Arturo Tozzi, MD, PhD, AAP
Center for Nonlinear Science, University of North Texas
1155 Union Circle, #311427
Denton, TX 76203-5017 USA
James F. Peters, Dr., Professor
Department of Electrical and Computer Engineering, University of Manitoba
75A Chancellor's Circle
Winnipeg, MB R3T 5V6 CANADA
The recent paper from Moore IV et al suggests that cues from different senses are integrated at very early levels of central processing, in a densely coupled system equipped with multisensory interactions occurring at all temporal and spatial stages. In such a multifaceted framework, a theorem from the far-flung branch of algebraic topology, namely, the Borsuk-Ulam theorem (BUT), comes into play, and is going to be game-changing. The theorem states that two opposite points on a sphere, when projected on a one-dimension lower circumference, give rise to a single point containing their matching description (Borsuk, 1933; Marsaglia, 1972; Moura and Henderson, 1996). Another less technical definition is: if a sphere is mapped continuously into a plane set, there is at least one pair of antipodal points having the same image; that is, they are mapped in the same point of the plane (Beyer and Zardecki, 2004). The two antipodal points can be used not just for the description of simple topological points, but also for the description of broad-spectrum phenomena: i.e., either two antipodal shapes, or functions, or signals, or vectors. To make an example, a “point” may be described as a collections of signals or a surface shape, where every shape maps to another antipodal one. It means that signal shapes can be compared (Weeks, 2002; Peters, 2014). If you thus evaluate physical and biological phenomena instead of “signals”, BUT leads naturally to the possibility of a region-based, not simply point-based, geometry.
The Borsuk-Ulam theorem nicely applies to multisensory integration: two environmental stimuli from different sensory modalities display similar features, when mapped into cortical neurons. To make an example, an observer stands in front of the surrounding environment. A violin player is embedded in the environment. The observer perceives, through his different sense organs such as ears and eyes, the sounds and the movements produced by the player. According to the BUT dictates, the violin player stands for an object embedded in a three dimensional sphere. The two different sensory modalities produced by the player (sounds and movements) stand for the antipodal points on the sphere’s circumference. Even if objects belonging to antipodal regions can either be different or similar, depending on the features of objects (Peters, 2015), however the two antipodal points must share the same features. In our case, both sounds and movements come from the same object embedded in the sphere, i.e the violin player. The two antipodal points project to a three-dimensional layer, the brain cortex - where multisensory neurons lie - and converge into a single multimodal signal, which takes into account the features of both. According to the dictates of the Borsuk-Ulam theorem, the single point contains therefore a “melted” messages from the two modalities.
In conclusion, BUT provides a general topological mechanism which explains the elusive phenomenon of multisensory integration. The question here is: what for? What does a topologic reformulation add in the evaluation of multimodal integration? The invaluable opportunity to treat elusive mental phenomena as topological structures allows us to describe brain activity in the language of powerful analytical tools, such as combinatorics, homology theory, functional analysis (Matoušek, 2003). Embracing the BUT theory of multisensory integrations means that the “real” neuronal activity can be described as paths or trajectories on “abstract” structures (called “topological manifolds”). It takes us into the powerful realm of algebraic topology, where the abstract metric space (a projection of the environment’s real geometric space) can be easily assessed.
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