar 

  • Huys R, Daffertshofer A, Beek PJ (2004) Multiple time scales and multiform dynamics in learning to juggle. Motor Control 8:188–212

    PubMed  Google Scholar 

  • Kantz H, Schreiber T (2004) Nonlinear time series analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • Kaplan DT, Furman MI, Pincus SM, Ryan SM, Lipsitz LA, Goldberger AL (1991) Aging and complexity of cardiovascular dynamics. Biophys J 59:945–949

    Article  PubMed  CAS  Google Scholar 

  • Kay BA (1988) The dimensionality of movement trajectories and the degrees of freedom problem: a tutorial. Hum Mov Sci 7:343–364

    Article  Google Scholar 

  • Kiemel T, Oie KS, Jeka JJ (2002) Multisensory fusion and the stochastic structure of postural sway. Biol Cybern 87:262–277

    Article  PubMed  Google Scholar 

  • Kyriazis M (2003) Practical applications of chaos theory to the modulation of human ageing: nature prefers chaos to regularity. Biogerontology 4:75–90

    Article  PubMed  Google Scholar 

  • Lake DE, Richman JS, Griffin MP, Moorman JR (2002) Sample entropy analysis of neonatal heart rate variability. Am J Physiol Regul Integr Comp Physiol 283:789–797

    Google Scholar 

  • Lipsitz LA (2002) Dynamics of stability: the physiologic basis of functional health and frailty. J Gerontol Biol Sci 57A:B115–B125

    Google Scholar 

  • Mandelbrot BB, van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422–437

    Article  Google Scholar 

  • Mégrot F, Bardy BG, Dietrich G (2002) Dimensionality and the dynamics of human unstable equilibrium. J Mot Behav 34:323–328

    Article  PubMed  Google Scholar 

  • Milton JG, Small SS, Solodkin A (2004) On the road to automatic: dynamic aspects in the development of expertise. J Clin Neurophysiol 21:134–143

    Article  PubMed  Google Scholar 

  • Newell KM, van Emmerik REA, Lee D, Sprague RL (1993) On postural stability and variability. Gait Posture 4:225–230

    Article  Google Scholar 

  • Newell KM, Slobounov SM, Slobounova ES, Molenaar PCM (1997) Stochastic processes in center-of-pressure profiles. Exp Brain Res 113:158–164

    Article  PubMed  CAS  Google Scholar 

  • Newell KM (1998) Degrees of freedom and the development of center of pressure profiles. In: Newell KM, Molenaar PCM (eds) Applications of nonlinear dynamics to developmental process modeling. Erlbaum, Hillsdale, NJ, pp 63–84

    Google Scholar 

  • Nienhuis B, Geurts AC, Duysens J (2001) Are elderly more dependent on visual information and cognitive guidance in the control of upright balance? In: Duysens J, Smits-Engelsman BC, Kingma H (eds) Control of posture and gait. NPI, Maastricht, pp 585–588

    Google Scholar 

  • Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from time series. Phys Rev Lett 45:712–716

    Article  Google Scholar 

  • Paillex R, So A (2005) Changes in the standing posture of stroke patients during rehabilitation. Gait Posture 21:403–409

    Article  PubMed  Google Scholar 

  • Pascolo PB, Marini A, Carniel R, Barazza F (2005) Posture as a chaotic system and an application to the Parkinson’s disease. Chaos Solitons Fractals 24:1343–1346

    Article  Google Scholar 

  • Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL (1994) Mosaic organization of DNA nucleotides. Phys Rev E 49:1685–1689

    Article  CAS  Google Scholar 

  • Peng C-K, Havlin S, Stanley HE, Goldberger AL (1995) Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 5:82–87

    Article  PubMed  CAS  Google Scholar 

  • Peterka RJ (2002) Sensorimotor integration in human postural control. J Neurophysiol 88:1097–1118

    PubMed  CAS  Google Scholar 

  • Pincus SM, Goldberger AL (1994) Physiological time-series analysis: what does regularity quantify? Am J Physiol Heart Circ Physiol 266:H1643–H1656

    CAS  Google Scholar 

  • Pincus SM (1991) Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA 88:2297–2301

    Article  PubMed  CAS  Google Scholar 

  • Rangarajan G, Ding M (2000) Integrated approach to the assessment of long-range correlation in time series data. Phys Rev E 61:4991–5001

    Article  CAS  Google Scholar 

  • Raymakers JA, Samson MM, Verhaar HJJ (2005) The assessment of body sway and the choice of the stability parameter(s). Gait Posture 21:48–58

    Article  PubMed  CAS  Google Scholar 

  • Richman JS, Moorman JR (2000) Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Circ Physiol 278:H2039–H2049

    PubMed  CAS  Google Scholar 

  • Riley MA, Balasubramaniam R, Turvey MT (1999) Recurrence quantification analysis of postural fluctuations. Gait Posture 9:65–78

    Article  PubMed  CAS  Google Scholar 

  • Riley MA, Turvey MT (2002) Variability of determinism in motor behavior. J Mot Behav 34:99–125

    PubMed  Google Scholar 

  • Rosenstein MT, Collins JJ, De Luca CJ (1993) A practical method for calculating largest Lyapunov exponents from small data sets. Phys D 65:117–134

    Article  Google Scholar 

  • Scholz JP, Schöner G (1999) The uncontrolled manifold concept: identifying control variables for a functional task. Exp Brain Res 135:382–404

    Article  Google Scholar 

  • Schöner G (1995) Recent developments and problems in human movement science and their conceptual implications. Ecol Psychol 7:291–314

    Article  Google Scholar 

  • Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds) Dynamical systems and turbulence. Springer, Berlin

  • Theiler J (1986) Spurious dimension from correlation algorithms applied to limited time-series data. Phys Rev A 34:2427–2432

    Article  PubMed  Google Scholar 

  • Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD (1992) Testing for nonlinearity in time series: the method of surrogate data. Phys D 58:77–94

    Article  Google Scholar 

  • Theiler J, Rapp PE (1996) Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram. Electroencephalogr Clin Neurophysiol 98:213–222

    Article  PubMed  CAS  Google Scholar 

  • Thurner S, Mittermaier C, Ehrenberger K (2002) Change of complexity patterns in human posture during aging. Audiol Neurootol 7:240–248

    Article  PubMed  Google Scholar 

  • Wade DT (1992) Measurement in neurological rehabilitation. Oxford University Press, Oxford

    Google Scholar 

  • Wiesenfeld K, Moss F (1995) Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature 373:33–36

    Article  PubMed  CAS  Google Scholar 

  • Wing A, Daffertshofer A, Pressing J (2004) Multiple time scales in serial production of force: a tutorial on power spectral analysis of motor variability. Hum Mov Sci 23:569–590

    Article  PubMed  Google Scholar 

  • Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D 16:285–317

    Article  Google Scholar 

  • Yamada N (1995) Chaotic swaying of the upright posture. Hum Mov Sci 14:711–726

    Article  Google Scholar 

  • Zatsiorsky VM, Duarte M (1999) Instant equilibrium point and its migration in standing tasks: rambling and trembling components of the stabilogram. Motor Control 3:28–38

    PubMed  CAS  Google Scholar 

Download references

Acknowledgments

This research was conducted while the first author was working on a grant of the Netherlands Organization for Health Research and Development (ZonMw grant 1435.0004).

Author information

Authors and Affiliations

  1. Faculty of Human Movement Sciences, Institute for Fundamental and Clinical Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT, Amsterdam, The Netherlands

    M. Roerdink, A. Daffertshofer, S. F. Donker, A. C. H. Geurts & P. J. Beek

  2. Department of Rehabilitation, Amsterdam Medical Centre, University of Amsterdam, Amsterdam, The Netherlands

    M. De Haart

  3. Department of Research, Development, and Education, St. Maartenskliniek, Nijmegen, The Netherlands

    M. De Haart & A. C. H. Geurts

  4. Department of Otorhinolaryngology, Vrije Universiteit Medical Centre, Amsterdam, The Netherlands

    S. F. Donker

  5. Department of Rehabilitation Medicine, University Medical Centre, St. Radboud, Nijmegen, The Netherlands

    A. C. H. Geurts

Authors
  1. M. Roerdink
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. De Haart
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Daffertshofer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. F. Donker
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. C. H. Geurts
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. P. J. Beek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Roerdink.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roerdink, M., De Haart, M., Daffertshofer, A. et al. Dynamical structure of center-of-pressure trajectories in patients recovering from stroke . Exp Brain Res 174, 256–269 (2006). https://doi.org/10.1007/s00221-006-0441-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00221-006-0441-7

Keywords

Search

Navigation