Download Hypothesized neural dynamics of working memory

Brain Research Bulletin, Vol. 50, No. 2, pp. 77–93, 1999
Copyright © 1999 Elsevier Science Inc.
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REVIEW ARTICLE
Hypothesized neural dynamics of working memory:
Several chunks might be marked simultaneously
by harmonic frequencies within an octave band
of brain waves
Robert B. Glassman
Department of Psychology, Lake Forest College, Lake Forest, IL, USA
[Received 11 November 1998; Revised 17 June 1999; Accepted 21 June 1999]
ABSTRACT: The capacity of working memory (WM) for up to
about seven simple items holds true both for humans and other
species, and may depend upon a common characteristic of
mammalian brains. This paper develops the conjecture that
each WM item is represented by a different brain wave frequency. The binding-by-synchrony hypothesis, now being
widely investigated, holds that the attributes of a single cognitive element cohere because electroencephalogram (EEG) synchrony temporarily unifies their substrates, which are distributed among different brain regions. However, thought requires
keeping active more than one cognitive element, or WM
“chunk,” at a time. If there is indeed a brain wave frequency
code for cognitive item-representations that are copresent
within the same volume of neural tissue, the simple mathematical relationships of harmonies could provide a basis for maintaining distinctness and for orderly changes. Thus, a basic
aspect of music may provide a model for an essential characteristic of WM. Music is a communicative phenomenon of “intermediate complexity,” more highly organized than the firing
patterns of individual neurons but simpler than language. If
there is a distinct level of neural processing within which the
microscopic physiological activity of neurons self-organizes
into the macroscopic psychology of the organism, it might
require such moderate complexity. Some of the obvious properties of music— orderly mixing and transitions among limited
numbers of signal lines—are suggestive of properties that a
dynamic neural process might need in order to organize and
reorganize WM markers, but there are a number of additional,
nonobvious advantageous properties of summating sinusoids
in music-like relationships. In particular, harmonies register a
stable periodic signal in the briefest possible time. Thus, the
regularity of summating sinusoids whose frequencies bear harmony ratios suggests a particular kind of tradeoff between
parallel and serial processing. When there are few copresent
waves, at EEG frequencies, this sort of parallel coding retains
behaviorally meaningful brief periods. A necessary companion
hypothesis is that the brain wave frequencies underlying WM
are confined to a single octave; that is, the upper and lower
bounds of the band are in the ratio of 2:1. This hypothesized
restriction, suggested by an empirical property of EEG bands
that has been widely reported but rarely commented upon, has
the important property of precluding spurious difference
rhythms. A restriction to an octave, of “harmonious” frequencymarkers for WM items, also seems consistent with a great deal
of behavioral data suggesting that WM comprises a rapidly
fading trace process in which only up to three or four itemrepresentations are strongly activated simultaneously. There is
also an additional, sequential renewal-or-revision process,
within which up to another three or four items are being actively
refreshed by rehearsal or replaced. Such serial processing may
involve a less stringent octave band crowding problem.
© 1999 Elsevier Science Inc.
KEY WORDS: Binding, Brain waves, Consciousness, EEG, Harmonic, Harmony, Short-term memory, Synchrony, Working
memory capacity.
MIND/BRAIN RELATEDNESS WITHIN WORKING
MEMORY CAPACITY
The Enigma of Modest Working Memory Capacity
A basic characteristic of cognition is that we can consciously
commit to little more than one organized behavior or line of
thought at a time [19,94]. Taking elementary items— digits, letters,
or short words—as expedient objects for quantitative measurement
of capacity, George Miller’s 1956 classic review characterized
what is now usually called working memory (WM), as having a
limited span for ordered recall of about seven independent, simple
items—“the magical number”([83]; see [8,110] for systematic
historical reviews of the terms “working memory” and“ short-term
memory”). This finding is easily replicated, and versions of this
* Address for correspondence: Prof. Robert B. Glassman, Department of Psychology, Lake Forest College, Lake Forest, IL 60045-2399, USA. Fax:
⫹1-(847)-735-6195; E-mail: [email protected]
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experiment are regularly found in software packages for teaching
undergraduate psychology laboratories (e.g. Mind Scope, Version
␤.2c West Publishing Company, 1991). Miller [83] also reviewed
many findings indicating that when people make quantitative judgments using a linear rating scale (“absolute judgment”), such
estimations are reliable only so long as we are required to use up
to 7 ⫾ 2 scale divisions. Remarkably, this constraint on numerosity of rating scale divisions for reliable quantitative categorization
holds almost independently of the particular dimension of experience or size of the interval of the dimension along which magnitudes are being estimated (e.g., length of line, pitch of tones).
Although Miller was circumspect in joining his considerations of
immediate memory capacity and rating scale capacity, the latter set
of findings concerns our ability to maintain a number of ordered
items in mind at once, and thus suggests an underlying property of
WM [45].
Other methods of measuring recent memory routinely yield
quantitative results not very different from those with ordered
recall tasks. When participants are asked to reproduce in any order
(method of “free recall”) an arbitrary list of 20, 30, or 40 unrelated
words after a single hearing, many individuals recall a total of
more than nine words. Yet across a group of participants only
about nine of the words, most of them from the end of the list
(“recency portion of the serial position curve”), stand out from the
rest in probability of recall, and only about four or five of these
words are recalled at better than 50% probability [89]. This is now
also a “textbook finding” about memory (e.g., [6], pp. 156 –158,
esp. Fig. 4-4).
Today, a prevailing judgment of cognitive psychologists is that
WM capacity is somewhat less than seven items. For simple verbal
items recalled with high reliability, WM capacity seems to be
closer to 6 [8,120]. “Complex span” procedures, in which simple
verbal or mathematical tasks are interspersed with the items to be
remembered yield WM capacity of two to five items [57]. During
any given moment in a typical laboratory procedure, WM seems to
briefly capture about three or four items at once in a trace, as
suggested by the recency portion of the serial position curve [89].
The remainder of the six, or 7 ⫾ 2, items are meanwhile undergoing refreshment as the participant sequentially rehearses as
many items as can be mentally articulated within the approximately 2-s span of the “phonological loop” ([8]; also see
[24,119,139]). When rehearsal was precluded by the requirement
to constantly update a WM trace, capacity was about four items
[138].
Interestingly, the recency portion of the serial position curve
comprises approximately the same numerousness as is evident in
simple perceptual experiments requiring a participant to report the
number of a briefly seen small set of dots, with no time to
sequentially count (about four or five dots; [83,123]; pp. 422– 424
of ref. [141]). Recent research on briefly flashed arrays of color
patches of lines at different orientations has revealed visual WM
capacity for up to four items, whether or not they comprise single
features or conjunctions of features [78]. Although much contemporary neurocognitive theory defines multiple subsystems of shortand long-term memory [72,110,129], the foregoing coincidences
of modest numerousness remain suggestive of a common neurocognitive source, or reason, for limitation of simultaneously graspable, separate elementary items.
Comparing Species in a Spatial WM Task: Persistence of 7 ⫾ 2
In our laboratory, to better link knowledge of human WM to
that of other species, we have tested human participants in versions
of the radial-arm maze, a device that has been widely used for
testing laboratory rats. Our procedure discourages strategic use of
GLASSMAN
response patterns, and thus presses the human participants to
contend with the same memory load as do rats in radial mazes. Our
findings, with probabilistic correction for guessing, indicate that
capacity for remembering locations that have just been visited falls
into the range 7 ⫾ 2; the quantitative results for humans and lab
rats are about the same with mazes having 8, 13, or 17 arms (i.e.,
numbers of sites to remember; [47,48]).
It is not known whether the modest capacity of WM for simple
items involves a constraint necessary for the inner logic of memory, for example whether it concerns manageable combinatorial
effects in organizing coherent knowledge (e.g., see [79,137];
briefly reviewed in [45]), or whether it is merely the best that
evolutionary natural selection has done so far. In either case we
may ask what attributes of the brain are most relevant.
Attributes needed for WM at the neural level. Which attributes
of the brain seem to have appropriate potential for quickly referencing stored information, maintaining and creating distinctions
among a small number of simultaneously active signals, and for
unifying some of those signals at a pace that might be relevant to
WM?
Associated Levels, or Scales, of Mind/Brain Dynamics
Organisms’ behaviors are organized in three dimensions of
space and one dimension of time. This obvious consideration
might be placed tendentiously alongside the most general facts
about the brain: It is an exquisite three-dimensional organization of
elements that are largely linelike, or one-dimensional, in their
microscopic structure. In some way, the linear temporal sequences
of electrical and chemical events that are embedded along those
physical lines must be causally related, comprehensively, to all
aspects of organisms’ behaviors.
To avoid a gappy, mystical dualism we must assume certain
kinds of connections between mind and brain, before we have
empirical knowledge of their character. Thus, we assume that there
are comprehensive causal links between the microscopic, relatively high-frequency scales of neuronal and subneuronal activity
and the more middle-sized, mid-frequency scales of thought, emotion, and behavior. Each of these levels has been the subject of a
great deal of research, yet they remain largely separated in our
understanding. Myriad neuroscientific findings remain only patchily connected to myriad data about the experimental psychology of
mind. The “engram” [15] remains sufficiently puzzling that some
particle physicists have been taken seriously in a speculative
excursion that neglects the middle levels and, with that, the explanatory connectedness across scales that science requires. By
placing neurobehavioral integration out of reach of neuroscience
and experimental psychology, and attributing it to subatomic quantum phenomena, they introduce an element of mysticism (see e.g.,
[67] for a critical review).
If the mind/brain problem is inherently unlucky for the scientific enterprise, it may be beyond heuristics. Then, achieving an
adequate description of a memory event will require piecing together knowledge of thousands of neural elements behaving in
diverse ways, not describable in any general terms. Such a description will be too complex to be intuitively satisfying or to
deserve to be called “understanding.” Here, nature may be complexly mazelike, or like an extremely obscure cryptogram, refractory to analogical leaps. Does scientific intuition end at the mind/
brain junction?
A Possible Intermediate Scale: Brain Waves
Alternatively, there may be physiologically identifiable, robust
transitional phenomena at an intermediate scale. During the past
several years some neuroscientific ferment has centered on possi-
WORKING MEMORY AND EEG HARMONIES
ble significances of oscillatory activity at different spatiotemporal
scales (e.g., [36]). For example, electroencephalogram (EEG) activity may play causal roles, in some of its manifestations, even if
it is epiphenomenal in others. Graded potentials and currents are
more ubiquitous aspects of neural physiology than are the action
potentials that they sometimes generate. So, also, are oscillatory
graded electrical phenomena necessarily more ubiquitous than the
trains of action potentials that they may generate.
The possible functionality of graded activity synchronized
across neural aggregates is controversial. Oscillations of neural
masses are well-known to be reliable correlates of such global
behavioral phenomena as sleep and attention, and the behavioral
effectiveness of electrical brain stimulation at natural oscillatory
frequencies suggests that such mass-oscillations are part of a
causal sequence ([64]; also reviewed in [50]). However, we do not
yet know a great deal about whether neural oscillations play a role
in more finely differentiated functions of cognition [32]. One
theory proposes that neuroelectric oscillations in resting wakefulness and in deep sleep serve the primary function of refreshing
synaptic memories across the brain ([71]; also [50]).
Binding-by-synchrony: A possible neural dynamic basis of a
single WM chunk. Synchrony of neuroelectric oscillations may
bind together neural activity that is broadly distributed among
brain systems, evoked by the various attributes of a stimulus object
or other cognitive item [73,84,85,117,126,134]; and see, e.g., [35],
for evidence conflicting with the binding-by-synchrony hypothesis). The gamma band of EEG has been proposed as the vehicle for
such binding [26]; however, recent evidence suggests a role of low
beta frequencies (13–21 Hz) in coherence of visual and motor
cortex during a tracking task, in humans [22]. The binding hypothesis is now being widely considered, at a time when neuroscientific
empirical techniques from macroscopic to microscopic levels of
analysis are well-prepared for it. Relevant examples are in coherence analyses of EEG activity [23,97,103] and in studies of active
and passive properties of dendrites, which suggest that they might
exploit subtle differentiations in timing and synchrony of incoming
signals, to yield local dynamic patterns for memory retrieval or
storage (e.g, [20,86,121,122]). A series of satellite symposia at the
Annual Society for Neuroscience meetings has been devoted
largely to these issues [36].
A possible mechanism of causal linkage, between widespread
EEG synchronies and more microscopic activation patterns at the
neuronal level, involves the concept of the “Hebbian neuron.” That
is, EEG synchrony implies spatial and temporal convergence of
excitation (or of inhibition) upon neurons in the areas of the brain
that are within its field. This suggests a means for neural structural
revisions, which might underlie the creation of new long-term
memories: “Neurons that fire together wire together” [14].
Binding-by-synchrony, with action potentials resulting as target neurons are pushed past their thresholds, is readily envisioned
on a foundation of classical integrate-and-fire neurons, which
respond to synaptic transmission with intradendritic passive summations of graded potentials. Active dendritic potentials [86,121]
may also be involved.
Conceivably, volume conduction [1] provides still another possible route for information processing across larger masses of brain
tissue. Volume conduction is like an electrical short-circuit, not
dependent upon the microanatomical routes of connecting fibers
and their synaptic contacts. However, ordinary synaptic transmission and dendritic electrodynamics may be so much stronger than
volume conduction that the latter is no more than an epiphenomenal signature, visible to a researcher reading an amplified EEG
recording from a point in the brain, but “invisible” to the tissue
itself at that point in the brain. Indeed, if short-circuiting by
volume conduction were too powerful, that would negate the very
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reason for being of the brain’s exquisite microscopic spatial organization and neurochemical specificities. Nevertheless, it seems
possible that a gentler influence of volume conduction might
sometimes act as an additive factor that achieves effectiveness by
working “at the margin,” or near threshold.
Informational combinatorics? Subtle volume conductive influences might act in summative or synergistic interactions with other
neural information coding processes, perhaps at temporarily sensitized particular locations, which thereby act as receivers of an
electrical rhythm “broadcast” across a volume of tissue. Such an
effect might be analogous to the various degrees of diffuse influence effected by neurochemical modulators and by tiny concentrations of hormones in the body [1]. A combinatorial effect of
diffuse activity that is no more than weakly informational might
have a disproportionately large, even exponential, effect on the
information capacity of the system of which it is a part [41]. For
example, if only two different state-markers were provided by
diffuse EEG activity (e.g., via two different typical frequencies,
amplitudes, or phases), that could double the number of possible
informational states of the system as a whole [41].
HARMONIC PRINCIPLES AND HOW THEY MIGHT
APPLY TO BRAIN WAVES
Octave EEG Bands: Informational Safe Zones
General knowledge about brain waves is consistent with the
inference that each band of EEG falls within approximately an
octave; that is, the high-end frequency is about twice the low-end
frequency. This seems particularly clear for theta (ca. 4 –7 Hz),
alpha (ca. 8 –14 Hz), and gamma (ca. 40 – 80 or 60 –90 Hz). (See,
e.g., [93] pp. 5–9; [37], for general description of traditional EEG
categories, and [134] or [26] for a description of the gamma band.)
Might such a band limitation have adaptive significance?
A possible reason for the apparent octave restriction is that any
set of two or more frequencies occurring at the same time within
an octave cannot create beats [98,114] nor any difference rhythms
of quadratic or higher order [101] which are also within that same
octave. A basic arithmetic principle is that for any two numbers in
the interval between x and 2x, the difference of the two numbers
cannot fall within the interval x to 2x, even when either of them is
raised to any exponent. Thus, if brain information processing uses
regular rhythms in time series as signals, the octave comprises a
safe zone, protected from confusions that could be caused by
emergent beats or other difference rhythms [43].
A brain wave octave of harmonies: Possible neural dynamic
basis of multiple WM chunks. Consideration of basic properties of
musical scales (e.g., [56,98,114]) suggests that the octave ratio
might naturally encompass a biological informational band that is
restricted to a small number of simple signals. While some present
theory and empirical testing concern the possibility of quasiperiodic, “chaotic” generators of separable components of EEG signals
[11,33,114] the simple mathematical relationships of harmonies
[98,114] remain unexplored as a possible foundation of signal
composition. Indeed, hypothesizing relevance to EEG of harmony
ratios seems a natural extension of the hypothesis of binding-bysynchrony for single percepts or concepts.
Brief Historical Note
Johannes Kepler’s fascination with harmonies and British
chemist John Newlands’ conjecture about octaves were followed,
respectively, by the discoveries of “Kepler’s Laws” describing the
periods and spacing of the planets ([76], p. 112; [96], pp. 70 –71),
and the periodic table, with its prominent groupings of eight
elements ([100], pp. 194 –195; [143], pp. 261–263). Historians of
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GLASSMAN
science are uncertain, however, about the issue of logical entailment, with these two cases. Speculation about “vibrations” as the
foundation of neural codes appears in Isaac Newton’s 1723 edition
of Principia (91); this was extended by David Hartley in a 1749
book [54]. Such speculation was rejected as untestable by Priestley
[112], but the essayist Coleridge [127,128] objected. Speculations
about neural vibrations recur historically in an 1884 book by
Henry Maudsley [80] and one by T. H. Huxley in 1880 [63].
Recently, Nunez [93] also has suggested exploiting knowledge of
the physics of music for understanding EEG wave phenomena (pp.
139 –144).
Higher-Level Binding of Memory Item-Markers: Simultaneous
Synchronies
The present hypothesis is as follows:
Specific EEG frequencies comprise the distinguishing marks for simple
items in WM, i.e., WM “chunks.” The capacity to hold several chunks in
mind at one time is limited by basic properties of oscillations when they are
simultaneously present in a medium. The simple ratio relationships of
harmonies, including octaves, are relevant to this limit.
Fundamental Problems of WM Design: Elemental Distinctness
and Miscibility
The natural functions of working memory are to serve the
ongoing organization and updating of cognition, and ongoing
sensory-motor interaction with the world. Whatever codes WM
uses must have powerful implications for the efficacy of such
commerce, and for survival. The brain, remarkably, conserves the
restriction of modest numerousness of elements in WM capacity,
across wide variations in size, duration, and dimensional ranges of
the referents of WM items, and surprisingly wide variations in the
pace of some of the tasks in which WM items are embedded
(reviewed in [45]).
If the brain follows the evolutionary principle of hierarchical
decomposability—also an aspect of engineering and of structured
computer programming [51], then the general problem of unifying
individual WM chunks must be solved at one sublevel, relieving
higher sublevels of the burden of micromanaging the composition
of the elements which interact as wholes at those higher levels. The
most general problem of interaction of elements in a system of
working memory must concern clarity of distinctness among elements. Because WM is the crucible for dynamic revision of cognition, its most general issues must also include the means of
interaction among elements: When do interactions suggest revising
of chunks and how or when is that task shifted elsewhere (“down
to a lower level”?) to enable WM to get on with the next moment’s
business?
More specifically, the nature of WM seems to require the
following four subtasks: (1) entrances of elements driven by
perception or by elicited associations; (2) exits of elements as a
result of one of the following—passive loss following attentional
neglect, exceeding the modest capacity, active rejection, or storage
in long-term memory; (3) new differentions of “smaller” elements;
and (4) new combinations of “larger” elements into single chunks.
As already noted, these fundamental issues of WM design may
imply a mind/brain research issue of immense complexity, with
diverse mechanisms. However, as suggested above, the temporal
structure of music, as well as a listener’s experience of interacting
musical lines, suggest the present harmonics model might be fertile
enough for possible future expansion beyond the present main
concern with the state of WM as it briefly holds a single complement of several chunks.
If each of the hypothesized chunk-signals in WM must be
optimally discriminable, then their frequencies might well have
FIG. 1. Illustration of the fact that maximizing the size of all subdivisions
of a line segment requires making all the segments equal. Both of the
illustrated line segments have the same average size of the five intervals.
evolved even spacing within the octave. For a fixed number of
divisions of a linear interval, any one subdivision can become
longer only at the expense of making another of the subdivisions
shorter by the same amount (Fig. 1). The maximal differences
among dividing points can be achieved via equal differences
among neighboring pairs. However, when the interval in question
is a range of frequencies of an oscillatory signal, the importance of
harmony ratios may supervene and force a compromise. A selfconsistent set of harmony ratios would allow the differences
between neighboring pairs of signals to be approximately, but not
exactly, equal.
Properties of Sinusoidal Waveforms and Harmonic
Relationships
There follows a basic review of properties of sinusoids and of
harmonic and harmony ratios, with indications of how these properties might be useful in tagging information. Among these, the
hypothesis that brain waves in harmony ratios serve as WM
chunk-markers entails an implication regarding the pace of cognition and a tradeoff between advantages of serial and parallel
processing.
The simplest waveform. Rhythms are a rudimentary form of
activity, organized linearly in time. Although a sinusoidal curve
has a beguiling shape, elementary trigonometry tells us that it is
generated simply as the projection on a line of a radius revolving
in a circle ([114], pp. 143–146). Sinusoids represent an interesting
abstract similarity among a large class of disparate physical phenomena in various media, including pendulums, sounds and other
mechanical vibrations, waves in water, aspects of some biological
rhythms, and the natural oscillation of a simple electrical circuit in
which a capacitor discharges through an inductor [62,118,140].
Brief overview of harmonic and harmony mathematical relations. It surprises the intuition that the simple sinusoid shape, in
various frequency, amplitude, and phase summations, can be used
to build any single-valued waveform. About 70 years ago, Dayton
Clarence Miller delightfully illustrated Fourier synthesis of a repeating profile of a woman’s face, using only the delicate mechanical generators of sinusoids then available (Fig. 2; from [82], pp.
119 –120). Synthesis of any waveform, in like manner, involves
harmonic exact multiples of the lowest (fundamental) frequency.
Harmonics and harmonies are the two related ways in which
rhythmic signals can have integer-ratio relations. Harmonics occur
when one signal frequency is a whole-number multiple of the other
and is therefore in a different octave; harmonies occur, when the
signals are related to each other by fractional multiples that are
low-number ratios. Prototypically, harmonies occur within the
same octave, although a doubling, halving, or any 2n multiple of
any of the notes in a musical harmony, which places that note in
a different octave from the rest, will remain harmonious with the
other notes. Harmonics of a given fundamental frequency are often
in the same octave with each other, with this octave above the
octave of the fundamental. This implies that all the harmonics of
a fundamental are also in harmony ratio relationships with each
other [98,113,114].
WORKING MEMORY AND EEG HARMONIES
FIG. 2. Reproduction from Dayton Clarence Miller’s book, The Science of
Musical Sounds, published in 1926. “Figure 94. Reproduction of a portrait
profile by harmonic analysis and synthesis” and “Fig. 95. Waveform
obtained by repeating a portrait profie.” The harmonic analyzer and synthesizer used by Miller were purely mechanical devices. The equation of
the curve is given on p. 119 of Miller’s text.
Harmonics: Further Explanation
A harmonic is an exact integer multiple of another frequency,
called the “fundamental.” For example, middle C (C4) has been
standardized at a frequency just under 262 Hz. The “first overtone,” or “second harmonic” of this frequency comprises twice as
many oscillations per second: 2 ⫻ 262 ⫽ 524 Hz (C5). The second
overtone, or third harmonic, is 3 ⫻ 262 ⫽ 786 Hz (G5), etc. For all
notes of a given musical instrument the distribution of relative
amplitudes of the set of harmonics is fairly constant, and that
amplitude distribution of harmonics comprises the distinctive timbre, which clearly differentiates a piano from a clarinet or a viola,
etc.
Harmonics must involve integer multiples because only integer
multiples can possibly stay synchronized with the fundamental
frequency. Thus, harmonics are required for a structured, regular
waveform. Figure 3 illustrates this point with a sinusoidal fundamental (y ⫽ sin x) and an equal-amplitude harmonic three times
the frequency of the fundamental (y ⫽ sin 3x). When the two
signal sources are summated the harmonic always rides at the same
phase positions on the fundamental. Even if these two waves had
different relative phases, the exact harmonic relationship of 3:1
would ensure that the resulting wave shape would repeat itself over
and over indefinitely, at the same frequency as the one in this
illustration. Helmholtz’s 1885 classic exposition, based theoretically on his use of Fourier mathematics and empirically on mechanical resonators that did not permit direct writeout of waveforms, remains among the most extensive and lucid [56].
Harmonies: Further Explanation
Tones an octave apart (frequency ratio of 2/1) sound highly
consonant, so much so that they are sometimes mistaken for each
other. What happens when tones are closer in pitch? The maximum
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FIG. 3. Illustration of the synchrony that can occur only with waves that
bear a harmonic relationship with each other. In this case (sin X) and (sin
3X) are plotted individually and in summation.
possible harmonious spacing that is bounded by both ends of an
octave involves the highest possible whole-number ratio of frequencies that is lower than the octave ratio of 2/1. Thus, the
“perfect fifth” (for example C to G), comprises a ratio between
frequencies of 3/2 (e.g., G4/C4 ⬇ 392 Hz/261 Hz ⬇ 3/2. The next
lower ratio between small integers is 4/3, or the “perfect fourth”
(e.g., C to F); this pairing also sounds highly consonant so long as
the frequencies are above about 150 Hz [114]. The smallest ratio
that is ordinarily considered to sound consonant when both notes
are sounded simultaneously is the minor third, or 6/5 (e.g., C to
Eb). At closer ratios than this there is generally an unharmonious
fusion and roughness, or dissonance.
Cross-cultural consistencies and cultural variations. Significantly, these perceptions of harmony appear to be basic, rather
than accidents of cultural history. Cross-cultural constancies such
as octave equivalence [25] may reflect an innate characteristic of
the brain ([16]; also see [68] on “Pythagoreanism,” and [55].
Although the music of India is quite different from Western music,
in its uses of small intervals and glides, the octave interval (2/1
ratio) and fifth (3/2) play primary roles. Moreover, the scales in
Indian music, which comprise the foundations of improvisational
“ragas,” each have six defined notes between the anchoring tones
at either end of the octave [107]—as does the Western diatonic
scale.
Further evidence that the musical octave is of basic perceptual
significance arises from experiments, carried out with Western
listeners, in which attempts were made to “stretch the octave.”
This manipulation left the ordering among notes about as in the
standard octave but systematically corrupted their low-integer ratio
relationships. The resulting sounds turned out to be intractably
weird, and they could not be relearned as possessing consonance
([98], pp. 89 –92). Even such an experimentalist in 20-century
genres as the composer/conductor Leonard Bernstein believed that
musical perceptions reflect innate brain characteristics (as cited in
[2], p. 43).
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In his classic The Sense of Music, Victor Zuckerkandl ([142], p.
69) follows his explanation of the ancient Greeks’ discovery of the
effects of small-integer ratios of the lengths of musical strings,
with this comment:
As no ancient Mathematician-King is known to have constructed the
diatonic scale according to a mathematical order and to have imposed it by
decree on the peoples of the Western world, we must conclude that these
people, when choosing to adhere in their melodies to the intervals of the
diatonic order, unconsciously followed the Law of Number. Their singing,
certainly one of their most spontaneous activities, appears secretly governed by mathematical necessity.
The present paper does not take up the interesting additional
matter of how musical sophistication can lead to tolerance and
enjoyment of deviations from the basic harmony ratios, and
whether this fact suggests further relevance to neural modeling.
Harmonics Entailed by Harmonies
A property of consonant pairs of tones is that their low-integer
ratios entail more possibilities of harmonics of the two tones that
are in synchrony with each other, and more possibilities of relatively wide intervals among the remaining harmonics of the two
tones ([98], p. 83, esp. his Fig. 5-6). One possible relevance to
brain waves is that a brain wave frequency in the gamma band
might be a harmonic of a lower-frequency generator brain wave.
Possibly consistent with this idea are findings, currently being
interpreted in other ways than proposed here, that gamma waves
bearing WM item information are synchronized with, and embedded in, cycles of theta waves [66,77]. Other evidence suggests
gamma waves are harmonics of alpha [69]. Although harmonics
might also be a mere epiphenomenon, having no informational
significance, the following point explains why such a state of
affairs would represent a missed opportunity in the evolution of
cognitive brain functions.
Harmonics clarify a fundamental frequency. It seems unlikely
that natural selection would allow a phenomenon as robust as the
tendency of a wave to carry harmonics to remain epiphenomenal,
in a system in which waves bear information, because the presence
of harmonics clarifies the distinction between fundamental frequencies. In simple listening tasks, with musical tones having
timbre, the low-integer ratios among the fundamental frequencies
must be more precise for there to be an experience of harmony,
than if the fundamentals exist alone as pure tones [98]). This must
be because additional convergent, disequivocating information is
provided by the harmonics. Our ears can hear the sharper, clarifying “edges” or “contours” that they add to the shape of the
otherwise smooth sinusoid. These edges and contours are evident
in a literal sense when looking at a graphic representation of the
oscillation in which harmonics are present.
Do brain waves have timbre? The fact that the brain exploits an
advantage of timbre in the auditory system implies that there is a
biological evolutionary route to this feat of “engineering.” This
suggests that an analogous evolutionary route is available in any
living system where waves carrying information linearly summate
in a volume, for example in brain tissue at the scales of dendritic
potentials or of volume conduction. Thus a possible reason why
brain waves often depart from a purely sinusoidal shape is that
they have “timbre.” This might help them to establish a small set
of frequencies more quickly or reliably, or it might comprise a
code-component which enables additional information to be borne
in the “tonal color” of brainwaves, i.e., in waveshape recognition.
A possible basis for new chunking. Turning this consideration
around for a moment, it is conceivable that instances of coherent
activity at lower EEG frequencies may be signatures of integrative
merging of harmonics at higher frequencies. That is, as two or
GLASSMAN
more WM chunks enter into a newly forming concept, there may
emerge a common fundamental. This could happen if two or more
EEG signals glide together into a low-integer ratio relationship
from frequencies that previously bore “poorer harmonies” with
each other (i.e., either a nonrational relationship between any two
of the frequencies, or a ratio expressible only with large integers).
Following such harmonic merging, a subsequent doubling or other
multiplication of the emergent chunk-frequency could bring it up
into the octave where the WM chunk-markers are mixing. It would
be premature to pursue this more elaborate possibility of modeling
music-like transitions of brain waves in this paper. The important
point here is that the present hypothesis is fertile with testable
implications and additional theoretical possibilities. It is also worth
noting that frequency multiplication easily occurs in many, simple
natural and artificial physical systems.
The Virtual Pitch Illusion as a Model of an EEG Mechanism
A more concrete way to consider the informationally convergent aspect of harmonics is by analogy with the auditory illusion
of virtual pitch. The illusion arises from the fact, noted above, that
any set of harmonies must also comprise a set of harmonics of an
implied fundamental frequency one or more octaves below the set
of harmonious frequencies. This implied fundamental can often be
heard whether or not it is actually present. An analysis of the
“fundamental bass,” perhaps first offered by Rameau in 1722
([98], pp. 96 –101), accounts for this virtual pitch illusion. The
simultaneous presence of adjacent harmonics (e.g. fourth, fifth,
and sixth) is most effective in eliciting the effect ([113], p. 44),
perhaps because neighboring harmonics imply a particular fundamental unequivocally.
Our ear-brain system thus does something very much like a
Fourier analysis. In common experience, the virtual pitch illusion
enables us to follow a piece of music emerging from the speaker
of a small, inexpensive radio whose cutoff of lower frequences
eliminates the fundamental frequencies in which the melody was
originally recorded. Thus, as with certain visual phenomena such
as subreceptor vernier acuity, the perceptual performance of neural
aggregates operating over a time interval normally allows the
organism as a whole to “see through” a degraded signal to the
underlying reality. As with other contrived illusions, the illusion of
virtual pitch is representative of an underlying system propensity
that is an adaptive aspect of ordinary functioning.
In a human electrophysiological study, the harmonics of a
fundamental, indeed even with the fundamental missing, yielded a
more discriminating mapping to the auditory event-related potential N1 latency than did the respective fundamentals themselves
[104]. Thus, the auditory phenomenon of illusory, virtual pitch
serves as a model of a possible more general competency of the
brain to cope with simultaneously present rhythms. The phenomenon depends on simple quantitative relationships among the different frequencies, which seem as if they might readily be analyzed
at low levels of neural processing.
EVOLUTIONARY PLAUSIBILITY OF BRAIN WAVE
INFORMATION PROCESSING
Evolutionary, psychological, and neurological plausibility have
been prime considerations with models of cognitive binding (e.g.,
[61,124]). Because music is a form of sound-communication more
primitive than the symbol and grammar systems of natural languages it suggests a plausible inner “language of the brain.”
Similarly, the basic generative processes described by applying
chaos theory to dynamical systems have been considered by theorists to be sufficiently simple to be evolutionarily plausible as
possible sources of biological rhythms (e.g., [11]). However, one
WORKING MEMORY AND EEG HARMONIES
investigation of this possibility showed an inverse relationship
between dimensional complexity, or degrees of freedom, of the
EEG and WM load [115].
There are a variety of other, complex mathematical wave
functions which it might be possible for us to set aside from
consideration, as far as brain rhythms are concerned. For example,
waves are modulated in a variety of ways to code information in
radio signals; but like the wheel and many other inventions of the
human brain as a whole, most of these outcomes of engineering
genius probably do not have access routes via the raw biological
natural selection that preceded the emergence of human brains.
Evolutionary natural selection seems unlikely to desultorily find its
way to exploitation of such ingeniously engineered, naturally
improbable phenomena as FM sidebands [118,130,135].
Naturalness of Sinusoids for Biological Information Processing
What are some possible ways in which sinusoids ([60,88], and
[108], p. 70) might provide a basis for a naturally selected flexible
information processing system?
Any single-valued repeating wave is mathematically decomposable, by Fourier analysis, into a sum of sinusoids; likewise, any
such wave can be synthesized by addition of some set of sinusoids.
However, it is uncertain whether the physiological generation of
EEG waves involves a basically sinusoidal process. Barlow [9]
argues for viewing the triangle wave as basic. That view might be
considered consistent with “flush and fill” models of membrane
potential generation ([58], p. 28) and perhaps with widespread
empirical evidence of rather sharp peaks of sleep spindles (see,
e.g., [132,133]; also [46]). On the other hand, if phenomenological
waveform is a valid clue, EEG waves do often have a rounded,
somewhat sinusoidal appearance (e.g., [93], p. 6).
Continuous electrical waveforms with rounded peaks and/or
troughs occur in the brain both at more global and more microscopic scales. Even at phylogenetic levels below mammals and
birds, in whom spontaneous EEG reliably appears, there is a
tendency for inputs to induce a rhythmic brain response [12]
sometimes having an approximately sinusoidal appearance.
In many simple physical systems there is a natural tendency for
abrupt perturbations to drive a sinusoidal dynamic, when there is
rhythmic repetitiveness associated with range-limited energy kinetics. Familiar examples are in pushing a swing, or in the mechanical pendulum of a clock. Close analogs in electrical circuits
are easily constructed from passive elements, i.e., capacitors, inductors, and resistors [62]. Thus, the rounded shape of the space
and/or time distribution of many biological potentials is not an
improbable evolutionary contrivance but a natural outcome, likely
based largely on passive properties of membranes (see [125], p.
31; [43,121]).
Simple Combinatorial Systems?
Although neurons do not act in the same way as elements of
digital computers, some of the same information-theoretical concepts can be relevant. For example, the possibility that simple
signals such as sinusoids might be neural markers for WM items
raises the matter of range and resolution. Does a given sinusoidal
frequency in a particular communication line act in a simple binary
way, as an on-off event? Or can the amplitude of this signal
frequency, and some number of copresent others, vary meaningfully along a broader range of activation levels? One metaphor that
is familiarly applied to a biological information system is the
so-called “four-word language” of the genetic code. As in natural
language, the presence or absence of each “word” is essentially a
binary phenomenon, because each of the four chemical bases is
either present or absent at a given position on a chromosome.
83
Pinker [99] observes that this quality of speech and natural
language gives it the character of a “discrete combinatorial system”; that is, one that has the advantage of preserving the integrity
of the basic informational building blocks in the face of constant
copyings and mixings. In contrast, an analog communicative system runs greater risks of degradation of the information bearing
units, via gradual drifts or erosions. Perhaps, in like way, there is
some limited set of quantities in the brain that can reasonably be
characterized as a gene-like small “language of the brain.” One
such possibility is in the present hypothesis that an octave band of
brain waves contains WM item-markers. A small set of discrete
brain wave frequencies may wax and wane in amplitude, combinatorially. The character of a discrete combinatorial system might
be enhanced if the brain wave frequencies were standardized. On
the most general level, this might be the significance of the beta or
the gamma octave in mammals. More specific frequency settings,
within the hypothesized octave WM buffer, might occur as individual personality characteristics, or perhaps as opportunistic settings over brief task-related time intervals. The orderliness of a
piece of music is a model of this latter possibility. Within any brief
time interval there is a more highly restricted set of possible
frequencies contributing to a harmony than over the longer run,
which can include transitions between keys.
Sinusoids as Markers That Might Evoke Distributed
Representations
A simple sinusoid, in the neural oscillatory range of up to about
100 Hz [95] by itself cannot carry a great deal of information. By
comparison, radio carrier waves are available for amplitude, frequency, or phase modulation, because their high, radio frequency
entails a fine grain of wavelengths many times smaller than the
detailed informational patterns that are sculpted across them. For
example, the longest wavelength (coarsest grain) AM radio broadcast frequency in the United States is 550 KHz. Speech and
conscious thought take place at far fewer events per second. Our
spoken words can easily make a fully detailed impression across
the fine grain of 550,000 waves per second; thus, amplitude
modulation of radio carrier frequencies can easily support variations at the speed of thought.
An EEG gamma wave cannot possibly do anything comparable
to a radio carrier frequency. Thus, it seems that EEG activity might
carry information only in the sense of comprising markers; that is,
by acting as small sets of simple signals, which achieve higher
information capacity by playing across the ultradense spatial organization of the brain, somehow triggering and sometimes revising stored, distributed memory representations. Spatially organized brain waves are envisioned with the “binding” hypothesis
[126,134] and other conceptions of EEG codes [31,33]. An alternative metaphor to “markers” or “tokens” is that of “pointers,”
used by Baddeley ([7], p. 165) in discussing a function of WM
chunks.
The expression pointer, although derived from computer programming, is used here without intending to imply details of
mechanism that are like object-oriented languages, such as C⫹⫹;
(e.g., [60], pp. 159 –161), but only to suggest that the dynamic
neural attributes of WM need not themselves “contain” a great deal
of information. It is sufficient for these dynamic properties simply
to be capable of appropriately evoking activation of the systems
that do contain the information, while participating in appropriate
interactions. Although computer information processing is quite
different in many ways from brain information processing, the
situation here is somewhat analogous to a computer, insofar as a
sequence of thirty-two settings of 1s and 0s in a register is
informationally impoverished, in and of itself. However, such
84
information greatly gains in significance by virtue of the context
and sequence of events of which it is a part. This matter of minimal
requirements, for information-processing pointers, or markers, has
been explained well by Barnden [10] and by Hummel and Holyoak
[61], with particular reference to the longstanding controversy
about whether neural representations tend to be localized or distributed.
A contemporary mathematical alternative to traditional Fourier
analysis of waves into sinusoids is wavelet analysis. Although the
widespread availability of powerful computers has fueled the
growth of this form of mathematical research, wavelets may be
another example of an important mathematical engineering feat
that lies outside the direct access routes of biological natural
selection. Wavelet analysis is highly efficient at detecting a specific complicated signal waveshape; however, wavelets require
explicit scanning over locale and scale, and thus lack important
properties that are unique to sinusoids [59]. As explained next,
these special properties of sinusoids seem well-suited to achieve
error tolerances and necessary continuities in biological information systems.
Four Advantages of Rhythmic Repetition in Biological Signaling
Many biological systems are periodic [105], but so too are
many machines characterized by repetitive processes, e.g., continuously rotating wheels, reciprocating pistons, clock functions in
computers. Is there a reason to expect periodicities to be a more
basic and pervasive attribute of biological systems than of machines?
Continuities. At least some of the functional articulations of our
soft, wet neurons seem not to be machinelike. Every system that
has been studied at the single unit level shows a degree of broad
tuning ([14]; [126], p. 353) and functions distributed across neuronal ensembles [4,21]. Broad tuning has the virtue of ensuring
that all possibilities are covered within the realm of information
processing encompassed by a neuronal set, and that there are
functional adjacencies and overlaps among neuronal sets, thus
extending the brain’s representational reach. Interpolations readily
occur, and no new input falls into an empty “hole.” Broad tuning
thereby ensures a continuous landscape of sentience upon which
new associations or competencies may evolve gradually. This
physiological property of the brain’s microscopic elements helps
to explain the elusiveness of findings of discrete, pure localizations
of function when using the venerable ablation method in neuroscience or, indeed, other neurobehavioral methods [39]. Discussions continue about degrees of specificity and differentiation
versus diffuseness and equipotentiality [28,61,81].
As a “state of matter,” the brain’s high degree of spatial
differentiation makes it the quintessential opposite of a gas, yet
there is a certain “gaslike” degree of distributed processing across
anatomical and chemical substrates, or functional diffuseness.
Small lesions, even when encompassing all of some anatomically
or physiologically defined region, generally have yielded incomplete behavioral deficits in the hypothesized target function, and
often have yielded “side-effect” deficits in other functions. Functional localization is difficult to pin down, even when great care is
exercised in defining the function of interest and in distinguishing
between brain functions and behavioral functions [52]. Moreover,
even very broad ablations are generally followed by some degree
of recovery of the target function (e.g., [28,39,44]).
The brain’s partial diffusion of functions across its spatially
differentiated form suggests there may also be a functional advantage in some degree of temporal diffusion; this could be achieved
by means of signal repetition. Four reasons are given below why
repetition seems important at elementary levels of signaling, in a
GLASSMAN
biological informational system. Of these, the first two reasons
explain why sinusoidal wave properties are particularly serviceable.
1. Sinusoidal Form Invariance of Same-frequency Amplitude
Summations over Different Phases
Sinusoids are tolerant of phase errors. The moment-by-moment
amplitudes of sinusoids of a given frequency, coming from different sources and having different phase relations, sum to a sinusoid
of that same frequency. This striking, somewhat counterintuitive
property is proven by an elementary analytic geometry construction that develops the basic definition of a sinusoid as a rotating
radius of a circle projected onto a line (e.g., [65], pp. 38 – 43).
When the phase difference is sufficiently great at corresponding
x-axis positions, the two waves are going in opposite directions;
thus the net effect of point-by-point algebraic addition of amplitudes often results in subtraction, with the resultant wave at this
phase separation having a lower amplitude than either of the two
source waves. Through the diversity of possible additive and
subtractive combinations, the resultant always turns out, remarkably, not as an odd string of humps and troughs, but as a pure
sinusoid of the same frequency as its sources.
This property of sinusoids suggests permissiveness in cases of
small errors in the relative timing of sources. Alternatively, it
suggests a possible brain mechanism by which continuous variations of the phases of constant-frequency component signals might
modulate amplitude.
If the most elemental process is not amplitude addition but
phase addition, this can have a multiplier effect on amplitude,
according to Fukai [34], who argues that continuous subthreshold
signals in combination might thus yield a discrete resultant signal
when a threshold is crossed. Such an effect might enable a particular frequency to act in a binary way, as “on” or “off.”
2. Sinusoidal Form Invariance with Integration or
Differentiation
Responsiveness of neural elements often depends upon summations or, alternatively, on rates of change. This suggests that
there is significance for neural information processing, in the fact
that the integral or derivative of a sinusoid is a sinusoid of the same
amplitude, changed only in phase, by 90°. Usages of simple
sinusoidal messages, which involve responsiveness that tracks the
cumulating “power” of a wave (spatial and temporal summation,
or integral) or, conversely, its rate of increase or decrease (derivative) will pass on messages packaged in the same sinusoidal form.
This is a handy property for recursive systems. That is, the brain
can use the same “apparatus” if such integrating or differentiating
transforms recur, undergoing one or more stages of subsuming in
other messages. The same neural circuits can be used repeatedly in
loops mixed with convergence and divergence, without having to
change their input-output characteristics for each stage of processing of a given message. This “calculus-constancy” is an unusual
property for a mathematical function, and one that seems extremely useful for natural information-processing systems. Thus,
sinusoids in brains may comprise a particularly useful variety of
“interchangeable parts,” or dynamic modularity.
3. Reliability Maintenance
Rhythmic signals may automatically maintain reliable transmission of information as behavioral commitments are developing.
A rhythmically structured signal may compensate for the natural
distractibility of a biological information system, which changes
commitments from one function to another at different times. And
WORKING MEMORY AND EEG HARMONIES
this same mechanism that maintains reliability would also provide
a foundation for competitions between different inputs. One imaginable competition is the recruitment of a mass of brain tissue to
one or another frequency. Such a mechanism could underlie frequency glides and merges of harmonies, such as were hypothesized above. Evidence that processes of this general sort occur in
the brain, albeit on a slower time scale, is in the ”recruiting“ and
“augmenting” responses. These terms describe a crescendo of
cortical responsiveness that occurs to a rhythmically repeated
electrical stimulus to the thalamic nonspecific or specific nuclei.
Such artificially elicited spindle-shaped bursts of waves are
thought to reflect a normal brain function in wakefulness and sleep
([5], pp. 168 –171; [64]). At the much slower temporal scale of
ritualized behaviors of whole organisms, a recruitment of commitment that involves rhythmic repetition is illustrated by the gradual
development of vigorous responses in certain emotional communications of animals and humans, for example in reciprocal rhythmic movements of greeting, courtship, or aggressive signaling
[27]. Kavanau [71] has presented a theory of the role of EEG in
dynamic stabilization of neural circuitry (also [50]).
Extending a signal by the ploy of rhythmically repeating it
raises the possibility of resonance. Waxing of responsiveness
during the course of a number of input wave repetitions, becomes
a measure of certainty or of importance. Such a view is also an
aspect of the “adaptive resonance theory” of neural networks [53].
Waxing of responsiveness also requires neuronal systems to have
active or passive damping characteristics, to enable activity to
wane to a more neutral state of preparedness.
Such processes might work best with sinusoidal waves—with
receptivity of brain systems receiving the signals limited to an
octave, as explained above— because of the complications that
would be caused if the rhythms emerging from waxing and waning
modulation entered into the processing of information as spurious
additional signals. Perhaps that is one reason why EEG waves tend
shaped as spindles. Such gradual waxing and waning of the burst
feathers the onset and offset of the signal, and thus “mimics” the
statistical preconditioning of a periodic signal that is necessary to
avoid spurious high frequency readouts when doing a computerized Fourier analysis [38].
4. Temporal Extension Achieves Simultaneity of Signals with
Variable Onset Times
The pace of brain activity and the pace of behavior. Rhythmic
repetition extends a signal in time, helping to ensure coordination
of information converging from two or more separate sources. This
is a more general kind of error tolerance than that described above
under the heading of same-frequency summations. If evolution of
brain rhythms occurred with the precision of some engineered
artifacts, a single information-bearing EEG wave compounded
from several sources might occur at the very finest temporal grain
that is ever likely to be needed in behavior. However, safety factor
in reliability engineering was widely prefigured in biologically
evolved systems; therefore it seems likely that a minimal information-bearing wave would have to be at least a doublet or triplet
[3,17,40,70]. The additional need to summate signals with somewhat imprecise onset times would seem to require additional wave
repetitions. The evolution of the finest temporal grain for a compound neuroelectric event thus must determine a biophysical limit
to the overall pace of behavior.
Calculating an approximation. Possible corroboration of this
point is in the temporal relationship between EEG gamma waves
and the pace of behavior. EEG gamma frequency, possibly involved in binding of stimulus attributes, includes 40 Hz at the
low-frequency end. The single-wave duration for that frequency is
85
25 ms. At the behavioral level, a fast reaction-time, in a situation
requiring stimulus identification and choice, is on the order of 250
ms [29]. About 50 ms of the reaction time is taken up by action
potential travel time over the axonal distance from the eyes to the
cortex, and then down and out to the responding hand. Thus, by
allowing up to about eight wave repetitions (ca. 200 ms ⫼ 25 ms)
the gamma frequency may be a good “design choice.” It is about
an order of magnitude slower than the frequency of some of its
own “building blocks” or causal factors in a high neuronal firing
frequency, and an order of magnitude faster than the pace of a fast
behavior, for which it might be considered to comprise “building
blocks” [13]. If not gamma waves, but beta waves participate in
information processing, their lower frequency would allow only
half the room for error that gamma wave frequencies seem to
permit, but ballpark calculations such as the preceding suggest the
possibility that beta waves might also serve as chunk markers that
could underlie the known pace of behavior. These considerations
suggest a line of empirical research or additional literature review
that seeks relationships between behavioral pace and brain wave
frequencies across species, individuals, or behavioral states within
individuals.
CONSEQUENCES OF INCREASING NUMBERS OF
SIMULTANEOUS SIGNALS
Sensitivity to Coincident Peaking vs. Phase Transparency
In exaggeration of the phenomenon of beats (Fig. 4), sharp
summation peaks, separated by intervals having an envelope of
rather flattened amplitude, arise when a set of sinusoidal waves all
reach maximum synchronously. This occurs with the phase relationships of the cosine form (Fig. 5A). When the phases of the very
same set of waves are spread evenly over the cycle of the fundamental, the result is a more complicated-appearing waveform of
fairly regular amplitude (Fig. 5B). Similarly, Pierce ([98], p. 186,
Fig. 13-2) shows how different phase adjustments of summating
harmonics can lead either to sharp peaks or to an increasingfrequency ramp within a fairly constant envelope of amplitude; the
period of repetition of the ramp is equal to the fundamental
frequency.
Perhaps such effects are sometimes exploited in EEG information processing and sometimes not—just as they are sometimes
relevant to hearing and sometimes not. In hearing timbre, we are
quite sensitive to waveforms at lower frequencies, but for frequencies above about 200 or 400 Hz our hearing does not distinguish
among phase differences ([98], p. 187). That fortunate fact allows
for the possibility of music that comprises more than one line, or
more than one instrument playing the same line, i.e. polyphony,
harmonious chords, and accompaniment generally. Although it is
easy enough for a group of skilled musicians to synchronize on the
same beat on a time scale of large fractions of a second, they could
never synchronize on the much finer temporal scale of the hundredths or thousandths of a second consumed by the individual
sound waves produced by their instruments. Analogies in EEG
information processing can be imagined either involving waveform sensitivity or its opposite, phase transparency (linear additivity and decomposability), a characteristic that takes advantage
of the tendency of waves in water, air, and many other media to
pass through each other unchanged ([114], pp. 38 – 40, 143–144;
[140]). Slight phase adjustments of EEG harmonics might amplify
into a waveshape code. If there are threshold effects in phase
aggregations ([34]; [108], p. 45), then cosine-type, maximizing
summation peaks might yield a pulse code. An additional possibility is that seizures might result from a pathology of phase
adjustment that allows copresent EEG harmonics all to slip into a
cosine-like relationship.
86
GLASSMAN
FIG. 4. Illustration of beats, using sinusoidal simulations of gamma band
EEG frequencies. The frequency of beats is equal to the difference between
the frequencies of two waves whose amplitudes summate at every instant
of time. The x-axis in this graph, as in subsequent graphs, is labelled in
radians. If 2␲ radians are thought of as 1 s of time, then the waves are
plotted in cycles per second (Hz), as indicated by the coefficients of x. The
plot at lower right is at the beat frequency of the plot at upper right.
y ⫽ Sin 30x
y ⫽ Sin 30x ⫹ Sin 35x
y ⫽ Sin 35x
y ⫽ Sin 5x
Crowding and Roughness
Simultaneous maintenance of at least two items must be necessary to integrate information in WM. But, again, why is the
capacity of WM limited to no more than several items? Properties
of coexisting rhythms suggests drastically diminishing returns for
packing a band with simultaneous rhythms. There is a limit to how
crowded a band can become before the signals become confused
with each other. In hearing, a sensation of dissonance, or “roughness” occurs when two tones are closer in frequency than the
so-called “critical bandwidth.” This occurs at a ratio of less than
6/5 (the “minor third” such as C–Eb, for frequencies above 500 Hz
[98,113]). The critical band concept may also be relevant to the
brain’s ability to discriminate among the hypothesized information-bearing EEG waves. The critical bandwidth phenomenon in
audition depends to some degree upon mechanical properties of
the middle ear. EEG waves do not have to drive anything like the
ossicles and other masses of the the middle and inner ear, but
electrical inductive or capacitive reactances of brain tissue might
cause analogous distortions, leading to confusions among crowded
frequencies.
FIG. 5. (A) Summation of eight sinusoids, with equal differences (of 1)
between the frequencies of neighboring pairs of waves and with peaks all
beginning at the same phase position, i.e.,
冘
14
y⫽
Cos (ix)
i⫽7
The effect is as if a set of eight EEG waves, from 7 Hz to 14 Hz, summated
in the same volume of brain tissue. (B) The eight sinusoids that are
summated in this graph are the same individually as in Fig. 5A (i.e.,
simulations of 7, 8, 9, . . . , and 14 Hz), but in this graph their phases are
distributed over the cycle, so that corresponding points (e.g., peaks) do not
summate.
冘 冉
14
y⫽
Cos i x ⫹ 2 ␲
i⫽7
14 ⫺ i
7
冊
Crowding and Fusion: A Tradeoff Between Harmonies and
Melodies of the Brain?
It is more difficult to “hear out” two simultaneous tones that are
close in frequency, that is, to hear clearly that there are two of
them, than to discriminate between two successive tones. This
phenomenon occurs at closer frequency intervals than the critical
bandwidth, discussed above. For example, below 200 Hz, a difference of less than 15 Hz between two simultaneous tones leads
to an experience of dissonant “fused tones with roughness.” In
general, the difference between simultaneously sounded frequencies needed to avoid this experience of fusion is large: 7 to 30
times as great as the just noticeable difference for successive tones
[114].
As in the case of critical bands, the difference between the
auditory perceptual effects of simultaneous versus successive
sounds has sometimes been attributed to the mechanical limitations of the cochlea. However, at least a partial analogy seems
possible with electrical properties of brain tissue. Limitations of
frequency response, due to inertia and restricted elasticity in mechanical systems, may have partial analogies in electrical media,
due to capacitive, inductive, and resistive impedances.
Perhaps fusion of simultaneous tones sometimes involves entrainment of rhythms. When two oscillatory signals are very close
in frequency, one rhythm may capture, or entrain, the other. One
implication is that some pathologies of cognition may be due to
misregulation of intervals between brain wave frequencies. For
WORKING MEMORY AND EEG HARMONIES
87
FIG. 6. Six harmonious pairs of sinusoids ranging from the minor-third through the fifth. Top row (from left to
right): Fifth (3/2), Fourth (4/3), Major Third (5/4). Bottom row (from left to right): Major Sixth (5/3), Minor Sixth
(8/5), Minor Third (6/5). The denominator of each frequency ratio is the wavelength of the waveform resulting
from summing the two frequencies. The harmonies that most people consider more consonant tend to have both
shorter summated wavelengths and a simpler waveform. For example, compare the Minor Third (6/5, bottom right)
to the Major Third (5/4, top right) immediately above it; the wave patterns look similar but the minor summated
wavelength is longer. Also compare the Minor Third (6/5) to the more complicated waveform of the Minor Sixth
(8/5), immediately to its left. (Curves were plotted using Mathematica software, Version 2.2.1. The equation for
the Fifth was y ⫽ cos 2␲x ⫹ cos [(1.5)(2␲x)]).
example, small WM capacity within some modality or category of
information, or excessively loose associative fusions of ideas, may
be mediated by excessively wide fusion intervals or entrainment
sinks for EEG rhythms in certain brain areas. On the positive side,
entrainment might be a handy property for a system that uses a
limited number of frequency-encoded messages, each having components converging from different sources. Entrainment might
provide a kind of “touch-up,” or automatic correction, of slight
mismatches, like frets in a dial. This remarkable property of
rhythms occurs in many media and at different time scales, including mechanical clocks hanging and ticking together on a wall,
biological rhythms of organisms living together [116], and many
others [87].
The following consideration suggests a more basic reason why
interacting simultaneous periodic signals might best be highly
limited in number.
Harmonies Comprise Simple, Quick Repetition Patterns
If cognitive information processing does involve EEG harmonies, as hypothesized here, then inherent limits to WM capacity,
and limits to the degree of parallel information processing that
WM capacity entails, are implied by the periods of regularity of
summated harmonious frequencies. The sums of more harmonious
combinations of pure sinusoids comprise simpler compound waveforms, which have briefer periods. Figure 6 illustrates this point
with graphs of six harmonious pairs of sinusoids ranging from the
minor third (6/5), through the fifth (3/2). As shown, the lowerinteger ratio relationships create a compound wave that cycles with
a higher frequency, or shorter wavelength. The smaller the integers
in the ratio between the two frequencies, the more frequently a
simple summed waveform repeats itself exactly.
The experience of hearing roughness in the presence of two
audio frequencies having less than critical band separation is,
indeed, phenomenologically suggestive of the lack of a short
period of regularity. The somewhat “unsettled” feeling with dissonant tones may thus arise because there is a time limit, or
perhaps temporal function of fundamental frequency, for the level
of cognition at which settling of information into a regular pattern
is experienced as a harmonious, smooth sound.
Hypothetical numerical example. Using the gamma band for an
example, suppose there is frequency variation over an EEG octave
of 40 – 80 Hz (gamma band) within an individual, and that at any
given time, preferred values of the frequency variations bear
harmony ratios. Thus, if 40 is the lowest frequency in a particular
individual at a given time, an additional wave of 50 Hz would
carry a 5/4 (major 3rd) ratio.
Periods of regularity. How long would the system have to wait
to register regularity in a compound signal of 40 plus 50 Hz? These
two frequencies beat at 10 Hz. Thus, the time between two beats
is 0.1 s. A minimum of two complete beat cycles (in this example
0.2 s) is necessary to indicate rhythmic regularity. If this system
has a typical biological safety factor of at least 2 [3,40,70], the
minimal time to register stationarity reliably is 0.4 s.
Behaviorally, this duration is reminiscent of a rather long
choice reaction time. The human capacity to grasp about four items
simultaneously in a strong trace, as discussed earlier, would require a frequency spacing of about this magnitude if the items
were, indeed, represented by frequencies that had to fit within an
octave. If each frequency bore this ratio with the next, the values
would be 40, 50, 62.5, and 78.125 Hz; however, this oversimplifies, and contracts the necessary range, because a set of more than
two frequencies will not be harmoniously self-consistent if each
frequency merely bears the same ratio, locally, with the adjacent
frequency.
Capacity of an Octave Band
Again consider the analogy from hearing, of our greater reluctance to accept close frequencies as consonant in harmonies (simultaneous tones) than in melodies (successive tones). For example, the major scale splendidly occurs in its simple sequential
eight-note pattern in Beethoven’s Egmont Overture, and about 8
min from the beginning of the second movement of his Third
Piano Concerto. But only the most assertive of modern composers
would dare the dissonance of sounding all eight of these notes at
88
GLASSMAN
FIG. 7. Plots of the summated four frequencies of the major triad (left) and minor triad (right), both with tonic
completion of the octave. The higher magnification top curve shows about one and a half cycles of the major triad
while four full cycles can be seen in the lower magnifications bottom curve. (Plotted using Mathematica software,
Version 2.2.1. The equation of the major triad is y ⫽ Cos 2␲x ⫹ Cos [(1.25)(2␲x)] ⫹ Cos [(1.5)(2␲x)] ⫹
Cos 4␲x.)
the same time. Only with lesser crowding of frequencies within the
octave can there be a harmonious chord of about three to five
notes. These numbers are the same as the recency capacity in
verbal free recall experiments, capacity for tachistoscopic perception of numbers of dots, and other lower estimates of WM capacity, as discussed earlier. Thus, the harmonic capacity of an octave
is approximately of the same numerousness as the lower estimates
of WM capacity, which concern how many items can be apprehended at once in the passive trace component of WM.
As noted above in the subsection on cross-cultural consistencies,
although modern Western music (e.g. Stravinsky, Bartok, Webern) as
well as nonwestern musical styles make use of a variety of intervals
that deviate from the simpler consonances of classical (and traditional
popular and folk) Western music, there is a more universal tendency
to limit the permissible number of intervals within an octave that are
allowed to occur within any one piece and to spread these intervals
over the whole octave; for example, this is true of the seven-interval
octave of the Indian raga [107].
Further Analysis of Periods of Regularity
An example of a populous harmonious set of notes, restricted to
an octave, is the 5-note dominant seventh chord (with tonic doubling of the octave); an example, in the key of C is C-E-G-Bb-C⬘
(with middle C as the lowest frequency, respectively 261.63,
329.63, 392.00, 466.16, and 523.25 Hz). By shading the highest
frequency (C⬘) infinitesimally less than an octave interval from the
fundamental (C), our theoretical nervous system might gain the
hypothesized advantages of harmonious relationships within an
octave, without risk of confounding by spurious difference frequencies within this “safe zone.” The following illustration uses
the simpler harmony of the four-note major triad with tonic doubling of the octave, e.g., C-E-G-C⬘, comparing its regularity period
with the less consonant, corresponding minor triad, C-Eb-G-C⬘.
Figure 7 is a plot of both of these sets of four summed
sinusoidal waveforms, presented at two different magnifications.
As the figure shows, the major triad harmonic relationship leads to
a relatively brief compound cycle time of four times the fundamental frequency. The minor triad on the other hand comprises a
summed waveform with less obvious symmetries and simple regularities, in which the duration of a compound cycle is fully ten
times that of the lowest of the tones. The longer period of regularity suggests a reason why in hearing music most ordinary
listeners experience somewhat lesser consonance with a minor
chord than a major chord. Correspondingly, frequency ratios as in
the minor triad would involve more sluggish registration of compound signal in the EEG information-marking system postulated
here. (Although the Fig. 7 plots simply use the cosine phase
relation, the respective regularity periods of any other set of phase
relations, for these same sets of frequencies, would be the same.)
What is the general rule for how these compound cycle times
arise? Two ways of deriving the cycle times are in terms of beat
frequencies or in terms of greatest common divisors.
Beats: Implications for period of regularity. Any two waves
that are within the same octave will beat against each other at a
frequency that is equal to the difference in frequencies of the two
waves; for example waves at the alpha frequencies of 9 Hz and 8
Hz would yield a wave at 1 Hz. To compare the major and minor
triads, assume a fundamental frequency for “C” of 60 Hz, in the
gamma EEG range. This value will allow us the convenience of
whole numbers in thinking about the frequencies implied by different harmonic ratios. Figure 8 represents the frequencies in the
major and minor triad as positions along a line. In addition to each
of the source tones at the specified harmonic ratios, the beatfrequency difference tones are shown under the lines representing
the major and the minor triads.
Beats are physical phenomena, appearing not only in a mathematical plot of summated waves but in any physical measurement
of such waves. Therefore, beats between all source tones, in every
paired combination, are additionally accompanied by higher-order
beats, resulting from beat-frequencies beating against other beat
frequencies. Figure 8 shows one representative frequency resulting
from each such beat-combinatoric difference. In the case of the
WORKING MEMORY AND EEG HARMONIES
89
FIG. 8. Schematization of the frequency intervals of the major and minor triads, with tonic completion of the
octave.
major triad, the smallest such difference is 15, while with the
minor triad the smallest difference is 6. These numbers are, respectively, 1/4 and 1/10 of the frequency of the 60 Hz fundamental, and that is the origin of the respective repetition rates of the
compound waves illustrated in Fig. 7.
Inspection of Fig. 8 suggests that the smallest difference (or
difference of differences) is simply the greatest common divisor of
the frequencies participating in each of the two illustrated chords.
The following simple algebraic analysis shows this to be true in
general, and provides a rule for considering the regularity period of
any set of waves. Alongside each step in the proof is an example
from the preceding illustration of harmonic relationships.
Algebra of beats: Greatest common divisor. Consider a set of
numbers that have a common divisor, c:
共cA兲, 共cB兲, 共cD兲, and 共cE兲.
[1]
Look at the ratio between two of these:
cA/cB ⫽ A/B
e.g., 75/60 ⫽ 共15 ⫻ 5兲/共15 ⫻ 4兲 ⫽ 5/4
[2]
Also look at the difference between the same two terms:
cA ⫺ cB ⫽ c共 A ⫺ B兲
Now, by analogy with finding higher-order beats, we calculate the
difference of the above differences:
关c共D ⫺ E兲兴 ⫺ 关c共 A ⫺ B兲兴 ⫽ c关共D ⫺ E兲 ⫺ 共 A ⫺ B兲兴
[5]
It is evident that this algebraic process (related to the Euclidian
algorithm; [92], pp. 4 –9) is general, with c always factoring out as
a common divisor. Thus, differences of differences will come out
equal to the greatest common divisor. Recall, the implication for
harmonies is that the cycle duration of a compound wave, which
results from (linearly) summing a set of sinusoids, is equal to the
greatest common divisor of the frequency of all the sinusoids.
Carrying too many WM chunks at once will weigh you down.
These mathematical underpinnings of the experience of roughness
or dissonance in music show why crowding an octave of brain
waves with more frequencies might result in diminishing returns in
information processing. Packing in additional signals would mean
they must be spaced more closely, leading to a longer beat cycle,
or stationarity period. In music, the “just” scale tuning has the
lowest-number ratios (Table 1). However, if sinusoids at all eight
of these frequencies were present at once, the greatest common
divisor would be 24 times the period of the tonic, a long period of
e.g., 75 ⫺ 60 ⫽ 共15 ⫻ 5兲 ⫺ 共15 ⫻ 4兲
⫽ 15共5 ⫺ 4兲 ⫽ 15
[3]
This shows that the difference between the two terms is a multiple of
the common divisor (c). Since c can be any common divisor, it can be
the greatest common divisor. Similarly, for another example,
cD ⫺ cE ⫽ c共D ⫺ E兲
e.g., 120 ⫺ 75 ⫽ 共15 ⫻ 8兲 ⫺ 共15 ⫻ 5兲
⫽ 15共8 ⫺ 5兲 ⫽ 共15 ⫻ 3兲 ⫽ 45.
[4]
TABLE 1
RATIOS OF EACH NOTE TO THE FUNDAMENTAL (TONIC) IN THE
KEY OF C, AND THE RATIO EACH SUCH NOTE BEARS WITH THE
ADJACENT NOTE
Note-Example
C
D
E
F
G
A
B
C⬘
Ratio to fund.
Local ratio
1
1
9/8
9/8
5/4
10/9
4/3
6/15
3/2
9/8
5/3
10/9
15/8
9/8
2
16/15
90
regularity. This is 2.4 times longer than the period of the minor
triad illustrated earlier and six times that of the expeditious major
triad. Thus, if harmonic brain wave pointers are part of the foundation of WM, holding seven or eight chunks simultaneously,
without shifting to the rapid-successive mode characteristic of
rehearsal cycling, would add the burden of a long holding time
before the grouping could be registered. If a gamma brain wave
octave of 40 – 80 Hz performs the WM function, then a single
compound cycle for a summation of the seven (or eight) notes of
the major scale (e.g., CDEFGAB[C’]) would take 0.6 s. The need
to wait for two or more such cycles to be certain of the content of
one complete cycle then implies a sluggish organism, who requires
a waiting time of at least 1.2 s before being able to identify the
large set of information in WM. This contrasts with a waiting time
for the major triad of 0.2 s for two cycles, a more behaviorally
meaningful value, on the order of magnitude of a typical behavioral reaction time.
Additional Empirical Implications
Behavioral decision times. The above argument suggests a
particular mathematical function, for behavioral recognition or
decision times, when simple items are incrementally added to
WM. Because it depends on ratios, this prediction is independent
of whether the above examples, using hypothetical gamma frequencies, are correct estimates. Additional factors may have an
impact; for example, the motor reaction time for indicating what
was recognized may add a constant to the decision latency for each
WM load. One obvious next step to pursue this implication of the
present hypothesis might be to use the Sternberg paradigm, for
measuring decision reaction time with scanning of short-term
memory systematically loaded with between 1 and 6 items. However, much interpretive controversy has surrounded the large numbers of results obtained with variations of the Sternberg procedure,
during three decades of its use in cognitive psychology experimentation and theory (e.g., [8], pp. 199 –203). Reviewing that
work is beyond the scope of this paper.
Spectral analysis of brain waves. The foregoing points obviously also suggest it would be worthwhile to seek empirical
evidence of copresent brain waves in low-integer ratio relationships, using Fourier decomposition techniques on raw signals
taken from a single site and also comparing signals taken from
different brain regions at the same time. Finding empirical evidence of frequency distributions suggesting such harmonic combinations, would encourage additional theoretical and empirical
efforts along the present lines. Clearly, the theory predicts a
correlation between number of items in WM and number of
identifiable harmonic frequencies in the EEG. Because gammaband EEG has generally required recording directly from the brain,
using animals with implanted electrodes, it would be fortunate if
the beta band, easily recordable from scalp electrodes in humans,
showed the predicted harmony effects.
Other hypothesized roles of slower, EEG alpha or theta waves
have been proposed, as the fundamental frequency of gamma
waves [69], or to sequentially cycle temporarily maintained representations of a sequence of working memory items [42,66,77].
These authors have provided calculations to suggest plausibility in
terms of single neuronal timing and other physiological parameters. The many empirical correlations of large amplitude slow EEG
waves (“synchrony”) with behavioral inhibition or inaction [49,50]
may not be accurate indicators of functions of these waves at other
amplitudes and when there are more complex spatial distributions
with more varied frequencies or phases [73]. Thus, lower EEG
frequencies may also be involved in active information processing.
GLASSMAN
Further Theoretical Issues
This paper has dealt only in the most general terms with the
problems of how EEG markers might evoke chunks of memory
selectively as they are distributed across the brain in long-term
storage, and how such distributed long-term memory engrams
might be updated. As suggested above, the question of how transforms are effected between the massive parallelism of long-term
memory’s anatomical/chemical structure and the small degree of
parallelism, together with strong sequential aspect, of WM, may
lend itself to analogies with the modest parallelism of harmonies
and the serialism of melodies. (A different model that associates
neural patterns with music was previously proposed by Leng,
Shaw, and Wright; [75]).
Some empirical observations and models have been concerned
with the mapping of a time series to a spatial representation; for
example with hippocampal pyramidal neurons viewed as leaky
integrators [109]. Troyer, Levin, and Jacobs [136] demonstrate
how, in the cricket’s mechanoreception for discriminating patterns
of air currents, electrical and optical activity records can be coupled with a detailed database of the anatomy of dendritic structures, leading to a complete description of an ensemble code that
incorporates both spatial and brief temporal-sequence information
in a spatial array. Troyer et al. refer to work by Carr and Konishi
that illustrates how higher-order sensory maps may juxtapose
continuous codes for more than one parameter, one of which is a
spatial map of a time variable.
The mathematical formal equivalence of temporal and spatial
frequency functions might easily be exploited by means of sweep
operations somewhat analogous to those of bar-code readers, video
monitor rasters, etc. (For a mathematical review of principles of
computer graphics see, e.g., [30]) In one, almost literal neurobehavioral analogy, the tactile sensitivity of monkeys as they rub a
grating with a fingertip may also exploit simply coded conversions
between spatial stimulus properties and a temporal function of
activity in the peripheral sensorimotor neural systems, which then
reconverts back to a spatially coded representation in the brain
[74]. However, if there are EEG harmonies that function as hypothesized here, we need to try to understand these processes at a
deeper level than raw conversion between temporal and spatial
isomorphisms.
Pursuing the analogy with our musical sensibility, perhaps
there are varieties of ways in which melodious transitions of
harmonies might be models of rules for informational transitions
within the brain. The stylized, relative simplicity of music, by
comparison with speech and language, is consistent with the possibility that music is an externalized, embellished version of a
more primitive “language of the brain” [102] or middle-scale
processing dynamic in animals and humans, intermediate in complexity and functionally transitional, between the firing patterns of
single neurons and the behavioral patterns of a whole organism.
Addendum about a recent popularization. It is not obvious
whether the hypothesis developed here implies other cognitive
properties of music than those discussed above. Experimental
attempts to use music as a catalyst for aspects of cognitive functioning (the so-called “Mozart Effect” ) have yielded mixed results
[90,106,111,131]. Reliable positive results might suggest that the
mystery of why there is a human affinity for auditory patterns
having a musical structure has an answer in some basic property of
the brain. If that is true, that hypothetical property may or may not
comprise the characteristics of brain waves hypothesized in the
present paper. In evaluating popularizations, such as the idea that
music is good for cognition and health [18]— even when they
court the “New Age” attitude—we should be neither naively
accepting nor overly critical.
WORKING MEMORY AND EEG HARMONIES
91
ACKNOWLEDGEMENTS
I am grateful to Rami Levin for her careful reading of this paper,
leading to a number of clarifications of musicological conceptualization
and terminology. Steven Galovich provided a valuable comment on the
algebraic analysis of octave capacity; the mathematics has also benefitted
from conversations with David Cochran and Nader Nazmi. Forest Hansen
gave encouragement and musical knowledge during earlier phases of this
work. I thank four anonymous reviewers for valuable suggestions and
additional sources. I thank Anne Houde for a bibliographic source, Donald
Meyer for a correction of musical terminology, and Jared Konie for
catching an error in calculation.
This work was supported in part by a Lake Forest College sabbatical
leave and other faculty research funding.
23.
24.
25.
26.
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