Simplest Model of Nervous System. I. Formalism

Abstract

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.

Making Sense of Neural Networks in the Light of Evolutionary Optimization

“…Immediately, the spike-like behavior over the neuronal degree of freedom followed from a minimization of K1I1+K2I2, the part of I that depended on this degree of freedom. 27,28 This derivation of a spike-like behavior of neurons should be generalizable to the general case considered in this work. It is worth noting that when considering equations ( 29) and (31) [in the latter case, taking into account equations ( 32) and ( 42)] both integrals can become negligible if Pstat draws close to a product of Dirac delta functions over the neuronal degrees of freedom xj included in the sets S1 or S2 as defined above.…”

“…Necessary and sufficient constraints on u and ℚ might be introduced by demanding that a stationary solution to the Fokker-Planck equation Pstat exists for dynamic equations ( 4) with f defined from u and ℚ by equation (18). The analysis of the toy model 27,28 demonstrates that the mere existence of u with necessary properties like condition ( 43) is not sufficient for the existence of Pstat, despite equation (15). With some values of ℚ (for example, the above-mentioned unacceptable solution for the toy model) the stationary distribution may diverge.…”

“…It can be used for an exhaustive evolutionary optimization of the properties of neurons and neural networks. 27,28 We hypothesize that the key properties of nervous systems and biological neural networks can be understood and predicted by solving the mathematical problem of the optimization of the evolutionary fitness as a functional of the two above-mentioned functions. In this paper, we outline what a mathematical apparatus of such a theory might look like and discuss possible complications along this path.…”

Simplest Model of Nervous System. II. Evolutionary Optimization

“…Our exploration here is primarily confined to a simple model introduced in the previous paper. 1 It is our aspiration that this endeavor, with its findings and solutions, will pave the way for broader and more intricate inquiries in this domain. 2…”

“…In this endeavor, we tackle a simple case of this quandary. Building upon the foundation laid in our preceding companion paper, 1 we employ a simple model of the nervous system and its mathematical scaffolding. By formulating and subsequently resolving an optimization problem aligned with this model, we attempt to mirror the evolutionary optimization process.…”