Stochastic oscillations in neural models with static and dynamic Hopf bifurcation parameters

Here, we review how and when stochastic oscillations arise in stochastic dynamical model neural systems with supercritical Hopf bifurcations, and point to relevant mathematical results. We focus on four models of neural activity, all of which display supercritical Hopf bifurcations. To address neural plasticity we study two-time, slow–fast versions of these models in which the bifurcation parameter is slowly changing. We display with simulations the effect of noise on the dynamics of both base system (fixed bifurcation parameter) and slow–fast system (changing bifurcation parameter) models. Adding noise to a slow–fast model eliminates the bifurcation delay and induces what we term ‘noisy-delay-cycles’ during what would be the delay period in a deterministic solution. We measure both quasi-cycles, when the slow parameter is below the Hopf point, and noisy-delay-cycles during the delay period above the Hopf point. We also compute the power spectral density for each of the models, showing that all are versions of Lorentzians with peaks at the inflection points, representing the model-generated oscillations. We conclude that, given the similarities of the local dynamics of these models near the Hopf point under moderate noise, there is little reason to favour one model over another when studying the behaviour of large groups of neurons, i.e., when used as neural mass models, particularly as part of whole-brain network models.