Compartmental-Reaction Diffusion Framework for Microscale Dynamics of Extracellular Serotonin in Brain Tissue

Abstract

Abstract. The neurotransmitter serotonin (5-hydroxytryptamine) shapes the plasticity and integration of network dynamics in the entire brain. In contrast to axons that have well-defined targets and form classical synapses, serotonin-releasing (serotonergic) axons often produce random-walk-like trajectories, accumulate at extremely high densities in many brain regions, and can support the “wired” and “volume” modes of neurotransmission. Evidence suggests that their special properties lie at the functional core of neural tissue, but the dynamics of serotonin in the extracellular space remain poorly understood because of major experimental limitations on the relevant (sub-micrometer/millisecond) scales. Here we construct a mathematical framework that captures the coupled reaction–diffusion dynamics of serotonin in realistic tissue microenvironments, enabling quantitative predictions of how firing frequency, fiber geometry, and uptake kinetics shape extracellular serotonin profiles. Relevance to Life Sciences. Our results demonstrate that serotonin release/reuptake sites (varicosities) form diffusively coupled microdomains whose collective activity determines the spatiotemporal organization of extracellular serotonin in neural tissue. Simulations show that increasing firing frequency or local varicosity density leads to the emergence of extracellular “serotonin reservoirs” between active sites, with concentrations spanning the physiologically realistic nanomolar to micromolar range. These reservoirs are sustained by the nonlinear saturation of serotonin transporters that conform to Michaelis–Menten kinetics and mediate serotonin reuptake into the axons. The model predicts that transitions between behavioral states are associated with distinct spatial profiles of extracellular serotonin due to diffusion-mediated interactions among firing and nonfiring fibers. It can also be extended to understand the microscale effects of antidepressants such as selective serotonin-reuptake inhibitors (SSRIs). Mathematical Content. We develop and analyze a two-dimensional compartmental-reaction diffusion system in which each serotonergic varicosity and its local neighborhood is represented as an active compartment exchanging serotonin with the extracellular space. Using strong localized perturbation theory, we derive an asymptotically equivalent system of coupled nonlinear integro-ODEs that accurately captures diffusive coupling while being computationally efficient. We investigate the period-averaged steady-states that arise under periodic firing and establish analytical bounds on local serotonin concentrations using Jensen’s inequality. Closed-form estimates for long-term spike maxima and minima are obtained and validated numerically. A fast marching-scheme solver based on sum-of-exponentials kernel approximations allows efficient simulation of networks. Lastly, our comparison between Michaelis–Menten and linear uptake kinetics highlights the importance of nonlinear transporter saturation in shaping steady-state serotonin fields. This framework provides a mathematically rigorous foundation for future models incorporating multiple interacting species, stochastic firing, and glial or endothelial regulation of neurotransmitter signaling.