Noisy neural systems with static and dynamic Hopf bifurcation parameters

Abstract Stochastic neural oscillations may be quasi-cycles or stochastic limit cycles. Here we discuss how and when these arise in stochastic dynamical systems with Hopf bifurcations, and point to relevant mathematical results. We describe a new type of oscillation, quasi-limit-cycles, which are limit cycles stirred up by noise during time periods called ‘bifurcation delays’ in deterministic models. We define a class of models of neural activity all of which display Hopf bifurcations in their base (fixed bifurcation parameter) deterministic form. We then introduce a two-time, slow-fast, version of our model class, in which the bifurcation parameter, instead of being fixed, changes on a time scale slower than the time scale of the state variables. We demonstrate the effects of the slowly changing bifurcation parameter on the deterministic dynamics of the state variables, in particular ‘delayed bifurcation.’ We find a new understanding about the length of the delay. Most importantly, we display with simulations the effect of noise on the dynamics of both base system and slow-fast system models in our class. Adding noise to a slow-fast model eliminates the bifurcation delay and induces what we call ‘quasi-limit-cycles’ during the delay period. We measure the sizes of the quasi-limit-cycles and show that they are closely related to the sizes of the limit cycles that would arise from the same values of the bifurcation parameter in the deterministic base system. We conclude that, given the similarities of the dynamics of these models 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. Author summary Recordings of brain activity display noisy periodic oscillations. Many computational models of this oscillatory activity have what is called a ‘Hopf bifurcation.’ This is a point in the dynamic phase space at which solutions to the deterministic versions of the models change from having a stable fixed point to having a stable limit cycle. In this paper we define a new class of models of oscillatory brain activity all of which have Hopf bifurcations. We compute both deterministic and stochastic path solutions to the models. These solutions give additional insight into an intriguing phenomenon of these models when a parameter slowly changes, called ‘delayed bifurcation,’ as well as expanding our knowledge of their stochastic dynamics. In particular, we define and measure a new type of noisy oscillation, called ‘quasi-limit-cycles,’ that occur during a bifurcation delay in stochastic solutions. The stochastic versions of these models are roughly equally useful as neural mass models. Competing Interest Statement The authors have declared no competing interest.