Robert Cacioppo

We develop a theory of neural organization in which percepts are modeled as probabilistic structures on directed acyclic graphs of time-stamped subthreshold oscillatory states, and affective valence is defined by entropy dynamics on these structures. Each graph is assigned two pattern functors: a bound functor, whose elements are Bayesian networks compatible with the graph's causal organization, and an unbound functor, whose elements are arbitrary distributions on the corresponding vertex-state spaces. These functors are organized in a set-valued presheaf topos over the category of directed acyclic graphs with ancestral inclusions, so that restriction corresponds to marginalization. Within this setting, percept-types are subfunctors, percepts are compatible families of local distributions, and the entropic valence determines a time-dependent Grothendieck topology governing which local patterns count as admissible for perceptual integration. The resulting internal intuitionistic logic provides a formal language for describing local endorsement, obstruction, and amalgamation of perceptual structure. We argue that this framework gives a principled account of how perceptual coherence and affective valence may jointly arise from mesoscopic cortical field dynamics.

Version 6 (May 10, 2026) We develop a mathematical framework for neural computation in which percepts are modeled as probabilistic structures on directed acyclic graphs of subthreshold oscillatory (STO) activity, and affective valence is defined in terms of entropy dynamics on these structures. A central problem in the study of consciousness is how structured subjective experience, including both perceptual organization and affective tone, arises from distributed physical processes in the brain. Directed acyclic graphs (DAGs) representing time-stamped STO activity are assigned two pattern functors: a bound functor, yielding Bayesian networks compatible with the causal organization, and an unbound functor, yielding arbitrary vertex-wise distributions. These constructions form objects in the set-valued presheaf topos where the base category is the category of DAGs with ancestral inclusions. Within this setting, the internal intuitionistic logic of the topos provides a formal language for describing perceptual and computational organization. We propose that affective valence induces a Grothendieck topology on this topos, regulating which locally defined probabilistic structures admit amalgamation. This yields a unified framework in which logical, probabilistic, and affective aspects of neural processing are integrated. We argue that this framework provides a principled account of how perceptual coherence and affective valence may jointly emerge from mesoscopic cortical field dynamics, offering a mathematically grounded approach to the relationship between physical processes and conscious experience.

Version 2 (April 2026). Two new sections, 2.2 and 6.5, are added, which address TECC's interpretation of the neural basis for valence and surgical unconsciousnesss. We develop a mathematical framework for neural computation in which percepts are modeled as probabilistic structures on directed acyclic graphs of subthreshold oscillatory (STO) activity, and affective valence is defined in terms of entropy dynamics on these structures. A central problem in the study of consciousness is how structured subjective experience, including both perceptual organization and affective tone, arises from distributed physical processes in the brain. Directed acyclic graphs (DAGs) representing time-stamped STO activity are assigned two pattern functors: a bound functor, yielding Bayesian networks compatible with the causal organization, and an unbound functor, yielding arbitrary vertex-wise distributions. These constructions form objects in the set-valued presheaf topos where the base category is the category of DAGs with ancestral inclusions. Within this setting, the internal intuitionistic logic of the topos provides a formal language for describing perceptual and computational organization. We propose that affective valence induces a Grothendieck topology on this topos, regulating which locally defined probabilistic structures admit amalgamation. This yields a unified framework in which logical, probabilistic, and affective aspects of neural processing are integrated. We argue that this framework provides a principled account of how perceptual coherence and affective valence may jointly emerge from mesoscopic cortical field dynamics, offering a mathematically grounded approach to the relationship between physical processes and conscious experience.

Version 3 (April 2026). We develop a mathematical framework for neural computation in which percepts are modeled as probabilistic structures on directed acyclic graphs of subthreshold oscillatory (STO) activity, and affective valence is defined in terms of entropy dynamics on these structures. A central problem in the study of consciousness is how structured subjective experience, including both perceptual organization and affective tone, arises from distributed physical processes in the brain. Directed acyclic graphs (DAGs) representing time-stamped STO activity are assigned two pattern functors: a bound functor, yielding Bayesian networks compatible with the causal organization, and an unbound functor, yielding arbitrary vertex-wise distributions. These constructions form objects in the set-valued presheaf topos where the base category is the category of DAGs with ancestral inclusions. Within this setting, the internal intuitionistic logic of the topos provides a formal language for describing perceptual and computational organization. We propose that affective valence induces a Grothendieck topology on this topos, regulating which locally defined probabilistic structures admit amalgamation. This yields a unified framework in which logical, probabilistic, and affective aspects of neural processing are integrated. We argue that this framework provides a principled account of how perceptual coherence and affective valence may jointly emerge from mesoscopic cortical field dynamics, offering a mathematically grounded approach to the relationship between physical processes and conscious experience.