# THE MULTIDIMENSIONAL BRAIN

**Hidden symmetries (and symmetry breaks) in the brain**

Symmetries are widespread invariances underlining countless systems, including the brain. A symmetry break occurs when the symmetry is present at one level of observation, but “hidden” at another level. In such a general framework, a concept from algebraic topology, namely the Borsuk-Ulam theorem (BUT), comes into play and sheds new light on the general mechanisms of nervous symmetries. BUTtells us that we can find, on an n-dimensional sphere, a pair of opposite points that have same encoding on an n-1 sphere. This mapping makes it possible to describe both antipodal points with a single real-valued vector on a lower dimensional sphere. Here we argue that this topological approach is useful in the evaluation of hidden nervous symmetries. This means that __symmetries can be found when evaluating the brain in a proper dimension__, while they disappear (are hidden or broken) when we evaluate the same brain in just one dimension lower. In conclusion, we provide a topological methodology for the evaluation of __the most general features of brain activity, i.e., the symmetries__, cast in a physical/biological fashion that has the potential to be operationalized. PDF

**The brain activity takes place in higher dimensions: not just a figure of speech!**

Tozzi A. 2019. The multidimensional brain. Physics of Life Reviews. doi: https://doi.org/10.1016/j.plrev.2018.12.004.

__Brain activity takes place in three spatial-plus time dimensions. This rather obvious claim has been recently questioned__ by papers that, taking into account the big data outburst and novel available computational tools, are starting to unveil a more intricate state of affairs. Indeed, various brain activities and their correlated mental functions __can be assessed in terms of trajectories embedded in phase spaces of dimensions higher than the canonical ones__. In this review, I show how further dimensions __may not just represent a convenient methodological tool__ that allows a better mathematical treatment of otherwise elusive cortical activities, __but may also reflect genuine functional or anatomical relationships__ among real nervous functions. I then describe __how to extract hidden multidimensional information from real or artificial neurodata series__, and make clear how our mind dilutes, rather than concentrates as currently believed, inputs coming from the environment. Finally, I argue that __the principle “the higher the dimension, the greater the information”__ may explain the occurrence of mental activities and elucidate the mechanisms of human diseases associated with dimensionality reduction. PDF

**My Response to a Nobel Prize on the possibility of the multidimensional brain**

In this intriguing paper, __the Authors define a concept as "a set of CONVEX (i.e., positive curvature) regions__ of similar stimuli". __Such regions might also display other types of curvatures, such as CONCAVE ones__. Indeed, several studies point towards many biological and physical dynamics taking place in negative-curvature phase spaces: this is because trajectories on hyperbolic manifolds allow a more manageable treatment of many of the required equations, such as, e.g., the Fokker-Plank ones. Further, parallel transport from Euclidean spaces to concave manifold allows the assessment of nervous multidimensional dynamics in terms of symmetry breaks, and the latter, i.e., a successful approach borrewed from physics, would be very useful in the description and categorization of higher-dimensional manifolds. Linked to the issue of the multidimensional brain and nervous symmetries, stands the fundamental question raised by the Authors: "how a continuous code can be extended to map additional dimensions"? In order to answer, the "evidence of topological representations of spaces in rodents and humans" paves the way to the use of an algebraic topological tool, i.e., the Borsuk-Ulam theorem: __provided a function is continuous (in this case, "spatially specific cells provide a continuous code"),__ a single feature in one dimension (say, a sports car) maps to two features with matching description in a dimension higher (two sports cars, which might be slightly different, e.g., in their emotional, or cognitive content). In other words, __when I see a cat in my surrounding 3D environment, I perceive not just__ the 3D image of the real cat in front of me, but also many multidimensional features of the cat in my mind (emotional: "how tender!", cognitive: "this is a Feline", and so on). Therefore, the use of the Borsuk-Ulam theorem allows us to build symmetric, higher-dimensional topological spaces where mental activity might take place, and to calculate their thermodynamic constraints (given the link between symmetries, informational entropies and topological manifolds). PDF

**The fourth dimension of brain activity: an hypersphere in the brain**

Current advances in neurosciences deal with the functional architecture of the central nervous system, paving the way for general theories that improve our understanding of brain activity. From topology, a strong concept comes into play in understanding brain functions, namely, the 4D space of a “hypersphere’s torus”, undetectable by observers living in a 3D world. 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. Our thoughts exhibit similar behaviour, i.e. __the unique ability to connect past, present and future events in a single, coherent picture__ as if we were allowed to watch the three screens of past-present-future “glued” together in a mental kaleidoscope. Here we hypothesize that brain functions are embedded in a imperceptible fourth spatial dimension and propose a method to empirically assess its presence. Neuroimaging fMRI series can be evaluated, looking for the topological hallmark of the presence of a fourth dimension. Indeed, there is a typical feature which reveal the existence of a functional hypersphere: the simultaneous activation of areas opposite each other on the 3D cortical surface. Our suggestion - substantiated by recent findings - that __brain activity takes place on a closed, donut-like trajectory__ helps to solve long-standing mysteries concerning our psychological activities, such as mind-wandering, memory retrieval, consciousness and dreaming state. PDF

**The proof of further nervous dimensions: 4D maximal nucleus cluster in multiD brain**

We introduce a novel method for the measurement of information level in fMRI (functional Magnetic Resonance Imaging) neural data sets, based on image subdivision in small polygons equipped with different entropic content. We show how this method, called maximal nucleus clustering (MNC), is a novel, fast and inexpensive image-analysis technique, independent from the standard blood-oxygen-level dependent signals. MNC facilitates the objective detection of hidden temporal patterns of entropy/information __in zones of fMRI images generally not taken into account by the subjective standpoint of the observer__. This approach befits the geometric character of fMRIs. The main purpose of this study is to provide a computable framework for fMRI that not only facilitates analyses, but also provides an easily decipherable visualization of structures. This framework commands attention because it is easily implemented using conventional software systems. In order to evaluate the potential applications of MNC, we looked for the presence of a fourth dimension’s distinctive hallmarks in a temporal sequence of 2D images taken during spontaneous brain activity. Indeed, recent findings suggest that __several brain activities, such as mind-wandering and memory retrieval, might take place in the functional space of a four-dimensional hypersphere__, which is a double donut-like structure undetectable in the usual three dimensions. We found that the Rényi entropy is higher in MNC areas than in the surrounding ones, and that these temporal patterns closely resemble the trajectories predicted by the possible presence of a hypersphere in the brain. PDF

**How to build a 4D computer, ALIAS how to simulate a multidimensional brain**

**Through quaternionic networks**:

Tozzi A. 2020. Quaternion neural networks and the multidimensional brain (electronic response to: Li Y, Wang H. 2018. Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control. PLOS One, https://doi.org/10.1371/journal.pone.0198297).

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

. the nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

**Through the Quantum Hall effect: **

Relationships among near set theory, shape maps and recent accounts of the Quantum Hall effect pave the way to neural networks computations performed in higher dimensions. We illustrate the operational procedure to build a real or artificial neural network able to detect, assess and quantify a fourth spatial dimension. We show how, starting from two-dimensional shapes embedded in a 2D topological charge pump, it is feasible to achieve the corresponding four-dimensional shapes, which encompass a larger amount of information. Synthesis of surface shape components, viewed topologically as shape descriptions in the form of feature vectors that vary over time, leads to a 4D view of cerebral activity. This novel, relatively straightforward architecture permits to increase the amount of available qbits in a fixed volume. PDF

**Topodynamics of metastable brains: a survey of the applications of the Borsuk-Ulam theorem to neuroscience. **

Tozzi A, Peters JF, Fingelkurts AA, Fingelkurts AA, Marijuán PC. 2017. Topodynamics of metastable brains. Physics of Life Reviews, 21, 1-20. https://dx.doi.org/10.1016/j.plrev.2017.03.001.

The brain displays both the anatomical features of a vast amount of interconnected topological mappings as well as the functional features of a nonlinear, metastable system at the edge of chaos, equipped with a phase space where mental random walks tend towards lower energetic basins. Nevertheless, with the exception of some advanced neuro-anatomic descriptions and present-day connectomic research, very few studies have been addressing the topological path of a brain embedded or embodied in its external and internal environment. Herein, __by using new formal tools derived from algebraic topology__, we provide an account of the metastable brain, based on the neuro-scientific model of Operational Architectonics of brain-mind functioning. We introduce a “topodynamic” description that shows how __the relationships among the countless intertwined spatio-temporal levels of brain functioning can be assessed in terms of projections and mappings__ that take place on abstract structures, equipped with different dimensions, curvatures and energetic constraints. Such a topodynamical approach, apart from providing a biologically plausible model of brain function that can be operationalized, is also able to tackle the issue __of a long-standing dichotomy__: it throws indeed __a bridge between the subjective__, immediate datum of the naïve complex of sensations and mentations __and the objective, quantitative__, data extracted from experimental neuro-scientific procedures. Importantly, it opens the door to a series of new predictions and future directions of advancement for neuroscientific research. PDF

**Projectionism & brain manifolds: the philosophy beyond our approach**

This paper (formally a response to the comments of nine highly qualified commenters to our paper: " topodynamics of metastable brains ") introduces a novel paradigm in neuroscience, __termed "projectionism"__, which assesses projections and mappings among different functional brain dimensions and phase spaces. We describe recently published papers that confirm our general framework. Furthermore, we compare brain symmetries with the predictive coding that stands for __a sort of Kant a priori located in in our brains__. We illustrate __the "unreasonable power" of topology__ in neuroscience, which allows __a rationalistic but testable top-down inquiry of the brai__n activity, in order to mathematically assess the physical and biological dynamics of the human nervous system. We also propose __possible biochemical correlates of a brain fourth dimension__, with clues provided by… LSD intake. Also, we suggest fresh mathematical approaches to brain topological dynamics, introducing novel theorems and __proposing complex functional nervous spaces__ very different from the classical Euclidean ones. We close our paper with a novel computational scenario that takes into account __the tenets of neural Darwinism__. PDF

**The curse of dimensionality: increasing dimensions, volumes decrease!**

__How to avoid the curse of dimensionality, when assessing multidimensional phase spaces?__ Apart from the canonical techniques used to achieve the “blessing of dimensionality”, another, novel approach is available: the Borsuk-Ulam theorem, which has been already widely used in physics, biology and neuroscience. PDF

**Neural energy, entropies, information: a topological account via Borsuk-Ulam Theorem**

Recent approaches to brain phase spaces reinforce __the foremost role of ____symmetries and energy requirements in the assessment of nervous activity__. Changes in thermodynamic parameters and dimensions occur in the brain during symmetry breakings and transitions from one functional state to another. Based on topological results and string-like trajectories into nervous energy landscapes, we provide a novel method for the evaluation of energetic features and constraints in different brain functional activities. We show how __abstract approaches__, namely the Borsuk-Ulam theorem and its variants, __may display real, energetic physical counterparts__. When topology meets the physics of the brain, we arrive at a general model of neuronal activity, in terms of multidimensional manifolds and computational geometry, that has the potential to be operationalized. PDF

**Increase in complexity in the brain: the cat in your mind is multidimensional**

Contrary to common belief, the brain appears __to increase the complexity from the perceived object to the idea of it__. Topological models predict indeed that: a) increases in anatomical/functional dimensions and symmetries occur in the transitionfrom the environment to the higher activities of the brain, and b) informational entropy in the primary sensory areas is lower than in the higher associative ones. To demonstrate this novel hypothesis, we introduce a straightforward approach to measuring island information levels in fMRI neuroimages, via Rényi entropy derived from tessellated fMRI images. This approach facilitates objective detection of entropy and corresponding information levels in zones of fMRI images generally not taken into account. We found that the Rényi entropy is higher in associative cortices than in the visual primary ones. This suggests that __the brain lies in dimensions higher than the environment and that it does not concentrate, but rather dilutes messages coming from external inputs__. PDF **See also this movie**

**Rènyi entropy & nervous shadows: when a 2D shadow encompasses more information that the corresponding 3D object**

Tozzi A. 2015. Neural code & power laws. SCTPLS Newsletter, April, 7-10.

A two-dimensional shadow may encompass more information than its corresponding three-dimensional object. Indeed, if we rotate the object, we achieve a pool of observed shadows from different angulations, gradients, shapes and variable length contours that make it possible for us to increase our available information. Starting from this simple observation, we show how __informational entropies might turn out to be useful in the evaluation of scale-free dynamics in the brain__. Indeed, brain activity exhibits a scale-free distribution that leads to the variations in the power law exponent typical of different functional neurophysiological states. Here we show that modifications in scaling slope are associated with variations in Rényi entropy, a generalization of Shannon informational entropy. From a three-dimensional object’s perspective, by changing its orientation (standing for the cortical scale-free exponent), we detect different two-dimensional shadows from different perception angles (standing for Rènyi entropy in different brain areas). We show how, starting from known values of Rènyi entropy (easily detectable in brain fMRIs or EEG traces), it is feasible to calculate the scaling slope in a given moment and in a given brain area. Because changes in scale-free cortical dynamics modify brain activity, this issue points towards novel approaches to mind reading and description of the forces required for transcranial stimulation. PDF