Thalamus & Related Systems 2 (2003) 145–152
Natural logarithmic relationship between brain oscillators Markku Penttonen a , György Buzsáki b,∗ a
Department of Neurobiology, A.I. Virtanen Institute for Molecular Sciences, University of Kuopio, P.O. Box 1627, FIN-70211 Kuopio, Finland b Center for Molecular and Behavioral Neuroscience, Rutgers, The State University of New Jersey, 197 University Avenue, Newark, NJ 07102, USA Accepted 6 December 2002
Abstract Behaviorally relevant brain oscillations relate to each other in a specific manner to allow neuronal networks of different sizes with wide variety of connections to cooperate in a coordinated manner. For example, thalamo-cortical and hippocampal oscillations form numerous frequency bands, which follow a general rule. Specifically, the center frequencies and frequency ranges of oscillation bands with successively faster frequencies, from ultra-slow to ultra-fast frequency oscillations, form an arithmetic progression on the natural logarithmic scale. Due to mathematical properties of natural logarithm, the cycle lengths (periods) of oscillations, as an inverse of frequency, also form an arithmetic progression after natural logarithmic transformation. As a general rule, the neuronal excitability is larger during a certain phase of the oscillation period. Because the intervals between these activation phases and the temporal window of activation vary in proportion to the length of the oscillation period, lower frequency oscillations allow for an integration of neuronal effects with longer delays and larger variability in delays and larger areas of involvement. Neural representations based on these oscillations could therefore be complex. In contrast, high frequency oscillation bands allow for a more precise and spatially limited representation of information by incorporating synaptic events from closely located regions with short synaptic delays and limited variability. The large family of oscillation frequency bands with a constant relation may serve to overcome the information processing limitations imposed by the synaptic delays. © 2003 Elsevier Science Ltd. All rights reserved. Keywords: Natural logarithm; Brain oscillators; Thalamo-cortical; EEG; Rhythms; Alpha, gamma and theta oscillations
1. Introduction There are numerous oscillations in the brain of all species, ranging from very slow oscillations with periods of tens of seconds to very fast oscillations with frequencies exceeding 1000 Hz (Bullock, 1997). After extensive empirical work and subsequent power spectral or autocorrelation analyses, these oscillations have been divided into several frequency bands. All oscillations are behavioral state-dependent and numerous oscillation bands are present in the different stages of one wake–sleep cycle. Some waking state oscillations support cognitive functions such as perception, learning and memory (Singer, 1993). In contrast, some sleep-related oscillations may take part in the formation of permanent memories (Buzsáki, 1989; Steriade, 1999). Previous works have suggested some specific relationship between two or more oscillation bands. For example, during sleep, cortical ∗ Corresponding authors. Tel.: +1-973-353-1080x3131; fax: +1-973-353-1280. E-mail addresses:
[email protected] (M. Penttonen),
[email protected] (G. Buzs´aki).
delta and spindle oscillations are phase-related to a slower oscillation (Steriade et al., 1993a). Furthermore, in an alert animal, hippocampal gamma and theta oscillations occur together with a reliable correlation between their frequencies and phase–power relationship (Bragin et al., 1995; Buzsáki et al., 2003). However, with the exception of a somewhat arbitrary division of human brain rhythms for pragmatic reasons (The International Federation of Societies for Electroencephalography and Clinical Neurophysiology, 1974; Steriade et al., 1990), there has not been a systematic attempt to relate brain oscillations to each other. Therefore, we examined whether the various known oscillations in the mammalian brain present a continuum, or whether they correspond to quantal events with some predictable relationship among them. From our analysis a general principle emerged: discrete oscillation bands form a geometric progression on a linear frequency scale and thus a linear progression on a natural logarithmic scale. Furthermore, the numerous brain oscillators fill all frequency bands from ultra slow to ultra-fast frequencies without major gaps.
1472-9288/03/$ – see front matter © 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1472-9288(03)00007-4
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M. Penttonen, G. Buzs´aki / Thalamus & Related Systems 2 (2003) 145–152
2. A brief overview of oscillations in the mammalian brain Slow, supra-second, oscillations can be grouped into four ranges: oscillations with periods of longer than 15 s (
