Tension-resolution patterns seem to play a dominant role in shaping our emotional experience of music. In traditional Western music, these patterns are mainly expressed through harmony and melody. However, many contemporary musical compositions employ sound materials lacking any perceivable pitch structure, rendering the two compositional devices useless. Still, composers like Tristan Murail or Gérard Grisey manage to implement the patterns by manipulating spectral attributes like roughness and inharmonicity. However, in order to understand the music of theirs and the other proponents of the so-called “spectral music,” one has to eschew traditional categories like pitch, harmony, and tonality in favor of a lower-level, more general representation of sound—which, unfortunately, music-psychological research has been reluctant to do. In the present study, motivated by recent advances in music-theoretical and neuroscientific research into a the highly related phenomenon of dissonance, we propose a neurodynamical model of musical tension based on a spectral representation of sound which reproduces existing empirical results on spectral correlates of tension. By virtue of being neurodynamical, the proposed model is generative in the sense that it can simulate responses to arbitrary sounds.

The mechanism of seizure emergence and the role of brief interictal epileptiform discharges (IEDs) in seizure generation are two of the most important unresolved issues in modern epilepsy research. We found that the transition to seizure is not a sudden phenomenon, but is instead a slow process that is characterized by the progressive loss of neuronal network resilience. From a dynamical perspective, the slow transition is governed by the principles of critical slowing, a robust natural phenomenon that is observable in systems characterized by transitions between dynamical regimes. In epilepsy, this process is modulated by synchronous synaptic input from IEDs. IEDs are external perturbations that produce phasic changes in the slow transition process and exert opposing effects on the dynamics of a seizure-generating network, causing either anti-seizure or pro-seizure effects. We found that the multifaceted nature of IEDs is defined by the dynamical state of the network at the moment of the discharge occurrence.

The general idea of nonstationarity of brain activity or dependence of the dynamics on some, potentially unobserved, temporally changing or fluctuating parameter, has been familiar in the neuroscience community in contexts such as sleep dynamics or epileptology for a long time. However, recently it has been attracting increasing attention in the context of functional brain network analysis. This seems as a natural development of the field—once that functional connectivity as computed under the simplifying stationarity assumption has been well established, it is only logical to try to detect changes in brain functional connectivity over time. In general, detecting such nonstationarities in a reliable fashion is a methodologically challenging task, as changes in estimates of functional connectivity over time may be also due to random fluctuations, rather than genuine changes of the process. There is a wide array of approaches to studying such nonstationarities documented in literature (Hutchison et al., 2013), and an important but often neglected general methodological step is assessing the results against an appropriate null model corresponding to stationary process. In the following, we give an illustrative example of how a typical nonstationarity analysis can generate spurious signs of nonstationary dynamics even when applied to stationary process. To show that this is not a purely theoretical issue, we closely follow the analysis procedure used in a recently published study by Betzel et al. (2012). We note that this particular paper have caught our attention by coincidence, while we believe the issue is pertinent to a substantial fraction of the literature. In their paper, (Betzel et al., 2012) deal with characterizing the dynamics of brain activity measured by EEG. In particular, Betzel et al. report the detection of rapid transitions between intermittently stable states, explicitly saying that “As predicted, fast (~100 ms) dynamics of whole-brain synchronization were observed during resting-state EEG,” documenting the typical fast (~100 ms) time scale of these states in Figure 6B of their paper (see also Figures 4, 5). Their argument is based on the following data-processing scheme: First, for each time point of filtered EEG data, a functional connectivity matrix is computed using pairwise synchronization likelihood values and the time points are clustered based on similarity of the corresponding functional connectivity matrices. Next, contiguous stretches of time points that are members of the same cluster are interpreted as corresponding to a duration of an atomic brain state. Finally, the brain-state-representing functional connectivity matrices are pooled across subjects and clustered based on their similarity to define higher-order states. Notably, the procedure applied by Betzel et al. is principally data-driven, rather than relying on some model testing or assumptions, and it includes band-pass filtering and sliding-window-like analysis. We therefore conjectured that the temporal structure of the observed functional connectivity dynamics might have been crucially affected by the procedure itself (as the authors tentatively admitted in their discussion, albeit unfortunately have not tested the results against stationary model data). To explore the viability of this alternative explanation, we applied a processing pipeline built according to the description given in the original manuscript to model data, consisting of 100 samples (each of length T = 2500 time points, representing mock 5 s epoch of EEG data) of a multivariate (N = 20) white noise process. The applied processing steps included application of frequency filtering (using elliptic filters corresponding to the four specified frequency bands; we applied zero-phase digital filtering by processing the input data in both the forward and reverse directions) and subsequent computation of the synchronization likelihood (Stam and van Dijk, 2002). The parameters of the synchronization likelihood l, m, w1, w2, nrec were set for each frequency band as in Betzel et al. (2012). The resulting functional connectivity matrices were clustered using the standard k-means clustering method (Lloyd, 1982). In Figure ​Figure11 you see that the typical duration of detected states closely corresponds to the distributions observed in the original paper (compare with Figures 6B, 4A,B in Betzel et al., 2012). In particular, the typical timescale is in the order of tens to hundreds of ms. Also, this time scale depends on the selected filtering in the same way as in the original work, with the time scales of the beta and theta bands markedly shorter and longer, respectively, than those of the broadband and alpha bands, the latter two being relatively close to each other. Figure 1 Temporal dynamics of synchronization likelihood (SL) networks generated from realizations of stationary processes: white noise (A,C,E) and correlated noise [linear stationary (FFT) surrogates from EEG data] (B,D,F). The top, middle and bottom images were ... Even though spatially and temporally independent (white) noise model used here is clearly not a realistic model for EEG data; such a simplistic stationary model reproduces the clustering time scales of the original paper with a surprising accuracy. Of course, due to spatial independence of the processes, it does not reproduce the spatiotemporal patterns corresponding to Figures 4A,B in Betzel et al. (2012). We have further repeated the procedure using multivariate Fourier transform surrogates generated from a single segment of EEG data (for more details on the data see Horacek et al., 2010. Such surrogates correspond to realizations of linear stationary process with conserved auto- and cross-correlation structure, see Prichard and Theiler, 1994). The results are shown in the right column of Figure ​Figure1.1. Moving from white noise to EEG surrogates, the time scales of the observed clustering hardly changed. However, as expected due to the introduced spatial dependence, the EEG surrogates show now a patchy spatiotemporal pattern Figures 1D,F more closely corresponding to those in the original paper. The similarity of the spatiotemporal patterns is of course only qualitative—range of differences may have arisen due to combination of different acquisition parameters as well as intra- and inter-individual variability. Note that we applied the basic k-means clustering method instead of the evolutionary-clustering algorithm from the original paper; insufficient detail of description of the procedure in the original paper made reproducing it prohibitively difficult. The value k = 3 was chosen for display of the clustering results, however, the results proved to be quite insensitive to the choice of k. Our numerical simulation above focused particularly on the observed time scales of the network states as obtained with the described analysis approach. One could indeed ask further, what evidence regarding “repertoire of states” can be provided by the detection of clusters per se—and whether the detection of (some) clusters could be merely a consequence of running a clustering algorithm. For a k-means clustering, the answer is obvious. Even for more complex approaches without fixed number of clusters such as the approach of Betzel et al. (2012), we conjecture that a repertoire could be observed even for a stationary process, however this depends on the details of applied analysis approach. In summary, we aimed to illustrate the proposition that spurious nonstationarity manifesting itself as alternation of network states may appear due to methodological issues even in stationary processes such as white noise. In our example, we showed that for instance the observation of clustering of time points (more precisely, temporal windows) into consecutive clusters (“states”) of duration in the order of several hundred milliseconds (the time scale of putative brain microstates) might be reproduced by white noise to a remarkable detail. Of course, this does not disprove the existence of such states—it just suggests the evidence may not be sufficient. From a wider perspective, one could see a parallel here with other examples of data analysis approaches that may lead to spurious observation of intriguing structures due to intrinsic bias of the methods—such as apparent signs of chaos in power-law spectra stochastic processes (Osborne and Provenzale, 1989) or small-world properties of functional connectivity graphs (Hlinka et al., 2012). Or, from an experimental point of view, with the role measurement artifacts such those as due to head motion might play in observed network properties (Hlinka et al., 2010; van Dijk et al., 2012).

Nonstationarity of neural dynamics is a ubiquitous property that is crucial to understanding many key phenomena of both healthy and diseased brain function, including circadian rhythms, dynamics of epileptic activity as well as cognitive processing. Detecting switching of brain states has recently become of growing interest in the human brain neuroimaging community. However, from the data analysis/modelling perspective the task is quite challenging, and competing approaches exist [1]. One widely adopted approach is the use of clustering methods in the temporal domain to detect temporally contiguous clusters of time points with a similar structure of some instantaneous property - e.g. neural activity or functional connectivity profile. While this approach may in principle help to explore the switching structure of brain dynamics, it comes with technical challenges related the presence of noise in both the dynamics and measurements. In particular, as we documented in a recent study [2], comparison of the results with an appropriate null hypothesis is necessary to avoid spurious detection of nonstationarity markers such as switching of neural network states. We document this danger by applying an example analysis pipeline used in [3] to simulated EEG datasets. The simulated data are generated as realizations of temporally white noise process (either spatially uncorrelated or spatially correlated in a pattern corresponding to real EEG data). In each case, one hundred realizations of a 5 seconds long epoch of N = 20 'electrodes' (each of 2500 time points corresponding to 2ms sampling rate). A k-means clustering algorithm with k = 2 to 10 is applied to cluster the instantaneous synchronization likelihood matrix estimates with parameters as in [3]. The key observation stable across all setting is that the applied typical network switching analysis pipeline leads to spurious discovery of a multitude of network states in the stationary process realizations, with dominant state duration timescales of several tens to hundreds milliseconds qualitatively similar to the original results reported in [3]. These results suggest that observations of network switching should be always cautiously interpreted and tested against appropriate null models.

Music gives rise to some of the strongest emotional experiences in our lives. Both musicological and music-psychological evidence converge on the theory that music achieves this by a sophisticated play with our expectancies [1]. The extent to which a pitch/chord is expected in the context of a musical sequence is closely related to its perceived and also to how much it apparently attracts other pitches/chords in the sequence (see [2] and the references thereof). The phenomenon wherein more stable pitches attract the less stable ones is called tonality. Over the eons, virtually every musical culture has created its own set of musical scales. Generally, each scale imposes a hierarchy of stability on all pitches sounded in its context [3]. We call such a hierarchy a tonal hierarchy. In [4] a formula for computing tonal stability of arbitrary pitch in a given context was proposed based on a network of neural oscillators. Even though high correlations with tonal hierarchies obtained experimentally were achieved by tuning the single parameter of the formula, there were some notable discrepancies between predictions of the formula and the psychological data. These might be explained by drastic approximations made on the way from the complete neurodynamical model to the formula which were necessitated by the complexity of the model. For instance, all interactions within the network were neglected. Nevertheless, this line of research clearly demonstrates the potential of neurodynamics to serve as a basis for derivation of a general theory of tonality applicable even to contemporary art music which is our main interest. Indeed, the neurodynamical model from [4] makes no assumptions whatsoever regarding its inputs. What is currently missing is a model of a suitable form so that such a theory can be derived from it without drastic approximations. We aim to fill this gap. It has become a standard in neuroscience community to approach neurodynamics from a dynamical perspective. Unfortunately, none of the approaches to analytical treatment of oscillatory dynamical we have reviewed proved suitable for our purposes. Consequently, inspired by the so-called memory evolutive neural systems [5], we are currently developing a neurodynamical model based on category theory. More specifically, we model the neural rhythms present in the auditory system during stimulation with a tone/chord as a category with objects representing the rhythms and morphisms representing the resonant interactions among them. Response to a sequence of stimuli is modelled using partial functors between such categories. The preliminary results are quite promising: from a very simple model we managed to derive a formula which predicts the ordering of pitches according to tonal stability within a major scale almost perfectly. Moreover, due to its nature, our model yields an almost mechanistic explanation of tonality. This, in our opinion, is as important a feature of any model as a good fit to data.