Thursday, January 4, 2018

Multifractal research

Continuing this post, here is some research on the topic:

Top Cogn Sci. 2012 Jan;4(1):51-62. doi: 10.1111/j.1756-8765.2011.01162.x. Epub 2011 Oct 24.
Multifractal dynamics in the emergence of cognitive structure.
Dixon JA1, Holden JG, Mirman D, Stephen DG.
Author information
Abstract
The complex-systems approach to cognitive science seeks to move beyond the formalism of information exchange and to situate cognition within the broader formalism of energy flow. Changes in cognitive performance exhibit a fractal (i.e., power-law) relationship between size and time scale. These fractal fluctuations reflect the flow of energy at all scales governing cognition. Information transfer, as traditionally understood in the cognitive sciences, may be a subset of this multiscale energy flow. The cognitive system exhibits not just a single power-law relationship between fluctuation size and time scale but actually exhibits many power-law relationships, whether over time or space. This change in fractal scaling, that is, multifractality, provides new insights into changes in energy flow through the cognitive system. We survey recent findings demonstrating the role of multifractality in (a) understanding atypical developmental outcomes, and (b) predicting cognitive change. We propose that multifractality provides insights into energy flows driving the emergence of cognitive structure.
Copyright © 2011 Cognitive Science Society, Inc.
http://journal.frontiersin.org/article/10.3389/fphys.2015.00088/full
Front. Physiol., 19 March 2015 
Demystifying cognitive science: explaining cognition through network-based modeling
Emma K. Soberano and Damian G. Kelty-Stephen, Grinnell College, Grinnell, IA, USA
Even more stunning has been the evidence that the sand/rice-pile model may not just be fractal but, in fact, multifractal: it may exhibit several power-law forms at once (Tebaldi et al., 1999; Cernak, 2006; Bonachela and Muñoz, 2009), making it more complex. This multifractal wrinkle in the self-organization narrative may be exactly what's needed to help cognitive science play by more ordinary scientific rules. Observation of multifractal fluctuations offers the possibility that fractal fluctuations might interweave and spread into one another (Halsey et al., 1986). Where we might once have envisioned anatomical parts each with their own mysterious capacities, there may be less rigidly defined regions engaging in ongoing exchange of fractal and multifractal fluctuations.
The sharing of multifractal fluctuations has empirical anchoring in behaviors extending beyond the brain. Network analyses such as vector autoregression (VAR; Sims, 1980) allow us to depict the flow of information across nodes in full-body network, even from measurements of living, breathing organisms. […] The sharing of multifractal fluctuations may underwrite body-wide coordinations in ways that only network analyses have revealed.
Exciting as simulations may be, we see more promise in this latter attempt to draw from fractal statistics and matrix algebra to help us probe the full-body network. Distributing cognition across the body is still not repaying the loans of intelligence, but it may diminish the borrowed principal. Network modeling thus allows us to envision behavior—real, observed behavior—as the time-varying mixture of an extended field of multifractal fluctuations. Through this lens, behavior begins to require much less faith and much more like generic physical processes. Cognitive science need not ask to play by different rules or to start with different assumptions. On the contrary, network science might allow cognitive science operate on the same playing field as other sciences, whether sciences of living systems or otherwise.
Scaling in executive control reflects multiplicative multifractal cascade dynamics
Damian G. Stephen1*, Jason R. Anastas2 and James A. Dixon2,3,4
Self-organized criticality (SOC) purports to build multi-scaled structures out of local interactions.
Evidence of scaling in various domains of biology may be more generally understood to reflect multiplicative interactions weaving together many disparate scales. The self-similarity of power-law scaling entails homogeneity: fluctuations distribute themselves similarly across many spatial and temporal scales. However, this apparent homogeneity can be misleading, especially as it spans more scales. Reducing biological processes to one power-law relationship [monofractal] neglects rich cascade dynamics. We review recent research into multifractality in executive-function cognitive tasks and propose that scaling reflects not criticality but instead interactions across multiple scales and among fluctuations of multiple sizes. [...]
Executive control is a general phenomenon of biological systems whose explanation lies in generic principles of complexity, rather than specifically cognitive mechanisms (Van Orden, 2010). However, the evidence of scaling in executive control does not point simply to SOC-like dynamics. Like many other aspects of living systems (Ivanov et al.,2001; Plotnick and Sepkoski, 2001; Ihlen and Vereijken, 2010), executive control is better understood through multiplicative multifractal cascade dynamics.


Interaction-dominant dynamics in human cognition: Beyond 1/ƒα fluctuation.
Ihlen, Espen A. F.; Vereijken, Beatrix
Journal of Experimental Psychology: General, Vol 139(3), Aug 2010, 436-463. http://dx.doi.org/10.1037/a0019098

It has been suggested that human behavior in general and cognitive performance in particular emerge from coordination between multiple temporal scales. In this article, we provide quantitative support for such a theory of interaction-dominant dynamics in human cognition by using wavelet-based multifractal analysis and accompanying multiplicative cascading process on the response series of 4 different cognitive tasks: simple response, word naming, choice decision, and interval estimation. Results indicated that the major portion of these response series had multiplicative interactions between temporal scales, visible as intermittent periods of large and irregular fluctuations (i.e., a multifractal structure). Comparing 2 component-dominant models of 1/fα fluctuations in cognitive performance with the multiplicative cascading process indicated that the multifractal structure could not be replicated by these component-dominant models. Furthermore, a similar multifractal structure was shown to be present in a model of self-organized criticality in the human nervous system, similar to a spatial extension of the multiplicative cascading process. These results illustrate that a wavelet-based multifractal analysis and the multiplicative cascading process form an appropriate framework to characterize interaction-dominant dynamics in human cognition. This new framework goes beyond the identification of 1/fα power laws and non-Gaussian distributions in response series as used in previous studies. The present article provides quantitative support for a paradigm shift toward interaction-dominant dynamics in human cognition. (PsycINFO Database Record (c) 2012 APA, all rights reserved)


Ecological Psychology
Volume 25, Issue 3, 2013
Special Issue: A Cognitive Science Slam in Honor of Guy Van Orden
Notes on a Journey From Symbols to Multifractals: A Tribute to Guy Van Orden
Abstract
Rejecting traditional cognitive science put us in a bind. On the one hand, traditional cognitive science is our heritage; our curiosity about the big questions of cognition led us initially to invest in the conventional approaches. On the other hand, we eventually became dissatisfied with the fundamentals of traditional cognitive science. Rather than criticize from the sidelines, we struggled for a new way to address the same problems with a new explanatory framework. Guy Van Orden spurred us forward on 2 counts. First, his work inspired us to consider fractal scaling as a new framework for exploring change in cognitive structure. Second, his provocative contrast between pink and white noises as diagnostic of interactions and components, respectively, intrigued us. Our struggle for a new direction became a struggle to understand what Guy meant and how his ideas might translate within our research domains. Guy helped us to forge a perspective that would have surprised us before, namely, the perspective that cognitive and, more generally, biological structure reflects turbulent flows structured over many different scales with multifractal fluctuations.
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