A Linear Analysis of Coupled Wilson-Cowan Neuronal Populations

Let a neuronal population be composed of an excitatory group interconnected to an inhibitory group. In the Wilson-Cowan model, the activity of each group of neurons is described by a first-order nonlinear differential equation. The source of the nonlinearity is the interaction between these two groups, which is represented by a sigmoidal function. Such a nonlinearity makes difficult theoretical works. Here, we analytically investigate the dynamics of a pair of coupled populations described by the Wilson-Cowan model by using a linear approximation. The analytical results are compared to numerical simulations, which show that the trajectories of this fourth-order dynamical system can converge to an equilibrium point, a limit cycle, a two-dimensional torus, or a chaotic attractor. The relevance of this study is discussed from a biological perspective.

Measuring complex behaviors of local oscillatory networks in deep brain local field potentials

BACKGROUND

Multiple oscillations emerging from the same neuronal substrate serve to construct a local oscillatory network. The network usually exhibits complex behaviors of rhythmic, balancing and coupling between the oscillations, and the quantification of these behaviors would provide valuable insight into organization of the local network related to brain states.

NEW METHOD

An integrated approach to quantify rhythmic, balancing and coupling neural behaviors based upon power spectral analysis, power ratio analysis and cross-frequency power coupling analysis was presented. Deep brain local field potentials (LFPs) were recorded from the thalamus of patients with neuropathic pain and dystonic tremor. t-Test was applied to assess the difference between the two patient groups.

RESULTS

The rhythmic behavior measured by power spectral analysis showed significant power spectrum difference in the high beta band between the two patient groups. The balancing behavior measured by power ratio analysis showed significant power ratio differences at high beta band to 8-20 Hz, and 30-40 Hz to high beta band between the patient groups. The coupling behavior measured by cross-frequency power coupling analysis showed power coupling differences at (theta band, high beta band) and (45-55 Hz, 70-80 Hz) between the patient groups.

COMPARISON WITH EXISTING METHOD

The study provides a strategy for studying the brain states in a multi-dimensional behavior space and a framework to screen quantitative characteristics for biomarkers related to diseases or nuclei.

CONCLUSIONS

The work provides a comprehensive approach for understanding the complex behaviors of deep brain LFPs and identifying quantitative biomarkers for brain states related to diseases or nuclei.