Simplest Model of Nervous System. I. Formalism

This study presents a novel, highly simplified model of the nervous system, inspired by one hypothetical scenario of its origin. The model is designed to accommodate both mathematical derivations and numerical simulations, offering a template for studying generalized principles and dynamics beyond the specifics of the referenced origin scenario. The model offers a holistic perspective by treating the nervous system and the environment (in their simplest forms) as parts of one system and, together with a companion paper, notes the key role of evolutionary factors (in this model, predator evasion) in shaping the properties of the nervous system. To emphasize these fundamental principles, some aspects, such as the highly dimensional nature of the networks or detailed molecular mechanisms of their functioning, are omitted in the current version. Analytically, the model facilitates insights into the stationary distribution as a solution to the Fokker-Planck equation and the corresponding effective potential and rotation (solenoidal) terms. Numerically, it generates biologically plausible (given its high abstraction) solutions and supports comprehensive sampling with limited computational resources. Noteworthy findings from the study include limitations of the commonly used weak noise approximation and the significance of rigorous mathematical analysis over heuristic interpretations of the potential. We hope that this abstract model will serve as a fruitful tool for better understanding a complete set of principles for modeling nervous systems.