A theory of 1/f noise in human cognition

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Abstract

The brain is probably the most interesting example of a complex network having 1/f variability as determined through the analysis of EEG time series and magnetoencephalogram recordings. Herein we develop a theory of 1/f noise of human cognition to explain the recent experimental observations that increasing the difficultly of cognitive tasks accelerates the transition from observed 1/f noise to white noise in decision-making time series.

Introduction

There is widespread conviction in the scientific community [1] that 1/f noise is the signature of complexity, where the spectrum of the phenomenon studied is given by S(f)1/fη and the power-law index falls in the interval 0.5<η<1.5. In this paper we shall adopt a more extended definition with 0<η<3. The index η was thought to be rigorously η=1 when it was first discovered by Schottky [2] at the turn of the last century in the study of electrical conductivity and consequently given the name 1/f noise. But subsequent studies in a variety of fields have shown a rather broad range for the power-law index [1]. The fascination with this phenomenon is due in part to its ubiquitous nature and the large number of theories used to explain it. It is useful to list some of the complex processes that produce 1/f noise in order to gain perspective on how pervasive this phenomenon is. Dutta and Horn [3] wrote the first comprehensive review on the variety of condensed matter systems demonstrating this effect and were followed a few years later by Weissman [4] with an equally impressive review of its physical manifestations. Other physical networks displaying 1/f noise include optical photon counting systems [5], vacuum tubes [6] along with resistors and semiconductors [7], [8], [9]. More recently biomedical phenomena have proven to be equally rich in 1/f noise, which appears in body movement such as walking [10], [11], postural sway [12], [13] and movement in synchronization with external stimulation such as a metronome [14], [15]; also in physiologic networks as manifest in heart rate variability [16], [17], [18], [11], human vision [19] and the dynamics of the human brain [20], [21], [22], [23]. The complexity of social phenomena is also not immune giving rise to 1/f noise involving the interface between neuroscience and economics [24] and the willingness to delay rewards [25], [26], [27]. Perhaps among the more surprising phenomena manifesting 1/f noise is music [28], [29], [30], the visual arts [31], [32], [33] and the quenching of chronic intractable pain [34]. We emphasize that the list of disciplines manifesting 1/f noise is not complete and the references quoted are intended to be representative and not exhaustive.

The number of mathematical models that have been used to explain the generation of 1/f noise is nearly as broad as the number of disciplines in which the phenomena are observed. A recent strategy for modeling the dynamic variability captured by the 1/f spectrum appears to be independent of the specific mechanisms generating it and involves fractal statistics in one way or another. This approach actually began with fractionally integrated white noise [35]; it was extended to fractal Brownian motion [36]; it was further modified to fractal shot noise [37]; it was shown to be related to the statistical mechanism of subordination [4], [38], [39]; and it was subsequently related to fractal renewal processes [40]. This historical trajectory has now targeted non-stationary and non-ergodic stochastic processes as the natural way to describe the complexity captured in 1/f noise [1].

Of special interest are the results illustrated in Refs. [1], [41] about models that connect the emerging non-ergodic properties to the cooperation of units that can be interpreted as interacting neurons. When isolated each neuron makes a choice between 1 and 1, the waiting time in either of these two states is given by an exponential distribution density. As a result of cooperation a kind of global two-state network is created, and as an effect of cooperation the waiting-time distribution density in one of the two states is no longer an exponential function, but is a renewal non-ergodic non-Poisson process, with a virtually infinite mean time. Another example of dynamic units that obey Poisson statistics in isolation and generate a non-Poisson global process is found in the field of the random growth of surfaces [42]. From an experimental point of view, the prediction that interactions convert Poisson units into a global non-Poisson renewal process, including the emergence of 1/f noise, has been confirmed by experiment [43] where the role of interacting units (neurons) is played by liquid crystal defects.

Herein we address a particularly interesting set of experiments relating to 1/f noise in human cognition [44], [45] and develop a non-stationary, non-ergodic statistical model for their description. However, we do not make direct use of the dynamic arguments considered previously [1], [41], [43] and herein we remain at a phenomenological level. As we shall see, this perspective makes it possible to contribute to the long-standing debate among psychologists on the reliability of the Weber–Fechner law, and as to whether subjective time can be expressed as a logarithmic or a power-law function of physical time [46], [47], [48].

In the first group of experiments subjects were asked to reproduce from memory a given spatial or temporal interval and the errors associated with their estimates were recorded as a time sequence. Gilden et al. [44] determined that the fluctuations in the error had an inverse power-law spectrum, even though they had Gaussian statistics, and pointed out that the predictability of such phenomena is intermediate between white noise (no correlation in time) and Brownian motion (no correlation between increments). These authors postulated two mechanisms to explain their observations: an internal clock that mediates the judgement of time passage and a motor program that actuates responses. They associated 1/f noise with perceptual judgement and white noise with motor control; the 1/f noise dominates at low frequency but is eventually lost in the white noise fluctuations of motor control at high frequencies. This dual mechanism model was used by Correll [45] over a decade latter to explain the variability observed in latencies in the performance of simple computer tasks.

Correll [45] designed a group of experiments in which pictures of a person holding an object were flashed on a computer screen for a fixed short time interval. The subjects were given instruction to: press a button indicating a gun if one was seen, or no gun if a neutral object, such as a flashlight, was seen. The pictures flashed had an equal number of Blacks and Whites and the purpose of the experiment was to use analysis of the time interval response in identifying the objects to quantify stereotypical bias. Our purpose here is not to go into the details of this interesting experiment, for that we refer the reader back to the original paper; here we merely address the results of analysis. The model used for the response times consists of two separate pieces: the judgement piece in which the subject decides whether the person on the computer screen had a gun and the motor activation piece in which the gun or no-gun button is pushed. Here again cognition is modeled by 1/f noise and Correll [45] provides qualitative discussions of a number of alternative mechanisms that could account for its generation. He demonstrates that shuffling the data transforms the spectrum from 1/f0.8 to 1/f0 indicating that the spectrum is a consequence of the long-range order in the underlying process and therefore depends on the ordering of the data points.

It is found that the properties of 1/f noise depend on the nature of the task being performed. In particular when simple cognitive tasks are made more complicated such as by requiring additional memory the slope of the 1/f noise decreases [49], [50]. To test this observation Correll [45] divided his experiment into two studies. In study 1, the participants were instructed to shoot all armed targets and to indicate don’t shoot for all unarmed targets. In study 2 the participants performed a simple task in which they were asked to classify objects on the screen as either guns or tools. Prior to the presentation of the target, participants were primed with either a Black or a White face. Some of the participants (the control group) did not receive any instruction. A second group of subjects were instructed “not to take race into account” and the participants of a third group were instructed to “take race into account”. In each study, participants who made efforts to take racial information into account produced less 1/f noise than the participants who made less effort. The main goal of the present paper is to explain why making decisions with effort has the effect of producing less 1/f noise.

The outline of this paper is as follows. In Section 2 we mathematically model the two mechanisms proposed by Gilden [44] to explain 1/f noise in human cognition and used by Correll [45] to explain the latency time spectra in his experiments on racial bias. The model is based on the Weber–Fechner law. In Section 3 we discuss a more general law, of which the Weber–Fechner one is a special case. Section 4 illustrates the results of recent theoretical work that has been addressing the 1/f noise issue from the new perspective of non-ergodic renewal events using the dichotomous noise idealization. In Section 5 we show theoretically and numerically that the new theoretical prescription, based on crucial events, does not require the dichotomous assumption and can be realized by surrogate sequences that are similar to those produced by the psychological tests. In Section 6 we show that the effect of mental effort is to weaken the intensity of the 1/f noise component of the spectrum. Section 7 is devoted to concluding remarks and some plans for future work.

Section snippets

Phenomenological modeling of 1/f noise

It is worthwhile to point out that the model mechanism postulated by Gilden et al. [44] is an old one, and exists in many forms. It was first suggested around 1962 and developed in some detail by Eisler in 1975 [47]. Grondin [48] has an excellent review article describing these models and many other aspects of subjective duration perception up to 2001. The contributions of Gilden et al. [44] are limited to the 1/f problem, and they slightly modified the standard model, proposing that an

Beyond the Weber–Fechner law

It is interesting to notice that according to Norwich [46] to get a unifying formula compatible with both the semilog law of Weber and Fechner and the power law of Plateau, Brentano and Stevens [46], the response F to a stimulus I should be written as F=12kln(1+βIn/t). The parameters k, n, β and t are constants and we refer the readers to Ref. [46] for details on their meaning. Here we limit ourselves to noticing that using the arguments of Ref. [46] we should replace Eq. (2) with τm=Tmln(1+(τcT

Spectra

We should use the following definition of spectrum of the sequence ξ(t): Sξ(f)=limL1L|0Lexp(i2πft)ξ(t)dt|2, where L is the sequence time length. Due to the non-stationary nature of the condition μ2, it is convenient to adopt the definition: Sξ(f)=1L|0Lexp(i2πft)ξ(t)dt|2. As we shall see, for μ2, Sξ(t) turns out to be dependent on L and tends to vanish for L. In this case it would more accurate to use the term periodogram rather than spectrum. However, for the sake of homogeneity with

Beyond the dichotomous assumption

The theory of Ref. [53], and the numerical calculations of Fig. 1 as well, rest on the study of dichotomous fluctuations. This is equivalent to assuming that all the data points between two consecutive crucial times have the same value, drawn from a bimodal distribution, namely, only two possible values, with the same probability. The time distance between two consecutive data points is identified with the integration time step Δt, whose real value depends on the performed task (for instance

Transition from 1/f to white noise

In this Section we show that the Weber–Fechner prescription of Eq. (12), in addition to being compatible with the generation of 1/f noise, can be used to explain the result of Correll’s experiment, while the generalized expression of Eq. (12) can do that only when α=1, namely, when it coincides with the Weber–Fechner prescription.

Concluding remarks and plans for future research work

The precision of where 1/f noise emerged was enhanced by Gilden et al. [44] who made an average over six subjects and Kello et al. [20] who made averages over 18 subjects. The results illustrated in Correll’s Fig. 5, on the contrary, refer to single realizations, which are affected by strong fluctuations. Nevertheless, the fitting procedure adopted led Correll [45] to the correct conclusion that challenging tasks have the effect of generating a flat white noise spectrum. Our Fig. 5 shows that

Acknowledgments

We warmly thank two unknown referees for remarks and suggestions that led us to turn the original version into a much richer and more attractive illustration of the cognition origin of 1/f noise. M.B. and P.G. thank ARO (grant W911NF-08-1-0117) and Welch (grant B-1577) for financial support.

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