Espen A. F. Ihlen

Introduction to multifractal detrended fluctuation analysis in Matlab

Fractal structures are found in biomedical time series from a wide range of physiological phenomena. The multifractal spectrum identifies the deviations in fractal structure within time periods with large and small fluctuations. The present tutorial is an introduction to multifractal detrended fluctuation analysis (MFDFA) that estimates the multifractal spectrum of biomedical time series. The tutorial presents MFDFA step-by-step in an interactive Matlab session. All Matlab tools needed are available in Introduction to MFDFA folder at the website www.ntnu.edu/inm/geri/software. MFDFA are introduced in Matlab code boxes where the reader can employ pieces of, or the entire MFDFA to example time series. After introducing MFDFA, the tutorial discusses the best practice of MFDFA in biomedical signal processing. The main aim of the tutorial is to give the reader a simple self-sustained guide to the implementation of MFDFA and interpretation of the resulting multifractal spectra.

Interaction-dominant dynamics in human cognition: Beyond 1/ƒα fluctuation

It has been suggested that human behavior in general and cognitive performance in particular emerge from coordination between multiple temporal scales. In this article, we provide quantitative support for such a theory of interaction-dominant dynamics in human cognition by using wavelet-based multifractal analysis and accompanying multiplicative cascading process on the response series of 4 different cognitive tasks: simple response, word naming, choice decision, and interval estimation. Results indicated that the major portion of these response series had multiplicative interactions between temporal scales, visible as intermittent periods of large and irregular fluctuations (i.e., a multifractal structure). Comparing 2 component-dominant models of 1/f(alpha) fluctuations in cognitive performance with the multiplicative cascading process indicated that the multifractal structure could not be replicated by these component-dominant models. Furthermore, a similar multifractal structure was shown to be present in a model of self-organized criticality in the human nervous system, similar to a spatial extension of the multiplicative cascading process. These results illustrate that a wavelet-based multifractal analysis and the multiplicative cascading process form an appropriate framework to characterize interaction-dominant dynamics in human cognition. This new framework goes beyond the identification of 1/f(alpha) power laws and non-gaussian distributions in response series as used in previous studies. The present article provides quantitative support for a paradigm shift toward interaction-dominant dynamics in human cognition.

The influence of center-of-mass movements on the variation in the structure of human postural sway

The present article investigates the influence of center-of-mass movements on the variation of the structure in human postural sway. Twelve healthy younger persons performed 60s quiet standing, 60s relaxed standing, and 10 min relaxed standing on two force plates. Center-of-pressure (CoP) and gravitational line (GL) profiles were calculated from the ground reaction forces and moments. The temporal variation of CoP structure was calculated by the local scaling exponent h(t) and a Monte Carlo surrogate test was used to identify phase couplings between temporal scales. The range of variation of h(t) was significantly larger in relaxed standing compared to quiet standing (p<0.00001) and highly correlated with the range of GL movements (r>0.76, p<0.001). However, the variation in h(t) was not generated by the GL movements because the CoP-GL traces was close to identical variation in h(t) (r>0.95, p<0.00001). The Monte Carlo surrogate test indicated the presence of intermittent phase couplings between the temporal scales of both CoP traces and the CoP-GL residuals in the periods with GL movements. The present results suggest that human posture is controlled by intermittent phase coupling of the CoP and GL movements. Furthermore, the investigation of the variation in CoP structure might extend existing theories of changes in postural control for example older persons and patients with a neurodegenerative disease.

Multifractal formalisms of human behavior

With the mounting realization that variability is an inevitable part of human behavior comes the need to integrate this phenomenon in concomitant models and theories of motor control. Among other things, this has resulted in a debate throughout the last decades about the origin of variability in behavior, the outcome of which has important implications for motor control theories. To date, a monofractal formalism of variability has been used as the basis for arguing for component- versus interaction-oriented theories of motor control. However, monofractal formalism alone cannot decide between the opposing sides of the debate. The present theoretical overview introduces multifractal formalisms as a necessary extension of the conventional monofractal formalism. In multifractal formalisms, the scale invariance of behavior is numerically defined as a spectrum of scaling exponents, rather than a single average exponent as in the monofractal formalism. Several methods to estimate the multifractal spectrum of scaling exponents - all within two multifractal formalisms called large deviation and Legendre formalism - are introduced and briefly discussed. Furthermore, the multifractal analyses within these two formalisms are applied to several performance tasks to illustrate how explanations of motor control vary with the methods used. The main section of the theoretical overview discusses the implications of multifractal extensions of the component- and interaction-oriented models for existing theories of motor control.

The complexity of daily life walking in older adult community-dwelling fallers and non-fallers

Complexity of human physiology and physical behavior has been suggested to decrease with aging and disease and make older adults more susceptible to falls. The present study investigates complexity in daily life walking in community-dwelling older adult fallers and non-fallers measured by a 3D inertial accelerometer sensor fixed to the lower back. Complexity was expressed using new metrics of entropy: refined composite multiscale entropy (RCME) and refined multiscale permutation entropy (RMPE). The study re-analyses data of 3 days daily-life activity originally described by Weiss et al. (2013). The data set contains inertial sensor data from 39 older persons reporting less than 2 falls and 32 older persons reporting two or more falls during the previous year. The RCME and the RMPE were derived for trunk acceleration and velocity signals from walking epochs of 50s using mean and variance coarse graining of the signals. Discriminant abilities of the entropy metrics were assessed using a partial least square discriminant analysis. Both RCME and RMPE successfully distinguished between the daily-life walking of the fallers and non-fallers (AUC>0.8) and performed better than the 35 conventional gait features investigated by Weiss et al. (2013). Higher complexity was found in the vertical and mediolateral directions in the non-fallers for both entropy metrics. These findings suggest that RCME and RMPE can be used to improve the assessment of fall risk in older people.

Multifractal analyses of response time series: A comparative study

Response time series with a non-Gaussian distribution and long-range dependent dynamics have been reported for several cognitive tasks. Conventional monofractal analyses numerically define a long-range dependency as a single scaling exponent, but they assume that the response times are Gaussian distributed. Ihlen and Vereijken (Journal of Experimental Psychology: General, 139, 436–463, 2010) suggested multifractal extensions of the conventional monofractal analyses that are more suitable when the response time has a non-Gaussian distribution. Multifractal analyses estimate a multifractal spectrum of scaling exponents that contain the single exponent estimated by the conventional monofractal analyses. However, a comparison of the performance of multifractal analyses with behavioral variables has not yet been addressed. The present study compares the performance of seven multifractal analyses. The multifractal analyses were tested on multiplicative cascading noise that generates time series with a predefined multifractal spectrum and with a structure of variation that mimics intermittent response time variation. Time series with 1,024 and 4,096 samples were generated with additive noise and multiharmonic trends of two different magnitudes (signal-to-noise/trend ratio; 0.33 and 1). The results indicate that all multifractal analysis has individual pros and cons related to sample size, multifractality, and the presence of additive noise and trends in the response time series. The summary of pros and cons of the seven multifractal analyses provides a guideline for the choice of multifractal analyses of response time series and other behavioral variables.

Phase-dependent changes in local dynamic stability of human gait

Several methods derived from nonlinear time series analysis have been suggested to quantify stability in human gait kinematics. One of these methods is the definition of the maximum finite time Lyapunov exponent (λ) that quantifies how the system responds to infinitesimal perturbations. However, there are fundamental limitations to the conventional definition of λ for gait kinematics. First, exponential increase in initial perturbations cannot be assumed since real-life perturbations of gait kinematics are finite sized. Second, the transitions between single and double support phase within each stride cycle define two distinct dynamical regimes that may not be captured by a single λ. The present article presents a new method to quantify intra-stride changes λ(t) in local dynamical stability and employs the method to 3D lower extremity gait kinematics in 10 healthy adults walking on a treadmill at 3 different speeds. All participants showed an intra-stride change in λ(t) in the transition between single and double support phase. The intra-stride change reflected an both a increase and decrease in λ(t) at heel strike and toe off, respectively, with increased gait speed. Furthermore, a close relationship was found between the intra-stride change in standard deviation of foot velocity in the anterior-posterior direction and the intra-stride change of the initial perturbations. The present results indicate that local dynamical stability has gait phase-dependent changes that are not identified by conventional computation of a single λ.

Older adults have unstable gait kinematics during weight transfer

The present article investigates gait stability of healthy older persons during weight transfer. Ten healthy older persons and ten younger persons walked 10 min each on a treadmill at 3 different gait speeds. The intra-stride change in gait stability was defined by the local divergence exponent λ(t) estimated by a newly developed method. The intra-stride changes in λ(t) during weight transfer were identified by separating each stride into a single and double support phase. The intra-stride changes in λ(t) were also compared to changes in the variation of the gait kinematics, i.e., SD(t). The healthy older persons walked at the same preferred walking speed as the younger persons. However, they exhibited significantly larger λ(t) (p<0.001) during weight transfer in the double support phase. Local divergence was closely related to intra-stride changes in SD(t) of the feet in the anterior-posterior direction. Furthermore, a high correlation was found between local divergence and the variation in step length and step width for both older (R>0.67, p<0.05) and younger persons (R>0.67, p<0.05). The present results indicate that the gait kinematics of older adults are more dynamical unstable during the weight transfer compared to younger persons. Furthermore, a close relationship exists between intra-stride changes in dynamical stability and variation in step length and step width. Further work will validate the results of the present study using real-life perturbations of the gait kinematics of both younger and older adults.

A comparison study of local dynamic stability measures of daily life walking in older adult community-dwelling fallers and non-fallers

In the present study we compared the performance of three different estimations of local dynamic stability λ to distinguish between the dynamics of the daily-life walking of elderly fallers and non-fallers. The study re-analyses inertial sensor data of 3-days daily-life activity originally described by Weiss et al. (2013). The data set contains inertial sensor data from 39 older persons who reported less than 2 falls and 31 older persons who reported two or more falls the previous year. 3D-acceleration and 3D-velocity signals from walking epochs of 50s were used to reconstruct a state space using three different methods. Local dynamic stability was estimated with the algorithms proposed by Rosenstein et al. (1993), Kantz (1994), and Ihlen et al. (2012a). Median λs assessed by Ihlen׳s and Kantz׳ algorithms discriminated better between elderly fallers and non-fallers (highest AUC=0.75 and 0.73) than Rosenstein׳s algorithm (highest AUC=0.59). The present results suggest that the ability of λ to distinguish between fallers and non-fallers is dependent on the parameter setting of the chosen algorithm. Further replication in larger samples of community-dwelling older persons and different patient groups is necessary before including the suggested parameter settings in fall risk assessment and prediction models.

The Discriminant Value of Phase-Dependent Local Dynamic Stability of Daily Life Walking in Older Adult Community-Dwelling Fallers and Nonfallers.

The present study compares phase-dependent measures of local dynamic stability of daily life walking with 35 conventional gait features in their ability to discriminate between community-dwelling older fallers and nonfallers. The study reanalyzes 3D-acceleration data of 3-day daily life activity from 39 older people who reported less than 2 falls during one year and 31 who reported two or more falls. Phase-dependent local dynamic stability was defined for initial perturbation at 0%, 20%, 40%, 60%, and 80% of the step cycle. A partial least square discriminant analysis (PLS-DA) was used to compare the discriminant abilities of phase-dependent local dynamic stability with the discriminant abilities of 35 conventional gait features. The phase-dependent local dynamic stability λ at 0% and 60% of the step cycle discriminated well between fallers and nonfallers (AUC = 0.83) and was significantly larger (p < 0.01) for the nonfallers. Furthermore, phase-dependent λ discriminated as well between fallers and nonfallers as all other gait features combined. The present result suggests that phase-dependent measures of local dynamic stability of daily life walking might be of importance for further development in early fall risk screening tools.