Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness

Abstract

Consciousness is a central issue in neuroscience, however, we still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates. In this review, 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 which include (i) category; (ii) inclusion functor and expansion functor; and, most importantly, (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 temporospatial theory of consciousness (TTC). We conclude that CT, especially the application of the notion of natural transformation, highlights that we need to go beyond NCC and unravels questions that need to be addressed by any future neuroscientific theory of consciousness.

From Shorter to Longer Timescales: Converging Integrated Information Theory (IIT) with the Temporo-Spatial Theory of Consciousness (TTC)

“…[ 1 , 2 , 42 ]. TTC does not consider, as a starting point, neuronal activity and its relationship among different regions at a definite moment in time; rather, it stresses the relevance of the temporal dynamics of neuronal activity at different points in time as an intrinsic feature of consciousness [ 21 , 23 , 43 , 44 ]. This temporal dimension can be conceived as an empirical specification of the background conditions that influence the shape of current activity in the cause-effect structure of the PSC predicted by IIT [ 32 , 45 ].…”

The neural correlates of consciousness under the free energy principle: From computational correlates to computational explanation

“…Finding arbitrary mappings between neural states and conscious experiences can help illuminate how consciousness arises in human beings, but it will not lead to a deep understanding of consciousness. Chalmers (2000, p. 23) suggests that a non-arbitrary mapping could be established if not only neural states themselves are mapped to states of consciousness, but if also relations between neural states are mapped to relations between conscious states (a recent approach pursuing this idea using category theory is presented in Northoff et al, 2019). Drawing on such a mapping, one can extrapolate from existing NCCs to predict how certain changes in neural activity will change the correlated conscious experience (a similar idea, though not using the notion of an NCC, is explored in Churchland, 2005).…”

The neural correlates of consciousness under the free energy principle: From computational correlates to computational explanation

“…The mapping can be arbitrary in the sense that there seems to be no epistemically necessary connection between certain neural states and states of consciousness; however, such mappings will be non-arbitrary in the sense that the connection must be general and lawful: states of an NCC, as defined by Chalmers, must be sufficient for states of states and conscious experiences can help illuminate how consciousness arises in human beings, but it will not lead to a deep understanding of consciousness. Chalmers (2000) suggests that a non-arbitrary mapping could be established if not only neural states themselves are mapped to states of consciousness, but if also relations between neural states are mapped to relations between conscious states (a recent approach pursuing this idea using category theory is presented in Northoff, Tsuchiya, & Saigo, 2019). Drawing on such a mapping, one can extrapolate from existing NCCs to predict how certain changes in neural activity will change the correlated conscious experience (a similar idea, though not using the notion of an NCC, is explored in Churchland, 2005).…”