Analysis and extension of exact mean-field theory with dynamic synaptic currents

Neural mass models such as the Jansen-Rit system provide a practical framework for representing and interpreting electrophysiological activity (1-6) in both local and global brain models (7). However, they are only partly derived from first principles. While the post-synaptic potential dynamics in NMM are inferred from data and can be grounded on diffusion physics (8-10), Freeman's "wave to pulse" sigmoid function (11-13) is used to transduce mean population membrane potential into firing rate rests on a weaker theoretical standing. On the other hand, Montbrio et al (14, 15) derive an exact mean-field theory from a quadratic integrate and fire neuron model under some simplifying assumptions (MPR), connecting microscale neural mechanisms and meso/macroscopic phenomena. The MPR model can be seen to replace Freeman's sigmoid function with a pair of differential equations for the mean membrane potential and firing rate variables, providing a mechanistic interpretation of NMM semi-empirical sigmoid parameters. In doing so, it sheds light on the mechanisms behind enhanced network response to weak but uniform perturbations: in the exact mean-field theory, intrinsic population connectivity modulates the steady-state firing rate transfer function in a monotonic manner, with increasing self-connectivity leading to higher firing rates. This provides a plausible mechanism for the enhanced response of densely connected networks to weak, uniform inputs such as the electric fields produced by non-invasive brain stimulation. Finally, we complete the MPR model by adding the equations for delayed post-synaptic currents, bringing together the MPR and NMM formalisms into a unified exact mean-field theory (NMM2) displaying rich dynamical features. As an example, we analyze the dynamics of a single population model, and a model of two coupled populations with a simple excitation-inhibition (E-I) architecture, showing it displays rich dynamics with limit cycles, period doubling, bursting behavior, and enhanced sensitivity to external inputs.