Simplest Model of Nervous System. IV. General Solution

In this paper, we extend our previous work on a simplified model of the nervous system by solving the general optimization problem for the evolutionary cost of the nervous system. This optimization takes into account constraints on the scales of membrane potential kinetics and sensory response function to ensure finite, biologically plausible solutions. Our analysis reduces the variational problem to a system of two integro-differential equations, which we solve asymptotically using series expansions. This study confirms the emergence of sharp finite neuronal spikes and robust sensory and motor response functions as evolutionarily optimal solutions. We note that, in principle, alternative evolutionary solutions with different biophysical interpretations might exist for this optimization problem. This work provides a rigorous mathematical framework bridging evolutionary optimization with the fundamental properties of nervous systems.