A categorical and entropic framework for Bayesian networks and consciousness

Beginning with spatiotemporal patterns of states at the vertices of a finite directed acyclic graph where a directed edge indicates an influence of the domain state on the codomain state, we construct a Grothendieck topos defined on sites for the category of finite directed acyclic graphs. We suggest this as a computational framework for cortical activity. Since a topos expresses a constructive logic through its objects and morphisms, we posit that evolutionary pressure selected for increasingly better approximations in biological neural networks for the critical objects of this topos, allowing for greater expression of its logic to guide behavior. We also posit that the potential of a quale is expressed as a Bayesian network comprised of a spatiotemporal pattern of states (subthreshold membrane oscillations) produced by action potentials. We propose that the intensity of the subjective experience of a quale depends on the temporal rate of change of an entropy in the neuronal ensemble resulting from a change in the correlations between the states in the pattern.