Abstract
Neural mass models such as the Jansen-Rit or Wendling systems provide a practical framework for representing and interpreting electro-physiological activity (1–6) in both local and global brain models (7, 8). However, they are only partly derived from first principles. While the post-synaptic potential dynamics are inferred from data and can be grounded on diffusion physics (9–11), Freeman’s “wave to pulse” (W2P) sigmoid function (12–14), used to transduce mean population membrane potential into firing rate, rests on a weaker theoretical standing. On the other hand, Montbrió et al (15, 16) derive an exact mean-field theory (MPR) from a quadratic integrate and fire neuron model under some simplifying assumptions, thereby connecting microscale neural mechanisms and meso/macroscopic phenomena. The MPR model can be seen to replace Freeman’s W2P sigmoid function with a pair of differential equations for the mean membrane potential and firing rate variables—a dynamical relation between firing rate and membrane potential—, providing a more fundamental interpretation of the semi-empirical NMM sigmoid parameters. In doing so, we show 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 W2P relation 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. This new, dynamic W2P relation also endows the neural mass model with a form of “inertia”, an intrinsic delay to external inputs that depends on, e.g., self-coupling strength and state of the system. Next, we complete the MPR model by adding the second-order equations for delayed post-synaptic currents and the coupling term with an external electric field, bringing together the MPR and the usual NMM formalisms into a unified exact mean-field theory (NMM2) displaying rich dynamical features. In the single population model, we show that the resonant sensitivity to weak alternating electric field is enhanced by increased self-connectivity and slow synapses.
Significance
Several decades of research suggest that weak electric fields influence neural processing. A long-standing question in the field is how networks of neurons process spatially uniform weak inputs that barely affect a single neuron but that produce measurable effects in networks. Answering this can help implement electric field coupling mechanisms in neural mass models of the whole brain, and better represent the impact of electrical stimulation or ephaptic communication. This issue can be studied using local detailed computational models, but the use of statistical mechanics methods can deliver “mean-field models” to simplify the analysis. Following the steps of Montbrió et al (15, 16), we show that the sensitivity to inputs such a weak alternating electric field can be modulated by the intrinsic self-connectivity of a neural population, and produce a more grounded set of equations for neural mass modeling to guide further work.
Competing Interest Statement
Giulio Ruffini works for and is a co-founder of Neuroelectrics, a company dedicated to the creation of non-invasive brain stimulation solutions.
Footnotes
Added clarifications for the different unit systems used in the models. + corrected plots for resonance.
↵∗ One may be tempted to replace the last equation by z = a2 βφ − 2az − a2 s, but this is redundant. The change of variables
and then lead to , and the β factor pops up in the second NMM2 equation, changing J → βJ. This is also clear from the point of view of the L operator.
