Comparison between an exact and a heuristic neural mass model with second-order synapses

Abstract

Neural mass models (NMMs) are designed to reproduce the collective dynamics of neuronal populations. A common framework for NMMs assumes heuristically that the output firing rate of a neural population can be described by a static nonlinear transfer function (NMM1). However, a recent exact mean-field theory for quadratic integrate-and-fire (QIF) neurons challenges this view by showing that the mean firing rate is not a static function of the neuronal state but follows two coupled nonlinear differential equations (NMM2). Here we analyze and compare these two descriptions in the presence of second-order synaptic dynamics. First, we derive the mathematical equivalence between the two models in the infinitely slow synapse limit, i.e., we show that NMM1 is an approximation of NMM2 in this regime. Next, we evaluate the applicability of this limit in the context of realistic physiological parameter values by analyzing the dynamics of models with inhibitory or excitatory synapses. We show that NMM1 fails to reproduce important dynamical features of the exact model, such as the self-sustained oscillations of an inhibitory interneuron QIF network. Furthermore, in the exact model but not in the limit one, stimulation of a pyramidal cell population induces resonant oscillatory activity whose peak frequency and amplitude increase with the self-coupling gain and the external excitatory input. This may play a role in the enhanced response of densely connected networks to weak uniform inputs, such as the electric fields produced by noninvasive brain stimulation.

Complex spatiotemporal oscillations emerge from transverse instabilities in large-scale brain networks

“…Overall, the large-scale brain model with NG-NMMs displays a mechanism towards the onset of irregular states analogous to the Jansen case, despite the different nature of the two models [89]. For instance, we did not include synaptic dynamics in Eq (36), i.e., neurons receive instantaneous delta-like pulses.…”

Complex spatiotemporal oscillations emerge from transverse instabilities in large-scale brain networks

“…Overall, the large-scale brain model with NG-NMMs displays a mechanism towards the onset of irregular states analogous to the Jansen case, despite the different nature of the two models [85]. For instance, we did not include synaptic dynamics in Eq (36), i.e., neurons receive instantaneous delta-like pulses.…”

Intensity- and frequency-specific effects of transcranial alternating current stimulation are explained by network dynamics

“…Neural entrainment becomes weaker as network size and number of synapses decrease (figure S7). Entrainment relies on slight shifts in spike timings at single cell level that are amplified by network effects, such as number of neurons and synaptic connections affected by the weak exogenous electric field [5,32,73].…”

“…However, an isolated single PY cannot account for all tACS effects. To study the network-level effect of tACS, many modeling studies have utilized mean-field/neural mass models [5,[32][33][34][35], as well as spiking neuronal networks of single-compartment point neurons [15,36,37]. These models demonstrated that an applied oscillating electric field causes network resonance in neuronal populations [5,[32][33][34][35][36][37][38].…”

“…To study the network-level effect of tACS, many modeling studies have utilized mean-field/neural mass models [5,[32][33][34][35], as well as spiking neuronal networks of single-compartment point neurons [15,36,37]. These models demonstrated that an applied oscillating electric field causes network resonance in neuronal populations [5,[32][33][34][35][36][37][38]. Thus, stimulation frequencies that are closer to the endogenous frequency are more effective at entraining the network.…”