Elsevier

Brain Research

Volume 1335, 4 June 2010, Pages 91-102
Brain Research

Research Report
When frequencies never synchronize: The golden mean and the resting EEG

https://doi.org/10.1016/j.brainres.2010.03.074Get rights and content

Abstract

The classical frequency bands of the EEG can be described as a geometric series with a ratio (between neighbouring frequencies) of 1.618, which is the golden mean. Here we show that a synchronization of the excitatory phases of two oscillations with frequencies f1 and f2 is impossible (in a mathematical sense) when their ratio equals the golden mean, because their excitatory phases never meet. Thus, in a mathematical sense, the golden mean provides a totally uncoupled (‘desynchronized’) processing state which most likely reflects a ‘resting’ brain, which is not involved in selective information processing. However, excitatory phases of the f1- and f2-oscillations occasionally come close enough to coincide in a physiological sense. These coincidences are more frequent, the higher the frequencies f1 and f2. We demonstrate that the pattern of excitatory phase meetings provided by the golden mean as the ‘most irrational’ number is least frequent and most irregular. Thus, in a physiological sense, the golden mean provides (i) the highest physiologically possible desynchronized state in the resting brain, (ii) the possibility for spontaneous and most irregular (!) coupling and uncoupling between rhythms and (iii) the opportunity for a transition from resting state to activity. These characteristics have already been discussed to lay the ground for a healthy interplay between various physiological processes (Buchmann, 2002).

Introduction

Despite the massive interest in the way the human brain is processing information and despite a strong focus on brain research during the past decades, the foundation of a global ‘brain theory’ is not insight. The current consensus about brain processes may be characterized by the observation that the brain functions in a massively parallel and distributed manner. Thus, one of the fundamental questions is, which processes underlie the communication between largely distributed brain structures.

One possible scenario is that oscillations play an important role for the communication between different brain structures and networks. EEG oscillations reflect rhythmic fluctuations in the excitability of the dendritic tree of a large number of neurons (cf. Logothetis et al., 2001). During the high excitability phase, neurons are very likely to fire, whereas during the low excitability phase activity is reduced or even suppressed (cf. Fries, 2005, Mehta et al., 2002). This rhythmic synchronization facilitates simultaneous activation of common target cells and enhances (or enables) the spread of neural firing through a cell assembly (cf. Basar, 1999a, Basar, 1999b, Salinas and Sejnowski, 2001, Varela et al., 2001). As a consequence, oscillations provide time windows of synchronized neural activity that plays an important role for cognitive processes (for a review cf. e.g., Klimesch et al., 2007a). Because different frequencies appear to be related to the timing of different neuronal assemblies (as activated parts of a network), the general assumption is that the functional interplay between units of different assemblies is reflected by a special type of oscillatory coupling (cf., e.g., Varela et al., 2001) that is based on the synchronization of the excitatory phases of different frequencies (f1, f2). There are different EEG measures that are capable to analyse this type of cross-frequency (CF) phase coupling (or synchronization). Prominent examples are (i) cross-frequency phase φ1(t) and φ2(t) synchronization, (ii) coupling between amplitude envelopes A1(t) and A2(t), and (iii) phase-amplitude coupling (e.g., between φ1(t) and A2(t)). Phase synchronization may occur for harmonic frequency relationships, phase-amplitude coupling for frequency relationships larger than 3 (e.g. Lakatos et al., 2005, Roopun et al., 2008b).

In this study we focus on a specific problem that arises from CF coupling: spurious CF synchronization. Depending on the numerical ratio between f1 and f2, the excitatory phases of f1 and f2 will meet (i) frequently and on the basis of a regular pattern (as is the case for harmonic frequency relationships) or (ii) infrequently (as for irrational relationships). In the first case, CF coupling is assumed to play an important, active role for information processing. The second case, however, is associated with an inactive brain state that is characterized by a lack of selective information processing and selective coupling between frequencies. Spurious CF coupling may occur in this case and will represent an ‘unreliable’ or ‘erroneous’ mechanism because it is susceptible to randomly occurring ‘unwanted’ synchronization between frequencies.

Based on the finding that different frequencies f1 and f2 will never synchronize (in a mathematical sense) if the frequency relationship for f1, f2 (f2 > f1) is irrational, we show that the golden mean as the ‘most irrational’ number provides the most irregular pattern of excitatory phase meetings (in a physiological sense). As a consequence, we are able to demonstrate that there is no other frequency relationship that is better capable of avoiding spurious CF coupling. Searching for a frequency relationship that prevents spurious synchronization was motivated by theoretical considerations (Weiss and Weiss, 2003) and by recent physiological findings reporting that neighbouring EEG frequency bands form a geometric series (Roopun et al., 2008a, Roopun et al., 2008b).

Section snippets

Results

We provide evidence for the assumption that the usual distinction of frequency bands in the EEG form a geometric series with a ratio of (1+5)/21.618, which is the golden mean g. Based on this we show that excitatory phase meetings of these oscillations are impossible in a mathematical sense and most irregular in a physiological sense.

Discussion

We have shown that excitatory phase meetings based on a golden mean frequency relationships that are not only as rare as possible but occur in the most irregular pattern possible (Box 4 and Box 5), since the golden mean among other irrational numbers has the largest distance to any rational number in the interval [1, 2]. We assume that this most irregular pattern of coincidences characterizes the resting state of the brain, in which no selective information processing takes place.

Excitatory

Experimental procedure

For our mathematical analysis, we applied basic axioms and theorems from simple algebra, statistics and stochastics, as well as probability theory and probability logic.

Acknowledgments

This research was in part funded by the DOC fFORTE programme of the Austrian Academy of Science (No. 22192) supporting Belinda Pletzer and by project P 21503-B18 of the Austrian Science Fund supporting Wolfgang Klimesch.

References (20)

There are more references available in the full text version of this article.

Cited by (61)

  • Tracking transient changes in the intrinsic neural frequency architecture: Oxytocin facilitates non-harmonic relationships between alpha and theta rhythms in the resting brain

    2021, Psychoneuroendocrinology
    Citation Excerpt :

    In line with this notion, the here adopted method for assessing cross-frequency dynamics (i.e., based on the calculation of transient numerical ratios between peak frequencies of two brain rhythms) allows quantifying short-lived changes in the neural frequency architecture that mathematically enable cross-frequency coupling or decoupling (Klimesch, 2018; Pletzer et al., 2010). In the theoretical accounts by Klimesch (2012, 2013) and Pletzer et al. (2010), the EEG frequency architecture is proposed to entail ‘optimal’ frequency domains for facilitating cross-frequency coupling (i.e., based on harmonic numerical ratios), as well as for facilitating controlled cross-frequency decoupling (i.e., based on non-harmonic numerical ratios approximating the golden mean 1.618…). In line with prior work (Rodriguez-Larios and Alaerts, 2019, 2021), the current study confirmed a proportionally high incidence of cross-frequency relationships approximating the non-harmonic ‘decoupled’ 1.6:1 cross-frequency state during resting-state, thereby providing further experimental support that this configuration forms a prevalent physiological state within the intrinsic EEG frequency architecture.

  • Online Effects of Beta-tACS Over the Left Prefrontal Cortex on Phonological Decisions

    2021, Neuroscience
    Citation Excerpt :

    As in our previous study, we chose 16.18 Hz as control frequency since the ratio of 16.18 Hz and 10 Hz minimizes the probability of synchronization. The control frequency, considered as frequency of no interest, was defined as 10*1.618 stimulation in the sense of the “golden mean of frequencies”, where random cross-frequency synchronization events are assumed to be least frequent (Pletzer et al., 2010), and the sensations of tACS in the beta range are similar to those of tACS in the alpha range (compared to tACS in the theta or gamma range) (Kanai et al., 2008). Statistical analyses were performed in R 4.0 RC, 2016 (URL https://www.R-project.org/.)

  • Coupling and Decoupling between Brain and Body Oscillations

    2019, Neuroscience Letters
    Citation Excerpt :

    This is a simple but interesting fact, showing that cross frequency phase coupling can be considered a ‘two step’ process: Two oscillations shift their frequencies (m, n) in a way that their ratio is an integer (r = m/n = integer number), which then, in a second step, invites phase coupling. On the other hand, if the ratio produces an irrational number (r = m/n = irrational number), stable phase coupling is disabled [7,8]. As an illustration, let us assume three sine oscillations, one with 10 Hz, another with 20 Hz, and a third with g*10 = 16.18… Hz, where g is an irrational number, termed golden mean (g = 1.618….).

View all citing articles on Scopus
View full text