Abstract
We introduce the concept of spatial and temporal complexity with emphasis on how its fractal characterization for 1D, 2D or 3D hemodynamic brain signals can be carried out. Using high-resolution experimental data sets acquired in animal and human brain by noninvasive methods – such as laser Doppler flowmetry, laser speckle, near infrared, or functional magnetic resonance imaging – the spatiotemporal complexity of cerebral hemodynamics is demonstrated. It is characterized by spontaneous, seemingly random (that is disorderly) fluctuation of the hemodynamic signals. Fractal analysis, however, proved that these fluctuations are correlated according to the special order of self-similarity. The degree of correlation can be assessed quantitatively either in the temporal or the frequency domain respectively by the Hurst exponent (H) and the spectral index (β). The values of H for parenchymal regions of white and gray matter of the rat brain cortex are distinctly different. In human studies, the values of β were instrumental in identifying age-related stiffening of cerebral vasculature and their potential vulnerability in watershed areas of the brain cortex such as in borderline regions between frontal and temporal lobes. Biological complexity seems to be present within a restricted range of H or β values which may have medical significance because outlying values can indicate a state of pathology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Norris, V., Cabin, A. and Zemirline, A. (2005) Hypercomplexity. Acta Biotheor 53, 313–330.
Krone, G., Mallot, H., Palm, G. and Schuz, A. (1986) Spatiotemporal receptive fields: A dynamical model derived from cortical architectonics. Proc R Soc Lond B Biol Sci 226, 421–444.
Grassberger, P. and Procaccia, I. (1983) Measuring the strangeness of strange attractors. Physica D 9, 189–208.
Kaplan, D. and Glass, L. (1997) Understanding Nonlinear Dynamics. Springer-Verlag, New York.
Falconer, K. (1990) Fractal geometry: Mathematical Foundations and Applications. Wiley, Chichester, New York.
Eke, A., Herman, P., Kocsis, L. and Kozak, L. R. (2002) Fractal characterization of complexity in temporal physiological signals. Physiol Meas 23, R1–38.
Beran, J. (1994) Statistics for Long Memory Processes. Chapman and Hall, New York.
Bassingthwaighte, J., Liebovitch, L. and West, B. (1994) Fractal Physiology. Oxford University Press, New York, Oxford.
Mandelbrot, B. (1983) The Fractal Geometry of Nature. WH Freeman, San Francisco.
Avnir, D., Biham, O., Lidar, D. and Malcai, O. (1998) Is the geometry of nature fractal? Science 279, 39–40.
Mandelbrot, B. (1985) Self-affine Fractals and Fractal Dimension. Phys Scripta 32,257–260.
Dunn, A. K., Bolay, H., Moskowitz, M. A. and Boas, D. A. (2001) Dynamic Imaging Of Cerebral Blood Flow Using Laser Speckle. J Cereb Blood Flow Metab 21, 195–201.
Eke, A. (2003) Fractal, chaos, physiological complexity. In: Studia Physiologica (Series Ed.: A. Juhasz-Nagy) 13,1–157, Scientia Kiado, Budapest.
Eke, A., Herman, P., Bassingthwaighte, J. B., Raymond, G. M., Percival, D. B., Cannon, M., Balla, I. and Ikrenyi, C. (2000) Physiological time series: Distinguishing fractal noises from motions. Pflugers Arch 439, 403–415.
Eke, A., Herman, P. and Hajnal, M. (2006) Fractal and noisy CBV dynamics in humans: Influence of age and gender. J Cereb Blood Flow Metab 26, 891–898.
Herman, P. and Eke, A. (2006) Nonlinear analysis of blood cell flux fluctuations in the rat brain cortex during stepwise hypotension challenge. J Cereb Blood Flow Metab 26,1189–1197.
Herman, P., Kida, I., Sanganahalli, B., Hyder, F. and Eke, A. (2005) Fractal correlation structure in fMRI data of rat brain. J Cereb Blood Flow Metab 25, S379.
Herman, P., Kocsis, L., Portöro, I. and Eke, A. (2007) Heterogenous response in CBF during autoregulation: A non-invasive laser speckle study in the rat brain cortex. Brain‘07. The 23rd International Symposium on Cerebral Blood Flow, Metabolism and Function, Osaka, Japan.
Davies, R. B. and Harte, D. S. (1987) Test for Hurst effect. Biometrika 74, 95–101.
Bassingthwaighte, J. B. and Raymond, G. M. (1995) Evaluation of the dispersional analysis method for fractal time series. Ann Biomed Eng 23, 491–505.
Cannon, M., Percival, D. B., Caccia, D., Raymond, G. M. and Bassingthwaighte, J. B. (1997) Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series. Physica A 241, 606–626.
Eke, A., Herman, P., Bassingthwaighte, J. B., Raymond, G. M., Balla, I. and Ikrenyi, C. (1997) Temporal fluctuations in regional red blood cell flux in the rat brain cortex is a fractal process. Adv Exp Med Biol 428,703–709.
Turcotte, D. L., Malamud, B. D., Guzzetti, F. and Reichenbach, P. (2002) Self-organization, the cascade model, and natural hazards. Proc Natl Acad Sci USA 99, 2530–2537.
Waliszewski, P. (2005) A Principle Of Fractal-Stochastic Dualism And Gompertzian Dynamics Of Growth And Self-Organization. Biosystems 82, 61–73.
Eke, A. and Herman, P. (1999) Fractal analysis of spontaneous fluctuations in human cerebral hemoglobin content and itsoxygenation level recorded by NIRS. Adv Exp Med Biol 471, 49–55.
Chance, B., Anday, E., Nioka, S., Zhou, S., Hong, L., Worden, K., Li, C., Murray, T., Ovetsky, Y., Pidikiti, D. and Thomas, R. (1998) A novel method for fast imaging of brain function, non-invasively, with light. Opt. Express 2, 411–423.
Intaglietta, M. (1990) Vasomotion and flowmotion – physiological mechanisms and clinical evidence. Vasc Med Rev 1, 101–112.
Schroeter, M. L., Schmiedel, O. and von Cramon, D. Y. (2004) Spontaneous low-frequency oscillations decline in the aging brain. J Cereb Blood Flow Metab 24, 1183–1191.
Nilsson, H. and Aalkjaer, C. (2003) Vasomotion: Mechanisms and physiological importance. Mol Interv 3, 79–89, 51.
Miklossy, J. (2003) Cerebral hypoperfusion induces cortical watershed microinfarcts which may further aggravate cognitive decline in Alzheimer’s disease. Neurol Res 25, 605–610.
Jorgensen, L. and Torvik, A. (1969) Ischemic cerebrovascular diseases in an autopsy series. 2. Prevalence, location, pathogenesis, and clinical course of cerebral infarcts. J Neurol Sci 9, 285–320.
Yong, S. W., Bang, O. Y., Lee, P. H. and Li, W. Y. (2006) Internal and cortical border-zone infarction: Clinical and diffusion-weighted imaging features. Stroke 37, 841–846.
Eke, A. (1993) Multiparametric imaging of microregional circulation over the brain cortex by video reflectometry. Adv Exp Med Biol 333, 183–191.
Eke, A., Hutiray, G. and Kovach, A. G. (1979) Induced hemodilution detected by reflectometry for measuring microregional blood flow and blood volume in cat brain cortex. Am J Physiol 236, H759–768.
Herman, P., Kocsis, L. and Eke, A. (2001) Fractal branching pattern in the pial vasculature in the cat. J Cereb Blood Flow Metab 21, 741–753.
Hyder, F., Kida, I., Behar, K. L., Kennan, R. P., Maciejewski, P. K. and Rothman, D. L. (2001) Quantitative functional imaging of the brain: Towards mapping neuronal activity by bold fMRI. NMR Biomed 14, 413–431.
Acknowledgments
The authors gratefully acknowledge the contribution of Ms. Andrea Mile to the human LED Imager study and Drs. Fahmeed Hyder, Ikuhiro Kida and Basavaraju G. Sanganahalli to the MRI study. This work was supported by the Hungarian Research Foundation (OTKA) by its grants T016953, T34122 and the High Performance Computing of the Hungarian National Information Infrastructure Development Program.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Humana Press, a part of Springer Science+Business Media, LLC
About this protocol
Cite this protocol
Herman, P., Kocsis, L., Eke, A. (2009). Fractal Characterization of Complexity in Dynamic Signals: Application to Cerebral Hemodynamics. In: Hyder, F. (eds) Dynamic Brain Imaging. METHODS IN MOLECULAR BIOLOGY™, vol 489. Humana Press. https://doi.org/10.1007/978-1-59745-543-5_2
Download citation
DOI: https://doi.org/10.1007/978-1-59745-543-5_2
Publisher Name: Humana Press
Print ISBN: 978-1-934115-74-9
Online ISBN: 978-1-59745-543-5
eBook Packages: Springer Protocols