The fractal brain: scale-invariance in structure and dynamics

Distinctive regional patterns of dynamic neural noise in cortical activity

Objective. Neurons exhibit deterministic behavior influenced by stochastic cellular or extracellular components. Estimating this random component is challenging due to unknown underlying deterministic dynamics. In this study, we aim to estimate the neural random component, termed intrinsic dynamic neural noise, from experimental time series without prior assumptions on the underlying neural model.Approach. The method relies on the nonlinear approximate entropy profile and was evaluated using synthetic data from Izhikevich's models and simulated calcium dynamics driven by dynamical noise. We then applied the method to experimental time series from calcium imaging in mice and zebrafish brain regions, as well as electrophysiological data from a 128-channel cortical probe in anesthetized rats.Main results. The results show region-specific behavior, with higher dynamic neural noise in the somatosensory cortex of mice and anterior telencephalic area of zebrafish. Furthermore, neuronal stochasticity is greater in genetically encodedCa2+indicators than inCa2+dyes, and neural noise increases with recording depth.Significance. These findings offer insights into neural dynamics and suggest dynamic noise as a key biomarker.

Development of brain metastable dynamics during the equivalent of the third gestational trimester

Metastability, a concept from dynamical systems theory, provides a framework for understanding how the brain shifts between various functional states and underpins essential cognitive, behavioural, and social function. While studied in adults, metastability in early brain development has only received recent attention. As the brain undergoes dramatic functional and structural changes over the third gestational trimester, here we review how these are reflected in changes in brain metastable dynamics in preterm, preterm at term-equivalent and full-term neonates. We synthesize findings from EEG, fMRI, fUS, and computational models, focusing on the spatial distribution and temporal dynamics of metastable states, which include functional integration and segregation, signal predictability and complexity. Despite fragmented evidence, studies suggest that neonatal metastability develops over the equivalent of the third gestational trimester, with increasing ability for integration-segregation, broader range of metastable states, faster metastable state transitions and greater signal complexity. Preterms at term-equivalent age exhibit immature metastability features compared to full-terms. We explain and interpret these changes in terms of maturation of the brain in a free energy landscape and establishment of cognitive functions.

Multifractal dynamic changes of spontaneous brain activity in psychiatric disorders: Adult attention deficit-hyperactivity disorder, bipolar disorder, and schizophrenia

It is one of the strategies to study the complexity of spontaneous fluctuation of brain neurons based on resting-state functional magnetic resonance imaging (rs-fMRI), but the multifractal characteristics of spontaneous fluctuation of brain neurons in psychiatric diseases need to be studied. Therefore, this paper will study the multifractal spontaneous brain activity changes in psychiatric disorders using the multifractal detrended fluctuation analysis algorithm based on the UCLA datasets. Specifically: (1) multifractal characteristics in adult attention deficit-hyperactivity disorder (ADHD), bipolar disorder (BP), and schizophrenia (SCHZ); (2) the source of those multifractal characteristics. Results showed that for adult ADHD, BP, and SCHZ, all 6 functional brain regions exhibit multifractal characteristics, and the multifractal spectrum shows a reduction in bell-shaped asymmetry, unlike the intensity of healthy control (HC) asymmetry. Besides, compared with HC, the multifractal sources of all functional brain regions were fat-tail probability distribution and the long-range dependence correlation, but the intensity of fat-tail probability distribution was decreased and the long-range dependence correlation was increased. The results provide a reference for further understanding the complexity of spontaneous fluctuation of neurons in psychiatric disorders.

Kovács BB, Novák T, Erdélyi M. Cluster parameter-based DBSCAN maps for image characterization

Single-molecule localization microscopy techniques are one of the most powerful methods in biological studies, allowing the visualization of nanoclusters. Cluster analysis algorithms are used for quantitative evaluation, with DBSCAN being one of the most widely used. Clustering results are extremely sensitive to the initial parameters; thus, several methods including DBSCAN maps, have been developed for parameter optimization. Here, we introduce cluster parameter-based DBSCAN maps, which are directly applicable to measured datasets. These maps can be used for image characterization and parameter optimization through sensitivity studies. We show the applicability of these maps to simulated and measured datasets and compare our results with the recently implemented lacunarity analysis for SMLM measurements.

Fast updating feedback from piriform cortex to the olfactory bulb relays multimodal identity and reward contingency signals during rule-reversal

While animals readily adjust their behavior to adapt to relevant changes in the environment, the neural pathways enabling these changes remain largely unknown. Here, using multiphoton imaging, we investigate whether feedback from the piriform cortex to the olfactory bulb supports such behavioral flexibility. To this end, we engage head-fixed male mice in a multimodal rule-reversal task guided by olfactory and auditory cues. Both odor and, surprisingly, the sound cues trigger responses in the cortical bulbar feedback axons which precede the behavioral report. Responses to the same sensory cue are strongly modulated upon changes in stimulus-reward contingency (rule-reversals). The re-shaping of individual bouton responses occurs within seconds of the rule-reversal events and is correlated with changes in behavior. Optogenetic perturbation of cortical feedback within the bulb disrupts the behavioral performance. Our results indicate that the piriform-to-olfactory bulb feedback axons carry stimulus identity and reward contingency signals which are rapidly re-formatted according to changes in the behavioral context.

Universality of ecological memory for local and global net ecosystem exchange, atmospheric CO2, and sea surface temperature

Modeling global net ecosystem exchange is essential to understanding and quantifying the complex interactions between the Earth's terrestrial ecosystems and the atmosphere. Emphasizing the inter-relatedness between the global net ecosystem exchange, global sea surface temperature, and atmospheric CO 2 levels, intuitively suggests that all three systems may exhibit collective environmental memory. Motivated by this, we explicitly identified a collective memory function and showed a similar non-Markovian stochastic behavior for these systems exhibiting superdiffusive behavior in short time intervals. We obtained the values of the memory parameter, μ , and the characteristic frequencies, ν , for global net ecosystem exchange (GNEE) ( μ = 0.94 ± 0.03 , ν = 0.67 ± 0.08 / m o . ), global sea surface temperature (GSST) ( μ = 0.68 ± 0.11 , ν = 0.30 ± 0.18 / m o . ), and atmospheric CO 2 ( μ = 0.78 ± 0.08 , ν = 0.66 ± 0.13 / w k . ). The values of the memory parameter are within the range, 0 < μ < 1 , and thus all three systems are in the superdiffusive regime. We emphasize, further, that these results were consistent with our previous analyses at the ecosystem level (i.e. Great Barrier Reef) suggesting scale invariance for these phenomena. Thus, the observed superdiffusive behavior operating at different scales suggests universality of the collective memory function for these systems.

Evaluating and Comparing Measures of Aperiodic Neural Activity

Neuro-electrophysiological recordings contain prominent aperiodic activity - meaning irregular activity, with no characteristic frequency - which has variously been referred to as 1/f (or 1/f-like activity), fractal, or 'scale-free' activity. Previous work has established that aperiodic features of neural activity is dynamic and variable, relating (between subjects) to healthy aging and to clinical diagnoses, and also (within subjects) tracking conscious states and behavioral performance. There are, however, a wide variety of conceptual frameworks and associated methods for the analyses and interpretation of aperiodic activity - for example, time domain measures such as the autocorrelation, fractal measures, and/or various complexity and entropy measures, as well as measures of the aperiodic exponent in the frequency domain. There is a lack of clear understanding of how these different measures relate to each other and to what extent they reflect the same or different properties of the data, which makes it difficult to synthesize results across approaches and complicates our overall understanding of the properties, biological significance, and demographic, clinical, and behavioral correlates of aperiodic neural activity. To address this problem, in this project we systematically survey the different approaches for measuring aperiodic neural activity, starting with an automated literature analysis to curate a collection of the most common methods. We then evaluate and compare these methods, using statistically representative time series simulations. In doing so, we establish consistent relationships between the measures, showing that much of what they capture reflects shared variance - though with some notable idiosyncrasies. Broadly, frequency domain methods are more specific to aperiodic features of the data, whereas time domain measures are more impacted by oscillatory activity. We extend this analysis by applying the measures to a series of empirical EEG and iEEG datasets, replicating the simulation results. We conclude by summarizing the relationships between the multiple methods, emphasizing opportunities for reexamining previous findings and for future work.

Rényi entropy-complexity causality space: a novel neurocomputational tool for detecting scale-free features in EEG/iEEG data

Scale-free brain activity, linked with learning, the integration of different time scales, and the formation of mental models, is correlated with a metastable cognitive basis. The spectral slope, a key aspect of scale-free dynamics, was proposed as a potential indicator to distinguish between different sleep stages. Studies suggest that brain networks maintain a consistent scale-free structure across wakefulness, anesthesia, and recovery. Although differences in anesthetic sensitivity between the sexes are recognized, these variations are not evident in clinical electroencephalographic recordings of the cortex. Recently, changes in the slope of the power law exponent of neural activity were found to correlate with changes in Rényi entropy, an extended concept of Shannon's information entropy. These findings establish quantifiers as a promising tool for the study of scale-free dynamics in the brain. Our study presents a novel visual representation called the Rényi entropy-complexity causality space, which encapsulates complexity, permutation entropy, and the Rényi parameter q. The main goal of this study is to define this space for classical dynamical systems within theoretical bounds. In addition, the study aims to investigate how well different time series mimicking scale-free activity can be discriminated. Finally, this tool is used to detect dynamic features in intracranial electroencephalography (iEEG) signals. To achieve these goals, the study implementse the Bandt and Pompe method for ordinal patterns. In this process, each signal is associated with a probability distribution, and the causal measures of Rényi entropy and complexity are computed based on the parameter q. This method is a valuable tool for analyzing simulated time series. It effectively distinguishes elements of correlated noise and provides a straightforward means of examining differences in behaviors, characteristics, and classifications. For the iEEG experimental data, the REM state showed a greater number of significant sex-based differences, while the supramarginal gyrus region showed the most variation across different modes and analyzes. Exploring scale-free brain activity with this framework could provide valuable insights into cognition and neurological disorders. The results may have implications for understanding differences in brain function between the sexes and their possible relevance to neurological disorders.

Fractals in Neuropsychology and Cognitive Neuroscience

The fractal dimension of cognition refers to the idea that the cognitive processes of the human brain exhibit fractal properties. This means that certain patterns of cognitive activity, such as visual perception, memory, language, or problem-solving, can be described using the mathematical concept of fractal dimension.The idea that cognition is fractal has been proposed by some researchers as a way to understand the complex, self-similar nature of the human brain. However, it's a relatively new idea and is still under investigation, so it's not yet clear to what extent cognitive processes exhibit fractal properties or what implications this might have for our understanding of the brain and clinical practice. Indeed, the mission of the "fractal neuroscience" field is to define the characteristics of fractality in human cognition in order to differently characterize the emergence of brain disorders.

Box-Counting Fractal Analysis: A Primer for the Clinician

This chapter lays out the elementary principles of fractal geometry underpinning much of the rest of this book. It assumes a minimal mathematical background, defines the key principles and terms in context, and outlines the basics of a fractal analysis method known as box counting and how it is used to perform fractal, lacunarity, and multifractal analyses. As a standalone reference, this chapter grounds the reader to be able to understand, evaluate, and apply essential methods to appreciate and heal the exquisitely detailed fractal geometry of the brain.

Multifractal long-range dependence pattern of functional magnetic resonance imaging in the human brain at rest

Long-range dependence is a prevalent phenomenon in various biological systems that characterizes the long-memory effect of temporal fluctuations. While recent research suggests that functional magnetic resonance imaging signal has fractal property, it remains unknown about the multifractal long-range dependence pattern of resting-state functional magnetic resonance imaging signals. The current study adopted the multifractal detrended fluctuation analysis on highly sampled resting-state functional magnetic resonance imaging scans to investigate long-range dependence profile associated with the whole-brain voxels as specific functional networks. Our findings revealed the long-range dependence's multifractal properties. Moreover, long-term persistent fluctuations are found for all stations with stronger persistency in whole-brain regions. Subsets with large fluctuations contribute more to the multifractal spectrum in the whole brain. Additionally, we found that the preprocessing with band-pass filtering provided significantly higher reliability for estimating long-range dependence. Our validation analysis confirmed that the optimal pipeline of long-range dependence analysis should include band-pass filtering and removal of daily temporal dependence. Furthermore, multifractal long-range dependence characteristics in healthy control and schizophrenia are different significantly. This work has provided an analytical pipeline for the multifractal long-range dependence in the resting-state functional magnetic resonance imaging signal. The findings suggest differential long-memory effects in the intrinsic functional networks, which may offer a neural marker finding for understanding brain function and pathology.

Sharp detection of oscillation packets in rich time-frequency representations of neural signals

Brain oscillations most often occur in bursts, called oscillation packets, which span a finite extent in time and frequency. Recent studies have shown that these packets portray a much more dynamic picture of synchronization and transient communication between sites than previously thought. To understand their nature and statistical properties, techniques are needed to objectively detect oscillation packets and to quantify their temporal and frequency extent, as well as their magnitude. There are various methods to detect bursts of oscillations. The simplest ones divide the signal into band limited sub-components, quantifying the strength of the resulting components. These methods cannot by themselves cope with broadband transients that look like genuine oscillations when restricted to a narrow band. The most successful detection methods rely on time-frequency representations, which can readily show broadband transients and harmonics. However, the performance of such methods is conditioned by the ability of the representation to localize packets simultaneously in time and frequency, and by the capabilities of packet detection techniques, whose current state of the art is limited to extraction of bounding boxes. Here, we focus on the second problem, introducing two detection methods that use concepts derived from clustering and topographic prominence. These methods are able to delineate the packets' precise contour in the time-frequency plane. We validate the new approaches using both synthetic and real data recorded in humans and animals and rely on a super-resolution time-frequency representation, namely the superlets, as input to the detection algorithms. In addition, we define robust tests for benchmarking and compare the new methods to previous techniques. Results indicate that the two methods we introduce shine in low signal-to-noise ratio conditions, where they only miss a fraction of packets undetected by previous methods. Finally, algorithms that delineate precisely the border of spectral features and their subcomponents offer far more valuable information than simple rectangular bounding boxes (time and frequency span) and can provide a solid foundation to investigate neural oscillations' dynamics.

Fluctuations in Upper and Lower Body Movement during Walking in Normal Pressure Hydrocephalus and Parkinson's Disease Assessed by Motion Capture with a Smartphone Application, TDPT-GT

We aimed to capture the fluctuations in the dynamics of body positions and find the characteristics of them in patients with idiopathic normal pressure hydrocephalus (iNPH) and Parkinson's disease (PD). With the motion-capture application (TDPT-GT) generating 30 Hz coordinates at 27 points on the body, walking in a circle 1 m in diameter was recorded for 23 of iNPH, 23 of PD, and 92 controls. For 128 frames of calculated distances from the navel to the other points, after the Fourier transforms, the slopes (the representatives of fractality) were obtained from the graph plotting the power spectral density against the frequency in log-log coordinates. Differences in the average slopes were tested by one-way ANOVA and multiple comparisons between every two groups. A decrease in the absolute slope value indicates a departure from the 1/f noise characteristic observed in healthy variations. Significant differences in the patient groups and controls were found in all body positions, where patients always showed smaller absolute values. Our system could measure the whole body's movement and temporal variations during walking. The impaired fluctuations of body movement in the upper and lower body may contribute to gait and balance disorders in patients.

Fast updating feedback from piriform cortex to the olfactory bulb relays multimodal reward contingency signals during rule-reversal

While animals readily adjust their behavior to adapt to relevant changes in the environment, the neural pathways enabling these changes remain largely unknown. Here, using multiphoton imaging, we investigated whether feedback from the piriform cortex to the olfactory bulb supports such behavioral flexibility. To this end, we engaged head-fixed mice in a multimodal rule-reversal task guided by olfactory and auditory cues. Both odor and, surprisingly, the sound cues triggered cortical bulbar feedback responses which preceded the behavioral report. Responses to the same sensory cue were strongly modulated upon changes in stimulus-reward contingency (rule reversals). The re-shaping of individual bouton responses occurred within seconds of the rule-reversal events and was correlated with changes in the behavior. Optogenetic perturbation of cortical feedback within the bulb disrupted the behavioral performance. Our results indicate that the piriform-to-olfactory bulb feedback carries reward contingency signals and is rapidly re-formatted according to changes in the behavioral context.

Multifractality in Surface Potential for Cancer Diagnosis

Recent advances in high-resolution biomedical imaging have improved cancer diagnosis, focusing on morphological, electrical, and biochemical properties of cells and tissues, scaling from cell clusters down to the molecular level. Multiscale imaging revealed high complexity that requires advanced data processing methods of multifractal analysis. We performed label-free multiscale imaging of surface potential variations in human ovarian cancer cells using Kelvin probe force microscopy (KPFM). An improvement in the differentiation between nonmalignant and cancerous cells by multifractal analysis using adaptive versus median threshold for image binarization was demonstrated. The results reveal the multifractality of cancer cells as a new biomarker for cancer diagnosis.

Inter-individual variability during neurodevelopment: an investigation of linear and nonlinear resting-state EEG features in an age-homogenous group of infants

Electroencephalography measures are of interest in developmental neuroscience as potentially reliable clinical markers of brain function. Features extracted from electroencephalography are most often averaged across individuals in a population with a particular condition and compared statistically to the mean of a typically developing group, or a group with a different condition, to define whether a feature is representative of the populations as a whole. However, there can be large variability within a population, and electroencephalography features often change dramatically with age, making comparisons difficult. Combined with often low numbers of trials and low signal-to-noise ratios in pediatric populations, establishing biomarkers can be difficult in practice. One approach is to identify electroencephalography features that are less variable between individuals and are relatively stable in a healthy population during development. To identify such features in resting-state electroencephalography, which can be readily measured in many populations, we introduce an innovative application of statistical measures of variance for the analysis of resting-state electroencephalography data. Using these statistical measures, we quantified electroencephalography features commonly used to measure brain development-including power, connectivity, phase-amplitude coupling, entropy, and fractal dimension-according to their intersubject variability. Results from 51 6-month-old infants revealed that the complexity measures, including fractal dimension and entropy, followed by connectivity were the least variable features across participants. This stability was found to be greatest in the right parietotemporal region for both complexity feature, but no significant region of interest was found for connectivity feature. This study deepens our understanding of physiological patterns of electroencephalography data in developing brains, provides an example of how statistical measures can be used to analyze variability in resting-state electroencephalography in a homogeneous group of healthy infants, contributes to the establishment of robust electroencephalography biomarkers of neurodevelopment through the application of variance analyses, and reveals that nonlinear measures may be most relevant biomarkers of neurodevelopment.