Arturo Tozzi, MD, PhD, AAP

Center for Nonlinear Science, University of North Texas, Denton, Texas 76203, USA

DOI: 10.13140/RG.2.1.2601.9045 

The recent, astonishing  advances in neuroimaging (Duffau, 2011; Kjeldsen et al., 2015) focus on establishing causal relations between specific aspects of neuronal function and dynamics.  However, simplified data analysis and data mining processes are required to deal with the huge amount of the gathered datasets.  In such a framework, the concept of  “n-sphere” - from the far-flung branch of multidimensional geometry - comes into play, and is going to be game changing.  A n-sphere is a n-dimensional structure embedded in a n+1 space, called a n+1-ball (Giblin, 2010).  To make some examples, a circumference (1-dimensional sphere) surrounds a cd-rom (2-dimensional ball), while a plastic surface (2-dimensional sphere) surrounds a beach ball (3-dimensional ball).  A 3-sphere (also called glome, or hypersphere) is a 3-dimensional elliptic surface, embedded in a 4-dimensional ball.  In other words, a glome is an object with positive curvature in an Euclidean 4-dimensional space, formed by all the points at constant distance from a fixed central point (Weeks, 1985). 

In our proof of concept experiment, we evaluated datasets obtained from different available neuroimaging techniques - such as fMRI or diffusion tensor imaging – .  We showed that brain function can be visualized like a hypersphere (see Figure fur further details):  in summary, dynamical cortical oscillations from activated/deactivated (on/off states) brain subareas become circular trajectories projected onto a 3-sphere.  

The use of hyperspheres in neuronal data analysis and data mining displays several advantages.  The entire brain function can be visualized by just looking at a single mathematical structure, the glome, which is easy to manage and standardize via computerized tools.  Further, the highly dynamical hypersphere conformation changes according to the underlying anatomical/functional activity and glomes of different shapes are correlated with different physiological and pathological brain states. It allows us to predict psychological states and/or diseases, based not only on the qualitative images of fMRI or other neuroimaging techniques, but also on the quantitative informations extrapolated from 3-sphere manifolds.   In summary, we propose a novel approach which allows us to cast biological brain activities in a mathematically informed fashion and which has the potential to be operationalized and assessed empirically.  





The oversimplified Figure, modified from Johnson (2015), describes how to build a 3-sphere from a 2-sphere. 

We choose at first an arbitrary fMRI image displaying cortical areas in a state of activation (blue and red spots on the brain surface).  The two brain emispheres are then “sphericized”, i.e projected on a 2-sphere, which is a simple 3-dimensional ball equipped with the three canonical axis x, y and z. The activated brain areas become points on the surface of  the 2-sphere (black and grey dots).  A technique, called Hopf fibration (Hopf, 2001), allows us to project the points from the 2-sphere onto a glome: every point on the 2-sphere matches a single circular trajectory on the 3-sphere, in an organized and symmetric way.  In mathematical terms, the circular trajectories on the 3-sphere match the compact, simply connected symplectic Lie group Sp(1) equipped with quaternionic 1x1 unitary matrices. 

The Figure at the bottom shows how the dynamical, everchanging circular trajectories of the hypershpere may take countless different conformations, depending on the specific underlying brain activity.   





1)       Duffau, H. 2011.  Brain Mapping.  From Neural Basis of Cognition to Surgical Applications. Springer-Verlag. Wien. 

2)       Giblin, P. 2010. Graphs, surfaces and homology.  Cambridge University Press, Cambridge. 

3)       Hopf, H. 2001. Collected papers/Gesammelte Abhandlungen. Springer-Verlag, Berlin & New York.

4)       Johnson, N. 2015.  A visualization of the Hopf fibration.

5)       Kjeldsen, H.D., Kaiser, M., Whittington. M.A. 2015. Near-field electromagnetic holography for high-resolution analysis of network interactions in neuronal tissue.  J. Neurosci. Methods 253, 1-9. doi: 10.1016/j.jneumeth.2015.05.016.

6)       Weeks, J. R. 1985.  The shape of space: how to visualize surfaces and three-dimensional manifolds.  Pure and Applied Mathematics Vol. 96.  Marcel Dekker, New York.