Non-linear stochastic model for dopamine cycle

Dopamine is a crucial neurotransmitter that plays a central role in various aspects of brain functions, including reward processing, motivation, learning, and movement control. Its intricate involvement in these biological processes has made it a subject of extensive research across multiple disciplines, ranging from neuroscience and psychology to computational modeling. In this paper the non-linear stochastic model describing synthesis, storage, release, uptake and metabolism of dopamine in dopaminergic nerve terminal of the rat striatum is introduced. The model is driven by 9-dimensional Brownian motion with constant coordinate intensities. Existence and uniqueness of a positive global solution x(t) = [x(1)(t), ... , x(9)(t)](tau) on a corresponding time-domain are proved and lower and upper bounds for specific moments of coordinate processes {x(i)(t), 0 <= t <= T} are derived. These results are used for calculation of bounds for intensities of driving processes and time horizons ensuring that expected values of coordinate processes stay in the same time-interval as the corresponding expected starting values. Furthermore, positivity preserving balance implicit method is used for simulation of coordinate processes in order to illustrate theoretical results.