Bifurcation analysis of neural mass models: Impact of extrinsic inputs and dendritic time constants

Neural mass models (NMMs) explain dynamics of neuronal populations and were designed to strike a balance between mathematical simplicity and biological plausibility. They are currently widely used as generative models for noninvasive electrophysiological brain measurements; that is, magneto- and electroencephalography (M/EEG). Here, we systematically describe the oscillatory regimes which a NMM of a single cortical source with extrinsic input from other cortical and subcortical areas to each subpopulation can explain. For this purpose, we used bifurcation analysis to describe qualitative changes in system behavior in response to quantitative input changes. This approach allowed us to describe sequences of oscillatory regimes, given some specific input trajectory. We systematically classified these sequential phenomena and mapped them into parameter space. Our analysis suggests a principled scheme of how complex M/EEG phenomena can be modeled parsimoniously on two time scales: While the system displays fast oscillations, it slowly traverses phase space to another qualitatively different oscillatory regime, depending on the input dynamics. The resulting scheme is useful for applications where one needs to model an ordered sequence of switching between qualitatively different oscillatory regimes, for example, in pharmacological interventions, epilepsy, sleep, or context-induced state changes.