Simplest Model of Nervous System. III. Partial Optimization

This paper extends our previous work on the simplest model of a nervous system by providing an asymptotic analysis of the evolutionarily optimal solution in this model. Building on the formalism and principles established earlier, we derive an asymptotic solution to the Fokker-Planck-Kolmogorov equation for a given dynamical equation for the state of the neuron. This solution provides the stationary probability distribution for the position of an organism in its environment and the state of its nervous system. Next, exact and asymptotic solutions for the optimal motor and sensory responses to the approach of a predator are derived. These results align with biological expectations and provide a robust mathematical framework for predicting the behavior of simple nervous systems.