Hidden symmetries (and symmetry breaks) in the brain

Tozzi A, Peters JF. 2016.  A Topological Approach Unveils System Invariances and Broken Symmetries in the Brain.  Journal of Neuroscience Research 94 (5): 351–65. doi:10.1002/jnr.23720.

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:

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

Tozzi A.  2018.  How a continuous code can be extended to map additional dimensions?  (electronic response to: Bellmund JLS, Gardenfors P, Moser EI, Doeller CF.  Navigating cognition: spatial codes for human thinking.  Science, 362 (6415), eeat6766). 

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

Tozzi A, Peters JF. 2016.  Towards a Fourth Spatial Dimension of Brain Activity.  Cognitive Neurodynamics 10 (3): 189–199. doi:10.1007/s11571-016-9379-z.

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

Peters JF, Ramanna S, Tozzi A, İnan E.  2017.  Bold-Independent Computational Entropy Assesses Functional Donut-Like Structures in Brain fMRI Images.  Front Hum Neurosci. 2017 Feb 1;11:38. doi: 10.3389/fnhum.2017.00038. eCollection 2017.

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


Don AP, Peters JF, Ramanna S, Tozzi A.  2020.  Topological View of Flows inside the BOLD Spontaneous Activity of the Human Brain.  Front. Comput. Neurosci.  DOI: 10.3389/fncom.2020.00034.

Spatio-temporal brain activities with variable delay detectable in resting-state functional magnetic resonance imaging (rs-fMRI) give rise to highly reproducible structures, termed cortical lag threads, that can propagate from one brain region to another. Using a computational topology of data approach, we found that Betti numbers that are cycle counts and the areas of vortex cycles covering brain activation regions in triangulated rs-fMRI video frames make it possible to track persistent, recurring blood oxygen level dependent (BOLD) signals. Our findings have been codified and visualized in what are known as persistent barcodes. Importantly, a topology of data offers a practical approach in coping with and sidestepping massive noise in neuro data, such as unwanted dark (low intensity) regions in the neighbourhood of non-zero BOLD signals. A natural outcome of a topology of data approach is the tracking of persistent, non-trivial BOLD signals that appear intermittently in a sequence of rs-fMRI video frames. The end result of this tracking of changing lag structures is a persistent barcode, which is a pictograph that offers a convenient visual means of exhibiting, comparing and classifying brain activation patterns.  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,

Through the Quantum Hall effect: 

Tozzi A, Ahmad MZ, Peters JF.  2020.  Neural computing in four spatial dimensions.  Cognitive Neurodynamics.   

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.

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

Tozzi A, Peters JF, Fingelkurts AA, Fingelkurts AA, Marijuán PC.  2017.  Brain projective reality: novel clothes for the emperor.  Reply to comments on “Topodynamics of metastable brains”by Tozzi et al.  Physics of Life Reviews, 21, 46-55.

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 brain 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!

Tozzi A, Peters JF.  2019.  The Borsuk-Ulam theorem solves the curse of dimensionality: Comment on “the unreasonable effectiveness of small neural ensembles in high-dimensional brain” by Alexander N. Gorban et al. Physics of Life Reviews. 

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

From abstract topology to real thermodynamic brain activity.  Cognitive Neurodynamics, 11(3) 283–292. Doi:10.1007/s11571-017-9431-7. 

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

Peters JF, Tozzi A, Ramanna S, Inan E.  2017. The human brain from above: an increase in complexity from environmental stimuli to abstractions.  Cognitive Neurodynamics,11(4), 391–394.  DOI: 10.1007/s11571-0­17-9428-2.

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. 

Tozzi A, Peters JF, Cankaya MN.  2018. The informational entropy endowed in cortical oscillations.  Cognitive Neurodynamics, 12(5), 501-507.  DOI: 10.1007/s11571-018-9491-3. 

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