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Neuroscience Letters 653 (2017) 234–238

Contents lists available at ScienceDirect

Neuroscience Letters

jo ur nal ho me p age: www.elsev ier .com/ locate /neule t

esearch article

repetitive modular oscillation underlies human brain electricctivity

rturo Tozzia,f,∗, James F. Petersa,b,c,d, Norbert Jausovece

Computational Intelligence Laboratory, University of Manitoba, Winnipeg R3T 5V6 Manitoba, CanadaDepartment of Electrical and Computer Engineering, University of Manitoba, 75A Chancellor’s Circle, Winnipeg MB R3T 5V6, CanadaDepartment of Mathematics, Adıyaman University, 02040 Adıyaman, TurkeyDepartment of Mathematics, Faculty of Arts and Sciences, Adıyaman University, 02040 Adıyaman, TurkeyDepartment of Psychology, University of Maribor, Koroska Cesta 160, 2000 Maribor, SloveniaCenter for Nonlinear Science, Department of Physics, University of North Texas, Denton, Texas 76203, USA, 1155 Union Circle, #311427, Denton, TX6203-5017, USA

i g h l i g h t s

Modular functions are widespread in physics and biology.We examined a modular function in electric brain activity.We found a modular function, the J-function, embedded in human EEGs.

r t i c l e i n f o

rticle history:eceived 7 September 2016eceived in revised form 23 May 2017ccepted 23 May 2017vailable online 26 May 2017

eywords:-Function

a b s t r a c t

The modular function j, central in the assessment of abstract mathematical problems, describes elliptic,intertwined trajectories that move in the planes of both real and complex numbers. Recent clues suggestthat the j-function might display a physical counterpart, equipped with a quantifiable real componentand a hidden imaginary one, currently undetectable by our senses and instruments. Here we evaluatewhether the real part of the modular function can be spotted in the electric activity of the human brain. Weassessed EEGs from five healthy males, eyes-closed and resting state, and superimposed the electric traceswith the bidimensional curves predicted by the j-function. We found that the two trajectories matched

odular functionervous systemEGomplex numbersultidimensionality

in more than 85% of cases, independent from the subtending electric rhythm and the electrode location.Therefore, the real part of the j-function’s peculiar wave is ubiquitously endowed all over normal EEGspaths. We discuss the implications of such correlation in neuroscience and neurology, highlighting howthe j-function might stand for the one of the basic oscillations of the brain, and how the still unexploredimaginary part might underlie several physiological and pathological nervous features.

© 2017 Elsevier B.V. All rights reserved.

ignificance statement

Our results point towards the brain as ubiquitously equippedith j-function’s oscillations, which movements take place on thelane of the complex numbers. It means that there must be, in brainlectric activity, also a veiled complex part, which can be assessed

ith the help of imaginary numbers. The modular j-function pro-

ides further dimensions to the real numbers, in order to enlargeheir predictive powers: it suggests the possible presence of hid-

∗ Corresponding author.E-mail addresses: tozziarturo@libero.it, tozziarturo@libero.i7 (A. Tozzi),

ames.peters3@umanitoba.ca (J.F. Peters), norbert.jausovec@um.si (N. Jausovec).

ttp://dx.doi.org/10.1016/j.neulet.2017.05.051304-3940/© 2017 Elsevier B.V. All rights reserved.

den (functional or spatial) brain extra-dimensions. Furthermore,j-oscillations could be disrupted during pathologies, paving the wayto novel approaches to central nervous system’s diseases.

1. Introduction

The j-function describes intertwined, repetitive elliptic trajec-tories taking place on the so-called plane of the complex numbers [1].Such plane can be exemplified as a plot displaying on the x axis thereal numbers (e.g., 1, 2, −1, −2) and on the y axis the imaginary part

(e.g., a component that satisfies the equation i2 = −1). The intersec-tion of a real and an imaginary number in the graph is termed acomplex number (Fig. 1A). The role of complex numbers in mathe-matics is to enlarge the concept of the one-dimensional number line

A. Tozzi et al. / Neuroscience Letters 653 (2017) 234–238 235

Fig. 1. Plots of Klein j-functions and their different possible paths. Fig. 1A depicts the general scheme of the elliptic oscillations on the upper half plane of the complexnumbers. For sake of clarity, the locations of imaginary units and of real and complex numbers are provided. Fig. 1B and C shows the 2D plots achieved for j = 1728 and196,884, respectively. Note the different coarse-grained appearance of the two plots. Fig. 1D displays the 3D plot, in case of j = 196,844. The plots in Figs. 1B–D were obtainedusing Mathematica® 10.

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36 A. Tozzi et al. / Neuroscien

o the two-dimensional complex plane, allowing the description ofeal numbers with an additional coordinate. This trick is very usefulot just in mathematics, but also in the other disciplines, from fluidynamics to electrical engineering and computer graphics, fromignal analysis to quantum mechanics, and so on [2,3].

The j-function is plotted on the upper half of the complex plane,.g., it can be drawn just starting from positive imaginary numbers.he pattern depicted by the j-function is called modular, e.g. inimple terms, it looks like curved trajectories intersecting one eachther in a recurrent and fractal pattern. The j-function exhibits theeculiarity that, contrary to other modular functions that can beapped in all their points on planes, is equipped with cusps, e.g.,

ingle points where a trajectory starts to move backward.It has been speculated that modular functions might have physi-

al counterparts in our Universe, because they seem to be indirectlyinked with string theories that describe the microscopic featuresf cosmos [4–7]. In particular, it has been suggested that the j-unction might count the ways strings can oscillate at each one ofheir energy levels. Therefore, modular functions might be moreidespread than thought and could be an underlying, hidden

hythm that dictates the oscillatory activities of different physicalnd biological systems. In this paper we evaluated the possibilityhat brain electric rhythms encompass the j-function. We chose,mong the countless modular functions, the J one, because it isathematically easier to treat and because it has been widely

escribed in correlation with some physical phenomena. Indeed,-function and brain activity display, at first sight, many commoneatures: they are both modular, bot sinusoidal, both symmetric [8]nd both can be described in terms of Fourier transform. In such aein, we assessed EEG patterns collected from adult controls duringyes closed resting state, looking for the j-function’s features. Weemonstrated that the j-function is endowed in the very structuref human brain electric spikes, giving rise, at least in normal sub-ects, to reiterating oscillations equipped with a highly predictableattern. Furthermore, we discuss the implications, both in basiceuroscience and in medical science, of our finding.

. Methods

.1. Subjects

The sample included 5 right-handed male students, undergrad-ates (age = 21 years). They were recruited from a large pool of

ndividuals who participated in different experiments conductedt Department of Psychology, University of Maribor lab, in the last

years (e.g., [9]. The participants were tested with the WAIS-R testattery [10]; M = 100.60; SD = 1.14). The experiment was under-aken with the understanding and written consent of each subject.

.2. Procedure, EEG recording and quantification

The participants were seated in a reclining chair while their rest-ng eyes closed EEG was recorded for 5 min. Afterwards they solvedeveral tasks which were part of different studies described above.

Quick–Cap with sintered (Silver/Silver Chloride; 8 mm diame-er) electrodes was fitted to each participant. Using the Ten-twentylectrode Placement System of the International Federation, theEG activity was monitored over nineteen scalp locations (Fp1, Fp2,3, F4, F7, F8, T3, T4, T5, T6, C3, C4, P3, P4, O1, O2, Fz, Cz and Pz).ll leads were referenced to linked mastoids (A1 and A2), and around electrode was applied to the forehead. Vertical eye move-

ents were recorded with electrodes placed above and below the

eft eye. Electrode impedance was maintained below 5 k�. EEGas recorded using NeuroScan Acquire Software and a SynAmps RT

mplifier (Compumedics, Melbourne Australia), which had a band-

ters 653 (2017) 234–238

pass of 0.15–100.0 Hz. At cutoff frequencies the voltage gain wasapproximately −6 dB. The 19 EEG traces were digitized online at1000 Hz with a gain of 10× (accuracy of 29.80 �V/LSB in a 24 bitA to D conversion) and stored on a hard disk. Offline EEG analysiswas performed using NeuroScan Scan 4.5 software (Compumedics,Melbourne, Australia). IAF for each subject was determined basedon their resting EEG [11]. First, power spectra were calculated overthe entire epoch length of 11 s (11,132 data points) and automat-ically screened for artifacts. Second, an automatic peak detectionprocedure found the highest peak (maximum alpha power) in a7–14 Hz window (frequency steps of 0.09 Hz) separately for eachlead. This method was used because it is more adequate for studyingendophenotypic qualities during resting (eyes closed) EEG sessions[12,13] than the peak frequency center of gravity method proposedby [12], which is used for eyes open conditions. Third, IAF was com-puted for each channel and averaged over anterior (Fp1, Fp2, F3,F4, F7, F8, Fz), and posterior (T5, T6, P3, P4, O1, O2, Pz) locations(M = 9.88; SD = 0.34).

For our study, we examined artifact-free sequences of 5 s forevery eye closed resting individual, from three scalp locations: CZ-,FZ- and PZ-. The EEG traces were filtered for different frequencies,in order to elucidate whether the j-function could be endowed inmore than one brain rhythm. IAF between subjects did not dif-fer significantly, therefore the resting eyes-closed EEG recordingswere filtered in 5 fixed frequency bands: delta [1.0–3.9 Hz]; theta[4.0–7.9 Hz]; alpha [8.0–12.0 Hz]; beta [15.0–31.0 Hz] and gamma[32.0–49.0 Hz]. We used a band-pass Infinite Impulse Response(IIR) filter with a 48 dB/octave rolloff.

2.3. Looking for j-function in EEGs

The modular function j is the generator or hauptmodul of thegenus zero function field (Fig. 1). The j-function was defined [14]by

j(�) = 1728J(�), whose Fourier development is

j(�) = e−2�ir + 744 + 196884e2�ir + 21493760e4�ir + · · ·, where

I [�] > 0 is the half-period ratio and

J(�) = 427

[1 − �(�) + �2(�)

]3

�2(�)[1 − �(�)]2

is Klein’s absolute invariant and�(�) is the eliptic lambda function(Piezas)

�(�) ≡ϑ4

2

(e�ir

)

ϑ43

(e�ir

) , where

ϑi (0, q) are Jacobi theta functions and

q = e�ir is the nome and 1728 = 123.

In case of a j-function endowed in brain electric spikes, wewould be able to detect just the waves depicting the real part ofthe modular function, because the imaginary part lies in a complexdimension (either numerical, or functional) that is undetectableby our senses and our technological tools, unless you explicitlylook for it. In particular, our goal was to assess whether the EEGs’bidimensional lines match with the bidimensional real part ofthe j-function. At first, we built a bidimensional pattern of the j-function. The pattern (Fig. 2A) describes a curve conventionally

moving from T0 to T1, from left to right. The most importantparameter to take into account for our purposes is the distancebetween two peaks, that, being recursive, is the best testable fea-ture of two-dimensional j-curves. The frequency and the amplitude

A. Tozzi et al. / Neuroscience Letters 653 (2017) 234–238 237

Fig. 2. Bidimensional pattern of a j-function, displaying the real part of j-oscillations. The black line depicts the j-pattern correlated with EEGs traces.

Fig. 3. Relationships between the j-function pattern (blue waves) and normal EEG rhythms (black waves) in the FZ- derivation of a single male control. The predicted bluec alpha

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urves match with the normal EEG oscillations almost everywhere. Delta, theta and

iew for the beta and gamma waves, usually in the <1 mV range. Therefore, beta anemporal scale. (For interpretation of the references to colour in this figure legend,

f the sinusoidal oscillations are less important, because, being theodular function fractal, it can be detected at different spatial and

emporal coarse-grained scales. Likewise, because the EEG wavesre equipped with multiple, unpredictable deformation arising byultiple intertwined interactions, the j-pattern could be nested in

ifferent amplitudes and frequencies.We correlated the predicted j-pattern with the above men-

ioned EEG traces from adult controls. In particular, we evaluatedhe percentage of the two oscillations’ superimposition. We used a-student test for statistical analysis.

. Results

The predicted j-pattern, standing for the real part of the j-

unction, matched the EEG plots recorded from all the normalealthy, resting, eyes-closed males (Fig. 3). In particular, the mod-lar j-function is correlated with the distance among the peaks.

-function and EEG traces’ superimposition was found in all the sub-

traces have the same resolution for the y (microV) axis, which gives a rather unclearma frequencies are depicted in Figure at higher magnification and with a differentader is referred to the web version of this article.)

jects and derivations. Taking into account all the frequencies and allthe derivations, EEGs oscillations match the pattern of the predictedj-function in a high number of plots (Mean% ± SD: 87.6 ± 5.54).Although not significant, the slower waves (delta, theta and alpha)exhibited a slightly higher correlation (93% vs. 81%) than the fasterones (beta and gamma). It could depend on a more marked inter-twining of oscillations in faster frequencies, that partially interfereswith the detection of j-function’s typical patterns. No statistical dif-ferences were found among different electrodes localizations. Insum, we found that the real part of the j-function’s peculiar waveis ubiquitously endowed all over normal EEGs traces, independentof electric rhythms and electrode localizations.

4. Conclusions

We demonstrated that normal EEG’s waves encompass the realpart of a modular function. Our results suggest that the human braindisplays a pattern resulting from elliptic, intertwined oscillations

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[22] A. Tozzi, How to turn an oscillation in a pink one, J. Theor. Biol. 377 (2015)117–118, http://dx.doi.org/10.1016/j.jtbi.2015.04.018, Elsevier.

[23] B. Sengupta, A. Tozzi, G.K. Coray, P.K. Douglas, K.J. Friston, Towards a neuronalgauge theory, PLoS Biol. 14 (3) (2016) e1002400, http://dx.doi.org/10.1371/

38 A. Tozzi et al. / Neuroscien

esembling the waves described by the j-function. Such oscillationsndowed in the brain spikes result from the delicate interactionf the countless moving trajectories that are at the very heart ofodular functions. The presence in EEGs of the j-function, whichovements take place on the so-called plane of the complex num-

ers, means that there must be, in brain electric activity, also atill unexplored complex part. While current neuro-techniques areble to detect and describe just the real part of nervous function,.g., the part quantifiable through real numbers, the complex parts hidden from our observation, because it can be described just

ith the help of an imaginary component. It means that electric fea-ures linked with elusive brain functions, such as, e.g., long-distancenteractions, consciousness and so on, could be unnoticed in theEG traces. They could be assessable, if we add further parametersble to assess neural data sets not just in terms of real numbers, butlso of complex ones. In such a vein, it must be taken into accounthat EEG techniques are able to assess just the part of the j-functionhat lies very close to the x axis, therefore missing the extensive partf the upper half plane of complex functions that lies farther awayrom the axis of real numbers.

The j-function is a mathematical tool which provides theeal numbers with further dimensions, in order to enlarge theirredictive powers. Therefore, it could be hypothesized that theathematical superimposition between modular functions and

EG spikes might be correlated with the presence of further brainimensions, undetectable by our 3D analyses. Indeed, the possibil-

ty of hidden functional or spatial nervous dimensions has beenecently discussed [15]. The Authors described the occurrence,t least during spontaneous brain activity, of multi-dimensionaloruses, where trajectories take place in guise of particles travelingn donut-like manifolds.

Although the possible presence of nested (real and the imagi-ary) parts of the j-function has not yet been explored in physicalnd biological phenomena, there are good reasons for hypothe-izing that such j-oscillations could be widespread. For example,he Monstrous Moonshine conjecture suggests a puzzling relation-hip between one of the terms in the Fourier expansion of j(�),hich value is 19884, and the simple sums of dimensions of irre-ucible representation of the Monster group M, which is 19688316]. On the other hand, the Monster group has been correlatedith physical features underlying our Universe [17,5]. Further-ore, the j-function displays both (spatial) fractal-like structure,

nd (temporal) power laws, in touch with the description of manyhysical and biological systems (including the brain) in terms ofonlinear systems equipped with scale-free behavior [18–22]. Toive other examples, it has been recently proposed that symmetryreaks occur during brain function [8] and that the nervous activityight be described in terms of a gauge theory [23]. A gauge the-

ry states that systems’ general symmetries can be broken by localerturbations and then restored by the superimposition of gaugeelds. The modular function j, with its repetitive and highly con-erved pattern, is a very good candidate for the role of a general

ymmetry of the brain (and other systems).

Once established that the j-function’s is ubiquitously endowedll over normal EEGs traces during resting state, eyes open condi-ions, the further steps could be: a) to evaluate task related EEGs,

ters 653 (2017) 234–238

and b) to superimpose the j-pattern to nervous oscillations otherthan electric spikes, such as the ones detected by, e.g., fMRI orPET. Once elucidated the role of j-function in physiological brainoscillations, it will be worthwhile to look for possible j-function’salterations in pathological states. Indeed, it could be hypothesizedthat the recurring j-oscillations are disrupted during pathologies,paving the way to novel approaches to brain diseases.

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