The Fractal Geometry of the Brain (Advances in Neurobiology, 36)

Foreword to the 2nd Edition
Foreword to the 1st Edition
Preface to the 2nd Edition
Preface to the 1st Edition
Acknowledgments to the 2nd Edition
Acknowledgments to the 1st Edition
Contents
About the Editor
Part I Introduction to Fractal Geometry and Its Applications to Neurosciences
	1 The Fractal Geometry of the Brain: An Overview
		1.1 From The Fractal Geometry of Nature to Fractal Analysis in Biomedicine
		1.2 From Euclid to Fractal Metrology
		1.3 The Fractal Geometry of the Brain
		1.4 Fractal Dimensions and Neurosciences
		References
	2 Box-Counting Fractal Analysis: A Primer for the Clinician
		2.1 Fractal Analysis: Measuring Self-Similar Details
			2.1.1 Supplementary Measures in Fractal Analysis
		2.2 How Is a DF Calculated?
			2.2.1 Practical Points Relevant to the DF
				2.2.1.1 Not Unique
				2.2.1.2 Not Necessarily Exact, Infinite, and Known
				2.2.1.3 Density Vs. Dimension
			2.2.2 DFs in Neuroscience
		2.3 Box Counting
			2.3.1 Converting SB and NB to a DB
				2.3.1.1 Regression Lines
			2.3.2 Methodological Issues in Box Counting
				2.3.2.1 High Correlation Coefficients: Necessary but Not Sufficient
				2.3.2.2 Sampling Size, Location, and Orientation/Rotation Biases
				2.3.2.3 Box-Counting Solutions
		2.4 Lacunarity
			2.4.1 Calculating and Applying Lacunarity
			2.4.2 Understanding the DB and Λ
				2.4.2.1 Pattern Idiosyncrasies
				2.4.2.2 Applying Lacunarity and the DB in Complement
		2.5 Grayscale Volumes and Box Counting
		2.6 Subscanning
		2.7 Multifractal Analysis
			2.7.1 Reading the Dq Vs. Q Curve
			2.7.2 Reading the f(α) Curve
				2.7.2.1 Applying Multifractal Analysis
		2.8 The Validity of 2D Patterns from Four-Dimensional (4D) Reality
			2.8.1 Control and Calibration
		2.9 Conclusions
		References
	3 Tenets and Methods of Fractal Analysis (1/f Noise)
		3.1 Tenets and Methods of Fractal Analysis (1/f Noise)
		3.2 Statistical Terms: Parameter, Estimator, and Estimate
		3.3 Properties of 1/f Noise: Self-Similarity and Long Memory
			3.3.1 Memory
			3.3.2 Stationarity
		3.4 Fractal Parameters
			3.4.1 Hurst Coefficient
			3.4.2 Scaling Exponent (α)
			3.4.3 Power Spectra
			3.4.4 Power Exponent
			3.4.5 Differencing Parameter (d)
		3.5 Estimators of Fractal Parameters
		3.6 Identification of Fractal Noise in Empirical Settings
		3.7 Summary
		References
	4 Multifractal Analysis in Neuroimaging
		4.1 Introduction
		4.2 Tenets of Multifractal Analysis
		4.3 Methods of Multifractal Analysis
			4.3.1 Time Domain Methods
				4.3.1.1 Generalized Fractal Dimensions and the Multifractal Spectrum
				4.3.1.2 The “Sandbox” or Cumulative Mass Method
				4.3.1.3 Large Deviation Multifractal Spectrum
				4.3.1.4 Multifractal Detrended Fluctuation Analysis (MDFA)
				4.3.1.5 Multifractal Detrended Moving Average (MDMA)
			4.3.2 Time-Frequency Domain Methods
				4.3.2.1 Wavelet Transform Modulus Maxima (WTMM)
				4.3.2.2 Wavelet Leaders-Based Multifractal Analysis (WLMA)
				4.3.2.3 Multifractional Brownian Motion (mBm)
		4.4 Applications of Multifractal Analysis
			4.4.1 Electroencephalogram (EEG) Signals
			4.4.2 Brain Imaging
			4.4.3 Brain Network Analysis
		4.5 Conclusions
		References
	5 Fractal Time Series: Background, Estimation Methods, and Performances
		Abbreviations
		5.1 Introduction
		5.2 Fractal Geometry
			5.2.1 Self-Similarity
			5.2.2 Scaling
			5.2.3 From Euclidean Dimension to Fractal Dimension
			5.2.4 Fractal Dimension
		5.3 Fractality in Natural Objects
			5.3.1 Mathematical Formalization of Self-Similarity and Scaling
			5.3.2 Self-Affinity
			5.3.3 Power Law
			5.3.4 The Hurst Exponent
		5.4 Fractal Analysis of Time Series
			5.4.1 Overview of Methods to Generate and Estimate FDs in Synthetic Time Series
			5.4.2 Generating Synthetic Fractal Time Series
				5.4.2.1 Weierstrass Cosine Function (WCF)
				5.4.2.2 Takagi–Landsberg Function
				5.4.2.3 Fractional Brownian Motion (fBm)
				5.4.2.4 Fractional Gaussian Noise
			5.4.3 Methods to Estimate Fractal Dimensions
				5.4.3.1 Slope of Power Spectral Density
				5.4.3.2 Detrended Fluctuation Analysis
				5.4.3.3 Generalized Hurst and Range-Scale-Based Exponent
				5.4.3.4 Higuchi's Fractal Dimension
				5.4.3.5 Katz's Fractal Dimension
				5.4.3.6 Box-Counting Method
			5.4.4 Results and Discussion on the Methods of Estimating Fractal Dimensions on Synthetic Time Series
				5.4.4.1 Performances of Methods for Estimating Fractal Dimensions
				5.4.4.2 Effect of Sampling Frequency
				5.4.4.3 Effect of Signal Amplitude
				5.4.4.4 Effect of the Noise Level
		5.5 Conclusions
		References
Part II Fractals in Neuroanatomy and Basic Neurosciences
	6 Fractals in Neuroanatomy and Basic Neurosciences: An Overview
		6.1 What About the Brain?
		6.2 Fractals, Neurons, and Microglia
		6.3 The Brains and Trees
		6.4 Increase of the Fractal Dimension from “Too Smooth to Too Folded” Human Brains
		6.5 Neuronal Networks
		References
	7 Morphology and Fractal-Based Classifications of Neurons and Microglia in Two and Three Dimensions
		7.1 A Brief Introduction to Neurons and Microglia
			7.1.1 Neuronal and Microglial Morphology in Context
		7.2 Fractal Analysis of Neurons
			7.2.1 Fractal Analysis of Dendritic Arbors
			7.2.2 Methodological Issues
				7.2.2.1 Complementary Methods
				7.2.2.2 3D Analysis
		7.3 Microglia
		7.4 Future Directions
		References
	8 The Morphology of Brain Neurons: The Box-Counting Method in the Quantitative Analysis of 2D Images
		8.1 Introduction
			8.1.1 From Fractal Geometry Toward Fractal Analysis
			8.1.2 Fractal Analysis
		8.2 The Box-Counting Method
			8.2.1 Software for Box Counting
			8.2.2 Application on 2D Digital Images
			8.2.3 The Methodology
				8.2.3.1 Image Size and Resolution
				8.2.3.2 Image Rotation
				8.2.3.3 Image Representation
		8.3 Materials and Methods
			8.3.1 Samples from the Human Cerebellum
			8.3.2 Recording of Samples and Formation of an Image
			8.3.3 Quantifying the Image
			8.3.4 Statistics
		8.4 Results
			8.4.1 The Dentate Nucleus
			8.4.2 The Neostriatum
			8.4.3 The Olivary Nucleus
		8.5 Discussion
		8.6 Conclusions
		References
	9 Neuronal Fractal Dynamics
		9.1 Synapse Formation from the Perspective of Molecular and Cellular Biology
		9.2 Fractal Time Space in the Dynamic Process of Synapse Formation
		A.1 Appendix
			A.1.1 Entropy and Dynamics of Synapse Formation in Fractal Time Space
		References
	10 A Self-Similarity Logic May Shape the Organization of the Nervous System
		10.1 Introduction
		10.2 Structural Self-Similarity of the Nervous System
			10.2.1 Cell Level: Complex Geometry of Neurons and Glial Cells
			10.2.2 Tissue Level
				10.2.2.1 Central Nervous System
				10.2.2.2 Peripheral Nervous System
		10.3 A Self-Similarity Logic Drives the Functional Features of the CNS
			10.3.1 Interaction-Dominant Dynamics in the CNS
				10.3.1.1 The Concept of “Fringe”
				10.3.1.2 The Concept of “Lateral Inhibition”
			10.3.2 Remodeling Processes in the Nervous System
		10.4 Fractal Features and Pathological Processes in the Nervous System
		10.5 Concluding Remarks: A Place for Self-Similarity in a Global Model of the Nervous System?
		References
	11 Fractality of Cranial Sutures
		11.1 Biology of Skull Suture Development
		11.2 Definition of Fractal Patterns
		11.3 Models to Generate Fractal Pattern
			11.3.1 Geometric Model
			11.3.2 Eden Front
			11.3.3 Diffusion-Limited Aggregation
			11.3.4 Stochastic Differential Equation (SDE) Models
		11.4 Models of Skull Suture Fractal Pattern Formation
			11.4.1 Eden Collision Model
			11.4.2 Partial Differential Equation (PDE)-Based Model and the Koch Curve
			11.4.3 Mechanics-Based Model Using DLA
			11.4.4 SDE-Based Model at the Anterior Part of the Sagittal Suture
		11.5 Future Directions
			11.5.1 Experimental Verification of Theoretical Models
			11.5.2 Application of Fractal Aspect of Skull Suture
		References
	12 The Fractal Geometry of the Human Brain: An Evolutionary Perspective
		12.1 Introduction
		12.2 Principles of Brain Evolution
			12.2.1 Evolution of the Cerebral Cortex
			12.2.2 Mechanisms of Cortical Folding
			12.2.3 Scaling of the Primate Neocortex
		12.3 The Fractal Geometry of Convoluted Brains
			12.3.1 Principles of Scaling
			12.3.2 Fractal Scaling of the Neocortex
		12.4 Fractal Principles of Neural Wiring
			12.4.1 Neocortical Wiring
			12.4.2 Neural Network Communication
			12.4.3 Limits to Information Processing
		12.5 Concluding Remarks
		References
	13 Fractal Analysis in Clinical Neurosciences: An Overview
		13.1 Fractals in Neurosurgery and Neural Systems
		13.2 Clinical Neurology and Cerebrovascular System
		13.3 Neuroimaging
		13.4 Neurohistology, Neuropathology, and Neuro-oncology
		13.5 Fractal-Based Time-Series Analysis in Neurosciences
		13.6 Neuro-ophthalmology, Pain Medicine, Cognitive Sciences, Neuropsychology, and Psychiatry
		13.7 Limitations of Application of Fractal Analysis into Clinical Neurosciences
		13.8 The “Black Box”
		References
	14 Fractals, Pattern Recognition, Memetics, and AI: A Personal Journal in the Computational Neurosurgery
		14.1 From Fractals to Artificial Intelligence
		14.2 Fractals, Artificial Intelligence, and Cognitive Neurosciences
		14.3 Toward the Augmented Decision-Making of the Future
		14.4 Limitations and Future Perspectives
		References
	15 Clinical Sensitivity of Fractal Neurodynamics
		15.1 Introduction
		15.2 Physiological Processes
			15.2.1 Sleep
			15.2.2 Anesthesia
			15.2.3 Maturation and Healthy Aging
		15.3 Neurological Disorders
			15.3.1 Epilepsy
			15.3.2 Stroke
			15.3.3 Alzheimer's Disease
		15.4 Psychiatric Disorders
			15.4.1 Depression
			15.4.2 Schizophrenia
		15.5 Neuromodulation
			15.5.1 Photostimulation and Microwaves Stimulation
			15.5.2 Electrical and Magnetic Stimulation
			15.5.3 Transcranial Electrical Stimulation (tES)
			15.5.4 Repetitive Transcranial Magnetic Stimulation (rTMS)
		15.6 Other Modalities than EEG
			15.6.1 MEG
			15.6.2 fMRI
		15.7 Conclusions
		References
	16 Fractal Dimension Analysis in Neurological Disorders: AnOverview
		16.1 Introduction
		16.2 Geometric Fractal Analysis Applied to Neuroscience
		16.3 Relationship Between Fractal Dimension and Healthy Subjects
			16.3.1 Aging and Development
			16.3.2 Cognition
		16.4 Fractal Analysis and Neurological Disorders
			16.4.1 Alzheimer's Disease (AD)
			16.4.2 Frontotemporal Dementia
			16.4.3 Multiple Sclerosis
			16.4.4 Parkinson's Disease (PD)
			16.4.5 Epilepsy
			16.4.6 Autism Spectrum Disorder (ASD)
			16.4.7 Attention Deficit Hyperactivity Disorder (ADHD)
			16.4.8 Brain Tumors
			16.4.9 Vascular Disease
		16.5 Conclusion
		References
	17 Fractal Dimension Studies of the Brain Shape in Aging and Neurodegenerative Diseases
		17.1 Introduction
			17.1.1 Anatomical Shape Features of Interest
			17.1.2 Fractal Dimension Methods
		17.2 Fractal Dimension Studies of the Brain Shape
			17.2.1 Aging
			17.2.2 Alzheimer's Disease
			17.2.3 Amyotrophic Lateral Sclerosis
			17.2.4 Frontotemporal Dementia
			17.2.5 Epilepsy
			17.2.6 Multiple Sclerosis
			17.2.7 Multiple System Atrophy
			17.2.8 Spinocerebellar Atrophy
			17.2.9 Stroke
			17.2.10 Huntington's Disease
			17.2.11 Parkinson's Disease
		17.3 Discussion
		References
	18 Fractal Analysis in Neurodegenerative Diseases
		18.1 Alzheimer's Disease and Vascular Dementia
			18.1.1 Fractal Dimension—A Classifier for the AD Pathology
			18.1.2 Imaging and Fractal Analysis in AD
		18.2 Other Neurodegenerative Diseases
		18.3 Conclusion
		References
	19 Fractal Analysis of the Cerebrovascular System Pathophysiology
		19.1 Introduction
		19.2 Cerebral Autoregulation as a Feedback Loop
		19.3 Variability and Complexity
		19.4 Methodology of Variation and Fractal Analysis
		19.5 Hurst Coefficient HbdSWV
		19.6 Spectral Index ß
		19.7 Spectral Exponent α
		19.8 Fractal Analysis of Human CBF
		19.9 Decomplexification
		19.10 Frequency-Dependent CBF Variability
		19.11 Fractal Analysis of the Cerebral Microvasculature
		19.12 Conclusions
		References
	20 Fractals and Chaos in the Hemodynamics of IntracranialAneurysms
		20.1 Introduction
		20.2 Fractal Patterns in Time-Dependent Flows
		20.3 Basic Concepts Demonstrated on a Simplified 2D Case
		20.4 Measuring Chaotic Quantities from Residence Times
		20.5 Appearance of Chaotic Flow Inside Intracranial Aneurysms
		20.6 Concluding Remarks
		References
	21 Fractal-Based Analysis of Arteriovenous Malformations (AVMs)
		21.1 Introduction
		21.2 Neuroimaging of AVMs
		21.3 AVMs' Angioarchitecture Morphometrics
		21.4 Computational Fractal-Based Analyses of AVMs
			21.4.1 AVMs' Fractal Dimension
			21.4.2 Fractal Dimension of the Nidus and Its Relevance in Radiosurgery
		21.5 Limitations
		21.6 Computational Techniques for the Automatic Nidus Identification
		21.7 Conclusion
		References
	22 Fractals in Neuroimaging
		22.1 Introduction
		22.2 Fractals in Brain Magnetic Resonance Image Classification
		22.3 Other Applications of Fractal Analysis in Neuroimaging
		22.4 Conclusion and Future Perspective
		Appendix A: Appendix: Fractal Analysis Techniques
			Range-Scale-Based Hurst Exponent
			Detrended Fluctuation Analysis
			Generalized Hurst Exponent
		References
	23 Computational Fractal-Based Analysis of MR Susceptibility-Weighted Imaging (SWI) in Neuro-Oncology and Neurotraumatology
		23.1 Introduction
		23.2 Technical Aspects of SW Imaging
		23.3 SWI in Neuro-Oncology
			23.3.1 Morphometrics and Fractal-Based Analysis of SWI in Brain Tumors
		23.4 Future Perspective of SWI in Neurotraumatology
		23.5 Limitations
		23.6 Conclusion
		References
	24 Texture Estimation for Abnormal Tissue Segmentation in Brain MRI
		24.1 Introduction
		24.2 Background Review
			24.2.1 Fractal (PTPSA) Texture Feature Extraction
			24.2.2 Multifractal Brownian Motion (mBm) Process and Feature Extraction
		24.3 Methodology
			24.3.1 Preprocessing
			24.3.2 Feature Extraction, Fusion, Ranking, and Selection
			24.3.3 Classification with Random Forest
		24.4 Results and Discussion
		24.5 Conclusion and Future Work
		References
	25 Multifractal Analysis of Brain Tumor Interface in Glioblastoma
		25.1 Introduction
		25.2 Image Selection and Segmentation
		25.3 Multifractal Analysis
		25.4 Results and Discussion
			25.4.1 Multifractal Analysis of One-Dimensional Ordered Series
			25.4.2 Detrended Fluctuation Analysis of Two-Dimensional Tumor Interface Data
			25.4.3  Generalized Hurst Exponents and Singularity Spectra
		25.5 Conclusions
		References
	26 Fractal-Based Analysis of Histological Features of Brain Tumors
		26.1 Introduction
		26.2 Fractal Morphometry of Tissue Complexity
			26.2.1 Fractal Dimension Estimation
			26.2.2 Related Work
		26.3 Automated Histopathological Image Analysis
			26.3.1 Image Preparation
			26.3.2 Pre-processing and Focal Regions Segmentation
			26.3.3 Feature Extraction and Classification
			26.3.4 Qualitative Enhancement and Grading Results
		26.4 Characterizing Tissue via Fractal Properties
		26.5 Quasi-Fractal Texture Representation
		26.6 Multifractality Analysis
			26.6.1 Assessing Fractal Texture Heterogeneity
			26.6.2 Performance Under Tissue Distribution Variation
			26.6.3 Automated Classification Using Multiresolution Fractal Features
		26.7 Diagnostic Challenges and Future Perspectives
		26.8 Conclusion
		References
	27 Computational Fractal-Based Analysis of Brain Tumor Microvascular Networks
		27.1 Introduction
		27.2 Brain Tumors and Vascularization
			27.2.1 Immunohistochemistry
		27.3 Morphometrics of Microvascularity
			27.3.1 Euclidean-Based Parameters
			27.3.2 Image Analysis
		27.4 Fractal-Based Morphometric Analyses of Microvessels
			27.4.1 Microvascular Fractal Dimension (mvFD)
			27.4.2 Local Fractal Dimension and Local Box-Counting Dimension
		27.5 Fractal-Based Analysis of the Angio-Space in Brain Pathology
		27.6 Limitations
		27.7 Future Perspectives and Conclusion
		References
	28 Fractal-Based Morphometrics of Glioblastoma
		28.1 Introduction
		28.2 Fractal Analysis of Pretreatment MRI Provides Prognostic Information for Glioblastoma
			28.2.1 Thresholding Morphometric Values to Discriminate Survival
			28.2.2 Morphometrics as Continuous Prognostic Variables
		28.3 Reliability of Lacunarity and Fractal Dimension
			28.3.1 Intraclass Coefficients of Morphometrics on Repeated Segmentations
			28.3.2 Impact of MRI Resolution on Morphometrics
		28.4 Conclusion
		References
	29 Percolation Images: Fractal Geometry Features for Brain Tumor Classification
		29.1 Introduction
		29.2 Percolation
		29.3 Materials and Methods
			29.3.1 Brain Tumors Datasets
			29.3.2 Dataset Pre-processing
			29.3.3 Method Overview
			29.3.4 Classification
			29.3.5 Performance Evaluation
		29.4 Results and Discussion
		29.5 Conclusion
		References
	30 On Multiscaling of Parkinsonian Rest Tremor Signals and Their Classification
		30.1 Introduction
		30.2 Multifractal Detrended Fluctuation Analysis for Nonstationary Time Series
		30.3 Evidence of Multiscaling in Parkinsonian Rest Tremor Velocity Signals
		30.4 Concluding Remarks and Future Research Perspectives
		References
	31 Fractal Phototherapy in Maximizing Retina and Brain Plasticity
		31.1 Introduction in the Problems of Neuroprotection and Neurorehabilitation
			31.1.1 The Retina and Brain Neurodegenerative Diseases
			31.1.2 Key Problems of Axon Regeneration
		31.2 Retina Neuroplasticity and Aspects of Visual Rehabilitation
			31.2.1 Retinal Plasticity in Degenerative Disorders
			31.2.2 Achievements and Problems of Visual Rehabilitation
			31.2.3 Methods of Visual Rehabilitation Based on Neuroplasticity
			31.2.4 Biofeedback Training Systems for Visual Rehabilitation
			31.2.5 Other Methods of Visual Recovery
		31.3 Neuroplasticity and Problems of Neurorehabilitation
			31.3.1 Central Nervous System Plasticity
			31.3.2 Brain Plasticity During Aging and Neurodegenerative Diseases
		31.4 Natural and Artificial Fractals in Rehabilitation Therapy
			31.4.1 Plasticity-Based Stimulation Therapy Using Natural and Artificial Fractals
				31.4.1.1 Stimulation Therapy with Sensory Stimuli
				31.4.1.2 Environmental Enrichment
				31.4.1.3 Fractality of Nature and Man
				31.4.1.4 Fractality of Music and Fine Arts
			31.4.2 Perspectives of Fractal Stimulation Development
				31.4.2.1 Objective Restrictions on Neurorehabilitation and the Theory of “Fractality of Sensations”
				31.4.2.2 Observation of Artificially Generated Fractal Images
				31.4.2.3 Sensory Signals Control the Dynamics of Walking
				31.4.2.4 Experience in the Use of Fractal Phototherapy in Patients with Glaucoma
				31.4.2.5 Objective Studies of the Effect of Fractal Optical Stimulation on the Structure and Functional Activity of the Retina
		31.5 Conclusion Remarks
		References
	32 Fractal Similarity of Pain Brain Networks
		32.1 Brain Dynamics of Pain Perception
		32.2 Organization of Pain Brain Networks
		32.3 First-Level Nociceptive Processing
			32.3.1 How Does the Brain Encode Noxious Events?
			32.3.2 An Insular Pattern
		32.4 Second-Level Attentional Control and Cross-Modality Sensory Integration
		32.5 Third-Level Conscious Pain and Cognitive Control
		32.6 Pre-stimulus Brain State Influence How Pain Is Perceived
		32.7 Large-Scale Characterization of Brain Activity Can Predict Evoked Pain Intensity
		32.8 Disruption of Brain Fractal Patterns in Chronic Pain
			32.8.1 Fractal Dimension of Temporally Recurrent Brain Microstates
		32.9 Conclusion
		References
	33 Fractal Neurodynamics
		33.1 EEG-Derived Neurodynamics Assessment in the Human Brain
		33.2 Neurodynamics as Local Cortical Signature: A Spectral Estimate
			33.2.1 Spectral Features in Resting Wakefulness
			33.2.2 Spectral Features During Sleep
		33.3 Fractal Neurodynamics: Properties and Estimate Methods
			33.3.1 Evidence of Existing Scale-Free, Fractal Properties within EEG Signals
			33.3.2 Fractal Dimension Estimation Methods
				33.3.2.1 Higuchi Fractal Dimension (HFD)
		33.4 Neurodynamics as Local Cortical Signature: A Fractal Estimate
			33.4.1 Distinct Cortical Areas Exhibit Typical Complexity Relationship During Wakefulness
			33.4.2 Cortical Parcels Hold Typical Fractal Characteristics During Sleep
		33.5 Conclusions
		References
	34 Fractal Analysis of Electrophysiological Signals to Detect and Monitor Depression: What We Know So Far?
		34.1 Introduction: Neuroimaging Findings Related to Depression
		34.2 Physiological Complexity Dovetails Neuroimaging Findings About Depression
		34.3 Depression Detection Based on EEG
		34.4 Detecting Depressions' Hidden Cardiovascular Risks Based on ECG
		References
	35 Point of Care Testing (POCT) in Psychopathology Using Fractal Analysis and Hilbert Huang Transform of Electroencephalogram (EEG)
		35.1 Introduction
		35.2 Higuchi Fractal Dimension (HFD)
			35.2.1 EEG Fractal Analysis in Psychiatry
		35.3 Sample Entropy
			35.3.1 Sample Entropy in Psychiatry
		35.4 Empirical Mode Decomposition
			35.4.1 Pre-processing EEG
			35.4.2 Hilbert Huang Transform (HHT)
			35.4.3 HFD from HHT
			35.4.4 HFD-HHT as a Measure in Monitoring Relaxation
		35.5 EEG Complexity in a Healthy Control
		35.6 EEG Complexity in Major Depressive Disorder (MDD)
			35.6.1 Methods
			35.6.2 Results and Discussion
		35.7 Conclusion
		References
	36 Advances in Understanding Fractals in Affective and Anxiety Disorders
		36.1 Introduction
		36.2 Fractals and Affective Disorders
		36.3 Fractals and Anxiety Disorders
		36.4 Fractals in Affective and Anxiety Disorders Treatments
		36.5 Conclusions
		References
	37 EEG Complexity Analysis of Brain States, Tasks and ASD Risk
		37.1 Introduction
		37.2 Nonlinear Complexity of EEG
		37.3 Nonlinear Complexity EEG Measurement Methods
		37.4 Evolving Applications of Nonlinear Complexity EEG Analysis
		37.5 The Emergence of Nonlinear Complexity Biomarker Diagnostics
		37.6 Current Nonlinear Complexity Analysis Trends
		37.7 Practical Considerations of EEG Complexity Analysis Research
		37.8 Applied EEG Complexity Analysis of Resting State Data – Study One Results
		37.9 Applying EEG Complexity Analysis to Brain Task Data – Study Two Results
		37.10 EEG Complexity Measures of Cognitive States Show Heterogeneous ASD Risk with Sequence Clustering – Study Three
		37.11 Conclusion
		References
	38 Fractals in Neuropsychology and Cognitive Neuroscience
		38.1 The Fractal Dimension of Cognition
			38.1.1 Fractal Dimensions of Executive Functions
			38.1.2 Fractal Dimensions of Human Perception and Visuospatial Processing
			38.1.3 Fractal Dimensions of Language
		38.2 Fractal Dimensions of Brain Disorders
		38.3 Conclusions
		References
	39 Computational and Translational Fractal-Based Analysis in the Translational Neurosciences: An Overview
		39.1 How to Compute Fractals in Clinical Neurosciences
		39.2 Fractals in the Translational Neurosciences
		39.3 Fractals and Artificial Intelligence
		39.4 Conclusive Remarks: Toward a Unified Fractal Model of the Brain?
	40 ImageJ in Computational Fractal-Based Neuroscience: Pattern Extraction and Translational Research
		40.1 Introduction
		40.2 What Is ImageJ?
			40.2.1 Removing Barriers with Free, Open-Source Software
			40.2.2 Shaping Computational Fractal-Based Neuroscience
				40.2.2.1 Making Fractal Analysis Accessible and Customizable
		40.3 Where Does IJ Fit in Fractal-Based Neuroscience Today?
		40.4 Pattern Extraction
			40.4.1 Pattern Types
			40.4.2 Extraction Methods
				40.4.2.1 Built-in Functions
				40.4.2.2 Tracing Plug-Ins
				40.4.2.3 Thresholding
		40.5 Conclusion
		References
	41 Fractal Analysis in MATLAB: A Tutorial for Neuroscientists
		41.1 MATLAB Packages and Toolboxes for Fractal Analysis
		41.2 MATLAB Examples: Fractal Dimension Computation for 1D, 2D, and 3D Sets
			41.2.1 Fractal Dimension of an EEG
			41.2.2 Brain MRI Fractal Dimension of the Gray Matter with FracLab
			41.2.3 Fractal Dimension Computation of an MRI Volume of the Brain White Matter with a Boxcount-Based MATLAB Script
			41.2.4 Local Fractal Dimension Computation of a Cortical Surface with UJA-SHFD
		41.3 Other Software for Fractal Analysis
		41.4 Conclusions
		References
	42 Analyzing Eye Paths Using Fractals
		42.1 Introduction
		42.2 Eye Tracking
		42.3 Eye Movements
		42.4 Eye Path Representations
		42.5 String Edit
		42.6 Fractal Analysis
		42.7 Fractal Curves
		42.8 Fractal Dimension
		42.9 Uses and Applications
		42.10 Identifying Scanpath Outliers
		42.11 Comparing Scanpaths
		42.12 Spatial and Time Domain Analysis
		42.13 Recurrence Quantification Analysis and FD
		42.14 Scanpaths as Features in Neural Networks
		42.15 Discussion and Conclusion
		References
	43 Fractal Electronics for Stimulating and Sensing Neural Networks: Enhanced Electrical, Optical, and Cell Interaction Properties
		43.1 Introduction
		43.2 Fabrication of the Fractal Interconnects
		43.3 Functionality of the Fractal Interconnects
		43.4 The Biophilic Interface
		43.5 Conclusions
		References
	44 Fractal Resonance: Can Fractal Geometry Be Used to Optimize the Connectivity of Neurons to Artificial Implants?
		44.1 Introduction to Fractal Resonance
		44.2 The Geometric Origin of Neuron Fractality
		44.3 Fractal Dimension of the Branches
		44.4 Fractal Dimension of the Arbor
		44.5 Connectivity Analysis
		44.6 Conclusions
		References
	45 Fractal Fluency: Processing of Fractal Stimuli Across Sight, Sound, and Touch
		45.1 Introduction
		45.2 The Visual Complexity of Biophilic Fractals
		45.3 Fractal Fluency of Visual Fractals
		45.4 Fractal Aesthetics and Stress Reduction
		45.5 Beyond the Visual: Fractal Fluency in Touch and Sound?
		45.6 Fractal Applications
		45.7 Conclusions
		References
	46 Cognitive and Neural Representations of Fractals in Vision, Music, and Action
		46.1 Introduction
			46.1.1 Recursion in Human Cognition
		46.2 Neural Mechanisms of Recursive Hierarchical Embedding in Language
		46.3 Recursive Hierarchical Embedding in the Visual, Musical, and Motor Domains – Behavioral Research
		46.4 Recursive Hierarchical Embedding in the Visual, Musical, and Motor Domains – fMRI Research
			46.4.1 Acquisition Vs. Automatic Processing of RHE Structures
		46.5 Conclusion – Cognitive and Neural Bases of Fractal Cognition
		References
	47 Fractals in the Neurosciences: A Translational Geographical Approach
		47.1 Introduction
		47.2 Fractal-Based Computational Tools
			47.2.1 2D fractal Fragmentation Index
			47.2.2 3D Fractal Fragmentation Index
			47.2.3 Fractal Tentacularity Index
			47.2.4 Fractal Anisotropy Index
			47.2.5 Normalized Kolmogorov Complexity
		47.3 Application of FFI, FTI, FAI, and KC in the Analysis of Medical Images
		47.4 Patient Characteristics
		47.5 Conclusion
		References
	48 Fractal Geometry Meets Computational Intelligence: Future Perspectives
		48.1 Introduction
		48.2 Fractal Analysis and Brain Complexity
		48.3 Computational Intelligence Methods and the Challenge of Processing Nongeometric Input Spaces
		48.4 On the Interplay Between Fractal Analysis and CI Methods
		48.5 Future Perspectives and Concluding Remarks
		References
Index