Fuzzy approximate entropy analysis of resting state fMRI signal complexity across the adult life span
Introduction
Discovering how we age and identifying the proxies of successful ageing is becoming important as the population ages in the developed world. Biological systems theory suggests that physiological ageing is associated with a generalized loss of complexity in the dynamics of healthy systems and hypothesize that such loss of complexity leads to an impaired ability to adapt to physiologic stress, resulting in functional loss and deficit [1]. We have previously demonstrated that this is true in the brain using the blood oxygen level dependent (BOLD) fMRI signal [2]. However, the measurement of complexity in fMRI is difficult due to the limited temporal resolution and the length of the signal acquisitions possible (data length).
The subtle patterns and changes in the signals produced by the short and noisy sequences of real physiological data, from sophisticated biomedical systems are often undetected by nonlinear signal processing measures like correlation dimension [3] and Lyapunov exponent [4]. These nonlinear measures usually require a large data set [5] and assume that the signals produced by biomedical systems are time-invariant (output signal does not depend explicitly on time) [6]. Biomedical systems like the human brain have nonlinear and chaotic properties and the signals they produce are dynamic in nature [7]. Conceptually, the integral of the sum of all the positive characteristic Lyapunov exponents gives an estimate of the Kolmogorov-Sinai entropy (KS entropy) [8]. Also, the Lyapunov spectrum can be used to give an estimate of the rate of entropy production, fractal dimension and information dimension [9]. KS entropy was developed to classify deterministic dynamic systems by rates of information generation [10] where absence of noise and infinite data length are standard mathematical assumptions. As a result, it is compromised by noise and short data length [11]. Other signal processing measures such as spectral and autocorrelation analyses achieve minimal distinctions in both stochastic processes and noisy deterministic data sets [12]. The quantification and analysis of these undetected signal changes reflect underlying biological mechanisms and may provide fundamental insights into the nature of these processes and their understanding may lead to clinical and biomedical applications targeted at maintaining resilience or robustness.
To solve the problem of undetected signal changes in short and noisy datasets obtained from biomedical systems, Pincus proposed a family of statistics called approximate entropy (ApEn) [13], for measuring signal complexity. Here, complexity can be described as the presence of similar patterns in a signal. ApEn is defined as an approximation to the Kolmogorov complexity [14] and is in the same conceptual frame as KS entropy but with a different view of providing a widely applicable statistical formula to distinguish data sets that are composites of both deterministic and stochastic processes [11]. Given N data length and tolerance r, ApEn (m, r, N) is approximately equal to the negative average natural logarithm of the conditional probability that two sequences, which are similar for m points within the tolerance remain similar at the next point [13], [15]. ApEn has been applied to a wide range of biomedical signals such as hormone pulsatility [16], genetic sequences [17], respiratory patterns [18], heart rate variability [19], electrocardiograms [20], electroencephalograms (EEG) [21], magnetoencephalograms (MEG) [22] and functional magnetic resonance imaging data (fMRI) [2]. Correlation dimension and KS entropy employ embedding dimension m, like ApEn, but ApEn can be applied to biomedical settings where correlation dimension and KS entropy are either undefined or infinite with good replicability properties [11]. However, the ApEn algorithm counts each sequence as matching itself to avoid the occurrence of ln (0) in the calculations, which led to the bias of ApEn [11]. This bias causes ApEn to be heavily dependent on data length, makes it uniformly lower than expected for short data lengths and lack relative consistency.
Sample entropy (SampEn) was introduced as an improvement of ApEn where self-matches are excluded, i.e. vectors are not compared to themselves so as to reduce the bias of ApEn [23]. SampEn is the negative natural logarithm of the conditional probability that two sequences remain similar at the next point, where self-matches are not included in calculating the probability [23]. Hence, a lower value of SampEn also indicates more self-similarity and less complexity in the time series. SampEn is largely independent of data length and displays relative consistency over a broader range of possible parameters (m, r and N) under circumstances where ApEn does not [23]. Likewise, SampEn has been applied on a single scale and multi scale level in a number of biomedical signals such as EEG [24], MEG [25] and functional magnetic resonance imaging data (fMRI) [26], [14], [27], [28], [29]. In fact, due to its superior discriminatory ability it has become an established measure of signal complexity in most biomedical signals, especially fMRI signals which typically comprise short and noisy data sets. Irrespective of the superior properties SampEn exhibit in comparison to ApEn, it also has a limitation. SampEn (m, r, N) is not defined for signals if no template and forward match occurs (which may occur in signals with small r and N values) [23].
Recently, another improved version of approximate entropy that also measures signal complexity, called Fuzzy approximate entropy (fApEn) has been proposed [30], [15]. In fApEn, Zadeh's concept of “fuzzy sets” was incorporated into the standard ApEn, to obtain a fuzzy measurement of the similarity between the difference d|Xi, Xj|, of any pair, of corresponding m measurements of Xi and Xj based on their shapes. In comparison to standard ApEn and SampEn using independent, identically distributed (i.i.d) Gaussian and uniform noise, fApEn with respect to standard ApEn showed better monotonicity, relative consistency, and robustness to noise when characterising signals with different complexities [15]. In comparison to SampEn, fApEn was not limited by the tolerance, r as in the calculation of SampEn [30], [15]. Xie et al. [15] further characterised fApEn and ApEn with experimental electromyography (EMG) signals and found that fApEn significantly decreased during the development of muscle fatigue, which is a similar trend to that of the mean frequency of the EMG signal, while the standard ApEn failed to detect this change. Also, fApEn has been applied to other studies of EMG [31] and EEG [32]. To the best of our knowledge, the application of fApEn to analyse experimental BOLD fMRI signals remain unexplored.
Here, we investigate the performance and characteristics of fApEn on fMRI signal complexity and evaluate its potential for complexity analysis of fMRI data. To assess the appropriateness and effectiveness of fApEn for fMRI signal complexity analysis we compared it to the well-established SampEn analysis using resting state fMRI data of healthy adults with age ranging from 19 to 85 years. Here, we investigated the association between fApEn and age, and SampEn and age. BOLD fMRI signal is a good candidate for fApEn and SampEn analysis because it has poor temporal resolution (relatively few time points) and is inherently noisy.
The BOLD signal is an indirect measure of neural activity in the human brain. The haemodynamic response efficiency (HRE) is an index of the coupling between neural activity and vascular response [33]. Hemodynamic response in gray matter is higher than in white matter [33]. As a result of this, BOLD fMRI studies have mainly focused on gray matter. However, BOLD fMRI activations in white matter have been reported in many other studies and this continue to increase [34].
The characterization and analysis of a system's output complexity may give an indication of its health and robustness. [35], [36], [37]. Systems with complex output patterns are thought to be better able to adapt to perturbation and damage, minimising functional loss. Adaptability is the capacity to respond to unpredictable perturbation and stresses, and this is a defining feature of healthy function. A loss of adaptive capacity in complex physiological systems can be characterized as a manifestation of the degradation of multiple physiological processes that are normally responsible for healthy adaptation to daily stresses. The degradation of these processes with age and disease can be observed as a loss of complexity in the dynamics of complex physiological systems [1].
As we age there is a decline in cognitive abilities such as processing speed, memory, executive function and reasoning [38]. The basis for this decline is not well understood, although it is reasonable to assume that its pathological origin is the accumulation of a variety of age and disease specific pathologies. How the brain overcomes the effects of this pathological burden to maintain function is unclear. The complexity of fMRI signals using entropy has been shown to vary with age and may represent a decrease [26] or increase in the capacity to adapt to the accumulation of age-related pathologies [2], [39]. In the present analyses, we expect the complexity of the resting state fMRI signal measured by fApEn and SampEn to be consistent with the Goldberger/Lipsitz model for robustness [35], [36], [37], [1] where healthier and more robust systems exhibit more complexity in their physiological output and system complexity decrease with age. Also, we expect fApEn to exhibit at least similar performance and discriminatory ability to SampEn, a well-established method in fMRI signal complexity analysis [26], [14], [27], [28], [29].
Section snippets
Participants
The International Consortium for Brain Mapping (ICBM) resting state dataset made publicly available in the 1000 Functional Connectomes project was used for this investigation. The ages of the 86 healthy adults’ (41 males) ranged from 19 to 85 years (mean age of 44.19 ± 17.92 years). The study was approved by the local research ethics committee and subjects had no history of neurological or psychiatric disorders. Written informed consent was obtained from the subjects. Information regarding this
Results
The mean whole brain fApEn value of the male sample was not significantly (p > 0.05) higher than that of the female sample. When the general linear model analysis (GLM) was performed in SPSS, the GLM showed that there was a main effect of age (p < 0.001), there was no main effect of sex (p = 0.561) and there was no interaction between age and sex (p = 0.174). When the main effect of age was corrected for using the general linear model in SPSS, the mean whole brain fApEn difference between the
Discussion
In this study, we have presented a method for the implementation of fApEn on fMRI data. Here, we investigated the performance and characteristics of fApEn in comparison to SampEn on fMRI signal complexity. Our initial analysis, on a small sample of two groups of ten healthy younger adults and ten healthy older adults that are significantly (p < 0.001) different in age showed that fApEn demonstrated excellent performance in discriminating the younger from the older adults, while SampEn
Competing interests
There are no competing interests
Ethical approval
The International Consortium for Brain Mapping (ICBM) resting state dataset made publicly available in the 1000 Functional Connectomes project was used for this investigation. The study was approved by the local research ethics committee. Information regarding this dataset is available at https://www.nitrc.org/projects/fcon_1000/.
Acknowledgment
The authors would like to acknowledge the work of the International Consortium for Brain Mapping (ICBM) fMRI community in creating the resting state database and making it publicly available within the framework of the 1000 Functional Connectomes project (https://www.nitrc.org/projects/fcon_1000/). M.O. Sokunbi was supported by an MRC grant G1100629.
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