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Physica A 388 (2009) 4192–4204

Contents lists available at ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

A theory of 1/f noise in human cognition

Paolo Grigolini a,b, Gerardo Aquino c, Mauro Bolognaa,d, Mirko Luković e,f,∗, Bruce J. West g

a Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USAb Istituto dei Processi Chimico Fisici del CNR, Area della Ricerca di Pisa, Via G. Moruzzi, 56124, Pisa, Italyc Max-Planck Institute for the Physics of CompSems, Dresden, Germanyd Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 6-D, Arica, Chilee Dipartimento di Fisica ‘‘E. Fermi’’, Università di Pisa, Italyf INFM, Largo Pontecorvo 3, 56127 Pisa, Italyg Mathematics Division, US Army Research Office, Durham 27709, NC, USA

a r t i c l e i n f o

Article history:

Received 1 February 2009

Received in revised form 2 June 2009

Available online 21 June 2009

PACS:

02.50.Ey

05.40.Fb

05.45.Tp

87.85.dm

Keywords:

Decision making

1/f noise

Task difficulty

a b s t r a c t

The brain is probably the most interesting example of a complex network having 1/f vari-ability 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 re-

cent 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.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

There is widespread conviction in the scientific community [1] that 1/f noise is the signature of complexity, where thespectrumof the phenomenon studied is given by S(f ) ∝ 1/f η and the power-law index falls in the interval 0.5 < η < 1.5. Inthis paper we shall adopt amore extended definition with 0 < η < 3. The index ηwas thought to be rigorously η = 1whenit was first discovered by Schottky [2] at the turn of the last century in the study of electrical conductivity and consequentlygiven the name 1/f noise. But subsequent studies in a variety of fields have shown a rather broad range for the power-lawindex [1]. The fascinationwith this phenomenon is due in part to its ubiquitous nature and the large number of theories usedto explain it. It is useful to list some of the complex processes that produce 1/f noise in order to gain perspective on howpervasive this phenomenon is. Dutta and Horn [3] wrote the first comprehensive review on the variety of condensedmattersystems demonstrating this effect and were followed a few years later by Weissman [4] with an equally impressive reviewof its physical manifestations. Other physical networks displaying 1/f noise include optical photon counting systems [5],vacuum tubes [6] along with resistors and semiconductors [7–9]. More recently biomedical phenomena have proven to beequally rich in 1/f noise, which appears in body movement such as walking [10,11], postural sway [12,13] and movementin synchronization with external stimulation such as a metronome [14,15]; also in physiologic networks as manifest in

∗ Corresponding address: Dipartimento di Fisica ‘‘E. Fermi’’, Università di Pisa, Stanza 124 c/o Grigolini, Largo Pontecorvo 3, 56127 Pisa, Italy. Tel.: +39

050 2214 864; fax: +39 050 2214 333.

E-mail address: lukovic@df.unipi.it (M. Luković).

0378-4371/$ – see front matter© 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.physa.2009.06.024

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P. Grigolini et al. / Physica A 388 (2009) 4192–4204 4193

heart rate variability [16–18,11], human vision [19] and the dynamics of the human brain [20–23]. The complexity of socialphenomena is also not immune giving rise to 1/f noise involving the interface between neuroscience and economics [24]and the willingness to delay rewards [25–27]. Perhaps among the more surprising phenomena manifesting 1/f noise ismusic [28–30], the visual arts [31–33] and the quenching of chronic intractable pain [34]. We emphasize that the list ofdisciplines manifesting 1/f noise is not complete and the references quoted are intended to be representative and notexhaustive.

The number of mathematical models that have been used to explain the generation of 1/f noise is nearly as broad asthe number of disciplines in which the phenomena are observed. A recent strategy for modeling the dynamic variabilitycaptured by the 1/f spectrum appears to be independent of the specific mechanisms generating it and involves fractalstatistics in one way or another. This approach actually began with fractionally integrated white noise [35]; it was extendedto fractal Brownian motion [36]; it was further modified to fractal shot noise [37]; it was shown to be related to thestatistical mechanism of subordination [4,38,39]; and it was subsequently related to fractal renewal processes [40]. Thishistorical trajectory has now targeted non-stationary and non-ergodic stochastic processes as the natural way to describethe 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-ergodicproperties to the cooperation of units that can be interpreted as interacting neurons. When isolated each neuron makesa 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-timedistribution density in one of the two states is no longer an exponential function, but is a renewal non-ergodic non-Poissonprocess, with a virtually infinite mean time. Another example of dynamic units that obey Poisson statistics in isolation andgenerate a non-Poisson global process is found in the field of the random growth of surfaces [42]. From an experimentalpoint of view, the prediction that interactions convert Poisson units into a global non-Poisson renewal process, includingthe emergence of 1/f noise, has been confirmed by experiment [43] where the role of interacting units (neurons) is playedby liquid crystal defects.

Herein we address a particularly interesting set of experiments relating to 1/f noise in human cognition [44,45] anddevelop a non-stationary, non-ergodic statistical model for their description. However, we do not make direct use of thedynamic 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 theWeber–Fechner law, and as towhether subjective time can be expressed as a logarithmic or a power-law function of physicaltime [46–48].

In the first group of experiments subjects were asked to reproduce from memory a given spatial or temporal intervaland the errors associated with their estimates were recorded as a time sequence. Gilden et al. [44] determined that thefluctuations in the error had an inverse power-law spectrum, even though they had Gaussian statistics, and pointed out thatthe 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 internalclock that mediates the judgement of time passage and a motor program that actuates responses. They associated 1/f noisewith perceptual judgement and white noise with motor control; the 1/f noise dominates at low frequency but is eventuallylost in thewhite noise fluctuations ofmotor control at high frequencies. This dualmechanismmodelwas 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 computerscreen 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 Whitesand the purpose of the experiment was to use analysis of the time interval response in identifying the objects to quantifystereotypical bias. Our purpose here is not to go into the details of this interesting experiment, for that we refer the readerback to the original paper; here we merely address the results of analysis. The model used for the response times consistsof two separate pieces: the judgement piece in which the subject decides whether the person on the computer screen hada gun and the motor activation piece in which the gun or no-gun button is pushed. Here again cognition is modeled by1/f noise and Correll [45] provides qualitative discussions of a number of alternative mechanisms that could account forits generation. He demonstrates that shuffling the data transforms the spectrum from 1/f 0.8 to 1/f 0 indicating that thespectrum is a consequence of the long-range order in the underlying process and therefore depends on the ordering of thedata points.

It is found that the properties of 1/f noise depend on the nature of the task being performed. In particular when simplecognitive 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 wereinstructed to shoot all armed targets and to indicate don’t shoot for all unarmed targets. In study 2 the participants performeda simple task in which they were asked to classify objects on the screen as either guns or tools. Prior to the presentation ofthe target, participants were primed with either a Black or a White face. Some of the participants (the control group) didnot receive any instruction. A second group of subjects were instructed ‘‘not to take race into account’’ and the participantsof a third group were instructed to ‘‘take race into account’’. In each study, participants who made efforts to take racialinformation into account produced less 1/f noise than the participants who made less effort. The main goal of the presentpaper is to explain why making decisions with effort has the effect of producing less 1/f noise.

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The outline of this paper is as follows. In Section 2wemathematicallymodel the twomechanismsproposed byGilden [44]to explain 1/f noise in human cognition and used by Correll [45] to explain the latency time spectra in his experimentson racial bias. The model is based on the Weber–Fechner law. In Section 3 we discuss a more general law, of which theWeber–Fechner one is a special case. Section 4 illustrates the results of recent theoretical work that has been addressingthe 1/f noise issue from the new perspective of non-ergodic renewal events using the dichotomous noise idealization. InSection 5 we show theoretically and numerically that the new theoretical prescription, based on crucial events, does notrequire the dichotomous assumption and can be realized by surrogate sequences that are similar to those produced bythe psychological tests. In Section 6 we show that the effect of mental effort is to weaken the intensity of the 1/f noisecomponent of the spectrum. Section 7 is devoted to concluding remarks and some plans for future work.

2. 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 inmany forms. It was first suggested around 1962 and developed in some detail by Eisler in 1975 [47]. Grondin [48] has anexcellent review article describing these models and many other aspects of subjective duration perception up to 2001. Thecontributions of Gilden et al. [44] are limited to the 1/f problem, and they slightly modified the standard model, proposingthat an internal clock generates 1/f noise, through an unknown mechanism, whereas the motor response mechanismsgenerates white noise. Previously the standard model had both mechanisms generating white noise.

The main goal of the present paper is to explain the results of Correll’s experiment. This explanation requires, first ofall, that we establish a connection between the 1/f noise clock [44] and non-ergodic fluctuations. We make the simplestassumption possible regarding the decisions the brain makes and presume a cognitive state to be either +1 or −1. Weassume that there exist extended time intervals, called laminar regions, within which no significant event occurs. The brainassigns to each laminar region either the value+1 or the value−1,with equal probability, thereby generating a dichotomoussignal ξ(t). We can imagine the sign choice as being produced by a fair coin toss. The coin tossing events are called crucialevents. Thus, the occurrence of a crucial event can correspond to an abrupt transition from+1 to−1 or vice versa. However,crucial events can occurwithout any abrupt jump of the fluctuation ξ(t). In physical time asmeasured by a clock the intervalbetween two consecutive crucial events has the value τc . For these crucial events to generate 1/f noise, it is necessary that thewaiting-time distribution densityψ(τc) is an inverse power lawwith the value of the power-law indexμ as close as possibleto μ = 2. However, the brain assigns to the time durations of the laminar regions the values τm, which do not necessarilycoincide with the time duration of laminar regions measured by a clock. These mental time intervals are perceived as beingshorter than the corresponding clock times, and to have statistics corresponding to the typical exponential distributiondensity

φ (τm) = 1

Tpe− τm

Tp . (1)

The question of how to relate an experiential quantity to a quantitative measure spawned the discipline of psychophysics inthe middle nineteenth century. Here the experiential quantity is subjective mental time, the quantitative measure is clocktime and the question is how to relate the two. In the Encyclopedia ofMathematics Roberts [51] reviews the history of relatingpsychological scales to physical scales and here we adopt a modified version of Fechner’s Law to related mental time τm toclock time τc :

τm = Tm log

(1 + τc

Tc

). (2)

This equation is keeping with a similar relation identified by Takahashi et al. [27] as the Weber–Fechner law in a decision-making context and by Dehaene [52] in his argument for a science of a mental life. Thus, the distribution density ψ (τc) ofthe physical time intervals is related to the exponential distribution density of mental time intervals φ (τm) by means of

φ (τm) dτm = ψ (τc) dτc, (3)

which, establishes that themeasured quantity has equal probability of being in the intervals (τm, τm+dτm) and (τc, τc +dτc).Eq. (3), with some straightforward calculus, yields the hyperbolic distribution for the clock time intervals

ψ (τc) = (μ− 1) Tμ−1c

(Tc + τc)μ , (4)

with the parameter μ given by

μ = 1 + Tm

Tp. (5)

The time transformation of Eq. (3) preserve the normalization condition∫ ∞0ψ(τm)dτm = 1, thereby ensuring

∫ ∞0ψ(τc)dτc

= 1, as can be easily checked.The parameter μ is a property of the brain purported to remain constant throughout a psychological task. According to

the decision-making tasks in the intertemporal model [26] Tm � Tp, thereby yielding μ < 2. We reduce the number ofindependent parameters by setting Tp = Tc . Thus, the condition μ < 2 yields Tm < Tp, thereby making τm < τc , also when

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τc is much shorter than Tc . We believe that the action of crucial events is a source of non-ergodic behavior throughout theentire range 1 < μ < 3, as do other investigators [53,43]. To ensure this non-ergodic condition the parameter Tm can beeither smaller or larger than Tc , but strictly smaller than 2Tc .

The average clock time is given by

〈τc〉 =∫ ∞

0

τcψ (τc) dτc =∫ ∞

0

(τc + Tc)ψ (τc) dτc − Tc

∫ ∞

0

ψ (τc) dτc, (6)

which, using the explicit expression of Eq. (4) and the normalization condition, becomes

〈τc〉 = (μ− 1)Tμ−1c

∫ ∞

0

1

(τc + Tc)μ−1dτc − Tc . (7)

For μ > 2 we obtain

〈τc〉 = Tc

(μ− 2). (8)

In the case μ ≤ 2, as it is made clear by the first term on the right hand side of Eq. (7), the mean time 〈τc〉 diverges. Thisregion for the power-law index signals the emergence of a process incompatible with ergodicity and stationarity [54]. Theconditions of ergodicity and stationarity are assumed nearly everywhere in ordinary statistical physics, but are violated bycomplex networks, including the brain.

It should be noted that the time series {tn} with the time intervals τc (n) = tn+1 − tn, define the laminar region referredto above, and are derived from the hyperbolic distribution (4). The drawing of a number τc (j) does not have any correlationwith earlier or later drawings, thereby making the process renewal. Furthermore, we set μ < 3, thus making infinite thesecond moment of the clock time distribution, thereby generating in practice ergodicity breakdown [43], even if for μ > 2,a stationary condition exists.

It is important to stress that the distribution length of the laminar regions is perceived by the brain as corresponding to aPoisson process, as can be obtained from (1). The procedure described earlier can be illustrated in the following equivalentway. The mental time intervals τm are randomly selected using (1) to form a sequence at natural times n = 0, 1, 2, . . .withthe corresponding values ξd(n) = ±1, generated by tossing a fair coin. Then these times are converted into clock times τcof Eq. (4), which determine the time interval between the occurrence of two consecutive crucial events. The crucial events,namely, as seen earlier, the coin tossing events, take place at times tn. The fluctuations ξ(t) are defined by the piecewisecontinuous variable

ξ(t) = ξd(n), (9)

for tn ≤ t < tn+1. The corresponding diffusion process

x(t) =∫ t

0

dt ′ξ(t ′) (10)

is the realization at the level of mental processes of the Continuous Time RandomWalk (CTRW) theory [55]. We can also saythat this is a procedure to generate a process ξ(t) subordinated to the totally random process ξd(n), realized by repeatedlytossing a fair coin. In the mental time scale the subordination procedure generates a Poisson process and the correspondingdiffusion is ordinary.

3. 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 lawof Weber and Fechner and the power law of Plateau, Brentano and Stevens [46], the response F to a stimulus I should bewritten as

F = 1

2k ln

(1 + βIn/t

). (11)

The parameters k, n, β and t are constants and we refer the readers to Ref. [46] for details on their meaning. Here we limitourselves to noticing that using the arguments of Ref. [46] we should replace Eq. (2) with

τm = Tm ln

(1 +

(τc

Tc

)α), (12)

with α = 0.9 being the most commonly accepted exponent [47,48]. Using the same approach as that of the earlier sectionto connect the Poisson distribution of mental time to the distribution density of clock time, we obtain

ψ(τc) = (μ− 1)ατα−1

c

Tαc

Tαμc

(Tαc + ταc )μ. (13)

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Although this equation satisfies the normalization condition∫ ∞0

dτcψ(τc) = 1,whenα < 1 the function yields a divergenceat the time origin. When α > 1 the function vanishes for τc → 0. However, to establish the emergence of 1/f noise we haveto focus on the large time limit. It is straightforward to find that for τc → ∞ the function yields

ψ(τc) ∝ 1

τ νc, (14)

where

ν ≡ 1 + αTm

Tp. (15)

We have seen that the generation of 1/f noise requiresμ ≈ 2. Thus, we conclude that for values of α ≈ 1, only very modestchanges of the ratio Tm/Tp are required to meet the requirement that the process generates 1/f noise.

4. Spectra

We should use the following definition of spectrum of the sequence ξ(t):

Sξ (f ) = limL→∞

1

L

∣∣∣∣∫ L

0

exp(i2π ft)ξ(t)dt

∣∣∣∣2

, (16)

where L is the sequence time length. Due to the non-stationary nature of the condition μ ≤ 2, it is convenient to adopt thedefinition:

Sξ (f ) = 1

L

∣∣∣∣∫ L

0

exp(i2π ft)ξ(t)dt

∣∣∣∣2

. (17)

As we shall see, for μ ≤ 2, Sξ (t) turns out to be dependent on L and tends to vanish for L → ∞. In this case it would moreaccurate to use the term periodogram rather than spectrum. However, for the sake of homogeneitywith the current literatureon the ergodic breakdown associated to the condition μ ≤ 2 [53,54], we shall use the term spectrum. Note that the wholeinterval 1 < μ < 3 generates 1/f noise, according to the results of Ref. [53] which yield

Sξ (f ) ∝ 1

f η, (18)

with

η = 3 − μ. (19)

The non-stationary nature of 1/f noise has been stressed by Mandelbrot [56] and the most evident manifestation ofthis property is illustrated in Fig. 1: The longer the length L of the time series ξ (t), the less intense the correspondingpower spectrum. Margolin and Barkai [54] demonstrate that for a non-ergodic sequence the power spectral density variesinversely as a power of L. Their argument is rather extensive but the short version [53] is that the autocorrelation functionfor an irretrievably non-ergodic time series, that is, one for μ ≤ 2, is generated by the waiting-time distribution density

ψL(t) = 1

L2−μ1

(t + Tc)μ−1, (20)

through the negative time derivative of the finite time L autocorrelation function C(t, L)

ψL(t) = −dC(t, L)

dt. (21)

Consequently, using the generalized Onsager Principle of Ref. [53] we established that the average power spectral densitytakes the form⟨

Sξ (f )⟩ ∝ 1

L2−μ1

f η. (22)

As noted earlier, the parameterμ is the index for the clock time hyperbolic distribution between two consecutive events.Wenote that the ideal 1/f condition is generated by μ = 2, on the boundary between the μ > 2 and the μ < 2 condition. Thecondition μ ≤ 2 is non-stationary [54] and the condition μ > 2 is compatible with equilibrium, even if in practice it maygenerate non-ergodic statistics [43]. Note that the numerical results from simulating realizations of time series generatedby the hyperbolic distribution produces average power spectra recorded in Fig. 1 that are in complete agreement with (22).Moreover, the theory accounting for these results rests on the crucial event perspective [57] outlined in the previous section.

The psychological task of estimating interval sizes [44] establishes with sufficient accuracy that the inverse power law inthe spectrum is very close to η = 1. On the other hand, the results of psychological tasks [20,45] indicate that η can changefrom 1 to 0 and the work on attention-deficit disorder of Gilden and Hancock [58] also generates the unusual value η = 2.Therefore, on the basis of these experimental results, in accordance with an earlier remark, we extend our investigation to

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Fig. 1. These figures refer to the case where μ = 1.5 and T = 1. Left panel: This is the average of Sξ (f ) over a large number of realizations of length L.

From top to bottom, L = 106, L = 107, L = 108, L = 109, L = 1010. The full straight lines correspond to the theoretical prediction of (22). Right panel: Thiscurve aims to establish the accuracy of the prediction of (22) on the noise intensity proportional to 1/L2−μ (dashed line). This numerical result has been

obtained by changing L and keeping f fixed (f0 = 10−4).

the entire μ interval 1 ≤ μ ≤ 3, thereby leaving Tm free to move within the interval 0 ≤ Tm ≤ 2Tc . As a consequence ofthis assumption, in addition to (22) we also consider the traditional ergodic prescription for the spectrum:

Sξ (f ) ∝ Tμ−2c

f η, (23)

with η given in (19) and 2 < μ < 3. Note that in the case μ > 2 the intensity of the spectrum is virtually independentof L [53], but the departure from the ordinary fluctuation-dissipation theorem can still be detected with experimentalmeasurements [43]. In spite of the independence from L, the spectrum remains divergent as f → 0, although it becomesvery flat for μ close to the boundary 3.

An alternative derivation of the L-dependence of the spectrum may provide additional insight into the process. Theconversion of τm to τc by means of Fechner’s Law indicates that the brain generates 1/f noise, which implies the effectrecorded in (22). The latter equation is a clear indication of the non-stationary nature of 1/f noise when μ ≤ 2. In fact,according to Feller [59] the total number of events produced by a renewal network in the time interval t after the productionof an earlier event is proportional to

N(t) ∝(

t

Tc

)μ−1

. (24)

The intensity of the 1/f noise emerging from a time sequence of length L is assumed to be proportional to the number ofevents occurring per unit of time, namely, the noise intensity is proportional to

N(L)

L∝ 1

L2−μ, (25)

which explains the dependence of (22) and of Fig. 1 on the total time L. Note that the psychological task involving 1/f noisein cognition [44] (their Fig. 1A) yields results similar to those of Fig. 1, namely, curves with slope≈ −1, at a lower and lowerlevel as a consequence of tasks of ever increasing difficulty. From a qualitative point of view onemay be tempted to interpretthe results of Ref. [44] as a consequence of the fact that estimating shorter and shorter time lengths is a task of increasingdifficulty and that this is equivalent to increasing the time duration of the experiment, and consequently L. In the Gildenet al. [44] experiment the durations of the experiments (and thus the analyzed time series) for the various time intervals,Δt , used were (approximately): L = 4000 s using Δt = 10 s; L = 5000 s using Δt = 5 s; L = 1500 s using Δt = 1.5 s;L = 1000 s usingΔt = 1 s; L = 500 s usingΔt = 0.5 s; and L = 300 s usingΔt = 0.3 s. ForΔt = 10 s, the subjects wereasked to perform 400 trials. For the other time intervals,Δt , they were asked to perform 1000 trials. Also, each timing errorwas 4ms and hence negligible. Thus themore difficult the task, the shorter the experiment duration, not the longer. For thisreason, in Section 6 we shall adopt for the result of Correll’s experiments an explanation that can be used also to account forGilden’s results without using the incorrect recourse to L decreasing with the increase in task difficulty. It is worth stressingthat this explanation is based on the non-stationary approach to 1/f noise reviewed in this section.

5. 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

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Fig. 2. Left panel: A fluctuation, ξT (t), that is not dichotomous. Each laminar region is given a random integer value from the interval [0 : 100]. Rightpanel: The full line is the average power spectral density of 1000 realizations having the form shown in the left panel, obtained numerically usingμ = 1.8,

Tc = 1 and L = 105. The dotted straight line is the function 65/f 1.2 and it shows that the numerical result obeys (22).

from a bimodal distribution, namely, only two possible values, with the same probability. The time distance between twoconsecutive data points is identified with the integration time stepΔt , whose real value depends on the performed task (forinstance 500 ms in the case of the digit tapping experiment of Ref. [60]).

We want to prove that the theoretical prediction of Eqs. (18) and (19) holds true in a condition that is much moregeneral than the dichotomous one, for which it was originally proved [53,54]. To afford this theoretical proof, we note thatthe theory [53,54] rests on the non-stationary correlation function 〈ξ(t1)ξ(t2)〉 which can be expressed in a form that isindependent of the specific constant value used to fill a laminar region. Let f (t1, t2) = 1 if t1 and t2 are in the same laminarregion, and zero otherwise. Let Xn be the value of ξ in the nth interval. If t1 and t2 are in the same laminar region, then thesame Xn applies; otherwise two different Xn’s apply and they are independent. Then we obtain

〈ξ(t1)ξ(t2)〉 = 〈f (t1, t2)〉〈X2〉 + (1 − 〈f (t1, t2)〉)〈X〉2. (26)

Thus, the correlation function 〈ξ(t1)ξ(t2)〉 can be written in the form

〈ξ(t1)ξ(t2)〉 = 〈f (t1, t2)〉Var[X] + 〈X〉2, (27)

which proves that the spectral shape depends on 〈f (t1, t2)〉, and it is independent of the values of ξ selected to fill the laminarregions, which will only influence the spectral intensity. In Fig. 2 we show the case where the constant values of laminarregions are selected from a homogeneous distribution of integers in the interval [0, 100]. These numerical results fit thetheoretical prediction of Eq. (27).

In Fig. 3 we study a case where the laminar regions are assigned only two values, each of which is supplemented by awhite fluctuation. Finally, in Fig. 4 we show the dichotomous fluctuation ξ(t) plus a white noise fluctuation ξw(t), which isused in the numerical simulations of the next section. We see that in spite of the fact that the dichotomous assumptionmayseem to be an excessive idealization, the resulting fluctuation

ξT (t) = cξ ξ (t)+ cwξw (t) (28)

is qualitatively similar, for example, to those of Ref. [61]. Note that cξ and cw are constants, ξ (t) corresponds to the timeseries generated by (4) withμ < 2 and ξw (t) is white noise generated by a coin tossing procedure, namely, an uncorrelatedsequence of +1 and −1.

In conclusion, although the theory of Refs. [53,54] is based on the dichotomous assumption themain result is general andthe surrogate time series generated bywidely different prescriptions to fill the laminar regions fit the theoretical predictionsof Eqs. (18) and (19), as they should on the basis of Eq. (27). The addition of white noise does not affect the low-frequencyregion of the spectrum, and it is possible to generate time series that are qualitatively similar to those generated by the taskexperiments.

6. 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 thegeneration 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.

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Fig. 3. Left panel: A dichotomous signal that fluctuates between the constant values 3 and 4, added to white noise whose values are randomly selected

from the interval [−0.5, 0.5]. In other words, ξT (t) = 0.5ξ(t)+0.5ξw(t)+3.5. Right panel: The broken line is the average power spectral density of 1000

realizations having the form shown in the left panel, obtained numerically usingμ = 1.7, Tc = 1 and L = 107. The straight line is the function 0.0023/f 1.3,

showing that the numerical results fit well the prediction of (42). Note that in this case (43) is valid.

Fig. 4. Similar to Fig. 3. Left panel: A dichotomous signal that fluctuates between the constant values -1 and 1, added to white noise whose values are

randomly selected from the interval [−0.1, 0.1]. Here cξ = 1 and cw = 0.1, see (28). This fluctuation is the basis for the lower panels of Fig. 5. Right panel:The broken line is the average power spectral density of 100 realizations having the form shown in the left panel, obtained numerically using μ = 1.8,

Tc = 1 and L = 105. The straight line is the function 0.11/f 1.2 and it corresponds to (42) with the condition (43).

6.1. Physical and mental time

The choice of Eq. (12) affords the benefit of establishing a compromise between the logarithmic Fechner relationshipand the power relationship advocated by psychologists [48]. In fact, when τc � Tc , we obtain that the subjective timeis proportional to the physical time to the power of α, in line with Fig. 1 of Ref. [48]. In Section 3 we have seen that thisinteresting relationship does not conflict, in principle, with the emergence of a 1/f noise spectrum. However, on the basisof Eq. (12) it is difficult, if not impossible, to afford a phenomenological explanation of the Correll’s experiment. In fact, if weassume that Tc is made very large by the task effort, we do not generate a flat distribution density with times smaller thanthe sequence length L. This is so because the short-time distribution density is either divergent (α < 1) or empty (α > 1),if α = 1. The condition α = 1, on the contrary, corresponds to a flat distribution density of short times.

The connection between the flatness ofψ(t) at short times and Poisson statistics can be explained as follows. Accordingto the renewal theory literature [62] the age-specific failure rate, denoted by the symbol g(t), is expressed by

g(t) = ψ(t)

Ψ (t), (29)

where ψ(t) is the waiting-time distribution density, and Ψ (t) is the survival probability

Ψ (t) =∫ ∞

t

dt ′ψ(t ′). (30)

As a consequence

Ψ (t) = exp

(−

∫ t

0

g(t ′)dt ′). (31)

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4200 P. Grigolini et al. / Physica A 388 (2009) 4192–4204

y = -0.8045x - 3.3167

Log Frequency

y = 0.0073x - 2.8016

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-4 -3 -2 -1 1 20

log f

log

S T(f

)

Log Frequency-4 -3 -2 -1 1 20

Log Frequency-4 -3 -2 -1 1 20

y = 0.0186x - 3.4746

2

0

-2

-4

-6

-8

log

S T(f

)

2

0

-2

-4

-6

-8

log

S T(f

)

2

0

-2

-4

-6

-8-4 -3 -2 -1 1 20

log f-4 -3 -2 -1 1 20

log f-4 -3 -2 -1 1 20

Log

Pow

er

Log

Pow

er

Log

Pow

er

Fig. 5. This figure shows that the renewal theory approach to 1/f noise, with Tc increasing as a result of the task effort, is in remarkably good agreement

with the results of the experimental test of Ref. [45]. Top three panels: Experimental result obtained by Corell [45]. The three panels have been copied

here from his paper for the sake of clarity. (Left) The power spectral density graphs for participant A, who reported minimal effort to avoid bias. (Middle)

For participant A, again, after reshuffling the data to destroy the temporal sequence. (Right) The power spectral density for participant B who reported high

effort to avoid bias. Black data points represent log frequency below 0, which were used to calculate the 1/f slope; gray points represent high-frequency

waves, andwere excluded from 1/f analysis. Bottom three panels: These are the results of our numerical simulations.We show the power spectral density

of a single realization, ξT (t), given by (28), where μ = 2.2, cξ = 0.1 and cw = 0.01. (Left) The points represent the power spectral density of ξT (t) when

Tc = 1 and L = 107. The solid line corresponds to the theoretical prediction given by (42). (Middle) The power spectral density obtained after reshuffling

the values of ξT (t). The values of Tc and Lwere left unchanged. The horizontal line serves as a guide to show the absence of the 1/f component. (Right) The

power spectral density obtained after increasing the values of Tc and L to 1013 and 1012 respectively. To alleviate the computational load, we have assumed

that the complex noise ξ is dichotomous with Tc = 1013 and L = 1012 while the white noise, whose spectrum is known to be independent of L, has been

assigned the value L = 105. In this case also, the 1/f component of the spectrum vanishes.

It is straightforward to prove that the waiting-time distribution density ψ(τc) of Eq. (4), for tn+1 > t ≥ tn, is generated by

g(t) = r0

1 + r1(t − tn). (32)

Note that every time there is an event we reset the time origin to the time of occurrence of the last event (tn) beforethe observation time, t . Thanks to the renewal nature of the process, we define the times τc through the prescriptionτc(n) = tn+1 − tn. By substituting (32) into (31), after some algebra we obtain

Ψ (τc) =(

Tc

τc + Tc

)μ−1

, (33)

where Tc ≡ 1/r1 and μ − 1 = r0/r1, whose waiting-time distribution is Eq. (4). The derivation of Eq. (4) from (32) is donefor two reasons. Firstly, it serves the purpose of stressing that within our perspective the Weber–Fechner law is a renewalprocess. Furthermore, it affords the benefit of showing that on the time scale τc � Tc , g(t) ≈ r0, thereby generating theexponential distribution density

ψ(τc) = r0 exp(−r0τc). (34)

The above argument is not rigorous, because the renewal nature of the process makes it possible that a very long timemay appear immediately after a very short time. However, the numerical simulation yielding the results of Fig. 5 supportsthis conclusion. We expect that the emergence of Poisson statistics at short waiting times is not possible with α = 1.

As we shall see in Section 6.2, the emergence of Poisson statistics is caused by the task-effort induced enhancement ofTc , becoming larger than the time length L of the time series being analyzed. What is the reason for the task-effort induced

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P. Grigolini et al. / Physica A 388 (2009) 4192–4204 4201

enhancement of Tc? The authors of Ref. [41] found that Nn interacting neurons produce an intermittent fluctuation x(t) that

can be interpreted as the diffusion process x(t) = ∫ t

0dtζ (t) generated by thewhite noise ζ (t). The diffusion process is given

by the standard expression

p(x, t) = 1√4πDt

exp

(− x2

4Dt

). (35)

D is the diffusion coefficient, and is inversely proportional to the number of interacting neurons

D ∝ 1√Nn

. (36)

The crucial events ti correspond to the crossings of x = 0. Note that the diffusion coefficient D in the representation ofEq. (35) has the dimensions of inverse time. Thus, it is reasonable to assume that the number of crucial events per unit timeP(t) is derived from the distribution of zero-crossings

P(t) = Dp(0, t). (37)

The quantity P(t) is N(t)/t , where N(t) is the number of events produced up to time t . Using Eqs. (24) and (35) we obtainμ = 1.5 and time parameter value

Tc ∝ √Nn. (38)

Thus, the enhancement of Tc is determined by the number of neurons Nn involved in the task.

6.2. Explanation of Correll’s experiment

It seems to be well established that challenging tasks require the action of large but distinct sets of distributed neuronalgroups [63].We thereforemake the reasonable assumption that Tc increaseswith increasing task difficulty, an effect formallycaptured by Eq. (38). The time Tc has to be compared with the sequence length L, which is approximately equal to the totaltime duration of the psychological task. The time parameter Tc determines the length of the microscopic time scale, namelythe short time scale within which the time interval between two consecutive brain events shares the properties of ordinarystatistical physics and the signs of self-organized complexity are not yet manifest. When Tc � L, the lengths of the laminarregions within which the brain makes correlated decisions satisfy the condition τc � Tc corresponding to the distributiondensity [41] that is well approximated by the exponential

ψ (τc) ≈ μ− 1

Tcexp

[− (μ− 1)

τc

Tc

], (39)

which is derived from Eq. (34) by setting r0 = (μ− 1)Tc . Note that the condition Tc > L forces us to consider τc � Tc .

Notice that in the ideal condition μ = 2, which corresponds to the condition of maximum efficiency of communicationbetween complex networks [64], (39) becomes identical to the exponential distribution (1). This is not an accident, but rathera consequence of the fact that the task effort abolishes the distinction between mental and physical time thereby inducingthe brain to generate white noise rather than 1/f noise, as is evident in the numerical experiment depicted in Fig. 5.

It is worth noticing that we can use directly Eq. (12) to assess the effect of the increase of Tc . In fact, if Tc is anomalouslylarge (larger than L) and τc is kept small by the constraint τc � Tc , we obtain

τm = Tm

(τc

Tc

)α. (40)

In the case α = 1, this condition makes τc inherit the Poisson statistics of τm (see Eq. (1)). If α = 1 there is no exacttransmission of Poisson statistics from τm to τc .

As mentioned above, we need to establish the relation between the task difficulty and the time duration of the experi-ment. To determine this relation we focus on the work of Correll [45] and establish that the theoretical perspective of thispaper reproduces the results of his psychological experiments surprisingly well. We generate a realization of the time series(28). The spectrum corresponding to (28) is

ST (f ) = c2ξ Sξ (f )+ 4c2w (41)

in an obvious notation.

Consider the case of a simple task in which Tc � L. The total spectrum obtained from (41) is given by

ST (f ) ∝ c2ξ

L2−μ1

f 3−μ+ 4c2w (42)

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4202 P. Grigolini et al. / Physica A 388 (2009) 4192–4204

where we have again absorbed some constants, but this time into c2ξ . We note that the frequency set by the length of thetime series, f ≥ 1/L, has the spectral strength implying the condition

cξ√L � cw (43)

for the 1/f noise contribution to dominate.Now consider the case of a complicated task. Accomplishing a difficult task involves a very large number of neurons, and

consequently, the conjecture that Tc increases with the number of neurons Nn, may yield Tc � L. This condition implies theapproximate clock time distribution given by (39) and the associated total spectrum given by

ST (f ) ∝ c2ξr0

r20 + f 2+ 4c2w, (44)

where r0 = (μ− 1) /Tc . Using the condition Tc � L the spectrum generated by (39) with the small frequencies, but largerthan 10−4, is depicted by the lower-right frame of Fig. 5. Analytically, in this case we obtain the spectral strength at thefrequency set by the length of the time series

ST

(1

L

)∝ (μ− 1) c2ξ

L2

Tc+ 4c2w ≈ 4c2w (45)

implying the condition

cξ√L

√L

Tc� cw (46)

which establishes that for very large values of Tc the 1/f noise contribution becomes negligible compared with the motorterm. Thus, the increase of Tc dictated by the effort necessary to accomplish a difficult task has the effect of making 1/f noiseapparently vanish into the white noise background.

7. 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 subjectsand Kello et al. [20] who made averages over 18 subjects. The results illustrated in Correll’s Fig. 5, on the contrary, refer tosingle 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 showsthat the single realizations produce the same strong spectral fluctuations as do the psychological tasks [45]. However, theadoptions of surrogate sequences allows us to make averages on the same subject and the resulting mean values are shownto agree very well with the theoretical prediction of (42). The agreement of our Fig. 5 and Correll’s Fig. 2 affords compellingevidence that the brain operates in the anomalous region 1 < μ < 3. In fact, we have established that the task induces 1/fnoise reduction for μ < 2, in the non-stationary case.

In the ergodic case μ > 2 the spectrum (23) produces the total spectral strength at the frequency determined by thelength of the time series

ST

(1

L

)∝ c2ξ L

3−μTμ−2c + 4c2w. (47)

Thus, to generate a 1/f noise spectrum it is necessary for the parameters to satisfy the condition

c2ξ L3−μTμ−2

c � c2w (48)

and increasing Tc with task difficulty would have the effect of further increasing rather than decreasing the 1/f componentof the spectrum, in conflict with experimental observation [45]. However, if the effort-induced increase of Tc is sufficientlylarge as to make it exceed the sequence length L, then (47) no longer describes the total spectrum and we revert to (45) andto the predominantly white noise spectrum.

Therefore we conclude that the effort-reduced 1/f noise intensity is a property of the entire interval 1 < μ < 3. Thisconclusion leads us to consider the parameter μ as being free to depart from the ideal condition μ = 2 and to explore theentire range from 2 to 3, the boundary with ordinary statistical physics, as well as the entire non-ergodic region fromμ = 1to μ = 2.

The noise intensity reduction, revealed by psychological tasks [45,58], is in our perspective a consequence of the adoptionof crucial events for the foundation of 1/f noise in cognition, valid not only in the non-ergodic case μ < 2, but in the entireregion 1 < μ < 3, responsible for the divergence of the power spectrum at low frequency. Fechner’s Law affords attractivedirections to the understanding of psychophysical laws of elementary arithmetic [52] and explains the origin of 1/f η noise,with 0 < η < 2. At the same time the phenomenon of Tc increasing with the task effort explains why efforts of increasingdifficulty have the apparently contradictory effect of making the brain generate uncorrelated fluctuations [45]. This is a

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P. Grigolini et al. / Physica A 388 (2009) 4192–4204 4203

consequence of the fact that the brain, in making more difficult decisions rests on the coordinated action of larger numbersof neurons.

From an intuitive point of view one may argue that a difficult task is properly addressed by making a fixed numberof neurons work harder rather than increasing the number of neurons involved in a given task. This conjecture is correctand our group is obtaining [65] results confirming this property on the basis of the dynamic model of [41]. Here we limitourselves to noticing that the cooperation-induced global neuron has a fluctuating motion equivalent to that of a particle ina double-well potential. Increasing the number of neurons Nn is equivalent to reducing the intensity of the stochastic force1/

√Nn, whereas leaving the number of neurons fixed and increasing the cooperation strength has the effect of increasing the

barrier intensity. The resulting dynamics are determined by the ratio of the barrier intensity to the stochastic force intensity,and consequently the two effects are interchangeable.

There exists another interesting question to answer: Can a power-law index μ = 1.5 be generated from a model ofinteracting neurons? We are convinced that this depends on the topology of the complex neuronal network. However,all this is beyond the scope of the present paper that is essentially limited to phenomenological arguments. Within thelimits of a treatment that does not rest directly on the dynamics of a complex network of neurons, we conclude that theWeber–Fechner proposal of Eq. (12) with α = 1 affords a plausible explanation of Correll’s experiment. This is so because, aswe have seen in Section 6.2, it is essential that in the short time scale τc � Tc the waiting-time distribution is flat, in conflictwith the condition α = 1. We believe that there exists a close connection between the phenomenological structure of Eq.(12), and the class of possible neuron interactions. The parameters Tm, Tc andα of Eq. (12) depend also on the structure of thecomplex network within which the neurons are imbedded. Answering these remaining questions requires further research.

Acknowledgments

Wewarmly thank two unknown referees for remarks and suggestions that led us to turn the original version into amuchricher 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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