Mathematics and the brain –a category theoretic approach to go beyond the neural correlates of consciousness
Authors: Northoff Georg , Tsuchiya Naotsugu , Saigo Hayato 
Published: June 2019

Consciousness is a central issue in cognitive neuroscience. To explain the relationship between consciousness and its neural correlates, various theories have been proposed. We still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates though. Here, we provide a novel mathematical framework of Category Theory (CT), in which we can define and study the “sameness” between “different” domains of phenomena such as consciousness and its neural substrates. CT was designed and developed to deal with the “relationships” between various domains of phenomena. We introduce three concepts of CT including (i) category; (ii) inclusion functor and expansion functor; and (iii) natural transformation between the functors. Each of these mathematical concepts is related to specific features in the neural correlates of consciousness (NCC). In this novel framework, we will examine two of the major theories of consciousness: integrated information theory (IIT) of consciousness and temporo-spatial theory of consciousness (TTC). These theories concern the structural relationships among structures of physical substrates and subjective experiences. The three CT-based concepts, introduced in this paper, unravel some basic issues in our search for the NCC; while addressing the same questions, we show that IIT and TTC provide different albeit complementary answers. Importantly, our account suggests that we need to go beyond a traditional concept of NCC including both content-specific and full NCC. We need to shift our focus from the relationship between “one” neuronal and “one” phenomenal state to the relationship between a structure of neural states and a structure of phenomenal states. We conclude that CT unravels and highlights basic questions about the NCC in general which needs to be met and addressed by any future neuroscientific theory of consciousness.

Author summary

Neuroscience has made considerable progress in uncovering the neural correlates of consciousness (NCC). At the same time, recent studies demonstrated the complexity of the neuronal mechanisms underlying consciousness. To make further progress in the neuroscience of consciousness, we need proper mathematical formalization of the neuronal mechanisms potentially underlying consciousness. Providing a first tentative attempt, our paper addresses both by (i) pointing out the specific problems of and proposing a new approach to go beyond the traditional approach of the neural correlates of consciousness, and (ii) by recruiting a recently popular mathematical formalization, category theory (CT). With CT, we provide mathematical formalization of the broader neural correlates of consciousness by its application to two of the major theories, integrated information theory (IIT) and temporo-spatial theory of consciousness (TTC). Together, our CT-based mathematical formalization of the neural correlates of consciousness including its specification in the terms of IIT and TTC allows to go beyond the current concept of NCC in both mathematical and neural terms.