Periodic events as a subset of time-varying brain dynamics

Projects 1-3 argue that the characterization of non-stationary signal components permits not only the identification of what activity patterns occur, but also when and how frequently they appear in time. This supports a general shift away from static descriptions of signal averages towards a focus on the temporal aspects of macroscopic dynamics in neuroscience (Lurie et al., 2020). An intriguing alternative to the identification of rhythmic events (Project 1) is the use of data-driven latent state approaches that are increasingly used to investigate transient functional networks in fMRI and EEG/MEG (e.g., Karahanoglu & Van De Ville, 2015; Lurie et al., 2020; Taghia et al., 2018; Vidaurre, Abeysuriya, et al., 2018). For example, Hidden Markov Models allow estimation of a limited number of temporally recurrent latent states with e.g., specific spectral profiles (Vidaurre, Hunt, et al., 2018), which can subsequently be analyzed with regard to their transition probabilities, dwell times/durations and switching rates. These methods also successfully detect transient beta events (Heidema, Quinn, Woolrich, van Ede, & Nobre, 2020;

Quinn et al., 2019; Seedat et al., 2020), indicating their sensitivity to spectral features. Crucially, many of these models assume that only a single state is active at each time point, whereas rhythmic views of brain function often emphasize a simultaneous multiplexing of information at

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different temporal scales. While both approaches theoretically afford novel insights into time-varying neural dynamics, which approach proves more useful and/or accurate remains an interesting and open question. Alternative methods are also available to decompose neural time series into specific graphoelements, of which periodic patterns represent only a subset. Most prominently, matching pursuit (Chandran, Mishra, Shirhatti, & Ray, 2016; Mallat & Zhang, 1993) and dynamic mode decomposition (Brunton, Johnson, Ojemann, & Kutz, 2016) deconstruct signals into underlying ‘atoms’ or modes, either from a predefined dictionary or based on statistical covariation. An alternative nonparametric approach to pattern identification that is popular in genomics involves the testing of temporally binned time series against a specific morphology (e.g., including ‘sawtooth’ features) or a fixed library of waveforms (M. E. Hughes, Hogenesch, &

Kornacker, 2010; Michael et al., 2008; Thaben & Westermark, 2014). Similar to eBOSC (Project 1), such approaches may identify the contribution of specific graphoelements to neural time series.

Interestingly, despite the prevalent assumption of sinusoidal features in neuroscience, temporal dynamics may not be perfectly rhythmic, but instead systematically vary in form (Cole

& Voytek, 2017, 2019; Lozano-Soldevilla, 2018a; Schaworonkow & Nikulin, 2019). However, due to the mathematics of the Fourier algorithm, even non-sinusoidal shapes are separated into a combination of sinusoids, at a loss of morphological information. This overemphasis of sinusoidal features in spectral analysis (even when the signal does not warrant it) was noted early on (see also Rohracher, 1937): “Even though it may be possible to analyze the complex forms of brain waves into a number of different sine-wave frequencies, this may lead only to what might be termed a ‘Fourier fallacy,’ if one assumes ad hoc that all of the necessary frequencies actually occur as periodic phenomena in cell groups within the brain." (Jasper, 1948, p. 345; see also Rohracher, 1937)²³. Dedicated rhythm detection (Project 1) can assist in detecting periods that per se qualify as sinusoidal, but given the wavelet transform involved in their detection, does not provide strict

23Unfortunately, no English translation of Rohracher‘s concise discussion is yet available: “[W]enn jedoch die Tätigkeit eines Organs in einer charakteristischen Kurvenform des Spannungsverlaufes zum Ausdruck kommt, dann verliert die harmonische Analyse ihren Sinn;

denn sie löst die Kurve in Sinusschwingungen auf, also in Komponenten, von denen man nicht ohne weiteres annehmen darf, dass sie in Wirklichkeit bei der Entstehung der untersuchten Potentialschwankung beteiligt sind.” (Rohracher, 1937, p. 544). In coarse translation: “However, if the function of an organ is expressed in a characteristic shape of the voltage curve, then harmonic analysis loses its sense; it reduces the curve to sine waves, that is to components of which one may not without further ado assume that they are in reality involved in the generation of the examined potential fluctuation.“

39 evidence for it. As discussed in Project 1, a dedicated follow-up analysis that characterizes the time series morphology of rhythmic episodes (e.g., Cole & Voytek, 2019) may prove fruitful to further pursue such questions (e.g., Neymotin, Barczak, et al., 2020; Spyropoulos et al., 2020).

More generally, determining the sinusoidality of generative processes remains challenging given that a sinusoidal appearance at the scalp can arise from non-sinusoidal dynamics with different degrees of spatio-temporal mixing (Schaworonkow & Nikulin, 2019). This morphological uncertainty represents an ill-posed inverse problem, as sinusoidal events do not unequivocally suggest the presence of a sinusoidal generator, while a superposition of periodic generators can produce complex patterns (Lorincz et al., 2009). On a methodological note, entropy (Project 2) is invariant to the shape of repeating patterns, but captures the degree to which any pattern repeats.

Entropy applications may therefore prove advantageous when a strong assumption of sinusoidality is not warranted.

While analyses that are agnostic to time series shape remain productive, efforts toward better characterizing (quasi-)periodic or sinusoidal patters and deviations thereof (e.g., Cole &

Voytek, 2019) are poised to improve our understanding of the latent regimes that shape observed brain signals (e.g., Figure 1 in Breakspear, 2017). For example, a stable fixed-point attractor is expected to give rise to periodic signals, even in the presence of added noise that adds temporal amplitude and frequency fluctuations. In contrast, more complex patterns arise in the presence of a chaotic attractor (Breakspear, 2017)²⁴. While various circuit properties (e.g., spatio-temporal excitation-inhibition profiles) can instantiate periodic and chaotic regimes, large-scale models of neural circuits (e.g., Mejias et al., 2016; Neymotin, Daniels, et al., 2020; Robinson et al., 2001;

Schirner, McIntosh, Jirsa, Deco, & Ritter, 2018) can further constrain the space of biophysically-realistic implementations. Prior work (e.g., M. A. Sherman et al., 2016) has elegantly used such

24 Challenges regarding nonlinear contributions to measures such as entropy (highlighted in Project 2) are particularly relevant for inference of chaotic systems as suggested by neural mass models, given that “the presence of such nonlinear waveforms in macroscopic signals such as EEG would provide compelling support for these models and, more deeply, for the implicit assumption on which they rest: namely, that through synchrony, collective neuronal dynamics retain the nonlinearities present at the microscopic scale.” (Breakspear, 2017, p. 346). Similar to previous work (for reviews see Lancaster, Iatsenko, Pidde, Ticcinelli, & Stefanovska, 2018; Stam, 2005), Project 2 argues that such inference regarding non-linear contributions requires stringent null models, as provided by surrogate data.

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large-scale approaches to explain the emergence of transient beta events from an interaction of proximal and distal cortical input, while confirming model predictions using electrophysiological recordings. The methods highlighted here provide useful tools to further advance such efforts.

Together, Projects 1-3 encourage a wholistic perspective on features in both the time and frequency domains, in line with early conclusions that “[f]requency analysis, when related at all times to the original recording, is proving to be a useful adjunct to the electroencephalographer's armamentarium, if and when the various spectra thus obtained can receive adequate and valid interpretation.” (Jasper, 1948, p. 345). In the end, questions such as “When is fluctuating activity a rhythm? and How do we tell a real rhythm from an artifact of our analysis?”²⁵ (Bullock, 1997, p.

5) remain challenging and relevant.