In this paper, we introduce a novel technique of fMRI images analysis, called computational proximity method, i.e., nucleus clustering in Voronoï tessellations (Peters and Inan, 2016a). The images are subdivided in contiguous (without interstice or overlap) polygons, called the “Voronoï polygons.” They yield a density map, called “tessellation,” that makes it possible to make an objective measurement of the polygon areas' spatial distribution and helps to define “random,” “regular,” and “clustered” distributions (Duyckaerts and Godefroy, 2000; Edelsbrunner, 2014). Tessellations have been already used in neuroscience, i.e., to investigate spatial relations and connectivity between neural mosaics in the retina (Mozos et al., 2011) or to evaluate histological cortical sections (Peters et al., 2016) and pattern recognition (Hettiarachchi and Peters, 2017). In a Voronoï tessellation of an fMRI image, of particular interest is the presence of maximal nucleus clusters (MNC), i.e., zones with the highest number of adjacent polygons (Peters et al., 2016). MNC reveals regions of the brain, independent from blood-oxygen-level dependent (BOLD) signals, characterized by different gradient orientation and diverse functional dimensions (Saye and Sethian, 2011).

To evaluate the power and potentialities of this novel approach, we used the Voronoï tessellation technique combined with Rényi entropy, in order to test the brain-hypersphere hypothesis. Indeed, it has been recently hypothesized that brain activity is shaped in the guise of a functional hypersphere, which performs complicated 4D movements called “quaternionic” rotations (Tozzi and Peters, 2016a). They give rise to the so-called “Clifford torus,” a closed donut-like structure where mental functions might take place. The torus displays glued trajectories similar to a video game with spaceships in combat: when a spaceship flies off the right edge of the screen, it does not disappear but rather comes back from the left (Weeks, 2002). The human brain exhibits similar behavior, i.e., the unique ability to connect far-flung events in a single, coherent picture (Atasoy et al., 2016). During brain functions such as memory retrieval and mind-wandering, concepts flow from one state to another and appear to be “glued” together. It has also been recently proposed that the brain, when evaluated in the proper dimension (Kida et al., 2016), is equipped with symmetries in one dimension that disappear (said to be “hidden” or “broken”) in just one dimension lower (Tozzi and Peters, 2016b). A symmetry break occurs when the symmetry is present at one level of observation, but “hidden” at another level: it suggests that a 4D hypersphere could be equipped with symmetries, of great importance in order to explain central nervous system (CNS) activities, undetectable at the usual 3D cortical level.

Although we live in a 3D world with no immediate perception that 4D space exists at all, the brain hypersphere rotations can be identified through their “cross section” movements on a more accessible 3D surface, as if you recognized some object from its shadow projected on a screen. We may thus evaluate indirect clues of the undetectable fourth dimension, such hypersphere rotations' hallmarks or signs on a familiar 3D surface. Here we show that, in temporal fMRI series from spontaneous and evoked brain activity, MNC discloses the typical patterns of quaternionic rotations and hidden symmetries.

Here we describe the novel approach for fMRI neuroimages analysis. In brief, we show how fMRI images of the brain can be divided in small polygons, called tessellation regions, that display different levels of information. We elucidate how the differences amongst the small polygons make it possible for to evaluate otherwise hidden activities of the brain, and how such activities are the expression of transitory functional increases in nervous dimensions.

In all the 80 assessed images, either from intrinsic or extrinsic activity of the brain, we found that clusters of higher activity, which are equipped with higher Rényi entropy compared with the surrounding zones, are scattered throughout different nervous areas. Therefore, some micro-areas of a specific anatomical zones contain more information than the adjacent ones. In other words, the MNC approach detects regions in the brain with the higher Rényi entropy, compared with the surrounding ones. In 50% of the 80 analyzed images, more than one cluster is detectable.

Concerning the 64 images from spontaneous, intrinsic, resting state activity of the brain, the data analysis shows also that the MNC activity displays in 79.68% (51 of 64) the typical features of the Clifford torus' movements. This supports the hypothesis that a functional hypersphere occurs during resting state brain activity (Figure 7). At the beginning, the trajectory of spontaneous activity follows the median sections (see timeframes 0.1–0.3 in Figure 7), then moves toward the posterior part of the brain, where a reflexion of four trajectories along the lateral surfaces occurs (0.4). This pattern closely matches the one predicted by the model illustrating the quaternionic movements on a Clifford torus. The temporal sequence also show the hypersphere moves on the brain, and it moves relatively slowly. The hypersphere does not display a regular or continuous movement, rather it proceeds forward and then backward for a short time (time 0.5 and 0.6). From 0.7, the trajectory follows the patterns predicted by Figure 6.

In the four available temporal series of the 54 images of spontaneous activity from Mitra et al. (2015), the MNC movements follow a specular, repetitive temporal pattern of activation (Figures 7–9). For example, the pictures of the first and the last time display MNC activity with matching description. Note that, in the 64 rest-related images, the areas encompassing MNCs do not match those equipped with BOLD activity in 71.87% (46/64) (Figure 9).

Concerning the 16 images from evoked, task-related activity elicited by basic vision and object recognition (Mandelkow et al., 2016), they showed a widely distributed MNC pattern throughout the brain. Fifty percent (8/16) of the images displayed the pattern predicted by the presence of a hypersphere, against the above mentioned 79.68% of correspondence detected in images of intrinsic activity.

The results can be summarized as follows: MNCs sequences of brain region activations, apart from differences depending on slight methodological distinctions among the assessed papers, exhibit a stereotyped topographical firing pattern. The movements of a hypersphere are clearly detectable, and the MNC activity clearly displays antipodal points in a temporal sequence, independent from fMRI activation. Such features are more marked in images taken from intrinsic, rather than extrinsic, task-related activity of the brain.

There are two state-of-the-art approaches for understanding the communication among distributed brain systems using fMRI data. The first approach termed as dynamic causal modeling uses models of effective connectivity, while the second, called Granger causal modeling, uses models of functional connectivity (Friston, 2009, 2010). Our paper introduces a novel method, based on computational proximity (CP). This novelty can be expressed as follows: rather than being correlated with the “classical” BOLD activity, the CP method shows how spatial regions are correlated through their “proximity” (descriptive closeness to each other). From the experiments reported here, we are beginning to see different forms of brain function represented by the MNCs. We have demonstrated that computational proximity (i.e., strongly near nucleus mesh clusters) in 2D fMRI images is able to reveal hidden temporal patterns of Rényi entropy, enabling us to detect functional information from morphological data. The combination of tessellated images, topological framework provided by BUT, and Rényi entropy indicators of the information levels of image regions provide a systematic basis for the study of fMRIs. The validation of this approach to fMRI analysis stems from the mathematics of BUT and Rényi entropy. Central to this form of neural analysis is its introduction of tessellated fMRI images. Tessellation is a natural choice, since fMRI images by their nature both visual and geometric. There is a natural transition from a tessellated fMRI image to a consideration of those fMRI regions where the greatest changes occur. The greatest-change-regions are in the form of maximal nucleus clusters (MNCs) in each tessellated image. Detected MNCs in tessellated fMRI images are then subjected to analysis by BUT and the mechanics of Rényi entropy. The end result is a complete, verifiable, rigorous analysis fMRI images that are easily understood because of their visual character. The accuracy of the tessellated fMRI image approach stems from the computational geometry that leads to the detection of MNCs (Peters, 2016; Peters and Inan, 2016a; Peters et al., 2016; Tozzi and Peters, 2016b).

Here we have shown that a morphological analysis of simple 2D images taken from fMRI video frame sequences might give insights into the functional structure of neural processing. In a previous study, we evaluated the possible hints of a hypersphere on simple fMRI scans during resting state brain activity (Tozzi and Peters, 2016a). We showed how, due to the Borsuk-Ulam theorem (BUT), the fMRI activation of brain antipodal points could be a signature of 4D. The antipodal points predicted by BUT could be evaluated not just on images taken from fMRI studies, but also on datasets from other neuro-techniques, such as, for example, EEG. In the current study, we used a novel method, in order to confirm the data of the previous work with a completely different and more sophisticated approach. Indeed, looking at the sequences of maximal nucleus clusters and their entropy, we found experimental patterns compatible with the ones predicted by the hypersphere model. We detected on the 3D brain “shadows” of a 4D hypersphere rotating according to quaternion movements: these “hints” make it possible for us the possibility to visualize both the spatial arrangement and the movements of the corresponding Clifford torus. In other words, in particular during spontaneous brain activity, the apparently scattered temporal changes in MNCs follow a stereotyped trajectory which can be compared with the 4D movements of a hypersphere. Our study uncovered evidence of hypersphere during spontaneous activity, demonstrating that brain activity lies on a 3-sphere embedded in 4D space. How can be sure that the MNC reveals the presence of a brain hypersphere? Three cues talk in favor of this hypothesis. First, the MNC patterns, in particular during resting state fMRI data sets, closely resemble the theoretical trajectories predicted by Clifford torus movements. Second, temporal sequences of fMRI images display matching description, in agreement with the BUT dictates. Third, there is a difference in Rényi entropy between MNC and the surrounding zones, thereby pointing toward diverse levels of activity. The reproducibility of the hypersphere movements suggests that this organizational feature is essential to normal brain function. Because the Clifford torus incessantly changes its intrinsic structure, due to the different transformations of the quaternionic group, it is reasonable to speculate that each mental state corresponds to a different hypersphere's topological space. The concept of a hypersphere in the brain is also more noteworthy, if we frame it in the general picture of nervous symmetries (Tozzi and Peters, 2016b).

A shift in conceptualizations is evident in a brain theory of broken symmetries based on a hypersphere approach. It might be speculated that symmetries are hidden in a 3D dimension and restored in the higher dimension of the hypersphere, and vice versa. This means that brain functional and anatomical organization may be better assessed if one considers how certain hidden “symmetries,” essential to shaping brain gradients and activity, may only appear under the lens of higher-dimensional neural representations (Tozzi and Peters, 2016b), i.e., the hypersphere. We anticipate our essay to be a starting point for further evaluation, both in physiological and pathological conditions, of a neural fourth spatial dimension where other brain functions, such as perception, memory retrieval, might take place. It is also possible that every brain function displays a peculiar temporal pattern of such a novel entropic “activity.” In sum, we operationalized a novel, fairly inexpensive, image-analysis technique useful in detecting hidden temporal patterns in the brain. We can assess the spatial patterns described by MNCs in terms of entropy variations. While, by our “subjective” and “private” viewpoint, we tend to watch an image inferring the semantic parts in order to give it a meaning, MNCs allow the detection of the “objective” entropy, which do not necessarily correspond to the zones of the figure that we view as more significant. This means that MNCs provide a basis for quantifying high-yield information areas in fMRI image features that are normally “hidden” from our attention.

All the authors equally contributed to: study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical, material support, and study supervision.

The research has been supported by the Natural Sciences & Engineering Research Council of Canada (NSERC) discovery grants 185986 and 194376.