Critical fluctuations and coupling of stochastic neural mass models

Abstract

Neuroimaging and implanted medical devices give a view onto large scale oscillations in the brain that rapidly change in both power and synchrony at different frequencies. These measured time series reveal a projection to just one dimension for each location, a moving shadow of the complex underlying activity. For this reason mathematical models of the activity are essential to make full use of the time series. The flexible onset and cessation of oscillations may have an important role in allowing the brain rapidly to rearrange its effective network structure. The onset of pathological oscillations is also central to neurological disorders such as epilepsy and Parkinson's disease. It is in the regime close to linear instability that transitions to and from oscillation can readily occur, by mechanisms of bifurcation and multistability. Empirical evidence of critical fluctuations and critical slowing down in neural dynamics also suggest proximity to linear instability. This thesis develops novel methods to analyze stochastic dynamical models near the point of transition where oscillations start and stop, allowing to relate transitions in the models to changes in time series statistics that are accessible in clinical and experimental recordings. Taking as an example a widely used biophysical model of collective neural dynamics (the Jansen-Rit model) we show that near the point of oscillation onset at a supercritical Hopf bifurcation there can be a dramatic increase in autocorrelation length as well as power law scaling over four orders of magnitude in fluctuation size and duration, but that these time series indicators will depend sensitively on the direction in state space of input fluctuations and hence on which neuronal subpopulation is stochastically perturbed. In general near a supercritical Hopf bifurcation non-circular oscillations emerge on an arbitrarily curved two-dimensional surface embedded within the N-dimensional state space. This makes it difficult to determine exactly how noise will affect oscillation phase and amplitude. By applying Poincare normal form transformations to find a parameter-dependent coordinate system in which the unperturbed oscillations are simple circles in a flat plane, then using Stratonovich stochastic calculus to transform noise perturbations to the same coordinate system, this makes explicit the effect of noise on phase and amplitude. We give a criterion for this reduced model to be a good approximation, balancing noise intensity and center manifold stability. Optionally averaging the diffusion process in Fokker-Planck form gives a still simpler weak approximation of the oscillations in a symmetrical standard form that retains the leading order stochastic effects on phase and amplitude dynamics. We demonstrate by large scale numerical simulation of the stochastic differential equations that the oscillation time series statistics of interest are preserved by transformation to this symmetrical weak approximation. Close to instability, dynamic synchrony of networked oscillations can depend on both phase and amplitude changes of constituent oscillators. In computational neuroscience studies of network synchrony near instability, coupled systems of Hopf bifurcation normal forms and Generalized Hopf normal forms have been used successfully as minimal models with local oscillation onset occurring via bifurcation and bistability respectively. Such abstract models are often parameterized somewhat arbitrarily, as the individual flow in the state space is qualitatively the same for a range of parameters. But this assumption breaks down when the systems are not isolated. In this deterministic setting we demonstrate a second application of normal form transformations to constrain the parameters and coupling of a normal form network model based on a more detailed biophysical model. We show that using the Poincare normal form allows the simple model to match the collective synchronization behavior of the more detailed biophysical model, whereas using the topological normal form with symmetrical diffusive coupling does not. Ultimately the methods developed in this thesis allow to predict from a given biophysical model how time series statistics of oscillations will change as biological parameters are changed in experiment. This can assist in testing models empirically, to arrive at better models. After suitable models are evolved this will then allow diagnostic inference of brain state that cannot be observed directly, using clinical neuroimaging time series.