Expecting the Unexpected: Entropy and Multifractal Systems in Finance

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point's position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor's point of view, a rise in entropy is a signal of abnormal and possibly negative returns. This means he has to expect the unexpected and prepare for it. To explore this, we analyse the New York Stock Exchange (NYSE) U.S. Index as well as its constituents. Through this examination, we assess their multifractal characteristics and identify market conditions (bearish/bullish markets) using entropy, an effective method for recognizing fluctuating fractal markets. Our findings challenge conventional beliefs by demonstrating that price declines lead to increased entropy, contrary to some studies in the literature that suggest that reduced entropy in market crises implies more determinism. Instead, we propose that bear markets are likely to exhibit higher entropy, indicating a greater chance of unexpected extreme events. Moreover, our study reveals a power-law behaviour and indicates the absence of variance.

The modulatory effect of adaptive task-switching training on resting-state neural network dynamics in younger and older adults

With increasing life expectancy and active aging, it becomes crucial to investigate methods which could compensate for generally detected cognitive aging processes. A promising candidate is adaptive cognitive training, during which task difficulty is adjusted to the participants' performance level to enhance the training and potential transfer effects. Measuring intrinsic brain activity is suitable for detecting possible distributed training-effects since resting-state dynamics are linked to the brain's functional flexibility and the effectiveness of different cognitive processes. Therefore, we investigated if adaptive task-switching training could modulate resting-state neural dynamics in younger (18-25 years) and older (60-75 years) adults (79 people altogether). We examined spectral power density on resting-state EEG data for measuring oscillatory activity, and multiscale entropy for detecting intrinsic neural complexity. Decreased coarse timescale entropy and lower frequency band power as well as increased fine timescale entropy and higher frequency band power revealed a shift from more global to local information processing with aging before training. However, cognitive training modulated these age-group differences, as coarse timescale entropy and lower frequency band power increased from pre- to post-training in the old-training group. Overall, our results suggest that cognitive training can modulate neural dynamics even when measured outside of the trained task.

EEG variability: Task-driven or subject-driven signal of interest?

Neurons in the brain are seldom perfectly quiet. They continually receive input and generate output, resulting in highly variable patterns of ongoing activity. Yet the functional significance of this variability is not well understood. If brain signal variability is functionally relevant and serves as an important indicator of cognitive function, then it should be highly sensitive to the precise manner in which a cognitive system is engaged and/or relate strongly to differences in behavioral performance. To test this, we examined EEG activity in younger adults as they performed a cognitive skill learning task and during rest. Several measures of EEG variability and signal strength were calculated in overlapping time windows that spanned the trial interval. We performed a systematic examination of the factors that most strongly influenced the variability and strength of EEG activity. First, we examined the relative sensitivity of each measure to across-subject variation (within blocks) and across-block variation (within subjects). We found that the across-subject variation in EEG variability and signal strength was much stronger than the across-block variation. Second, we examined the sensitivity of each measure to different sources of across-block variation during skill acquisition. We found that key task-driven changes in EEG activity were best reflected in changes in the strength, rather than the variability, of EEG activity. Lastly, we examined across-subject variation in each measure and its relationship with behavior. We found that individual differences in response time measures were best reflected in individual differences in the variability, rather than the strength, of EEG activity. Importantly, we found that individual differences in EEG variability related strongly to stable indicators of subject identity rather than dynamic indicators of subject performance. We therefore suggest that EEG variability may provide a more sensitive subject-driven measure of individual differences than task-driven signal of interest.

Signal complexity indicators of health status in clinical EEG

Brain signal variability changes across the lifespan in both health and disease, likely reflecting changes in information processing capacity related to development, aging and neurological disorders. While signal complexity, and multiscale entropy (MSE) in particular, has been proposed as a biomarker for neurological disorders, most observations of altered signal complexity have come from studies comparing patients with few to no comorbidities against healthy controls. In this study, we examined whether MSE of brain signals was distinguishable across patient groups in a large and heterogeneous set of clinical-EEG data. Using a multivariate analysis, we found unique timescale-dependent differences in MSE across various neurological disorders. We also found MSE to differentiate individuals with non-brain comorbidities, suggesting that MSE is sensitive to brain signal changes brought about by metabolic and other non-brain disorders. Such changes were not detectable in the spectral power density of brain signals. Our findings suggest that brain signal complexity may offer complementary information to spectral power about an individual's health status and is a promising avenue for clinical biomarker development.

The Effects of the Combination of High-Intensity Interval Training with 3D-Multiple Object Tracking Task on Perceptual-Cognitive Performance: A Randomized Controlled Intervention Trial

The ability to process goal-related visual information while ignoring goal-irrelevant information is essential for the human attention system. The study aimed to investigate how perceptual–cognitive performance was affected during high-intensity interval training (HIIT) using a 3D-multiple object tracking (3D-MOT) task called Neurotracker (NT). In an experimental design, 42 healthy adults (age M = 23.3 SD = 2.94, VO₂max 52.8 ± 5.66 mL·kg⁻¹·min⁻¹) were randomly assigned to an intervention (HIIT + NT, NT, HIIT) or control group. NT performance (20 trials per session) was measured pre-and post-test (at 5, 15, and 25 min while running on the treadmill). The participants trained twice a week for a 4-week intervention period. There was a significant interaction effect between pre/post-test and groups regarding perceptual-cognitive performance, indicating similar enhancements in the HIIT + NT and the NT group during exercise. HIIT influences physical fitness but did not show any impact on perceptual–cognitive performance. Due to the specific NT task characteristics, improved physical abilities may not directly impact sport-specific perceptual-cognitive performance. Our findings suggest that training resulted in substantial task-specific gains. Therefore, combination training may be proposed as a training program to improve perceptual–cognitive, and physical performance in a time-efficient way.

Brain Entropy During Aging Through a Free Energy Principle Approach

Neural complexity and brain entropy (BEN) have gained greater interest in recent years. The dynamics of neural signals and their relations with information processing continue to be investigated through different measures in a variety of noteworthy studies. The BEN of spontaneous neural activity decreases during states of reduced consciousness. This evidence has been showed in primary consciousness states, such as psychedelic states, under the name of "the entropic brain hypothesis." In this manuscript we propose an extension of this hypothesis to physiological and pathological aging. We review this particular facet of the complexity of the brain, mentioning studies that have investigated BEN in primary consciousness states, and extending this view to the field of neuroaging with a focus on resting-state functional Magnetic Resonance Imaging. We first introduce historic and conceptual ideas about entropy and neural complexity, treating the mindbrain as a complex nonlinear dynamic adaptive system, in light of the free energy principle. Then, we review the studies in this field, analyzing the idea that the aim of the neurocognitive system is to maintain a dynamic state of balance between order and chaos, both in terms of dynamics of neural signals and functional connectivity. In our exploration we will review studies both on acute psychedelic states and more chronic psychotic states and traits, such as those in schizophrenia, in order to show the increase of entropy in those states. Then we extend our exploration to physiological and pathological aging, where BEN is reduced. Finally, we propose an interpretation of these results, defining a general trend of BEN in primary states and cognitive aging.

Entropy and the Brain: An Overview

Entropy is a powerful tool for quantification of the brain function and its information processing capacity. This is evident in its broad domain of applications that range from functional interactivity between the brain regions to quantification of the state of consciousness. A number of previous reviews summarized the use of entropic measures in neuroscience. However, these studies either focused on the overall use of nonlinear analytical methodologies for quantification of the brain activity or their contents pertained to a particular area of neuroscientific research. The present study aims at complementing these previous reviews in two ways. First, by covering the literature that specifically makes use of entropy for studying the brain function. Second, by highlighting the three fields of research in which the use of entropy has yielded highly promising results: the (altered) state of consciousness, the ageing brain, and the quantification of the brain networks' information processing. In so doing, the present overview identifies that the use of entropic measures for the study of consciousness and its (altered) states led the field to substantially advance the previous findings. Moreover, it realizes that the use of these measures for the study of the ageing brain resulted in significant insights on various ways that the process of ageing may affect the dynamics and information processing capacity of the brain. It further reveals that their utilization for analysis of the brain regional interactivity formed a bridge between the previous two research areas, thereby providing further evidence in support of their results. It concludes by highlighting some potential considerations that may help future research to refine the use of entropic measures for the study of brain complexity and its function. The present study helps realize that (despite their seemingly differing lines of inquiry) the study of consciousness, the ageing brain, and the brain networks' information processing are highly interrelated. Specifically, it identifies that the complexity, as quantified by entropy, is a fundamental property of conscious experience, which also plays a vital role in the brain's capacity for adaptation and therefore whose loss by ageing constitutes a basis for diseases and disorders. Interestingly, these two perspectives neatly come together through the association of entropy and the brain capacity for information processing.

Standard multiscale entropy reflects neural dynamics at mismatched temporal scales: What's signal irregularity got to do with it?

Multiscale Entropy (MSE) is used to characterize the temporal irregularity of neural time series patterns. Due to its' presumed sensitivity to non-linear signal characteristics, MSE is typically considered a complementary measure of brain dynamics to signal variance and spectral power. However, the divergence between these measures is often unclear in application. Furthermore, it is commonly assumed (yet sparingly verified) that entropy estimated at specific time scales reflects signal irregularity at those precise time scales of brain function. We argue that such assumptions are not tenable. Using simulated and empirical electroencephalogram (EEG) data from 47 younger and 52 older adults, we indicate strong and previously underappreciated associations between MSE and spectral power, and highlight how these links preclude traditional interpretations of MSE time scales. Specifically, we show that the typical definition of temporal patterns via "similarity bounds" biases coarse MSE scales-that are thought to reflect slow dynamics-by high-frequency dynamics. Moreover, we demonstrate that entropy at fine time scales-presumed to indicate fast dynamics-is highly sensitive to broadband spectral power, a measure dominated by low-frequency contributions. Jointly, these issues produce counterintuitive reflections of frequency-specific content on MSE time scales. We emphasize the resulting inferential problems in a conceptual replication of cross-sectional age differences at rest, in which scale-specific entropy age effects could be explained by spectral power differences at mismatched temporal scales. Furthermore, we demonstrate how such problems may be alleviated, resulting in the indication of scale-specific age differences in rhythmic irregularity. By controlling for narrowband contributions, we indicate that spontaneous alpha rhythms during eyes open rest transiently reduce broadband signal irregularity. Finally, we recommend best practices that may better permit a valid estimation and interpretation of neural signal irregularity at time scales of interest.

Sex differences in the relationship between cardiorespiratory fitness and brain function in older adulthood

We investigated sex differences in the association between a measure of physical health, cardiorespiratory fitness (CRF), and brain function using resting-state functional connectivity fMRI. We examined these sex differences in the default, frontoparietal control, and cingulo-opercular networks, assemblies of functionally connected brain regions known to be impacted by both age and fitness level. Healthy older adults ( n = 49; 29 women) were scanned to obtain measures of intrinsic connectivity within and across these 3 networks. We calculated global efficiency (a measure of network integration) and local efficiency (a measure of network specialization) using graph theoretical methods. Across all three networks combined, local efficiency was positively associated with CRF, and this was more robust in male versus female older adults. Furthermore, global efficiency was negatively associated with CRF, but only in males. Our findings suggest that in older adults, associations between brain network integrity and physical health are sex-dependent. These results underscore the importance of considering sex differences when examining associations between fitness and brain function in older adulthood. NEW & NOTEWORTHY We examined the association between cardiorespiratory fitness and resting state functional connectivity in several brain networks known to be impacted by age and fitness level. We found significant associations between fitness and measures of network integration and network specialization, but in a sex-dependent manner, highlighting the interplay between sex differences, fitness, and aging brain health. Our findings underscore the importance of considering sex differences when examining associations between fitness and brain function in older adulthood.

Aerobic exercise modulates transfer and brain signal complexity following cognitive training

Although recent evidence has demonstrated the potent effect of physical exercise to increase the efficacy of cognitive training, the neural mechanisms underlying this causal relationship remain unclear. Here, we used multiscale entropy (MSE) of electroencephalography (EEG)-a measure of brain signal complexity-to address this issue. Young males were randomly assigned to either a 20-day dual n-back training following aerobic exercise or the same training regimen following a reading. A feature binding working memory task with concurrent EEG recording was used to test for transfer effects. Although results revealed weak-to-moderate evidence for exercise-induced facilitation on cognitive training, the combination of cognitive training with exercise resulted in greater transfer gains on conditions involving greater attentional demanding, together with greater increases in cognitive modulation on MSE, compared with the reading condition. Overall, our findings suggest that the addition of antecedent physical exercise to brain training regimen could enable wider, more robust improvements.

Spatial complexity of brain signal is altered in patients with generalized anxiety disorder

BACKGROUND

Is it healthy to be chaotic? Recent studies have argued that mental disorders are associated with more orderly neural activities, corresponding to a less flexible functional system. These conclusions were derived from altered temporal complexity. However, the relationship between spatial complexity and health is unknown, although spatial configuration is of importance for normal brain function.

METHODS

Based on resting-state functional magnetic resonance imaging data, we used Sample entropy (SampEn) to evaluate the altered spatial complexity in patients with generalized anxiety disorder (GAD; n = 47) compared to healthy controls (HCs; n = 38) and the relationship between spatial complexity and anxiety level.

RESULTS

Converging results showed increased spatial complexity in patients with GAD, indicating more chaotic spatial configuration. Interestingly, inverted-U relationship was revealed between spatial complexity and anxiety level, suggesting complex relationship between health and the chaos of human brain.

LIMITATIONS

Anxiety-related alteration of spatial complexity should be verified at voxel level in a larger sample and compared with results of other indices to clarify the mechanism of spatial chaotic of anxiety.

CONCLUSIONS

Altered spatial complexity in the brain of GAD patients mirrors inverted-U relationship between anxiety and behavioral performance, which may reflect an important characteristic of anxiety. These results indicate that SampEn is a good reflection of human health or trait mental characteristic.

Changes in EEG multiscale entropy and power-law frequency scaling during the human sleep cycle

We explored changes in multiscale brain signal complexity and power-law scaling exponents of electroencephalogram (EEG) frequency spectra across several distinct global states of consciousness induced in the natural physiological context of the human sleep cycle. We specifically aimed to link EEG complexity to a statistically unified representation of the neural power spectrum. Further, by utilizing surrogate-based tests of nonlinearity we also examined whether any of the sleep stage-dependent changes in entropy were separable from the linear stochastic effects contained in the power spectrum. Our results indicate that changes of brain signal entropy throughout the sleep cycle are strongly time-scale dependent. Slow wave sleep was characterized by reduced entropy at short time scales and increased entropy at long time scales. Temporal signal complexity (at short time scales) and the slope of EEG power spectra appear, to a large extent, to capture a common phenomenon of neuronal noise, putatively reflecting cortical balance between excitation and inhibition. Nonlinear dynamical properties of brain signals accounted for a smaller portion of entropy changes, especially in stage 2 sleep.

Novel Behavioral and Neural Evidences for Age-Related changes in Force complexity

This study investigated age-related changes in behavioral and neural complexity for a polyrhythmic movement, which appeared to be an exception to the loss of complexity hypothesis. Young (n = 15; age = 24.2 years) and older (15; 68.1 years) adults performed low-level force-tracking with isometric index abduction to couple a compound sinusoidal target. Multiscale entropy (MSE) of tracking force and inter-spike interval (ISI) of motor unit (MU) in the first dorsal interosseus muscle were assessed. The MSE area of tracking force at shorter time scales of older adults was greater (more complex) than that of young adults, whereas an opposite trend (less complex for the elders) was noted at longer time scales. The MSE area of force fluctuations (the stochastic component of the tracking force) were generally smaller (less complex) for older adults. Along with greater mean and coefficient of ISI, the MSE area of the cumulative discharge rate of elders tended to be lower (less complex) than that of young adults. In conclusion, age-related complexity changes in polyrhythmic force-tracking depended on the time scale. The adaptive behavioral consequences could be multifactorial origins of the age-related impairment in rate coding, increased discharge noises, and lower discharge complexity of pooled MUs.

The relation between Scrabble expertise and brain aging as measured with EEG brain signal variability

Recent empirical work suggests that the dynamics of brain function, as measured by brain signal variability, differs between younger and older adults. We extended this work by examining how the relationship between brain signal variability and age is altered in the context of expertise. We recorded electroencephalography from Scrabble experts and controls during a visual word recognition task. To measure variability, we used multiscale entropy, which emphasizes the way brain signals behave over a range of timescales and can differentiate the variability of a complex system (the brain) from a purely random system. We replicated previously identified shifts from long-range interactions among neural populations to more local processing in late adulthood. In addition, we demonstrated an age-related increase in midrange neural interactions for experts, suggesting greater maintenance of network integration into late adulthood. Our results indicate that expertise-related differences in the context of age and brain dynamics occur across different timescales and that these differences are linked to task performance.

A practical comparison of algorithms for the measurement of multiscale entropy in neural time series data

There is a broad family of statistical methods for capturing time series regularity, with increasingly widespread adoption by the neuroscientific community. A common feature of these methods is that they permit investigators to quantify the entropy of brain signals - an index of unpredictability/complexity. Despite the proliferation of algorithms for computing entropy from neural time series data there is scant evidence concerning their relative stability and efficiency. Here we evaluated several different algorithmic implementations (sample, fuzzy, dispersion and permutation) of multiscale entropy in terms of their stability across sessions, internal consistency and computational speed, accuracy and precision using a combination of electroencephalogram (EEG) and synthetic 1/ƒ noise signals. Overall, we report fair to excellent internal consistency and longitudinal stability over a one-week period for the majority of entropy estimates, with several caveats. Computational timing estimates suggest distinct advantages for dispersion and permutation entropy over other entropy estimates. Considered alongside the psychometric evidence, we suggest several ways in which researchers can maximize computational resources (without sacrificing reliability), especially when working with high-density M/EEG data or multivoxel BOLD time series signals.

Exercise Training and Functional Connectivity Changes in Mild Cognitive Impairment and Healthy Elders

BACKGROUND

Effective interventions are needed to improve brain function in mild cognitive impairment (MCI), an early stage of Alzheimer's disease (AD). The posterior cingulate cortex (PCC)/precuneus is a hub of the default mode network (DMN) and is preferentially vulnerable to disruption of functional connectivity in MCI and AD.

OBJECTIVE

We investigated whether 12 weeks of aerobic exercise could enhance functional connectivity of the PCC/precuneus in MCI and healthy elders.

METHODS

Sixteen MCI and 16 healthy elders (age range = 60-88) engaged in a supervised 12-week walking exercise intervention. Functional MRI was acquired at rest; the PCC/precuneus was used as a seed for correlated brain activity maps.

RESULTS

A linear mixed effects model revealed a significant interaction in the right parietal lobe: the MCI group showed increased connectivity while the healthy elders showed decreased connectivity. In addition, both groups showed increased connectivity with the left postcentral gyrus. Comparing pre to post intervention changes within each group, the MCI group showed increased connectivity in 10 regions spanning frontal, parietal, temporal and insular lobes, and the cerebellum. Healthy elders did not demonstrate any significant connectivity changes.

CONCLUSION

The observed results show increased functional connectivity of the PCC/precuneus in individuals with MCI after 12 weeks of moderate intensity walking exercise training. The protective effects of exercise training on cognition may be realized through the enhancement of neural recruitment mechanisms, which may possibly increase cognitive reserve. Whether these effects of exercise training may delay further cognitive decline in patients diagnosed with MCI remains to be demonstrated.

The Effects of Physical Exercise and Cognitive Training on Memory and Neurotrophic Factors

This study examined the combined effect of physical exercise and cognitive training on memory and neurotrophic factors in healthy, young adults. Ninety-five participants completed 6 weeks of exercise training, combined exercise and cognitive training, or no training (control). Both the exercise and combined training groups improved performance on a high-interference memory task, whereas the control group did not. In contrast, neither training group improved on general recognition performance, suggesting that exercise training selectively increases high-interference memory that may be linked to hippocampal function. Individuals who experienced greater fitness improvements from the exercise training (i.e., high responders to exercise) also had greater increases in the serum neurotrophic factors brain-derived neurotrophic factor and insulin-like growth factor-1. These high responders to exercise also had better high-interference memory performance as a result of the combined exercise and cognitive training compared with exercise alone, suggesting that potential synergistic effects might depend on the availability of neurotrophic factors. These findings are especially important, as memory benefits accrued from a relatively short intervention in high-functioning young adults.