Many popular neural mass models for describing cortical population dynamics, such as Wilson- Cowan or Jansen-Rit, track an average activity without recourse to describing the degree of synchronisation or coherence within a population. This could of course be tracked within a large-scale model of synaptically interacting conductance based neurons, though at the expense of analytical tractability. Thus it is of interest to seek levels of description that provide a bridge between microscopic single neuron dynamics and coarse grained neural mass models, while preserving some notion of within-population coherence.
For a theta neuron choice of microscopic dynamics we can make use of the Ott-Antonsen ansatz  to find an exact mean field description of the population dynamics on a reduced invariant manifold. We consider an all-to-all coupled network of such neurons, allowing us to track the complex valued Kuramoto order parameter z. For one population, with realistic synaptic interactions, this amounts to 4 ordinary differential equations. Simulations have been carried out that prove our mean field reduction is a accurate representation of a large population of theta neurons.
Extensive bifurcation analysis has been carried out for a single population and for a two population excitatory-inhibitory (E-I) structure. As expected the addition of a second population allows for richer dynamics. The interplay between the excitatory and inhibitory synaptic interactions, as well as the background drive leads to very interesting behaviours. Our E-I model supports Hopf bifurcations, period doubling bifurcations, torus bifurcations and isola bifurcations.
A similar approach can be taken for a population of quadratic integrate-and-fire neurons . In this framework the mean field variable tracks the population firing rate. A simple transformation exists which allows us to switch between these two representations.
This work involves collaboration with colleagues in the Sir Peter Mansfield Magnetic Resonance Centre at Nottingham to use this new framework to understand the generation of beta-rhythms seen in motor cortex, and deliver a suitable model for understanding so-called beta-rebound. This is readily observed in MEG recordings whereby the initiation of hand movement causes a drop in the beta power band attributed to a loss of network synchrony. Early results suggest that our model is a suitable candidate model.
 T. Luke, E. Barreto, P. So, Neural Computation 25:3207-3234 (2013).
 A. Roxin, presented at workshop on Neurodynamics July 2014, Castro-Urdiales, Spain.