Skip to main content
SpringerLink
Log in

Fractal Geometry Meets Computational Intelligence: Future Perspectives

  • Chapter
  • First Online:
Book cover The Fractal Geometry of the Brain

Part of the book series: Springer Series in Computational Neuroscience ((NEUROSCI))

Abstract

Characterizations in terms of fractals are typically employed for systems with complex and multi-scale descriptions. A prominent example of such systems is provided by the human brain, which can be idealized as a complex dynamical system made of many interacting subunits. The human brain can be modeled in terms of observable variables together with their spatio-temporal-functional relations. Computational Intelligence is a research field bridging many nature-inspired computational methods, such as artificial neural networks, fuzzy systems, and evolutionary and swarm intelligence optimization techniques. Typical problems faced by means of Computational Intelligence methods include those of recognition, such as classification and prediction. Although historically conceived to operate in some vector space, such methods have been recently extended to the so-called non-geometric spaces, considering labeled graphs as the most general example of such patterns. Here we suggest that fractal analysis and Computational Intelligence methods can be exploited together in neuroscience research. Fractal characterizations can be used to (i) assess scale-invariant properties and to (ii) offer numeric, feature-based representations to complement the usually more complex pattern structures encountered in neurosciences. Computational Intelligence methods could be used to exploit such fractal characterizations, considering also the possibility to perform data-driven analysis of non-geometric input spaces, hence overcoming the intrinsic limits related to Euclidean geometry.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Amblard P-O, Michel OJJ. The relation between granger causality and directed information theory: a review. Entropy. 2012;15(1):113–43.

    Google Scholar 

  2. Bai L, Rossi L, Torsello A, Hancock ER. A quantum Jensen–Shannon graph kernel for unattributed graphs. Pattern Recog. 2015;48(2):344–55.

    Google Scholar 

  3. Banerji CRS, Severini S, Teschendorff AE. Network transfer entropy and metric space for causality inference. Phys Rev E. 2013;87(5):052814.

    Google Scholar 

  4. Bianchi FM, Livi L, Rizzi A, Sadeghian A. A granular computing approach to the design of optimized graph classification systems. Soft Comput. 2014;18(2):393–412.

    Google Scholar 

  5. Bicego M, Murino V, Figueiredo MAT. Similarity-based classification of sequences using hidden Markov models. Pattern Recog. 2004;37(12):2281–91.

    Google Scholar 

  6. Blythe DAJ, Haufe S, Müller K-R, Nikulin VV. The effect of linear mixing in the EEG on hurst exponent estimation. Neuroimage. 2014;99:377–87.

    Google Scholar 

  7. Bonissone PP. Soft computing: the convergence of emerging reasoning technologies. Soft Comp. 1997;1(1):6–18.

    Google Scholar 

  8. Brun L, Saggese A, Vento M. Dynamic scene understanding for behavior analysis based on string kernels. IEEE Trans Circ Syst Video Technol. 2014;24(10):1669–81.

    Google Scholar 

  9. Bullmore ET, Sporns O. The economy of brain network organization. Nat Rev Neurosci. 2012;13(5):336–49.

    Google Scholar 

  10. Bunke H, Riesen K. Recent advances in graph-based pattern recognition with applications in document analysis. Pattern Recog. 2011;44(5):1057–67.

    Google Scholar 

  11. Bunke H, Riesen K. Towards the unification of structural and statistical pattern recognition. Pattern Recog Lett. 2012;33(7):811–25.

    Google Scholar 

  12. Castillo O, Melin P, Pedrycz W. Design of interval type-2 fuzzy models through optimal granularity allocation. Appl Soft Comput. 2011;11(8):5590–601.

    Google Scholar 

  13. Ceroni A, Costa F, Frasconi P. Classification of small molecules by two-and three-dimensional decomposition kernels. Bioinformatics. 2007;23(16):2038–45.

    Google Scholar 

  14. Chen Y, Garcia EK, Gupta MR, Rahimi A, Cazzanti L. Similarity-based classification: concepts and algorithms. J Mach Learn Res. 2009;10:747–76.

    Google Scholar 

  15. Costa L d F, Rodrigues FA, Travieso G, Villas Boas PR. Characterization of complex networks: a survey of measurements. Adv Phys. 2007;56(1):167–242.

    Google Scholar 

  16. Crutchfield JP. Between order and chaos. Nat Phys. 2012;8(1):17–24.

    Google Scholar 

  17. Crutchfield JP, Feldman DP. Regularities unseen, randomness observed: levels of entropy convergence. Chaos: An Interdisc J Nonlinear Sci. 2003;13(1):25–54.

    Google Scholar 

  18. Daqing L, Kosmidis K, Bunde A, Havlin S. Dimension of spatially embedded networks. Nat Phys. 2011;7(6):481–4.

    Google Scholar 

  19. de Lange SC, de Reus M A, van den Heuvel MP. The Laplacian spectrum of neural networks. Front Comput Neurosci. 2013;7.

    Google Scholar 

  20. Dehmer M, Varmuza K, Borgert S, Emmert-Streib F. On entropy-based molecular descriptors: statistical analysis of real and synthetic chemical structures. J Chem Inf Model. 2009;49(7):1655–63.

    Google Scholar 

  21. Di Ieva A, Schmitz EM, Cusimano MD. Analysis of intracranial pressure: past, present, and future. Neuroscientist. 2013;19(6):592–603.

    Google Scholar 

  22. Di Ieva A, Grizzi F, Jelinek H, Pellionisz AJ, Losa GA. Fractals in the neurosciences, part I general principles and basic neurosciences. Neuroscientist. 2014;20(4):403–17.

    Google Scholar 

  23. Di Ieva A, Esteban FJ, Grizzi F, Klonowski W, Martín-Landrove M. Fractals in the neurosciences, part II clinical applications and future perspectives. Neuroscientist. 2015;21(1):30–43.

    Google Scholar 

  24. Duardo-Sánchez A, Munteanu CR, Riera-Fernández P, López-Díaz A, Pazos A, González-Díaz H. Modeling complex metabolic reactions, ecological systems, and financial and legal networks with MIANN models based on Markov-Wiener node descriptors. J Chem Inf Model. 2013;54(1):16–29.

    Google Scholar 

  25. Engelbrecht AP. Computational intelligence: an introduction. Hoboken: Wiley; 2007.

    Book  Google Scholar 

  26. Escolano F, Hancock ER, Lozano MA. Heat diffusion: thermodynamic depth complexity of networks. Phys Rev E. 2012;85(3):036206.

    Google Scholar 

  27. Fallani FDV, Richiardi J, Chavez M, Achard S. Graph analysis of functional brain networks: practical issues in translational neuroscience. Philos Trans R Soc B: Biol Sci. 2014;369(1653):20130521.

    Google Scholar 

  28. Fernández E, Jelinek HF. Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods. 2001;24(4):309–21.

    Google Scholar 

  29. Fernández E, Bolea JA, Ortega G, Louis E. Are neurons multifractals? J Neurosci Methods. 1999;89(2):151–7.

    Google Scholar 

  30. Fischer A, Suen CY, Frinken V, Riesen K, Bunke H. Approximation of graph edit distance based on Hausdorff matching. Pattern Recog. 2015;48(2):331–43.

    Google Scholar 

  31. Friedrich R, Peinke J, Sahimi M, Tabar MRR. Approaching complexity by stochastic methods: from biological systems to turbulence. Phys Rep. 2011;506(5):87–162.

    Google Scholar 

  32. Gallos LK, Makse HA, Sigman M. A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. Proc Natl Acad Sci. 2012;109(8):2825–30.

    Google Scholar 

  33. Gallos LK, Potiguar FQ, Andrade Jr JS, Makse HA. IMDB network revisited: unveiling fractal and modular properties from a typical small-world network. PLoS One. 2013;8(6):e66443.

    Google Scholar 

  34. Giuliani A, Krishnan A, Zbilut JP, Tomita M. Proteins as networks: usefulness of graph theory in protein science. Curr Protein Pept Sci. 2008;9(1):28–38.

    Article  PubMed  Google Scholar 

  35. Godwin D, Barry RL, Marois R. Breakdown of the brain’s functional network modularity with awareness. Proc Natl Acad Sci. 2015;201414466.

    Google Scholar 

  36. Hammer B, Hasenfuss A. Topographic mapping of large dissimilarity data sets. Neural Comput. 2010;22(9):2229–84.

    Google Scholar 

  37. Hancock ER, Wilson RC. Pattern analysis with graphs: parallel work at Bern and York. Pattern Recog Lett. 2012;33(7):833–41.

    Google Scholar 

  38. Haykin S. Neural networks: a comprehensive foundation. Upper Saddle River: Prentice Hall PTR; 2007.

    Google Scholar 

  39. Izakian H, Pedrycz W, Jamal I. Fuzzy clustering of time series data using dynamic time warping distance. Eng Appl Artif Intel. 2015;39:235–44.

    Google Scholar 

  40. Jang JSR. ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern. 1993;23:665–85.

    Google Scholar 

  41. Jizba P, Kleinert H, Shefaat M. R ́enyi’s information transfer between financial time series. Phys A: Stat Mech Appl. 2012;391(10):2971–89.

    Google Scholar 

  42. Karkare S, Saha G, Bhattacharya J. Investigating long-range correlation properties in EEG during complex cognitive tasks. Chaos Solitons Fractals. 2009;42(4):2067–73.

    Google Scholar 

  43. Kwapien J, Drożdż S. Physical approach to complex systems. Phys Rep. 2012;515(3):115–226.

    Google Scholar 

  44. Li B-G, Yu Z-G, Zhou Y. Fractal and multifractal properties of a family of fractal networks. J Stat Mech: Theory Exp. 2014;2014(2):P02020.

    Google Scholar 

  45. Liang Q, Karnik NN, Mendel JM. Connection admission control in ATM networks using survey-based type-2 fuzzy logic systems. IEEE Trans Syst Man Cybern. 2000;30:329–39.

    Google Scholar 

  46. Liu J-L, Yu Z-G, Anh V. Determination of multifractal dimensions of complex networks by means of the sandbox algorithm. Chaos: An Interdiscip J Nonlinear Sci. 2015;25(2):023103.

    Google Scholar 

  47. Livi L, Rizzi A. Graph ambiguity. Fuzzy Set Syst. 2013;221:24–47.

    Google Scholar 

  48. Livi L, Rizzi A. The graph matching problem. Pattern Anal Appl. 2013;16(3):253–83.

    Google Scholar 

  49. Livi L, Rizzi A, Sadeghian A. Optimized dissimilarity space embedding for labeled graphs. Inform Sci. 2014;266:47–64.

    Google Scholar 

  50. Livi L, Tahayori H, Sadeghian A, Rizzi A. Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions. Appl Soft Comput. 2014;17:79–89.

    Google Scholar 

  51. Livi L, Maiorino E, Pinna A, Sadeghian A, Rizzi A, Giuliani A. Analysis of heat kernel highlights the strongly modular and heat-preserving structure of proteins. Phys A: Stat Mech Appl. 2016;441:199–214.

    Google Scholar 

  52. Livi L, Rizzi A, Sadeghian A. Granular modeling and computing approaches for intelligent analysis of non-geometric data. Appl Soft Comput. 2015;27:567–74.

    Google Scholar 

  53. Livi L, Sadeghian A, Pedrycz W. Entropic one-class classifiers. IEEE Trans Neural Netw Learn Syst. 2015.

    Google Scholar 

  54. Livi L, Giuliani A, Sadeghian A. Characterization of graphs for protein structure modeling and recognition of solubility. Curr Bioinforma. 2016;11(1):106–14.

    Google Scholar 

  55. Lizier JT, Prokopenko M, Zomaya AY. Local measures of information storage in complex distributed computation. Inform Sci. 2012;208:39–54.

    Google Scholar 

  56. Maiorino E, Livi L, Giuliani A, Sadeghian A, Rizzi A. Multifractal characterization of protein contact networks. Phys A: Stat Mech Appl. 2015;428:302–13.

    Google Scholar 

  57. Marwan N, Carmen Romano M, Thiel M, Kurths J. Recurrence plots for the analysis of complex systems. Phys Rep. 2007;438(5):237–329.

    Google Scholar 

  58. Melin P, Castillo O. A review on the applications of type-2 fuzzy logic in classification and pattern recognition. Expert Syst Appl. 2013;40(13):5413–23.

    Google Scholar 

  59. Mendel JM. General type-2 fuzzy logic systems made simple: a tutorial. IEEE Trans Fuzzy Syst. 2014;22(5):1162–82.

    Google Scholar 

  60. Nauck D, Klawonn F, Kruse R. Foundations of Neuro-Fuzzy systems. New York: Wiley; 1997.

    Google Scholar 

  61. Oh S-K, Kim W-D, Pedrycz W, Seo K. Fuzzy radial basis function neural networks with information granulation and its parallel genetic optimization. Fuzzy Set Syst. 2014;237:96–117.

    Google Scholar 

  62. Pagola M, Lopez-Molina C, Fernandez J, Barrenechea E, Bustince H. Interval type-2 fuzzy sets constructed from several membership functions: application to the fuzzy thresholding algorithm. IEEE Trans Fuzzy Syst. 2013;21(2):230–44.

    Google Scholar 

  63. Pantic I, Dacic S, Brkic P, Lavrnja I, Jovanovic T, Pantic S, Pekovic S. Discriminatory ability of fractal and grey level co-occurrence matrix methods in structural analysis of hippocampus layers. J Theor Biol. 2015;370:151–6.

    Google Scholar 

  64. Papo D, Zanin M, Pineda-Pardo JA, Boccaletti S, Buldύ JM. Functional brain networks: great expectations, hard times and the big leap forward. Philos Trans R Soc Lond B: Biol Sci. 2014;369(1653):20130525.

    Google Scholar 

  65. Park H-J, Friston K. Structural and functional brain networks: from connections to cognition. Science. 2013;342(6158):1238411.

    Google Scholar 

  66. Pȩkalska E, Duin RPW. The dissimilarity representation for pattern recognition: foundations and applications. Singapore: World Scientific; 2005.

    Google Scholar 

  67. Prichep LS, Jacquin A, Filipenko J, Dastidar SG, Zabele S, Vodencarevic A, Rothman NS. Classification of traumatic brain injury severity using informed data reduction in a series of binary classifier algorithms. IEEE Trans Neural Syst Rehabil Eng. 2012;20(6):806–22.

    Google Scholar 

  68. Prokopenko M, Lizier JT. Transfer entropy and transient limits of computation. Sci Rep. 2014;4:5394.

    Google Scholar 

  69. Prokopenko M, Lizier JT, Price DC. On thermodynamic interpretation of transfer entropy. Entropy. 2013;15(2):524–43.

    Google Scholar 

  70. Rasouli G, Rasouli M, Lenz FA, Verhagen L, Borrett DS, Kwan HC. Fractal characteristics of human Parkinsonian neuronal spike trains. Neuroscience. 2006;139(3):1153–8.

    Google Scholar 

  71. Richiardi J, Achard S, Bunke H, Van De Ville D. Machine learning with brain graphs: predictive modeling approaches for functional imaging in systems neuroscience. IEEE Signal Proc Mag. 2013;30(3):58–70.

    Google Scholar 

  72. Riera-Fernandez P, Munteanu CR, Escobar M, Prado-Prado F, Martín-Romalde R, Pereira D, Villalba K, Duardo-Sanchez A, González-Díaz H. New Markov–Shannon entropy models to assess connectivity quality in complex networks: from molecular to cellular pathway, parasite–host, neural, industry, and legal–social networks. J Theor Biol. 2012;293:174–88.

    Google Scholar 

  73. Riesen K, Bunke H. Improving bipartite graph edit distance approximation using various search strategies. Pattern Recog. 2015;48(4):1349–63.

    Google Scholar 

  74. Rossi L, Torsello A, Hancock ER. Unfolding kernel embeddings of graphs: enhancing class separation through manifold learning. Pattern Recog. 2015;48(11):3357–70.

    Google Scholar 

  75. Rossi L, Torsello A, Hancock ER. Measuring graph similarity through continuous-time quantum walks and the quantum Jensen-Shannon divergence. Phys Rev E. 2015;91(2):022815.

    Google Scholar 

  76. Rozenfeld HD, Song C, Makse HA. Small-world to fractal transition in complex networks: a renormalization group approach. Phys Rev Lett. 2010;104:025701.

    Google Scholar 

  77. Rupp M, Schneider G. Graph kernels for molecular similarity. Mol Inform. 2010;29(4):266–73.

    Google Scholar 

  78. Russo R, Herrmann HJ, de Arcangelis L. Brain modularity controls the critical behavior of spontaneous activity. Sci Rep. 2014;4.

    Google Scholar 

  79. Sainath TN, Kingsbury B, Saon G, Soltau H, Mohamed A-r, Dahl G, Ramabhadran B. Deep convolutional neural networks for large-scale speech tasks. Neural Netw. 2014;64:39–48.

    Google Scholar 

  80. Schölkopf B, Smola AJ. Learning with kernels: support vector machines, regularization, optimization, and beyond. Cambridge, MA: MIT Press; 2002.

    Google Scholar 

  81. Seely AJE, Newman KD, Herry CL. Fractal structure and entropy production within the central nervous system. Entropy. 2014;16(8):4497–520.

    Google Scholar 

  82. Serletis D, Bardakjian BL, Valiante TA, Carlen PL. Complexity and multifractality of neuronal noise in mouse and human hippocampal epileptiform dynamics. J Neural Eng. 2012;9(5):056008.

    Google Scholar 

  83. Serratosa F, Cortés X, Solé-Ribalta A. Component retrieval based on a database of graphs for hand-written electronic-scheme digitalisation. Exp Syst Appl. 2013;40(7):2493–502.

    Google Scholar 

  84. Song C, Havlin S, Makse HA. Origins of fractality in the growth of complex networks. Nat Phys. 2006;2(4):275–81.

    Google Scholar 

  85. Stoop R, Saase V, Wagner C, Stoop B, Stoop R. Beyond scale-free small-world networks: cortical columns for quick brains. Phys Rev Lett. 2013;110(10):108105.

    Google Scholar 

  86. Tahayori H, Livi L, Sadeghian A, Rizzi A. Interval type-2 fuzzy sets reconstruction based on fuzzy information-theoretic kernels. IEEE Trans Fuzzy Syst. 2014.

    Google Scholar 

  87. Theodoridis S, Koutroumbas K. Pattern recognition. 4th ed. Waltham: Elsevier/Academic; 2008.

    Google Scholar 

  88. Tomida N, Tanaka T, Ono S, Yamagishi M, Higashi H. Active data selection for motor imagery EEG classification. IEEE Trans Biomed Eng. 2015;62(2):458–67.

    Google Scholar 

  89. van den Heuvel MP, Fornito A. Brain networks in schizophrenia. Neuropsychol Rev. 2014;24(1):32–48.

    Google Scholar 

  90. Wagner C, Hagras H. Toward general type-2 fuzzy logic systems based on zSlices. IEEE Trans Fuzzy Syst. 2010;18(4):637–60.

    Google Scholar 

  91. Warren Liao T. Clustering of time series data–a survey. Pattern Recog. 2005;38(11):1857–74.

    Google Scholar 

  92. West BJ. Fractal physiology, vol. 2. Oxford: Oxford University Press; 1994.

    Google Scholar 

  93. West BJ. Fractal physiology and chaos in medicine, vol. 16. Singapore: World Scientific; 2012.

    Google Scholar 

  94. Xiao B, Hancock ER, Wilson RC. Geometric characterization and clustering of graphs using heat kernel embeddings. Image Vision Comput. 2010;28(6):1003–21.

    Google Scholar 

  95. Zappasodi F, Olejarczyk E, Marzetti L, Assenza G, Pizzella V, Tecchio F. Fractal dimension of EEG activity senses neuronal impairment in acute stroke. PLoS One. 2014;9(6):e100199.

    Google Scholar 

  96. Zhang J, Tuo X, Yuan Z, Liao W, Chen H. Analysis of fMRI data using an integrated principal component analysis and supervised affinity propagation clustering approach. IEEE Trans Biomed Eng. 2011;58(11):3184–96.

    Google Scholar 

  97. Zhang Y, Zhou W, Yuan S. Multifractal analysis and relevance vector machine-based automatic seizure detection in intracranial EEG. Int J Neural Syst. 2015;0(0):1550020.

    Google Scholar 

  98. Zhou S-M, Garibaldi JM, John RI, Chiclana F. On constructing parsimonious type-2 fuzzy logic systems via influential rule selection. IEEE Trans Fuzzy Syst. 2009;17(3):654–67.

    Google Scholar 

  99. Zhu X, Gisbrecht A, Schleif F-M, Hammer B. Approximation techniques for clustering dissimilarity data. Neurocomputing. 2012;90:72–84.

    Google Scholar 

  100. Zhu X, Schleif F-M, Hammer B. Adaptive conformal semi-supervised vector quantization for dissimilarity data. Pattern Recog Lett. 2014;49:138–45.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Ryerson University, 350 Victoria Street, Toronto, ON, M5B 2K3, Canada

    Lorenzo Livi PhD & Alireza Sadeghian PhD

  2. Neurosurgery Unit, Faculty of Medicine and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Antonio Di Ieva

  3. Garvan Institute of Medical Research, Sydney, NSW, Australia

    Antonio Di Ieva

  4. Medical University of Vienna, Vienna, Austria

    Antonio Di Ieva

  5. University of Toronto, Toronto, ON, Canada

    Antonio Di Ieva

Authors
  1. Lorenzo Livi PhD
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alireza Sadeghian PhD
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Di Ieva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lorenzo Livi PhD .

Editor information

Editors and Affiliations

  1. Department of Surgery Division of Neurosurgery, University of Toronto St. Michael's Hospital, Toronto, Ontario, Canada

    Antonio Di Ieva

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media New York

About this chapter

Cite this chapter

Livi, L., Sadeghian, A., Di Ieva, A. (2016). Fractal Geometry Meets Computational Intelligence: Future Perspectives. In: Di Ieva, A. (eds) The Fractal Geometry of the Brain. Springer Series in Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3995-4_36

Download citation

Publish with us

Policies and ethics

Search

Navigation