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Spectral convergence in tapping and physiological fluctuations: coupling and independence of 1/f noise in the central and autonomic nervous systems
Michael Spivey
2014, Frontiers in Human Neuroscience
October 30, 2025
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Abstract
When humans perform a response task or timing task repeatedly, fluctuations in measures of timing from one action to the next exhibit long-range correlations known as 1/f noise. The origins of 1/f noise in timing have been debated for over 20 years, with one common explanation serving as a default: humans are composed of physiological processes throughout the brain and body that operate over a wide range of timescales, and these processes combine to be expressed as a general source of 1/f noise. To test this explanation, the present study investigated the coupling vs. independence of 1/f noise in timing deviations, key-press durations, pupil dilations, and heartbeat intervals while tapping to an audiovisual metronome. All four dependent measures exhibited clear 1/f noise, regardless of whether tapping was synchronized or syncopated. 1/f spectra for timing deviations were found to match those for key-press durations on an individual basis, and 1/f spectra for pupil dilations matched those in heartbeat intervals. Results indicate a complex, multiscale relationship among 1/f noises arising from common sources, such as those arising from timing functions vs. those arising from autonomic nervous system (ANS) functions. Results also provide further evidence against the default hypothesis that 1/f noise in human timing is just the additive combination of processes throughout the brain and body. Our findings are better accommodated by theories of complexity matching that begin to formalize multiscale coordination as a foundation of human behavior.
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ORIGINAL RESEARCH ARTICLE
published: 11 September 2014
doi: 10.3389/fnhum.2014.00713

HUMAN NEUROSCIENCE

Spectral convergence in tapping and physiological
fluctuations: coupling and independence of 1/f noise in the
central and autonomic nervous systems
Lillian M. Rigoli , Daniel Holman , Michael J. Spivey and Christopher T. Kello *
Cognitive and Information Sciences, University of California, Merced, CA, USA

Edited by:
José Antonio Díaz, Universidad de
Granada, Spain
Reviewed by:
Klaus Linkenkaer-Hansen,
Neuroscience Campus Amsterdam,
Netherlands
Vadim Nikulin, Charite University
Hospital, Germany
Gerardo Aquino, Imperial College
London, UK
*Correspondence:
Christopher T. Kello, Cognitive and
Information Sciences, University of
California, 5200 N. Lake Road,
Merced, CA 95343, USA
e-mail:
[email protected]
When humans perform a response task or timing task repeatedly, fluctuations in measures
of timing from one action to the next exhibit long-range correlations known as 1/f
noise. The origins of 1/f noise in timing have been debated for over 20 years, with
one common explanation serving as a default: humans are composed of physiological
processes throughout the brain and body that operate over a wide range of timescales,
and these processes combine to be expressed as a general source of 1/f noise. To test
this explanation, the present study investigated the coupling vs. independence of 1/f noise
in timing deviations, key-press durations, pupil dilations, and heartbeat intervals while
tapping to an audiovisual metronome. All four dependent measures exhibited clear 1/f
noise, regardless of whether tapping was synchronized or syncopated. 1/f spectra for
timing deviations were found to match those for key-press durations on an individual basis,
and 1/f spectra for pupil dilations matched those in heartbeat intervals. Results indicate a
complex, multiscale relationship among 1/f noises arising from common sources, such
as those arising from timing functions vs. those arising from autonomic nervous system
(ANS) functions. Results also provide further evidence against the default hypothesis that
1/f noise in human timing is just the additive combination of processes throughout the
brain and body. Our findings are better accommodated by theories of complexity matching
that begin to formalize multiscale coordination as a foundation of human behavior.
Keywords: complexity matching, long-range correlations, interdependent coordination, tapping, spectral analysis

INTRODUCTION
All behaviors of biological organisms can be viewed as phenomena of coordination, including human behaviors. Neurons
work together to create temporal patterns of neural activity, and
those patterns play important roles in motor activities leading to
overt human behaviors. Likewise, behaviors result in changes to
sensory and proprioceptive inputs that affect patterns of neural
activity. Thus coordination happens amongst the components of
brains and bodies, and also between brains, bodies, and their
environments.
Perhaps the most fundamental expression of coordination
in human behavior is found in the relative timing of events.
Movements of the eyes must be timed relative to those of the
hands to draw a picture (Huette et al., 2013); movements of hands
must be timed relative to those of the vocal tract to gesture during
speech (Kelly et al., 2010); and movements of the legs must be
timed with movement of the ball in soccer (Bartlett et al., 2007),
just to name a few examples. These are all exquisite phenomena
of timing and coordination, but it is often useful to study simpler
cases to formulate basic principles and theories.
From this perspective, one of the simplest cases of coordination occurs when brief, individual behaviors are timed in direct
relation to clearly delineated events in the environment—stepping

Frontiers in Human Neuroscience

on the gas or brake pedal in response to a traffic light, for instance,
or working on an assembly line to perform the same action
repeatedly on each widget being transported along a conveyor
belt. The experimental analogs to these illustrative examples are
simple response times to individual stimuli, and tapping in time
with a metronome. These experimental paradigms have been
used in thousands of studies, with response times figuring most
prominently in experimental psychology (Holden et al., 2009),
and tapping in motor control (Rosenbaum, 2009).
Despite the intimate relationship between timing and coordination, theoretical approaches to response times and tapping
times do not usually refer to coordination. Instead, response times
are usually studied in terms of information processing, where time
is theorized to reflect the number of processing steps needed to
identify each stimulus, and then choose and prepare a response
(Sternberg, 1998). Tapping times are usually studied in terms
of timing mechanisms (Georgopoulos, 2000), where questions
focus on the nature of internal clocks and their associated neural
machinery.
THE PUZZLE OF 1/f NOISE

Studies of information processing and clocks have led to many
advances in theories of cognitive and neural processes for decades,

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September 2014 | Volume 8 | Article 713 | 1

Rigoli et al.

1/f Complexity matching in metronome tapping

and this progress is likely to continue for some time. However, a
general property of response times and tapping times has been
established over the years, and it is not easily explained within
these paradigms. Responses and taps vary from one to the next,
even when there is no overt change in stimuli, the metronome,
or any other task conditions. This is not surprising in itself, given
that humans are not robots or machines in any traditional sense
of the words. One should expect a certain amount of imprecision
in human timing that could be described as “noise”.
The puzzling property concerns the kind of noise observed
in human timing, and many other fluctuations in biological and
complex systems. A default assumption of most statistical models
used in experimental psychology is that noise (i.e., error variance)
in repeated measures is Gaussian and uncorrelated. The term
“uncorrelated” means that current and previous measured values
of noise provide no information about future measurements. This
simplifying assumption is useful for statistical models, but we
know it is incorrect because memory is universal to all human
and biological systems—not memory as storage of information,
but the more general sense that system states are conditioned on
their past, and therefore carry some of their history forward in
time.
The memory inherent to human and biological systems suggests that noise in human timing should be correlated in some
way. For instance, homeostatic systems are often expected to
exhibit negative correlations in their fluctuations, as a result of
negative feedback. When synchronizing to a metronome, if one
tap falls behind the beat, the next tap would be adjusted earlier
in time, and vice versa. One can measure such negative feedback
by taking a time series of tap intervals, and a copy of the same
time series where values are shifted backward in time, and then
correlating the time series with itself at different lags. Such autocorrelation analyses show evidence to support the presence of
negative correlations in tapping data (Wing and Kristofferson,
1973), but negative correlations account for only a small amount
of the noise variance. Most of the noise in human response times
and tapping times exhibits positive auto-correlations (Pressing
and Jolley-Rogers, 1997).
In general, positive auto-correlations can be understood in
terms of hysteresis—simply put, systems are sluggish to change
their states. For instance, if a response is relatively fast on one
trial, then whatever system conditions caused the fast deviation
will still be in play on the next trial, at least to some degree.
This principle can explain positive auto-correlations in general,
but it is the specific pattern of auto-correlations that is puzzling.
In particular, they decay slowly as an inverse power law function
of increasing lag, C(k)∼1/kλ , where C() is the auto-correlation
function, k is the integer lag, and λ is the power law exponent.
This kind of positive auto-correlation is known as long-range
correlation because the power law indicates that all past states play
a role in determining any given current state. It is also known as
1/f noise because the auto-correlation function can be expressed
in the frequency domain as a relation between spectral power
and frequency, P(f )∼1/f α , where P() is the spectral function, f is
frequency, and α is the power law exponent. Exponents estimated
from timing data have varied across studies and conditions, but
most estimates have been near 1.

Frontiers in Human Neuroscience

The presence of 1/f noise has been reported in many studies
of human response times (Van Orden et al., 2003) and tapping
times (Ding et al., 2002), as well as other repeated measures of
human behavior (Gilden, 2001) and neural systems (Allegrini
et al., 2009). These reports have stirred up much debate. Some of
this debate has concerned the veracity of findings (Farrell et al.,
2006), with opponents arguing that observed auto-correlations
actually may not be long-range but short-range instead (i.e., fall
off exponentially with lag, instead of an inverse power law function). However, recent studies have compared these two statistical
models and found 1/f noise to better account for the data (Gilden,
2009).
Accepting that the noise in human timing follows a 1/f scaling
relation, most of the debate has focused on theoretical explanations. One reason for debating 1/f noise is that the theoretical
constructs of clocks and information processing yield no ready
insights in and of themselves. Certainly one can add mechanisms
to information processing models and clock models that exhibit
1/f noise, and this has been done (Torre and Wagenmakers, 2009).
Perhaps the most general mechanism thus far has been strategy
shifting (Diebold and Inoue, 2001), whereby a perturbation is
added to each response time or tapping time that reflects discrete
shifts among distinct plateaus in response strategies. The varying duration of these plateaus, and non-stationarity of shifting
among them, has been shown to yield 1/f noise under certain
parameterizations.
One problem with strategy shifting and similar accounts is
that they appear post hoc, in that they do not provide principled
answers as to why information processing and clock models
would include such processes. Another problem is that such
domain-specific processes are difficult to generalize to other
repeated measures of human activity that exhibit 1/f noise, such as
speech acoustics (Kello et al., 2008) and affect ratings (Delignières
et al., 2004), and repeated measures of human neural activity as
well (Linkenkaer-Hansen et al., 2001). We believe that progress
will continue to be made by improving and expanding domainspecific accounts related to clocks and information processing,
but here we investigate two domain-general accounts aimed at the
broader range of 1/f phenomena in human and biological systems.
TWO DOMAIN-GENERAL ACCOUNTS OF 1/f NOISE

The generality and ubiquity of 1/f noise has led some researchers
to formulate two classes of domain-general explanations: a process
summation account and an interdependent coordination account.
The process summation account is based on sums of processes
across various timescales, and the interdependent coordination
account is based on the interdependence of processes necessary
for coordination. Here we describe each in turn, and then present
an experiment designed to test these alternative accounts of 1/f
noise in human timing and physiology.
Regarding the process summation account, a 1/f-like signal
can be created by sampling from three or more uncorrelated
noises generated over different timescales and amplitudes (with
timescale inversely related to amplitude), and summing the samples together (Wagenmakers et al., 2004). A 1/f signal can be
similarly created by summing independent processes whose exponential decay rates span a range of timescales (Granger, 1980).

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September 2014 | Volume 8 | Article 713 | 2

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1/f Complexity matching in metronome tapping

In either case, the signal will be 1/f-like only within the range of
timescales sampled. Then again, 1/f noise in human behavior can
be observed only within a limited range of timescales due to limits
on measurement (Van Orden et al., 2005).
The human brain and body is composed of processes that
unfold over a wide range of timescales, from fast ion channel
dynamics to slower changes in neurotransmitters, cardiovascular,
and various homeostatic processes, and even slower changes in
hormones, circadian dynamics, and developmental processes in
general (Bassingthwaighte et al., 1994). A similar claim can be
made regarding cognitive processes, from the millisecond dynamics of perception, to the waxing and waning of attention that
may span seconds to minutes, to processes of decision-making,
planning, and learning that may span anywhere from seconds to
years (Ward, 2002). It seems quite plausible that any measurement of human behavior may be influenced by any combination
of these ongoing processes. If the magnitude of influence (i.e.,
amplitude) is generally inversely related to timescale, then one
would expect these processes to sum up to 1/f noise in repeated
measurements of response times, tapping times, and any other
measure of human behavior.
The interdependent coordination account is based on interactions among system components, rather than summations of
independent processes. The coordination of behavioral activity
requires interactions among whatever components and events are
being coordinated. The same is true for neural and physiological
activities, the difference being that the components and events are
different and reside on shorter spatial and temporal scales. We
can say further that interdependencies among system components
must strike a balance between too much and too little coupling as
a result of interactions (Kello and Van Orden, 2009). Too much
coupling would result in interlocked patterns of activity that are
unable to differentiate or adapt to changes in conditions. Too little
coupling would fail to support the emergence of coordination
patterns that extend in space and time. Instead, adaptive systems
exhibiting coordination need loosely coupled components that
support the formation of many different potential patterns of
activity.
The balance of coupling and its relationship to pattern formation has been formalized in statistical mechanics in terms
of metastability (Kelso, 1995), and the dynamics of interactions
that underlie metastability have been shown to produce 1/f noise
(Usher et al., 1995). Metastability appears to be a useful property for biological and behavioral systems in general, because it
endows them with an ability to respond and adapt to their everchanging conditions (Sasaki et al., 2007; Pinder et al., 2012).
On this account, 1/f noise reflects fluctuations across multiple
timescales that result from patterns being organized and reorganized across multiple timescales. Thus 1/f noise should be
a general property of any metastable system, including human
systems involved in response times and tapping times.
This approach to 1/f noise and other power laws in nature
was made famous by models of self-organized criticality (SOC;
Bak et al., 1987). The ubiquity of power laws in nature, like
1/f noise, led physicists to hypothesize that critical points may
be common attractors of complex systems in nature (Bak,
1996). Critical points are associated with (second-order) phase

Frontiers in Human Neuroscience

transitions in systems of many interdependent elements, where
dynamics take on unique properties of memory and symmetrybreaking (Stanley, 1987). Original models of SOC were criticized
as models of human behavior because they more closely resemble
models of avalanches, forest fires, and other physical complex
systems (Wagenmakers et al., 2005). However, a large body of
work has shown how SOC may be a functional principle of
neural networks and other physiological networks (see Kello,
2013).
The interdependent coordination account is similar to the process summation account, in that both provide a rationale for the
ubiquity of 1/f noise. However, they make different predictions
when it comes to taking multiple repeated measurements of
human behavior. Kello et al. highlighted this distinction between
accounts by measuring two aspects of key-press dynamics (Kello
et al., 2007). Participants made repeated simple responses to series
of visual cues, and both response times and key-press durations
were recorded. A key-press duration is the very brief period of
time (∼100–150 ms ) that a key remains in contact with its sensor
for a normal, ballistic keystroke. Both domain-general accounts
predict 1/f noise in both time series of measurements.
The accounts differ in whether they predict the same 1/f signal
to appear in each time series, or whether distinct 1/f signals
may arise from simultaneous yet distinct measures of behavior.
The process summation account predicts the same 1/f signal
because key-press response times and durations should draw from
roughly the same set of summed processes, especially at the larger
timescales (e.g., waxing and waning of attention and circadian
rhythms). The reasons are that the two measurements are inextricably paired for each keystroke, are produced by overlapping
sets of muscles, and effectively occur at the same time relative
to the timescales of 1/f noise spanning dozens and hundreds of
responses. It is difficult to hypothesize how these measurements
could tap into distinct sets of component processes spanning
the same timescales as 1/f noise. By contrast, interdependent
coordination holds that any given system or subsystem can exhibit
1/f noise on its own, or in coupling with other systems. The reason
is that interdependence can hold for components at all scales,
and criticality can create dynamics with long-range memory
(i.e., correlations) for any given subsystem. In other words, 1/f
noise is hypothesized to pervade the heterogeneous networks of
interacting processes in human systems.
Results from four experiments reported by Kello et al. (2007)
showed that key-press response times and durations were independent of each other, in terms of exhibiting 1/f noises that were
uncorrelated with each other, and also in terms of independently
manipulating the 1/f noise in response times while leaving keypress durations unaffected. The authors argued that the data
provided evidence against the process summation account, but
were consistent with the interdependent component account.
However, a subsequent reanalysis of these data indicated more
subtle, nonlinear relationships between the time series (Moscoso
del Prado Martín, 2011). Thus while the process summation
account is called into question, more experiments and analyses
are needed to investigate the nature of coupling and independence
among simultaneous measures of 1/f noises (e.g., Kello et al.,
2008).

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September 2014 | Volume 8 | Article 713 | 3

Rigoli et al.

1/f Complexity matching in metronome tapping

COMPLEXITY MATCHING AS MEASURED BY SPECTRAL CONVERGENCE

The present experiment and analyses were designed to further
investigate the nature of 1/f noise in human behavior by measuring coupling based on a recently formulated theoretical principle known as complexity matching (West et al., 2008; Aquino
et al., 2010, 2011). Theoretical analyses using statistical mechanics
have shown that, when two complex systems become coupled,
there is maximal rate of information exchange between them
when their power laws converge. This formalization of coupling is different from more standard measures like synchronization. Complexity matching between two signals does not
refer to phase relations—instead, it refers to convergence of
the two power spectra. Thus coupling in terms of complexity
matching means that each system retains its own distinct phase
dynamics, yet the systems affect the statistical character of each
other’s dynamics. This effect is equivalent to an exchange of
information between two given systems, in the sense of mutual
information.
Complexity matching is a theoretical construct general to all
complex systems, but it has already garnered some empirical
support in studies of dyadic coordination, which can be viewed
in terms of informational coupling between two human complex
systems. Marmelat and Delignières (2012) conducted an experiment in which each participant in a dyad swung a hand-held
pendulum, with instructions to swing in synchrony. Synchronization is a direct phase relation, but deviations from synchrony were
analyzed for 1/f noise. Results showed that the spectral shape
of 1/f noise for each member of a dyad converged to the extent
that coupling was facilitated by visual and physical contact. This
convergence could not be explained in terms of simple phase
relations because there were no reliable cross-correlations in the
time series of deviations from synchrony. Other more recent
experiments showed the same basic effect, but in the speech
signals of dyads engaged in conversation (Fusaroli et al., 2013;
Abney et al., 2014).
Dyadic coordination is one example of two interacting systems, but as we discussed at the outset, humans are composed
of many components across many scales that must coordinate in
order to function. Complexity matching suggests that the coordination of two subsystems in a single individual may manifest as
a convergence in their 1/f noise spectra when repeated measures
are taken. Evidence for spectral convergence in 1/f noise would
provide further evidence against the process summation account,
provided that this convergence was not simply a product of correlated time series. The present experiment tested this hypothesis
by measuring tapping deviations and key-press durations while
either synchronizing or syncopating with a metronome. Previous
studies have shown slightly stronger 1/f noise in timing deviations
during syncopation (Chen et al., 2001), so we varied tapping
between synchronization and syncopation to test whether an
effect on timing deviations would dissociate from an effect on keypress durations.
To further investigate coupling in terms of complexity matching, we wanted to compare 1/f noise in key-presses with other
fluctuations in physiological activity that either were or were not
responsive to the metronome. For the former, we presented a flash
of light with each auditory beat of the metronome, and measured

Frontiers in Human Neuroscience

fluctuations in the pupil dilation response across audiovisual
beats of the metronome. In the synchronization condition, pupil
and key-press responses occurred to the same stimuli, and roughly
at the same time. If this co-occurrence leads to coordination
between the neural and physiological systems underlying keypress and pupil responses, then we should observe coupling in
their 1/f noise signals. However, reflexive pupil dilation is coordinated by the autonomic nervous system (ANS), whereas learned
motor responses are coordinated by the central nervous system
(CNS). These two physiological systems may not measurably be
coupled when the body is at rest, as it is while sitting quietly
during a tapping task.
For physiological fluctuations that were not responsive to the
metronome, we measured heartbeat intervals. Resting heart rate
should not be driven by the negligible effort required to execute
each key-press. Yet healthy heartbeat intervals are known to
exhibit 1/f noise (Peng et al., 1995), and both heart rate and pupil
dilation are known to be coordinated by the ANS. Therefore, if the
ANS is not driven by the tapping task, then we expect 1/f noise in
pupil responses and heartbeat intervals to be coupled with each
other, but independent of 1/f noise in key-press durations and
timing deviations in tapping. The latter should be coupled with
each other through the CNS and the tapping task.
EXPERIMENTAL METHODS

Participants

Thirteen female and 13 male UC Merced students 18–30 years
of age participated in the experiment for course credit or as
volunteers. All reported having normal hearing and either normal
or corrected vision. Four participants were left-handed. Data from
two participants were removed due to equipment malfunction.
Apparatus

Pupil dilations were recorded using an Eye-Link II head mounted
video-based eye-tracker (SR Research Ltd.) with a temporal resolution of 500 Hz and a spatial resolution of 0.025◦ . The eye-tracker
uses two infrared LEDs mounted on the headband to illuminate
each eye, and pupil dilations were recorded from whichever
eye had the more accurate track. ECG samples were recorded
at 250 Hz using a Zephyr™ Bioharness 3 (Zephyr Technology,
Auckland, New Zealand) fastened around each participant as a
chest belt. Taps were recorded using a keyboard and MAX 6
(Cycling 74) experiment software. The audiovisual metronome
was presented using a 22-inch ThinkVision LCD monitor with
1280 × 1024 resolution, and Koss over-the-ear headphones. The
metronome consisted of a 200 ms tone played at a loud but
comfortable volume, synchronized with the display of a white
circle for 200 ms with 25 cm diameter on a blank screen viewed
from a 60 cm distance. A moderate level of light in the room was
held constant across all participants.
Procedure

Participants were instructed to sit quietly for 10 min at the
beginning of the experiment to allow the heart to settle to
its resting rate. Participants were instructed how to fasten the
Bioharness 3 to themselves, and the eye-tracker was calibrated
using the standard nine-point calibration method. Participants

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were randomly assigned to either the synchronization condition
(i.e., tapping in-phase with the metronome beats) or syncopation
condition (i.e., tapping in between the metronome beats). Participants were instructed that they would see and hear a metronome
beat presented at a constant, comfortable pace, and that they
should tap the spacebar either in time with the beat or in between
the beats. They were also instructed to keep their eyes fixated on
the screen for the duration of the experiment. Each participant
tapped to 1100 beats, which was set at a constant 800 ms interbeat interval. This interval was set to be within the range of the
healthy resting heart rate for young adults, and also to allow for
1100 beats to be administered in about 15 min.
Data pre-processing

The keyboard and heart rate apparatus directly produced time
series of key-press durations and heartbeat intervals. Timing deviations were computed by subtracting each key-press time from
its corresponding metronome beat. The interval timing of beats
was known with high precision, but the phase of the metronome
relative to key-press times was estimated for each participant. Any
error in this estimate was constant across each time series of keypresses, and therefore not a factor.
The eye-tracker produced a sampled time series of pupil
size that did not demarcate pupil responses to the flashes of
light. However, pupil dilation responses could be seen as a clear
waveform that rose and fell with roughly the same frequency as
the metronome. We wrote a simple signal processing algorithm
that found each peak value and trough value of the waveform.
The algorithm iterated through the sampled time series from
beginning to end, and determined “peak periods” and “trough
periods” relative to prior minima and maxima. Each peak period
started when the signal rose 100 units (approximately 5 µm
per unit) above the previous minimum, and ended when the
signal fell 100 units below its current peak value. Trough periods
were defined conversely, and minima below half the previous
maximum were discarded to remove eye blinks. The algorithm
produced one time series of peak values and a corresponding time
series of trough values for each participant. Analyses showed no
qualitative difference in results between peak and trough time
series, so here we report only analyses for peak dilation values.
The same trimming procedure was applied to all four time
series for each participant: values above and below 2.5 standard
deviations were removed. Then, if the remaining time series was
shorter than 1024 measurements, it was padded with mean values
to reach a length of 1024. If the remaining time series was longer
than 1024, an even amount of beginning and ending values were
trimmed to reach 1024 (with an extra value trimmed at the start
for odd numbers).

RESULTS
Each individual time series was submitted to spectral analysis,
and each resulting spectrum was logarithmically binned to create
nine estimates of spectral power in nine evenly spaced frequency
bins on a logarithmic scale (see also Thornton and Gilden, 2005).
Logarithmic binning ensures that the same amount of data goes
into each power estimate, and it also facilitates our spectral
matching analyses reported below.

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Mean spectra are plotted in Figure 1 for each of the four
dependent measures, separated by synchronization vs. syncopation. The graphs show that fluctuations for all measures in both
conditions followed a clear 1/f scaling relation. 1/f exponents
were estimated by fitting regression lines (and reversing their
signs to account for the inverse relationship) to spectra for individual participants: mean values combining the two metronome
conditions were 0.76 for timing deviations, 0.83 for key-press
durations, 0.90 for pupil dilations, and 0.81 for heartbeat intervals, where 1.0 is ideal 1/f noise. Estimated exponents for the
synchronization condition were not reliably different from the
syncopation condition—all t-tests were within-subjects and had
12−1 = 11 degrees of freedom, and all yielded t-values less than 1,
t (11) < 1. Thus we did not replicate a previous study showing
larger 1/f exponents for timing deviations when syncopating vs.
synchronizing to a metronome (Chen et al., 2001). However, we
found a trend in this direction (0.72 vs. 0.80, respectively), and
we used an audiovisual metronome whereas the previous study
used an audio-only metronome. Timing with pulsed visual signals
is known to be less accurate than for audio signals (Chen et al.,
2002), which might explain the small difference between our
results and previous results (but see Hove et al., 2010). In any case,
because there were no reliable effects of metronome condition, we
combined them in subsequent analyses.
To test for coupling among 1/f noises, we used a measure of
spectral convergence as an expression of complexity matching. In
particular, log power estimates were subtracted per frequency bin
for two given signals a and b, and the sum of their absolute values
served as our measure of spectral convergence:
| log Sf,a − log Sf,b |.
Ca,b =

Smaller values corresponded with more similar i.e., convergent
spectra. A measure of convergence was chosen over correlation of
estimated 1/f exponents because the former is sensitive to idiosyncrasies in the individual 1/f-like spectra that converge towards 1/f
in the average. Spectral Individual signals were compared because
convergence is hypothesized to occur for the motor processes and
ANS functions within individuals, as products of coordination,
rather than across individuals. We also compared spectral convergence with cross-correlation to test whether coupling could
be explained in terms of linear phase relations. Figures 2, 3 each
show two example signals from one participant in the syncopation
condition, along with two of the corresponding cross-correlation
functions and two pairs of spectra to visualize their differences.
These measures of coupling cannot be interpreted without a
baseline for comparison. With regards to spectral convergence,
a spectral difference of zero is the absolute maximal similarity,
but this measure does not have an inherent value or formula
corresponding to chance similarity. We created baselines from
surrogate pairings between signals from different participants.
In particular, for each original comparison between time series
A and B, a corresponding mean surrogate coupling was created
by pairing each original time series with all other participants,
i.e., 23 surrogate comparisons with A and 23 with B. Spectral
convergence values were averaged for each set of 56 surrogate
pairings to create a single baseline control for each pairing.

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FIGURE 1 | Logarithmically binned spectra plotted for each of the four dependent measures, separated by synchronization vs. syncopation, and
averaged across participants.

Comparisons between original and surrogate coupling values
showed a clear and consistent pattern of results: there was reliable spectral coupling between key-press timing deviations and
key-press durations, and also between peak pupil dilations and
heartbeat intervals. However, there were no reliable couplings
across key-press and ANS measures. To examine the two observed
effects of spectral convergence, Figure 4 plots the absolute log
differences as a function of frequency, averaged for originals and
baseline controls. Differences are generally greater in the lower
frequencies, which appears to be attributable to more overall
variability in spectral power relative to higher frequencies. Aside
from this effect, original pairings are seen to be more similar
to each other (i.e., smaller differences) compared with controls
across the range of measured frequencies, indicative of coupling
across timescales.
Statistical reliability of coupling was assessed using pairedsamples t-tests with Ca,b values for original pairings vs. their
yoked controls. Spectra for timing deviations were reliably more
similar to those for key-press durations compared with baseline
controls, t (23) = 2.52, p < 0.01, and the same was true for
pupil dilations and heartbeat intervals, t (23) = 2.18, p < 0.05.
No other comparisons for spectral convergence approached significance, all t (23) < 1. Mean Ca,b values (with standard errors)
for non-significant comparisons were the following for originals
and controls, respectively: 0.83 (0.06) and 0.81 (0.02) for timing

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deviations X pupil dilations, 0.81 (0.05) and 0.81 (0.02) for timing
deviations X heartbeat intervals, 0.88 (0.07) and 0.87 (0.03) for
key-press durations X pupil dilations, and 0.87 (0.04) and 0.87
(0.02) for key-press durations X heartbeat intervals. Altogether,
these tests show clear evidence of spectral convergence for keypress activity and for ANS activity, but not between the two.
Spectral convergence is not sensitive to the phase relation
between the signals because phase information is discarded by
spectral analysis. However, it is possible that phase relations
played a role in the observed effects because two highly correlated
signals (i.e., strong linear phase relation) will also have highly
similar spectra. Here we show that signals may appear to be
phase related, but that further analyses reveal these relations to be
purely spectral in nature. Linear cross-correlation is perhaps the
simplest and most common type of phase analysis, which tests
for phase relations across the range of available lags. Given that
we did not know a priori at what lag signals might be related, we
simply took the peak negative correlation as a measure of maximal
phase coupling. Preliminary analyses showed that peak negative
correlations were slightly stronger than peak positive correlations,
although both were weak: mean magnitudes varied within the
small range of 0.15–0.21 across pairwise comparisons, at mean
lags from about 140–340 beats apart. We report results with peak
negative correlations, but results were not qualitatively different
from peak positive correlations.

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FIGURE 2 | Time series of peak key-press response times and
durations for one participant (above), where the x-axis was the
sequence of over 1000 key-presses, and the y-axis is normalized
times and durations with 0 mean and showing +/−2.5 standard

deviations. Corresponding cross-correlation function and spectra are
shown below. The red dashed circle shows the peak negative
correlation, and the dashed lines between spectra show absolute log
differences.

FIGURE 3 | Time series of pupil dilation responses and heartbeat intervals for one participant, along with the corresponding cross-correlation
function and spectra. The red dashed circle shows the peak negative correlation, and the red dashed lines between spectra show absolute log differences.

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FIGURE 4 | Mean spectral differences |log(Sf,a )− log(Sf,b )| plotted as a function of frequency, for originals and surrogate controls, with standard
error bars.

We conducted the same baseline control analysis as for spectral convergence, and we found the same pattern of effects as
for spectral convergence: peaks were significantly more negative
between timing deviations and key-press durations compared
with baseline controls, t (23) = 1.82, p < 0.05, and the same was
true for pupil dilations and heartbeat intervals, t (23) = 2.18, p <
0.05. And again, no other comparisons for spectral convergence
approached significance, all t (23) < 1.93, p < 0.05. A summary of
the correlational and spectral coupling results is shown in Table 1,
which contains mean differences between coupling measures for
original pairings minus baseline controls, for all pairwise comparisons. The table shows that, for the two reliable comparisons,
differences from baseline were proportionally greater for spectral
coupling than for correlational coupling.
These results suggest that simple linear phase relations may
have contributed to the observed effects of spectral convergence,
but it is curious that peak lags were so far apart. We do not know
what type of phase coupling would explain phase relations offset
by 2–4 min and well over 100 responses. An alternate possibility

Table 1 | Mean correlational (top) and spectral (bottom) coupling
effects for all pairwise comparisons between the four dependent
measures (TD = Timing Deviations, KD = Key-Press Durations, PD =
Pupil Dilations).
Peak CC−Control
Timing Deviations
Key-Press Durations
Pupil Dilations
Heartbeat Intervals
C a,b –Control
Timing Deviations
Key-Press Durations
Pupil Dilations
Heartbeat Intervals

TD

KD

PD

0.036
−0.001
0.001

0.017
−0.010

0.032

0.829
−0.126
0.066

−0.094
−0.025

0.821

Statistically reliable effects are in bold, and the signs are reversed for correlational
effects for consistent interpretation with spectral effects.

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is that effects of spectral convergence can lead to spurious phase
coupling when measured using our surrogate baseline analysis.
We tested this alternative by using iterated amplitude adapted
Fourier transform (IAAFT; Theiler et al., 1992; Schreiber and
Schmitz, 1996), which scrambles phase relations in a given time
series while preserving its spectral properties. If there is truly
phase coupling, then cross-correlations for original comparisons
should be stronger than those for the corresponding scrambled
time series. Each surrogate pair had one original time series and
one scrambled time series, and each original time series was
paired with 100 scrambled series. We used paired-sampled t-tests
to compare each original peak cross-correlation with the mean of
its corresponding surrogate set.
Results from the IAAFT surrogate analysis revealed that there
was no reliable linear phase coupling among any of the four
dependent measures, as measured by peak cross-correlations. Surrogates were no different from originals for timing deviations and
key-press durations, t (23) = 1.5, p > 0.14, nor for pupil dilations
and heartbeat intervals, t (23) = 1.4, p > 0.17. The remaining
comparisons were all near t (23) ∼ 1.4 or less. The IAAFT surrogate
analysis provides additional evidence that the observed couplings
in key-press responses and in measures of ANS functions (but
not between the two) were expressed in terms of their power law
spectral distributions, and not their phase relations (see also Kello
et al., 2007).

DISCUSSION
The aim of the present experiment was to add to the body
of evidence on the origins of 1/f noise in human timing, particularly with respect to two domain-general explanations. We
employed a standard tapping task with synchronization and syncopation conditions, and we measured deviations in timing from
a metronome. Our contribution was to record and analyze three
additional repeated measures that varied in their relationship to
timing deviations and the metronome. All four measures exhibited clear 1/f noise, consistent with previous studies suggesting
that 1/f noise will manifest for any repeated measure of human

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behavior that is minimally perturbed and minimally constrained
from one measurement to the next (Kello et al., 2010).
Our goal in eliciting these 1/f signals was to examine the
relationships among them, as a way to test and elaborate upon
the process summation vs. interdependent coordination accounts.
The process summation account has served as a default explanation for many researchers over the years, in part because repeated
measures of human timing and other behaviors might plausibly
“pick up” on fluctuations in physiological and cognitive processes
ranging across spatial and temporal scales of the brain and body.
However, the idea of process summation has been called into
question by a number of recent results. Our findings cast further
doubt on this account because all four dependent measures exhibited distinct 1/f signals in terms of their phase relations—none
were reliably cross-correlated relative to IAAFT controls. These
findings are difficult to explain because at least some of these
measures should pick up on the same summation of processes,
which should result in reliable near-lag zero correlations. This is
not what we found.
One could argue that each of our dependent measures tapped
into a (mostly) distinct set of processes that each summed to produce distinct 1/f noises. However, while the 1/f signals had mostly
unique phase profiles, their spectra were not fully distinct. Instead
we found that spectra converged for timing deviations and keypress durations, and separately for pupil dilations and heartbeat
intervals. These results indicate that 1/f fluctuations in different
aspects of key-presses were coordinated across timescales, and
likewise for ANS activity.
Interdependent coordination is in a better position to accommodate these results. We started with the basic premise that
human timing is part and parcel with coordination, and that coordination requires a balanced, flexible coupling among whatever
components are being coordinated. Flexible coupling is hypothesized to support the soft-assembly of sensorimotor function, and
other types of biological and cognitive functions (Kello and Van
Orden, 2009). A defining feature of highly adaptive systems is that
their components can play multiple functional roles depending on
context. In order to take on these different roles, components need
to fall into different interdependent relationships under different
conditions.
It is challenging to understand how biological and cognitive
systems are so flexible. One valid and necessary approach is
to study very particular examples and develop domain-specific
theories to explain them. For instance, there are specific mechanisms of plasticity that re-organize sensorimotor maps in prism
adaptation studies (Redding et al., 2005) or amputation cases
(Sanes and Donoghue, 2000). However, it is equally valid and
necessary to study basic principles from which many or even
all mechanisms of sensorimotor function draw their flexibility.
Metastability is one such principle that explicitly predicts 1/f noise
to be a pervasive feature of systems of interdependent components
poised near critical points. Theories of SOC have been formulated
to explain why critical points appear to be so common to complex
systems.
Metastability can explain 1/f noise in all four dependent measures, and it is consistent with the finding that two and only
two pairs of these measures were coupled. However, the concept

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of metastability alone does not explain how spectral coupling
can occur across timescales distinct from any phase coupling,
nor does it explain the particular couplings of dependent measures that were observed. To explain the particular couplings
observed, we will ultimately need domain-specific theories of
manual sensorimotor control, and ANS function. For now, we can
say that timing deviations and key-press durations measured two
aspects of key-press dynamics that were coupled by the tapping
task, and that coupling between pupil dilation and heartbeat
intervals is “hard-wired” into the ANS. Moreover, tapping to
a metronome while at rest did not enforce any physiological
or informational demands on coupling between key-presses and
the ANS. We conjecture that these systems would couple under
more strenuous conditions, such as a sport with intense hand-eye
coordination.
Finally, to explain spectral coupling across timescales, we refer
to formal analyses of complexity matching that show maximal
information exchange between complex systems with convergent power laws, yet distinct phase portraits. It is reasonable
to assume that coordination is facilitated by maximal information exchange, and that key-press responses require information
exchange among neural and motor processes involved in depressing and releasing the key on each response. It is also reasonable to
assume that information must be exchanged among components
of the ANS. As mentioned in the Introduction section, we do not
mean information exchange in the sense of sending bits between
components and subsystems. Instead we mean that components
rely on each other to support sensorimotor and physiological
functions (Kello and Van Orden, 2009). These functions are
inherently multiscale, and hence the mutual interdependence that
underlies them must span a range of spatial and temporal scales.
Computational models based in metastability, such as critical
branching networks (Kello, 2013), are needed to express formal
theories of complexity matching in terms of neural, sensorimotor,
and cognitive functions.

ACKNOWLEDGMENTS
The experiment reported herein was approved by the UC Merced
Institutional Review Board, and conformed to the board’s regulatory standards. We thank the reviewers for their helpful comments
and suggestions for additional analyses.
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Testing for nonlinearity in time series: the method of surrogate data. Physica D
58, 77–94. doi: 10.1016/0167-2789(92)90102-s
Thornton, T. L., and Gilden, D. L. (2005). Provenance of correlations in psychological data. Psychon. Bull. Rev. 12, 409–441. doi: 10.3758/bf03193785
Torre, K., and Wagenmakers, E.-J. (2009). Theories and models for 1/f[beta] noise
in human movement science. Hum. Mov. Sci. 28, 297–318. doi: 10.1016/j.
humov.2009.01.001
Usher, M., Stemmler, M., and Olami, Z. (1995). Dynamic pattern-formation
leads to 1/f noise in neural populations. Phys. Rev. Lett. 74, 326–329. doi: 10.
1103/physrevlett.74.326
Van Orden, G. C., Holden, J. G., and Turvey, M. T. (2003). Self-organization of
cognitive performance. J. Exp. Psychol. Gen. 132, 331–350. doi: 10.1037/00963445.132.3.331
Van Orden, G. C., Holden, J. G., and Turvey, M. (2005). Human cognition and 1/f
scaling. J. Exp. Psychol. Gen. 134, 117–123. doi: 10.1037/0096-3445.134.1.117
Wagenmakers, E.-J., Farrell, S., and Ratcliff, R. (2004). Estimation and interpretation of l/f alpha noise in human cognition. Psychon. Bull. Rev. 11, 579–615.
doi: 10.3758/bf03196615
Wagenmakers, E. J., Farrell, S., and Ratcliff, R. (2005). Human cognition and a pile
of sand: a discussion on serial correlations and self-organized criticality. J. Exp.
Psychol. Gen. 135, 108–116. doi: 10.1037/0096-3445.134.1.108
Ward, L. M. (2002). Dynamical Cognitive Science. Cambridge, MA: MIT Press.
West, B. J., Geneston, E. L., and Grigolini, P. (2008). Maximizing information
exchange between complex networks. Phys. Rep. 468, 1–99. doi: 10.1016/j.
physrep.2008.06.003
Wing, A. M., and Kristofferson, A. B. (1973). Response delays and the timing of discrete motor responses. Percept. Psychophys. 14, 5–12. doi: 10.3758/bf03198607
Conflict of Interest Statement: The authors declare that the research was conducted
in the absence of any commercial or financial relationships that could be construed
as a potential conflict of interest.
Received: 05 May 2014; accepted: 26 August 2014; published online: 11 September
2014.
Citation: Rigoli LM, Holman D, Spivey MJ and Kello CT (2014) Spectral convergence in tapping and physiological fluctuations: coupling and independence of 1/f
noise in the central and autonomic nervous systems. Front. Hum. Neurosci. 8:713.
doi: 10.3389/fnhum.2014.00713
This article was submitted to the journal Frontiers in Human Neuroscience.
Copyright © 2014 Rigoli, Holman, Spivey and Kello. This is an open-access article
distributed under the terms of the Creative Commons Attribution License (CC BY).
The use, distribution or reproduction in other forums is permitted, provided the
original author(s) or licensor are credited and that the original publication in this
journal is cited, in accordance with accepted academic practice. No use, distribution
or reproduction is permitted which does not comply with these terms.

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Michael Spivey
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