Copyright © 2010 Elsevier Ltd All rights reserved.
Permissions & Reprints
Christopher T. Kello1, , Gordon D.A. Brown2, Ramon Ferrer-i-Cancho3, John G. Holden4, Klaus Linkenkaer-Hansen5, Theo Rhodes1 and Guy C. Van Orden4
1 Cognitive and Information Sciences University of California , Merced, 5200 North Lake Rd., Merced, CA 95343, USA
2 Department of Psychology, University of Warwick, Coventry CV4 7AL, United Kingdom
3 Department de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord, Edifici Omega, Jordi Girona Salgado 1-3, 08034 Barcelona, Catalonia, Spain
4 Center for Perception, Action and Cognition, Department of Psychology, University of Cincinnati, PO Box 210376, Cincinnati, OH 45221-0376, USA
5 Department of Integrative Neurophysiology, VU University Amsterdam, De Boelelaan 1085, 1081 HV Amsterdam, the Netherlands
Available online 1 April 2010.
Scaling laws are ubiquitous in nature, and they pervade neural, behavioral and linguistic activities. A scaling law suggests the existence of processes or patterns that are repeated across scales of analysis. Although the variables that express a scaling law can vary from one type of activity to the next, the recurrence of scaling laws across so many different systems has prompted a search for unifying principles. In biological systems, scaling laws can reflect adaptive processes of various types and are often linked to complex systems poised near critical points. The same is true for perception, memory, language and other cognitive phenomena. Findings of scaling laws in cognitive science are indicative of scaling invariance in cognitive mechanisms and multiplicative interactions among interdependent components of cognition.
The scaling law debate
Scaling laws in perception, action and memory
Scaling laws in reaction times and word frequencies
Scaling laws and criticality
Droga, essa idéia era minha...:
Another type of scaling law in memory comes from a classic free recall paradigm, yet was only recently discovered by drawing an analogy to studies of animal foraging behaviors . Birds, monkeys, fish and numerous other species have been reported to search for food in Lévy flight patterns , which have been hypothesized as effective search strategies because they cover more territory than, for example, a random walk with normally distributed steps . Searching for items or events in memory is like foraging, particularly in tasks such as free recall of members of a given semantic category (e.g. animals) in a given time period . Rhodes and Turvey  analyzed inter-response time intervals (IRIs) from this classic memory task, which are analogous to steps from one recalled item to the next. The authors found IRIs to be power-law distributed with exponents very similar to those found in animal foraging (Figure 2). These comparable results suggest that Lévy flights are generally adaptive across a variety of search ecologies. These results also illustrate how scaling laws can lurk unnoticed in data for decades, in the absence of theories and analytic techniques necessary to recognize them.
Complex Times for Earthquakes, Stocks, and the Brain's Activity
Christoph Kayser1, , and Bard Ermentrout2, ,
1 Max Planck Institute for Biological Cybernetics, Spemannstrasse 38, 72076 Tübingen, Germany
2 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
Available online 12 May 2010.
Refers to: The Temporal Structures and Functional Significance of Scale-free Brain Activity
Neuron, Volume 66, Issue 3, 13 May 2010, Pages 353-369,
Biyu J. He, John M. Zempel, Abraham Z. Snyder, Marcus E. Raichle
PDF (3392 K) | Supplementary Content
A new study by He et al. in this issue of Neuron shows that large-scale arrhythmic (1/f) brain activity contains nested temporal structure in the form of crossfrequency coupling. This suggests temporal organization in neural mass activity beyond oscillations and draws attention to ubiquitous but often ignored arrhythmic patterns in neural activity.
What do earthquakes, Dow-Jones, and brain activity have in common? Unpredictability first springs to mind, of course, but as researchers have long noticed, these and many other complex processes might actually share common patterns pertaining to long-range spatio-temporal correlations of the underlying quantities ([Kello et al., 2010] and [Jensen, 1998]). In addition, and as an intriguing study in this issue of Neuronillustrates (He et al., 2010), they might also share another level of temporal organization, whereby the phase of slower timescales predicts the amplitude of faster ones. This nesting of timescales might open a window onto the complex structure of neural activity, but also raises questions with regard to its universality.
In their new study, He et al. recorded electrocorticographic (ECoG) activity across several brain areas in human patients. To investigate the signal's temporal structure, they calculated the frequency spectrum, i.e., the distribution of amplitudes of individual frequency bands as a function of frequency. In concordance with previous studies, they described the frequency spectra using the power-law 1/fa, with the scaling factor adiffering between low (<1>1 Hz) frequency bands. When shown on logarithmic axes, such power-law scaling translates into a straight line with slope a, as illustrated in Figure 1A.
It is important to note the distinction between the spectral 1/fa shape and rhythmic oscillatory activity. Oscillatory activities with well-defined frequencies (e.g., theta, alpha, or gamma oscillations) are prevalent in neural networks and result in distinct peaks above the 1/fa background (Buzsaki, 2006) (cf. Figure 1A). Typically, such oscillations result from processes with well-defined intrinsic timescales and can be associated with defined networks such as thalamocortical or hippocampal loops. In contrast to this, activity characterized by a (straight) 1/fa spectrum is considered “arrhythmic,” as it does not reflect processes with identifiable timescales. Systems that generate perfect power-law spectra are also known as “scale-free,” since the underlying process or network possesses no distinguished scale ([Bak et al., 1987] and [Jensen, 1998]). Importantly, while oscillations have attracted wide interest and are matter of various speculations with regard to their meaning and function, the arrhythmic component of electric brain activity is often considered self-evident or uninteresting and hence ignored.
The stunning finding of He et al. is that even such supposedly arrhythmic brain activity has a complex temporal structure in the form of crossfrequency phase-amplitude coupling. Crossfrequency implies that the coupling involves two distinct frequency bands, and phase-amplitude implies that the amplitude of one band is dependent on the phase of the other. In particular, He et al. extracted band-limited components from their wide-band signals and found that the amplitude of the faster component depends on the phase of the slower one, as illustrated in Figure 1B. For their analysis they considered a range of frequency pairs and used statistical bootstrapping methods to validate the significance of phase dependency. Overall, they found that more than 70% of the electrodes contained frequency pairs with significant frequency coupling. Importantly, and to prove the importance of this phenomenon, they demonstrated the existence of crossfrequency coupling not only in resting state activity, but also during task performance and slow-wave sleep.