A TOPOLOGICAL APPROACH UNVEILS SYSTEM INVARIANCESAND BROKEN SYMMETRIES IN THE BRAIN

Arturo Tozzi (Corresponding author)

Center for Nonlinear Science, University of North Texas

1155 Union Circle, #311427

Denton, TX 76203-5017 USA

 

 

James F. Peters

Department of Electrical and Computer Engineering, University of Manitoba

75A Chancellor’s Circle

Winnipeg, MB R3T 5V6 CANADA

 

 

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 nervoussymmetries.   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. 

 

 Download here a previous version of the article published on J Neurosci Res, Feb 2016, DOI: 10.1002/jnr.23720