Review Reports - Foundations and Clinical Applications of Fractal Dimension in Neuroscience: Concepts and Perspectives

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors,

Kindly refer to the attached 'Comments to the Author' PDF file for comments.

Thank you.

Comments for author File: Comments.pdf

Author Response

Reviewer 1 Comments to Author

Dear Authors,

Kindly refer to the attached 'Comments to the Author' PDF file for comments.

Thank you.

Authors’ response

We thank the reviewer for the constructive and helpful comments, which have allowed us to further improve the clarity, focus, and presentation of the manuscript. We address each point below.

Reviewer’s comments

[Point 1] Please mention the type of review in the title of the manuscript.

Authors’ response

We have revised the title to explicitly reflect the review nature of the manuscript. The new title now reads: “Foundations and clinical applications of fractal dimension in neuroscience: concepts and perspectives”. This formulation clearly indicates that the article provides an integrative and conceptual overview of the field, consistent with a narrative review.

Reviewer’s comments

[Point 2] The introduction nicely explains fractal dimension (FD) and its superiority over the traditional Euclidean measures. However, the introduction lacks a clear and well-defined problem statement. It remains unclear what specific issues or knowledge gaps the current study seeks to address from this review. Therefore, the introduction section can be improved by highlighting the current issues and explaining why this review is necessary.

Authors’ response

We agree with the reviewer and have expanded the Introduction (lines 62-71) to explicitly articulate the current challenges motivating this review. In particular, we added a dedicated paragraph highlighting methodological heterogeneity, fragmented biological interpretation across modalities and scales, and the limited translational integration of fractal dimension (FD) measures. This addition clarifies the specific knowledge gaps addressed by the review and explains why a comprehensive synthesis is timely and necessary.

Reviewer’s comments

[Point 3] The manuscript would benefit from a clearer articulation of the research questions guiding the review. At present, it is not explicitly stated what specific questions the authors seek to address through this review. Clearly defining the primary research question(s) is essential to framing the study’s objectives.

Authors’ response

We thank the reviewer for this important suggestion. We have now explicitly stated the primary research questions guiding the review at the end of the Introduction (lines 82-91). These questions clearly frame the objectives, scope, and conceptual focus of the manuscript, addressing methodological interpretation, biological consistency across conditions, and the role of FD as a unifying multiscale biomarker.

Reviewer’s comments

[Point 4] Please spell ‘MRI’ in full when the abbreviation is used for the first time in the manuscript (line 68).

Authors’ response

This has been corrected. “Magnetic resonance imaging (MRI)” is now spelled out in full at its first occurrence in the manuscript.

Reviewer’s comments

[Point 5] Please use ‘FD’ for fractal dimension (line 68). Do the same throughout the manuscript, as the abbreviation has been introduced before (line 35).

Authors’ response

We have revised the manuscript to ensure consistent use of the abbreviation FD for fractal dimension after its initial definition. All instances where the full term was redundantly used have been corrected for consistency.

Reviewer’s comments

[Point 6] Did the authors create the images in Figure 1? If not, please cite the sources.

Authors’ response

We have clarified the origin of Figure 1 directly in the figure caption. The caption now explicitly states that the figure was created by the authors, ensuring full transparency regarding figure authorship.

We believe that these revisions substantially improve the clarity, focus, and framing of the manuscript, and we are grateful to the reviewer for comments that have strengthened the final version of the work.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The article provides a comprehensive, well-structured, and consistent review of the theoretical foundations, methods, and applications of fractal dimension in neuroscience. It includes clearly organized and objective tables and figures, as well as the addition of appendices outlining best practices and relevant future directions for the field.

Minor adjustments are recommended, such as reviewing and explicitly defining acronyms (for example, MRI in line 68). A few localized grammatical and typographical corrections are also necessary.

Another important point concerns self-citation. Although the authors are well recognized in the field, approximately 30% of the 32 references include the article’s authors. These self-citations are generally justified by their relevance and fit well within the text; however, in the context of a review article, this aspect may be perceived as sensitive. Therefore, the inclusion of additional independent references, when available, is recommended. In addition, several of these self-citations are accompanied by narratives such as “studies from our group,” and revisions to this type of wording are also suggested.

Overall, this is a timely and relevant manuscript that is suitable for publication after minor revisions.

Author Response

Reviewer 2 Comments to Author

Authors’ response

We thank the reviewer for the positive and careful evaluation of our manuscript and for the constructive suggestions.

Reviewer’s comments

Minor adjustments are recommended, such as reviewing and explicitly defining acronyms (for example, MRI in line 68). A few localized grammatical and typographical corrections are also necessary.

Authors’ response

We have revised the manuscript to ensure that all acronyms (e.g., MRI, EEG, fMRI, DTI) are explicitly defined at first mention. In addition, we corrected minor grammatical and typographical issues throughout the text.

Reviewer’s comments

Overall, this is a timely and relevant manuscript that is suitable for publication after minor revisions.

Authors’ response

We appreciate the reviewer’s comment regarding self-citation in the context of a review article. While the cited works by the present authors were included due to their direct relevance to the topics discussed, we have revised the text to avoid explicit author-centered phrasing (e.g., “studies from our group”) and replaced it with neutral formulations. Where appropriate, we have also added independent references to further contextualize these findings within the broader literature and reduce the perception of author bias. The new references added are:

-Fernández-Borkel T, Borkel LF, Rojas-Hernández J, Hernández-Álvarez E, Quintana-Hernández DJ, Ponti LG, Henríquez-Hernández LA. The Causal Role of Consciousness in Psychedelic Therapy for Treatment-Resistant Depression: Hypothesis and Proposal. ACS Pharmacol Transl Sci. 2025;8(8):2839-2847. doi: 10.1021/acsptsci.5c00445.

-Grosu GF, Hopp AV, Moca VV, Bârzan H, Ciuparu A, Ercsey-Ravasz M, Winkel M, Linde H, Mureșan RC. The fractal brain: scale-invariance in structure and dynamics. Cereb Cortex. 2023;33(8):4574-4605. doi: 10.1093/cercor/bhac363. Erratum in: Cereb Cortex. 2023;33(19):10475. doi: 10.1093/cercor/bhad335.

Hedderich DM, Bäuml JG, Menegaux A, Avram M, Daamen M, Zimmer C, Bartmann P, Scheef L, Boecker H, Wolke D, Gaser C, Sorg C. An analysis of MRI derived cortical complexity in premature-born adults: Regional patterns, risk factors, and potential significance. Neuroimage. 2020;208:116438. doi: 10.1016/j.neuroimage.2019.116438.

Karperien AL, Jelinek HF. Morphology and Fractal-Based Classifications of Neurons and Microglia in Two and Three Dimensions. Adv Neurobiol. 2024;36:149-172. doi: 10.1007/978-3-031-47606-8_7.

Marzi C, Giannelli M, Tessa C, Mascalchi M, Diciotti S. Toward a more reliable characterization of fractal properties of the cerebral cortex of healthy subjects during the lifespan. Sci Rep. 2020;10(1):16957. doi: 10.1038/s41598-020-73961-w.

Porcaro C, Diciotti S, Madan CR, Marzi C. Editorial: Methods and application in fractal analysis of neuroimaging data. Front Hum Neurosci. 2024;18:1453284. doi: 10.3389/fnhum.2024.1453284.

Porcaro C, Di Renzo A, Tinelli E, Di Lorenzo G, Parisi V, Caramia F, Fiorelli M, Di Piero V, Pierelli F, Coppola G. Haemodynamic activity characterization of resting state networks by fractal analysis and thalamocortical morphofunctional integrity in chronic migraine. J Headache Pain. 2020;21(1):112. doi: 10.1186/s10194-020-01181-8.

Reishofer G, Studencnik F, Koschutnig K, Deutschmann H, Ahammer H, Wood G. Age is reflected in the Fractal Dimensionality of MRI Diffusion Based Tractography. Sci Rep. 2018;8(1):5431. doi: 10.1038/s41598-018-23769-6.

Squarcina L, Lucini Paioni S, Bellani M, Rossetti MG, Houenou J, Polosan M, Phillips ML, Wessa M, Brambilla P. White matter integrity in bipolar disorder investigated with diffusion tensor magnetic resonance imaging and fractal geometry. J Affect Disord. 2024;345:200-207. doi: 10.1016/j.jad.2023.10.095.

We believe these revisions improve the balance, objectivity, and readability of the manuscript while preserving its scientific rigor.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript provides a comprehensive and timely overview of fractal dimension (FD) methodologies and their applications in neuroscience, spanning structural neuroimaging, neurodevelopment, neurodegenerative disorders, consciousness research, and psychiatric conditions. The authors successfully integrate methodological foundations with computational advances and translational clinical relevance. Overall, the writing is clear, detailed, and technically sound, and the topic is of high interest given the growing emphasis on quantitative measures of brain complexity.

That said, the manuscript would benefit from substantial reorganization. In its current form, methodological, clinical, and theoretical components are interwoven in a fragmented manner, which reduces readability and makes it challenging for readers to follow a coherent conceptual progression. This structure also obscures the relationships between methodological choices, empirical findings, and clinical interpretation.

Major Comments

Manuscript Structure and Logical Flow

Current issue:
The manuscript presents methodological foundations, structural applications, neurodegenerative disease findings, consciousness research, and theoretical considerations in a somewhat scattered sequence. For example, Alzheimer’s disease is discussed prior to a dedicated section on methodological reliability, and applications related to consciousness appear before integration with computational neuroscience and network theory.

Recommendation:
The manuscript would greatly benefit from reorganization into clearly delineated sections that follow a logical and pedagogically intuitive progression:

Introduction – Fractal dimension in neuroscience and review objectives
Methodological Foundations – FD estimation methods, preprocessing, scale selection, and reliability
Structural Neuroimaging Applications – Brain development, aging, and neurodevelopmental disorders
Neurodegenerative Diseases – Multiple sclerosis, Alzheimer’s disease, Parkinson’s disease, ALS, Huntington’s disease (including Table 2)
Neurophysiology and Consciousness – EEG/MEG, FDI, altered states, and psychiatric disorders
Integration with Computational Neuroscience and Network Analysis
Limitations and Future Directions
Conclusion
Appendices – Best practices and open research questions

This structure would align methodological discussions with their downstream applications, facilitating comprehension and strengthening the link between analytical choices and biological or clinical interpretation.

Methodological Section

The discussion of box-counting, Higuchi’s FD, and Katz’s FD is thorough and well-written. However, clarity would be enhanced by including a concise comparative table summarizing the advantages, limitations, and typical neuroscience applications of each method. Such a table would allow readers to quickly identify which FD approach is most appropriate for structural, temporal, or clinical analyses.

Clinical Relevance and Translational Impact

The manuscript appropriately highlights FD as a potential biomarker across multiple neurological and psychiatric conditions. To further strengthen the translational impact, the authors are encouraged to:

Explicitly link FD alterations to diagnostic, prognostic, and treatment-monitoring utility in each disease domain
Consider including summary schematics or figures that visually relate FD changes to disease stage, clinical severity, or longitudinal progression

This would clarify the added value of FD beyond conventional morphometric or signal-based measures.

Integration with Theoretical Neuroscience

The sections addressing consciousness, dynamical attractors, and network analysis are insightful but currently appear somewhat isolated from the rest of the manuscript. These topics could be better integrated by:

Linking them more explicitly to earlier methodological sections, or
Creating a dedicated subsection within the “Applications” section that illustrates how FD bridges empirical neuroimaging and neurophysiology with theoretical models of brain dynamics and information integration

Such integration would emphasize the role of FD as a unifying framework connecting data-driven and theoretical neuroscience.

Missing Areas for Inclusion

Several emerging areas could further strengthen the scope and impact of this review:

Microglial Morphology and Fractal Analysis
Fractal dimension has been widely used to quantify microglial activation states via morphological analysis. Inclusion of this topic would enhance the biological and translational relevance of the manuscript. In particular, the authors may consider discussing:

FD as a quantitative marker of microglial ramification and process complexity
Its relevance to neuroinflammatory mechanisms in multiple sclerosis, Alzheimer’s disease, and stroke

Diffusion MRI–Based Fractal Dimension
Applications of FD to diffusion MRI remain underrepresented. Emerging work applying fractal analysis to tractography, diffusion tensor imaging (DTI), and advanced microstructural diffusion metrics could be briefly reviewed to broaden the structural neuroimaging perspective.
Multifractal and Complexity Metrics in fMRI
The functional neuroimaging section would benefit from brief coverage of multifractal approaches, including multifractal detrended fluctuation analysis (MF-DFA) and temporal fractal dynamics in resting-state fMRI. Inclusion of these methods would provide a more complete view of complexity-based analyses in functional neuroscience.
Figures and Tables

Figures and tables are informative but could be repositioned to improve clarity:

Table 1 (FD estimation methods): place in the Methods section
Table 2 (FD alterations in neurodegenerative diseases): retain in the clinical applications section
Computational workflow figure: move to the Methods section or include as a schematic bridging methods and applications

Minor Comments

Some sections are lengthy and could benefit from subdivision into shorter subsections to improve readability.
Consider adding a schematic illustrating biological drivers of fractal changes (e.g., dendritic arborization, neuronal loss, demyelination).
Clarify instances where FD is used interchangeably to describe shape complexity, signal dynamics, and network topology; these should be explicitly distinguished.
Figures could be strengthened by including representative FD log–log plots, box-counting schematics, and EEG Higuchi FD examples.

Author Response

Reviewer 3 Comments to Author

Authors’ response

We sincerely thank the reviewer for the thorough, constructive, and insightful evaluation of our manuscript. We have carefully addressed all major and minor comments and substantially revised the manuscript to improve its structure, clarity, biological grounding, and translational relevance. Below, we provide a detailed point-by-point response.

Reviewer’s Major Comments

Manuscript Structure and Logical Flow

Current issue:

The manuscript presents methodological foundations, structural applications, neurodegenerative disease findings, consciousness research, and theoretical considerations in a somewhat scattered sequence. For example, Alzheimer’s disease is discussed prior to a dedicated section on methodological reliability, and applications related to consciousness appear before integration with computational neuroscience and network theory.

Recommendation:
The manuscript would greatly benefit from reorganization into clearly delineated sections that follow a logical and pedagogically intuitive progression:

Introduction – Fractal dimension in neuroscience and review objectives
Methodological Foundations – FD estimation methods, preprocessing, scale selection, and reliability
Structural Neuroimaging Applications – Brain development, aging, and neurodevelopmental disorders
Neurodegenerative Diseases – Multiple sclerosis, Alzheimer’s disease, Parkinson’s disease, ALS, Huntington’s disease (including Table 2)
Neurophysiology and Consciousness – EEG/MEG, FDI, altered states, and psychiatric disorders
Integration with Computational Neuroscience and Network Analysis
Limitations and Future Directions
Conclusion
Appendices – Best practices and open research questions

Author’s response

We fully agree with the reviewer and have substantially reorganized the manuscript, following the proposed structure almost verbatim. The revised version now follows a clear and coherent progression:

Introduction - Fractal dimension in neuroscience and review objectives
Methodological Foundations - FD estimation methods, preprocessing, scale selection, and reliability
Structural Neuroimaging Applications - Brain development, aging, and neurodevelopmental disorders
Neurodegenerative Diseases - Multiple sclerosis, Alzheimer’s disease, Parkinson’s disease, ALS, and Huntington’s disease (including Table 2)
Neurophysiology and Consciousness - EEG/MEG, FDI, altered states, and psychiatric disorders
Integration with Computational Neuroscience and Network Analysis
Limitations and Future Directions
Conclusion
Appendices - Best practices and open research questions

This reorganization aligns methodological foundations with downstream applications, improves readability, and clarifies the relationships between analytical choices, empirical findings, and biological or clinical interpretation throughout the manuscript.

Reviewer’s comments

Methodological Section

Authors’ response

This point has been fully addressed. The revised manuscript includes Table 1, which provides a concise comparative overview of the major FD estimation methods (box-counting, Higuchi’s FD, and Katz’s FD), explicitly summarizing their computational principles, strengths, limitations, and typical neuroscience applications.

To further enhance clarity, we now explicitly reference this table in the main text, emphasizing its role in facilitating method selection depending on data modality and research goals.

Reviewer’s comments

Clinical Relevance and Translational Impact

Explicitly link FD alterations to diagnostic, prognostic, and treatment-monitoring utility in each disease domain
Consider including summary schematics or figures that visually relate FD changes to disease stage, clinical severity, or longitudinal progression

This would clarify the added value of FD beyond conventional morphometric or signal-based measures.

Authors’ response

We thank the reviewer for this important suggestion and have strengthened the translational focus accordingly. In the revised manuscript, we added explicit translational summary paragraphs at the end of each major application domain:

Structural neuroimaging/neurodevelopment and aging (Section 3): highlighting diagnostic and prognostic relevance, early risk stratification, and differentiation between normative aging and pathological processes (lines 413-428).
Neurodegenerative disorders (Section 4): emphasizing early detection, disease staging, longitudinal monitoring, and added value beyond conventional volumetric MRI metrics (lines 536-555).
Neurophysiology and consciousness (Section 5): clarifying diagnostic and prognostic utility in disorders of consciousness and monitoring of state transitions under anesthesia or neuromodulation (lines 638-668).

These additions explicitly articulate the diagnostic, prognostic, and monitoring relevance of fractal dimension and clarify its added value beyond conventional morphometric or signal-based measures.

Reviewer’s comments

Integration with Theoretical Neuroscience

Linking them more explicitly to earlier methodological sections, or
Creating a dedicated subsection within the “Applications” section that illustrates how FD bridges empirical neuroimaging and neurophysiology with theoretical models of brain dynamics and information integration

Such integration would emphasize the role of FD as a unifying framework connecting data-driven and theoretical neuroscience.

Authors’ response

We fully agree and have now explicitly linked the theoretical discussion to the methodological framework. In particular, we added an integrative paragraph at the beginning of the “Neurophysiology and Consciousness” section (section 5, lines 567-573), explicitly referencing the FD estimation methods introduced in Section 2 and clarifying how box-counting and Higuchi-based approaches provide a common mathematical language connecting empirical neuroimaging and neurophysiological data with models of attractor dynamics, network organization, and information integration in theoretical neuroscience.

In addition, the dedicated section 6 “Integration with Computational Neuroscience and Network Analysis” now explicitly frames fractal dimension as a unifying framework bridging data-driven measurements with theoretical models of brain dynamics and information integration.

Reviewer’s comments

Missing Areas for Inclusion

Several emerging areas could further strengthen the scope and impact of this review:

Microglial Morphology and Fractal Analysis
Fractal dimension has been widely used to quantify microglial activation states via morphological analysis. Inclusion of this topic would enhance the biological and translational relevance of the manuscript. In particular, the authors may consider discussing:

FD as a quantitative marker of microglial ramification and process complexity
Its relevance to neuroinflammatory mechanisms in multiple sclerosis, Alzheimer’s disease, and stroke

Authors’ response

We thank the reviewer for highlighting this important point. In the revised manuscript, we have included a dedicated discussion within the “Neurodegenerative Disorders” section (section 4, lines 536-549) addressing fractal dimension as a quantitative marker of microglial ramification and activation states, thereby strengthening the biological grounding and multiscale translational relevance of fractal analysis.

Reviewer’s comments

Diffusion MRI–Based Fractal Dimension
Applications of FD to diffusion MRI remain underrepresented. Emerging work applying fractal analysis to tractography, diffusion tensor imaging (DTI), and advanced microstructural diffusion metrics could be briefly reviewed to broaden the structural neuroimaging perspective.

Authors’ response

We have addressed this by expanding the “Structural Neuroimaging Applications” section (Section 3, lines 413-421) to briefly review emerging applications of fractal analysis in diffusion MRI, including tractography, diffusion tensor imaging (DTI), and advanced microstructural diffusion metrics. We highlight the potential of FD to complement conventional diffusion measures in characterizing white matter complexity in development and disease.

Reviewer’s comments

Multifractal and Complexity Metrics in fMRI
The functional neuroimaging section would benefit from brief coverage of multifractal approaches, including multifractal detrended fluctuation analysis (MF-DFA) and temporal fractal dynamics in resting-state fMRI. Inclusion of these methods would provide a more complete view of complexity-based analyses in functional neuroscience.

Authors’ response

We have incorporated a new discussion within the “Neurophysiology and Consciousness” section (Section 5, lines 638-668) addressing multifractal approaches, specifically Multifractal Detrended Fluctuation Analysis (MF-DFA) and temporal fractal dynamics in resting-state fMRI. This addition provides a more complete view of complexity-based analyses in functional neuroscience and complements the electrophysiological and theoretical frameworks discussed elsewhere in the manuscript.

Reviewer’s comments

Figures and Tables

Figures and tables are informative but could be repositioned to improve clarity:

Table 1 (FD estimation methods): place in the Methods section
Table 2 (FD alterations in neurodegenerative diseases): retain in the clinical applications section
Computational workflow figure: move to the Methods section or include as a schematic bridging methods and applications

Authors’ response

We thank the reviewer for this suggestion. In the revised manuscript, figures and tables have been positioned accordingly:

Table 1 is now located in the “Methodological Foundations” section (line 113).
Table 2 is retained within the “Neurodegenerative Disorders” section as part of the clinical applications (line 479).
The computational workflow figure is placed in the “Methodological Foundations” section (line 178), where it serves as a schematic linking methodological approaches with downstream applications.

Reviewer’s Minor Comments

Some sections are lengthy and could benefit from subdivision into shorter subsections to improve readability.

Authors’ response

We agree that readability is an important consideration. In the revised manuscript, we improved readability by clarifying thematic transitions and strengthening paragraph-level organization, particularly at the beginning and end of major sections. Given the integrative and narrative nature of this review, we opted to preserve a streamlined section structure rather than introducing additional formal subsections, which could fragment the conceptual flow.

Reviewer’s Minor Comments

Consider adding a schematic illustrating biological drivers of fractal changes (e.g., dendritic arborization, neuronal loss, demyelination).

Authors’ response

We appreciate this suggestion and agree that such schematics can be valuable. In the revised manuscript, we have explicitly articulated the biological drivers of fractal dimension changes in the text, including dendritic arborization, synaptic loss, demyelination, microglial activation, and network reorganization. To avoid redundancy and excessive expansion of figures, we addressed these mechanisms textually rather than adding new schematic figures.

Reviewer’s Minor Comments

Clarify instances where FD is used interchangeably to describe shape complexity, signal dynamics, and network topology; these should be explicitly distinguished.

Authors’ response

We thank the reviewer for highlighting this important conceptual point. In the revised manuscript, we now explicitly distinguish between:

geometric or morphometric complexity (structural MRI, microglial morphology).
temporal signal dynamics (EEG and fMRI time series).
network or attractor dimensionality (connectomics and theoretical models).

We emphasize that while the mathematical concept of fractal dimension is shared, its biological interpretation depends on the underlying data structure.

Reviewer’s Minor Comments

Figures could be strengthened by including representative FD log–log plots, box-counting schematics, and EEG Higuchi FD examples.

Authors’ response

We agree that such illustrative figures can be pedagogically useful. However, given the review nature of the manuscript and space considerations, we elected not to include additional methodological example figures. Instead, we ensured that the computational principles underlying box-counting, Higuchi’s FD, and related approaches are clearly described in the text, and we direct readers to the original methodological references where representative plots and worked examples are provided in detail.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The responses are thorough, specific, and well aligned with the reviewer’s intent. The authors not only complied with most requests but also provided clear rationales where they chose alternative approaches. This response would generally be considered satisfactory and publication-ready, pending any minor editorial refinement.