The nervous system is a continuously fluctuating excitable medium. Nervous system dynamics unfold within the time interval between the presentation of a stimulus and the collection of a response in a standard laboratory-based cognitive task. How does one understand these cognitive dynamics via behavioral measurements? Cognitive scientists have identified a variety of paradigms to make dynamics available for measurement. One entry point into the dynamics of cognition is a trial series of response times (RTs)—sequences of RTs aligned in the temporal order in which the measurements were taken. There are many statistical analyses that can be applied to a trial series to determine if successive measurements bear any relation to each other, and if so, to assess the nature of that relationship. Figure 2 outlines the basic logic of one such procedure. It is called a power-spectral density analysis (or just spectral analysis for short).
Fractal 1/f Scaling
A fractal is a nested, statistically self-similar pattern. The pattern can be defined over space or time; the concern in cognitive science is typically with patterns of variation in cognitive performance over time. The patterns can be quantified with spectral analysis. Fractals are revealed when the frequency of the oscillations and their respective amplitudes are related by an inverse power-law scaling relation—a linear relation on double-logarithmic scales. The fractal pattern in Figure 2 is called ‘pink’ or 1/f noise. The pattern of variability in the observations that is expressed across short runs of several observations is echoed, statistically, across scores of observations, which are in turn echoed across hundreds, and even thousands of observations. Thus, pink noise represents a nested, statistically self-similar (correlated) pattern of fluctuation across widely ranging runs of successive measurements. A scaling exponent describes the relative coherence of the fractal pattern of correlation entailed in the series of measurements. Assuming appropriate statistical procedures were followed during the analysis, exponents near 1 signal a robust pattern of fractal scaling termed pink noise, exponents that are statistically equal to zero indicate the absence of fractal structure (i.e., randomness—no pattern) in the trial series.
Scaling relations were revealed in a wide range of cognitive performances and physiological measurements as described in previous reviews.66,94–96 Pink noise was reported in many standard cognitive tasks that measure RTs, such as simple reaction time, word naming, lexical decision, and other decision-based cognitive tasks.95–98 Similarly, judgment tasks such as temporal and spatial estimation tasks yield 1/f scaling.66,98,99 Explicit daily judgments of self-esteem and implicit measures of racial bias also yield 1/f scaling.65,100 Repetitive speech activity also yields pink noise in a large variety of the possible measures of speech that can be taken.101 Spectral analysis revealed that these data exhibited an unlikely but orderly dynamic relationship across the various timescales of fluctuation entailed in the signals.
As the early reports of 1/f scaling in cognitive performance began to accumulate, a concern was expressed that scientists could be mistaking short-range patterns of autocorrelation for 1/f scaling.98,102,103 The bulk of that initial skepticism was answered, however.44,104 The empirical patterns held up to rigorous statistical scrutiny, and fractal scaling, in one form or another, has emerged as the most representative and most likely description of the empirical reports.105,106
The patterns of 1/f scaling expressed in physical systems are typically relatively stable over time. By contrast, the strength and nature of the scaling relations that emerge across human activities tend to vary widely. Tasks that entail significant uncertainly from trial to trial—due either to variations in cognitive load97 or variability in the sequencing of the within-trial events, such as random inter-trial or inter-stimulus intervals—yield less robust but nevertheless reliable patterns of 1/f scaling. In contrast, repeatedly estimating the same temporal duration and RT tasks that use constant inter-trial intervals tend to yield robust and stable patterns of 1/f scaling. Differences in laboratory methodology explained some apparent discrepancies in the relative strength of the 1/f scaling that was observed in ostensibly identical tasks conducted in different laboratories.93,107–109
Given a relatively stable task context, changes from less robust to more robust 1/f scaling are associated with motor learning.110 Similarly, eye-movements while reading become more fluid and display more robust 1/f scaling the second time the same passage is read, as contrasted with the same measures taken during a first pass through the text.111 Dyslexic children display weaker patterns of 1/f scaling in their trial-series of word pronunciation times than age-matched, non-dyslexic controls. The dyslexic children's performance is more random and less fluid than their non-dyslexic counterparts.32 Dual-task (motor + cognitive) performance can also affect 1/f scaling. Kiefer et al.71 had participants walk on a treadmill or perform repeated temporal estimations, or perform both tasks concurrently. When performed separately, each task yielded clear 1/f scaling, but in the dual-task condition the variations in cognitive performance became essentially random (although the mean and amount of variability of the temporal estimates did not change compared to the single-task condition). Gait variability was unaffected by the concurrent cognitive task.
There are several competing explanations for 1/f scaling in cognition, but the only one that has gained any traction in predicting new phenomena is the straightforward hypothesis equating 1/f noise with evidence of coordinative activity across many temporal scales.30,106,108,112,113 Fractal patterns of variability in repeated measurements, such as 1/f scaling, constitute the empirical pattern that is symptomatic of the coupling that gives rise to coordination. Self-organizing physical, chemical, and biological systems exhibit 1/f scaling in their patterns of temporal evolution. This is consistent with the proposition that cognitive performance unfolds as a quasi-coordinated whole, a perspective that challenges the time-honored search for isolable components of mind. Moreover, when one considers that multiple measures of the same behavior can yield statistically independent ‘streams’ of 1/f scaling,101 the alternative hypothesis that each measurement of 1/f scaling derives from a corresponding component structure entails an absurd, never-ending proliferation of ad-hoc modules.42
Once scaling is identified in a system a natural working hypothesis is that the system may express additional forms of scaling. In fact, many additional scaling relations were identified in human performance. Several predate the identification of 1/f scaling by a century or more, while others were only recently described. Perhaps the first scaling relation identified in human performance is now called Stevens' law, and refers to the fact that to achieve algebraic changes in perceived stimulus magnitude human sensory systems (vision, audition, tactile sense, etc.) require objective changes in stimulation that follow a power law. The observed pattern highlights one key implication of scaling behavior. It is well known that a variety of anatomical and nervous system processes support any given sensory system, but functionally, the system appears to behave as a coordinated whole. The pieces of the system reveal little about the system's holistic behavior.
Another historic scaling relation, often called Zipf's law, refers to the fact that for most large samples of written text the relationship between the frequency of use of any given word is a power-law function of the relative usage rank-order of the word. The pattern is apparently expressed universally across many individual texts, corpora of aggregate text, and across both modern and ancient languages. Zipf's original hypothesis was that the scaling relation is a result of two general competing constraints on communication: Speakers emphasize easy-to-recall, frequent words, whereas listeners prefer distinctive, unambiguous, low-frequency items. Zipf saw the power law as a natural product of this competition.114,115
Scaling in RT Distributions
Distributions of RTs provide another entry point into scaling laws in cognition. Inverse power-law distributions entail a prominent skew, such that the probability of observing a particular event (a RT, for instance) is the inverse of the RT value raised to a scaling exponent α, i.e., p(RT) ≈ RT−α. An inverse power-law distribution entails a more dramatic positive skew than an exponential distribution, for example. An inverse power-law tail implies circular (feedback) coupling among processes that govern the system. Thus, power-law distributions are also associated with complex systems that coordinate their behavior across multiple temporal or spatial scales.
Dynamic systems accounts of cognitive activity rooted in the principle of circular causality predict the presence of power-law distributions of measures of cognitive performance.30 Recently, inverse power-law distributions were identified with the positive skew that is ubiquitous in the slow tails of RT distributions arising from laboratory-based cognitive tasks such as word recognition and decision-making. Statistical analyses of several extant large-scale RT databases demonstrated robust evidence for power-law behavior in the slow, stretched tails of RT distributions.116,117 The fact that inverse power-law scaling is intrinsic to both the time course of cognitive acts and the temporal patterns of correlation in RTs implicates a key role for dynamics in contemporary theoretical narratives of cognitive activity.
Scaling in Memory ‘Foraging’
Another power law hypothesis, put forward by Rhodes and Turvey,28 was inspired by animal foraging behavior. Remarkably, foraging behavior is generally well described in the terms of the far-from-equilibrium dynamics of contemporary thermodynamics.118,119 This source of hypotheses in thermodynamics is certainly credible given the fact that all living things are thermodynamic engines. Neuroimaging studies reflect that fact in their reliance on the BOLD signal of metabolism and glucose uptake, for example—the metabolic processes of the ‘thermodynamic’ brain.120,121
The foraging model predicted a Lévy distribution of temporal and spatial intervals of activity, which yields another power law. Animal memory is associated with foraging, and Rhodes and Turvey28 proposed that human symbolic memory could, in a sense, behave as an internalization and elaboration of foraging. They tested this idea by looking at the distribution of inter-recall-intervals in a free recall task. Participants recalled as many animals as possible in a 20-min time interval and the duration of the intervals between successive, recalled animal names was recorded. The resulting distribution of recall intervals followed a power-law with an exponent of α = 2, the predicted Lévy distribution.
Scaling relations are surprisingly common in human activities. They are associated not only with motor performance, where they were first identified, but also with perceptual, linguistic, and other cognitive performances—activities that in some sense are thought to set humans apart from other organisms. But the fact that scaling relations are observed in these activities seems to contradict the distinction and enhance the status of human activity as akin to the behavior of other natural systems. Rhodes and Turvey,28 for example, interpreted their findings as suggesting that memory is strongly influenced by external (environmental) structure, in much the same way as actual animal foraging is influenced by the distribution of food sources and geographical features in the environment. This interpretation is consistent with the claims of stronger versions of embodiment that cognition is not limited to the mind alone. Distributed cognitive systems are the result of informational couplings within and across agents, and between agents and their environments.52 In this approach, rather than processing and computation the new emphasis is on coordination and organization.86