Derivation of mean-field models from single neuron models subjected to dopamine modulation.

One of the main challenge in neuroscience is to model the behavior of the brain at different scales, both in physiological and pathological conditions. However, the inherent complexity of the brain make it really challenging to tackle the problem of studying the underlying brain dynamics directly. In an attempt to ease this challenge several techniques borrowed from physics have been exploited. One of these approach consist into derive the macroscopic dynamics of a population of neurons through mean field approximations. These models are based on the idea that, given the large number of neuron in a single population, we can focus our attention into describing the average activity of the group, instead that of each individual component. In this work we exploit the mean field derivation framework to achieve a system of closed equations for two slightly different network of adaptive quadratic integrate-and-fire (aQIF) neurons. The difference between the two resides in an heterogeneity term, which is considered respectively additive and multiplicative. In both cases, we extend the aQIF model, by explicitly considering the modulation effect due to the presence of dopamine in the extracellular space. Dopamine is a crucial neuromodulator that plays a pivotal role in several essential functions within the central nervous system, including regulation of movement, emotion, motivation, and reward. In health, dopamine facilitates motor control, as exemplified by its action in the basal ganglia, and contributes to the pleasure and reinforcement mechanisms that drive learning and behavior. Pathologically, dys-regulation of dopamine levels is implicated in various disorders. In the derivation, we first assume a population density approach, i.e. we consider the distribution of neurons over all possible states. Furthermore, we exploit the Lorentzian ansatz. We confront the derived mean field set of equations with simulations of the related network of all-to-all coupled neurons to identify whether it is able to correctly reproduce the same dynamical features.