How do thoughts flow in the brain? Current advances in neuroscience emphasize the role of energetic landscapes (Watanabe et al. 2014; Sengupta et al. 2013), a sort of functional linens equipped with peaks, valleys and basins made of free-energy, where thoughts move, following erratic and/or constrained trajectories (Afraimovich et al. 2013; Zare and Grigolini 2013; Fraiman and Chialvo 2012; Deco and Jirsa 2012). Such “dynamical systems” theories lack however an explanation of the possible physiological mechanisms underlying brain activity. General theories are thus needed to further improving our knowledge of central nervous system activity. This review, based on recent findings, introduces the concept of a spatial fourth dimension, where brain functions might take place, as a general device underlying our thoughts’ dynamics. In particular, we hypothesize that brain activity is shaped in guise of a “3-sphere” which performs 4D movements on the cortical layers, giving rise to a functional “Clifford torus” where mental operations might take place. We will discuss how thoughts follow constrained, donut-like trajectories along preferential functional railways embedded in the very structure of the human brain connectome. This essay comprises five sections and an appendix organized as follows. The very first section will be devoted to give as simple as possible explanations of concepts from the far-flung branch of topology. Because this method of investigation of the laws of biology diverges from what was previously familiar, we prefer to use a language addressed to a broad interdisciplinary audience which encompasses experts from different fields (neuroscientists, medical doctors, biologists, epistemologists). Sections two and three will aim to enclose the concept of the 3-sphere in the framework of brain activity, in order to demonstrate the feasibility of our theory and to sketch a four-dimensional brain model in a way that is empirically testable. Section four, taking into account suggestions from recent papers, will provide quantitative methods for an experimental assessment of our hypothesis, which could be of use to researchers that investigate network connectivity. The final section discusses how a brain hypersphere could lead to novel approaches to organized brain models, improving our understanding of cortical functions and paving the way to innovative research strategies. In order to show how our theory can be operationalized, we also provide an appendix as supporting information for technical readers. This latter material unpacks 3-spheres and fourth dimensions afforded by a mathematical point of view.

What is a hypersphere?

A n-sphere, also called Sn, is a n-dimensional structure embedded in a n + 1 space (Henderson 1996). For example, a 1-sphere (S1) is the one-dimensional circumference surrounding a 2-dimensional disk, while a 2-sphere (S2) is the 2-dimensional surface of a 3-dimensional space (a beach ball’s surface is a good illustration) (Henderson and Taimina 2001). In mathematical terms, a 3-sphere (S3), also called glome (from the Latin “glomus”, meaning ball of string), or generically hypersphere, is an artificial convex structure enclosed in a Euclidean 4-dimensional space called a 4-ball (Marsaglia 1972). The prefix “hyper” refers to 3- (and higher-) dimensional analogues of n-spheres. A 3-sphere is thus the surface of a 4-dimensional ball, while a 4-dimensional ball is the interior (the content) of a 3-sphere, in the same way as a bottle of water is made of a glass surface and a liquid content. How to build a glome? In topology, a glome can be attained by superimposing two 2-spheres (two beach balls) whose opposite edges are abstractly glued together: we obtain a structure called the Clifford torus (Manetti 2015). A Clifford torus reminds a lifebelt, with its donut-like shape (Fig. 1a), nevertheless with an important difference: the Clifford torus displays the same local geometry as an “ordinary” three-dimensional space, but its global topology is different (Willard 1970; Krantz 2009). Our torus is constantly in movement, because it is equipped with intricate rotations called quaternionic movements (Ozdemir and Özekes 2013; Lemaître 1948). Imagine a skein made of rope circles, where each rope takes a different rotation: the general shape of the skein is always the same, but an ever changing motion occurs inside it. The internal shape of the glome is in ceaseless activity, depending on the number of ropes taken into account and their trajectories. Complicated motions occur indeed on a hypersphere: four pairs of points opposite each other (called antipodal points 1, −1, k, −k, j, −j and i, −i) give rise to the “quaternion group” (Hart and Segerman 2014), which displays two possible types of reciprocal 4D rotations (Fig. 1b). For a mathematical treatment of hyperspheres, quaternions and the correlated Lie groups, see Appendix, Section 1. The glome, requiring four dimensions for its definition just as an ordinary sphere requires three, is not detectable in the usual spatial 3-dimensions and is thus very challenging to assess. Figures 1a–c show the possible ways to cope with a 3D visualization of a glome: observe how opposite sections of the rim fit together, rather than trying to visualize the whole thing at once the way you would visualize a common sphere. Each apparent line segment is really two, one arching upward into the third dimension and the other arching downward. To better understand the concept of a 3-sphere, the images should be watched during their complicated movements: a list of very useful videos is provided in Appendix, Section 1.

Fig. 1
figure 1

Different ways to depict a hypersphere. a How two 2-spheres glued together along their spherical boundary give rise to a donut-shaped Clifford torus. b Another way to depict a hypersphere: the superimposition of two 2-spheres (which circumferences are glued together) gives rise to a glome. Some of the quaternion rotations are depicted by the straight and curved arrows. c 3D projection of a hypersphere. The lines on the left enlarge in diameter, forming a circle of increasing circumference on the left surface of the 3D space. Conversely, on the opposite right side, the lines shrink and give rise to a circle of decreasing circumference on the right surface of the 3D space. See text for further details. The dotted lines and the black spheres depict some of the possible antipodal points predicted by the Borsuk Ulam Theorem (to give another example, J and −J are antipodal points in a)

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Why an hypersphere to explain brain activity?

Experimental and theoretical clues allow us to conjecture that the brain activities (at least some of them) are embedded in a torus lying on the surface of a hypersphere. The theoretical claims of brain multidimensionality are widespread (Tononi 2008; Papo 2014; Ponce-Alvarez et al. 2015; Schoenberg and Speckens 2015). Models characterized by dimensionality reduction have been used in the study of human central nervous system, particularly through full-brain auto-regressive modelling (Garg et al. 2011). It has been demonstrated that high dimensionality spontaneous activity structures—termed “lag threads”—can be found in the brain, consisting of multiple, reproducible temporal sequences (Mitra et al. 2015). Moreover, it has been shown that the exceedingly diverse nonlinear selectivity in single-neuron activity in the prefrontal cortex, which encodes distributed information about all task-relevant aspects, is a signature of high-dimensional neural representations: crucially, this dimensionality is predictive of animal behaviour as it collapses in error trials (Rigotti et al. 2013). In addition, recent findings suggest that nervous structures process information through topological as well as spatial mechanisms. For example, it is has been hypothesized that hippocampal place cells create topological templates to represent spatial information (Dabaghian et al. 2014). We stated that a glome displays a donut-like shape: it means that the trajectory followed by a particle inside the torus is closed. To make an example, if you walk along one of the curves of a 4-ball, you think are crossing a straight trajectory, and do not recognize that your environment is embedded in an higher dimension. The torus may be compared with a video game with biplanes in aerial combat: when a biplane flies off one edge of gaming display, it does not crash but rather it comes back from the opposite edge of the screen. Mathematically speaking, the display edges have been “glued” together. Our brain exhibits similar behaviour, i.e., the unique ability to connect past, present and future events in a single, coherent picture (Fingelkurts and Fingelkurts 2014; Nyberg et al. 2010), as if we were allowed to watch the three screens of past-present-future glued together in a mental kaleidoscope. The same occurs during other brain functions, e.g., memory retrieval, recursive imagination and mind wandering (Andrews-Hanna et al. 2014), in which concepts flow from a state to another and appear to be “glued” together. As stated above, the torus is naturally visualized intrinsically, by ignoring any extrinsic properties a surface may have: it is thought that all the movements onto a torus surface are performed just by trajectories internal to its structure. For example, take a sheet of paper and bend it into a half-cylinder: the extrinsic geometry of the paper has obviously changed, but the paper itself has not been deformed and its intrinsic geometry has not varied. What would you see if you lived in a closed 4-dimensional manifold? You should be able to see yourself, via the intrinsic structure provided by the glued surfaces of a hypersphere, in an otherwise unperceivable 4D space (Weeks 2002). In the same way, we humans perceive our thoughts intrinsically and naturally adopt “private”, subjective standpoints.

How to demonstrate the presence of a brain hypersphere?

Brains equipped with a hypersphere is a counter-intuitive hypothesis, since we live in a 3D world with no immediate perception that 4D space exists at all. We need thus to evaluate indirect clues of the undetectable fourth dimension, such as signs of the glome rotations on a familiar 3D surface. In other words, rotations of a 4D torus embedded in a 4-ball can be identified through their “cross section” movements on a more accessible 3D surface (Fig. 1c), as if you recognized an object just from its shadow projected on a screen. In Fig. 1c, the arrows illustrate the trajectories followed by the 4D quaternionic movements of a Clifford torus, when projected onto the surface of the 3D space in which it has been artificially embedded. Note that the arrows (representing the quaternions' flows) follow the external and medial surfaces of the 3D space in a way that is predictable. It must be emphasized that just one of the possible directions of the quaternion movements is displayed in Fig. 1c: the flow on a Clifford torus may indeed occur in each of the four planes. The small circle labelled T0 depicts one of the possible conventional starting points of the flow. In Fig. 1c we showed just the trajectory from right to left, starting from T0; however, also the opposite trajectory, from left to right, and countless others, can be exploited by the torus during its 4D movements.

The presence of a glome can be detected invoking the Borsuk-Ulam Theorem (BUT) from topology, which is characterized by very helpful features (Borsuk 1933; Matoušek 2003). BUT states that we can identify on a n-sphere two “antipodal points” (i.e., points directly opposite each other) (Fig. 2). To make an example, there always exist a pair of opposite points on the earth’s equator (a 1-sphere) with the same temperature. It is worth of mention that the two antipodal points display similar characteristics and have matching descriptions (Cohen 1973; Giblin 2010). Such two opposite points can be used not just for the description of simple topological structures, but can also be generalized to the description of two antipodal shapes or signals. Thus, if you take into account physical and biological phenomena instead of “points”, BUT leads naturally to the possibility of a region-based, not simply point-based, geometry, in which we view collections of signals as surface shapes, where one shape maps to another antipodal one (Peters 2014; Collins 2004; Borsuk 1958–1959; Borsuk 1969; Borsuk and Gmurczyk 1980). A point on S2 (which is embedded in R3) projects to two antipodal points on R4. It means that the projection from a lower dimension Rn (equipped with just one point) to an higher one Rn+1 gives rise to two opposite, symmetrical points. It also means that different phenomena (including brain activity) could be studied in terms of antipodal points, if we embed them in just one dimension higher than the usual one.

Fig. 2
figure 2

The Borsuk-Ulam theorem for different values of Sn. S1 depicts a circumference, S2 a common sphere, S3 a hypersphere, while R1 portrays a line, R2 a circumference and R3 a common sphere. Note that the two antipodal points in every sphere Sn project to a single point in the corresponding space Rn, and vice versa

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If we embed the brain in the 4D space of a Clifford torus, the presence of antipodal (opposite) points or shapes on 3D cortical surfaces could be evaluated (Fig. 3). We need to stress again that the hypersphere’s location displayed in Fig. 3 is just one of the countless possible: being the glome a functional structure equipped with many rotations and trajectories, it can be placed in different points of the brain surface. The antipodal points in the fourth dimension evoked by BUT can be viewed as brain signals opposite each other on the three-dimensional brain. For example, when a surface “fires”, the simultaneous firing of the opposite one is a proof of a perceivable “passing through” of the fourth dimension onto the brain 3D surface. The mechanism, elucidated in Fig. 3, is the following: the activation of a single point on the 3D brain S2 surface (Fig. 3a) leads to the activation of two antipodal points on the 4D brain S3 surface (Fig. 3b). In turn, the activation of two antipodal points on S3 leaves on the 3D brain S2 surface “hallmarks” (Fig. 3c) which can be detected by currently available neuroimaging techniques. For a mathematical treatment of the Borsuk-Ulam theorem, shapes, homotopies and their applications, see the Appendix, Section 2. In conclusion, the mathematical model of antipodal points can be casted in a biologically informed fashion, resulting in a framework that has the potential to be operationalized and assessed empirically.

Fig. 3
figure 3

The concept of hypersphere in the framework of brain functional activity. According to the Borsuk-Ulam theorem, the activation of a single point on the S2 brain surface (Fig. 1a) leads to the activation of two antipodal points (corresponding in this case to the quaternionic points J and −J) on the S3 brain surface (b). The simultaneous activation of the S3 antipodal points displayed in b can be also evaluated on the 3D cortical surface (c), provided the brain is embedded in a 3D space containing the 4D Clifford torus (the same 3D space described in Fig. 1c). The nomenclature is borrowed from c

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Are there proofs or clues of brain hypersphere?

Our brain hypersphere hypothesis could be corroborated either by prospective investigations, or by retrospective evaluation of published resting-state fMRI data, looking for the hallmarks of the hypothesized BUT. Why resting- state studies are more feasible for an evaluation of the possible presence of a glome in brain activity? Spontaneous oscillations are intrinsic, low-frequency fluctuations of cerebral activity which cannot be attributed to the experimental design or other explicit input or output (Fox and Raichle 2007). Among the networks exhibiting coherent fluctuations in spontaneous activity, the “default-mode network” (DMN) is worth of mentioning, because it includes functionally and structurally connected regions that show high metabolic activity at rest, but deactivate when specific goal-directed behavior is needed (Raichle et al. 2001). Spontaneous oscillations recapitulate the topographies of fMRI responses to a wide variety of sensory, motor and cognitive task paradigms, providing a powerful means of delineating brain functional organization without the need for subjects to perform tasks (Mitra et al. 2014). To evaluate our BUT hypothesis, we favour studies focused on intrinsic, instead of task-evoked activity, because the former is associated with mental operations that could be attributed to the activity of a glome—mental “screens” are glued together and the trajectories of particles (or thoughts!) follow the internal surface of a Clifford torus. For example, spontaneous brain activity has been associated with mind-wandering or day dreaming propensities (Kucyi et al. 2014), construction of coherent mental scenes, autobiographical memories (Conway and Pleydell-Pearce 2000), experiences focused on the future and dreaming state (Domhoff and Fox 2015). Recent evidence also suggests overlap between the DMN and regions involved in self- and other-related mental operations—such as affective and introspective processes (Amft et al. 2015; Philippi et al. 2015; Morewedge et al. 2014)—for a description of the terminology, see Andrews-Hanna et al. (2014). It has also been hypothesized that spontaneous functional connectivity patterns at rest might constitute a “signature of consciousness”, reflecting a stream of ongoing cognitive processes (Barttfeld et al. 2015). Spontaneous activity is highly variable among individuals, depending on local brain differences, somatosensory awareness, age span, race, culture and so on (Gorgolewski and al 2014; Sadtler et al. 2014). We speculate that such variability might be correlated with those differences in Clifford torus’ structure and movements illustrated above. A brain 3-sphere has the potential to represent a conceptual bridge, because it exhibits both anatomical/functional (spontaneous brain activity and DMN) and psychological correlates (spontaneous, deliberate, self-generated thoughts).

Several data (movies, figures and temporal series) extrapolated from different available experimental studies and/or meta-analyses describing brain spontaneous activity seem to corroborate our hypothesis. To make some examples:

  1. (a)

    High dimensionality spontaneous activity structures—“lag threads” -, consisting of multiple highly reproducible temporal sequences, showed the frequent occurrence of the antipodal points’ activation at different times (Mitra et al. 2015) (Fig. 4).

    Fig. 4
    figure 4

    Video frames, modified from Mitra et al. (2015), showing lag threads computed from real BOLD resting state rs-fMRI data in a group of 688 subjects, obtained from the Harvard-MGH Brain Genomics Superstruct Project (see Mitra et al. 2015 for further technical details). Note the widely diffused presence of BUT hallmarks (black lines) at different times and in different brain projections

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  2. (b)

    Decomposition of spontaneous brain activity into distinct fMRI co-activation patterns displayed a simultaneous activation of perisylvian and/or the opposite medial occipito-parietal areas (Liu et al. 2013) (Fig. 5a).

    Fig. 5
    figure 5

    Examples from functional neuroimaging real data showing how the predicted antipodal points can be correctly identified: given one point (a brain signal), there is a second point (another brain signal) at the opposite end of a straight line segment connecting them and passing for the center (white or black lines). a Decomposition of Spontaneous Brain Activity into Distinct fMRI Co-activation Patterns (Liu et al. 2013). b Different clusters during resting-state fMRI scanning, evaluated through innovation-driven co-activation patterns (called iCAPs) (Karahanoglu and Van De Ville 2015). c Significant meta-analytic clusters of fMRI temporal activation associated with mind-wandering and related spontaneous thought processes (Fox et al. 2015)

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  3. (c)

    Karahanoglu and Van De Ville (2015) decomposed resting-state fMRI using iCAPs, i.e. innovation-driven co-activation patterns. Their results show once again that opposite brain areas activate together: in particular, the pattern was cleary visible in bilateral posterior-frontal antipodal areas and in bilateral perisilvian antipodal areas (Fig. 5b).

  4. (d)

    Meta-analytic clusters of fMRI temporal activation associated with mind-wandering and related spontaneous thought processes (Fox et al. 2015) clearly displayed the simultaneous activation of two antipodal zones (Fig. 5c).

  5. (e)

    Unnoticed features of antipodal points’ activation can be found also in other data sets. For example, see Andrews-Hanna et al. (2014), Fox and Raichle (2007), Raichle (2009), Power et al. (2014), Mao et al. (2015), Harrison et al. (2015), Gusnard et al. (2001), Gravel et al. (2014), Barttfeld et al. (2015), Ajilore et al. (2013).

In sum, it is possible to find in literature several data confirming the occurrence of opposite brain zones’ activation, which is the sign predicted by BUT. The fMRI sequences of brain region activations, apart from differences depending on slight methodological distinctions among the papers, exhibited a stereotyped topographical firing pattern, such that brain loci are activated together with their opposite zones. As a result, highly reproducible topography and propagation through subsets of regions that are shared across multiple trajectories can be found in different data sets: it corroborates the prediction of BUT linked with the presence of a brain hypersphere.

What now?

Our review uncovered evidence of hypersphere in experimental fMRI series obtained during spontaneous activity, raising the possibility that (at least some) brain functions lie on a glome enclosed in 4D space. Brain activity is thus temporally driven by a functional 3-sphere, intrinsic to the brain and (probably) embedded in the very anatomical structure of the connectome. Indeed, to evaluate glomes in terms of framework for brain functions, we first need to identify potential brain signal loci where quaternion rotations might take place. The natural candidate is the spatially embedded network of the human connectome (Sporns 2011; Jarman et al. 2014), a non-stationary, highly dynamical structure (de Pasquale et al. 2015; Fox et al. 2015) characterized by complex topological features and an ever-changing geometry (Roberts 2015). Me may thus state that a 4D cap surrounds the brain, equipped with trajectories which follow quaternion movements along the nodes of the connectome. The reproducibility of the BUT hallmarks in different Authors suggests that this organizational feature is essential to normal brain physiology and function.

Further studies are needed to evaluate what happens when other techniques are used, e.g., EEG and diffusion tensor imaging. Does evoked, task-related activity exhibit the same features displayed by resting-state? Additional investigations will elucidate whether, following the stimulus onset (either external from the environment or internal from the body melieu and the brain), the multidimensional space outlined by cortical activity is invariant or reduced (Papo 2014). Human fMRI studies have recently revealed that a co-activation of sensory and fronto-parietal areas (in particular, simultaneous firing of activated antipodal points in somatosensory, lateral and medial frontal, posterior parietal and in the cingulate cortex) is crucial for conscious sensory perception in the several second time-scale of BOLD signal fluctuations (Hirvonen and Palva 2015).

If you are embedded in a 4D torus containing a doll, you see in front of you side-to-side reversal (and/or three-quarters turn) doll’s images. To be into a glome looks like to stand in front of a weird mirror: if you raise your left hand, you see in infinitely many screens countless copies of yourself raising the left hand, while other copies raise the right one. Because such neighboring images of the same object are related by glide reflections translations (Weeks 2002), it remains to be seen what the implications of the glome would have for consciousness, perception of time and nature of reality. Going a step further, our “deterministic” account of linear transformations needs to be contextualized, taking into account the suggestions of the brain as an energy-sparing, complex, nonlinear system equipped with attractors and/or random walks (Beggs and Timme 2012; Friston 2010; Zhang et al. 2015). Also the role of electromagnetic currents needs to be re-evaluated, i.e., do spikes contain the message, or, as recently suggested (Tozzi 2015), they serve other kinds of functions? For example, it has been proposed that features of a brain signal with spectral peaks in preferred bands (gamma, beta and so on) provide a basis for feature vectors in a 4D Euclidean space (Padhy et al. 2011). Further, the hypersphere, due to different movements of the quaternionic system, relentlessly varies its intrinsic structure: in this context, it is reasonable to speculate that each mental state corresponds to a different glome’s topological space. In conclusion, our original approach to brain activity, starting from such a completely new point of view, contains the promise of great fruitfulness in the mere fact that it directs us into new ways, opens out new perspectives and shows how it is possible to get quite a new light upon the old problems which are constantly being turned over and over in the same way. Indeed, a deeper knowledge of a fourth brain dimension via experimental confirm will lead to novel approaches to organized biological models, will improve our understanding of cortical functions and will also pave the way to innovative therapeutic strategies.