Spatio-temporal dynamics associated with a class of coupled membrane-bulk PDE-ODE models in one spatial dimension is analyzed using a combination of linear stability theory, numerical bifurcation software, and full time-dependent simulations. In our simplified 1-D setting, the mathematical model consists of two dynamically active membranes, separated spatially by a distance 2L, that are coupled together through a linear bulk diffusion field, with a fixed bulk decay rate. The coupling of the bulk and active membranes arises through both nonlinear flux boundary conditions for the bulk diffusion field together with feedback terms, depending on the local bulk concentration, to the dynamics on each membrane. For this class of models, it is shown both analytically and numerically that bulk diffusion can trigger a synchronous oscillatory instability in the temporal dynamics associated with the two active membranes. For the case of a single active component on each membrane, and in the limit L → ∞, rigorous spectral results for the linearization around a steady-state solution, characterizing the possibility of Hopf bifurcations and temporal oscillations in the membranes, are obtained. For finite L, a weakly nonlinear theory, accounting for eigenvalue-dependent boundary conditions appearing in the linearization, is developed to predict the local branching behavior near the Hopf bifurcation point. The analytical theory, together with numerical bifurcation results and full numerical simulations of the PDE-ODE system, are undertaken for various coupled membrane-bulk systems, including two specific biologically relevant applications. Our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur.

Synchronization is studied in a spatially-distributed network of weeklycoupled, excitatory neurons of Hodgkin-Huxley type. All neurons are coupled to each other synaptically with a fixed time delay and a coupling strength inversely proportional to the distance between two neurons. We found that a robust, noise-resistant phase clustering state occurred regardless of the initial phase distribution. This has not been shown in previous studies where similar clustering states were found only when the coupling was inhibitory. The spatial distribution of neurons in each synchronous cluster is determined by the spatial distribution of the coupling strength. Phase-interaction properties of the model neurons in the network are used to explain why can such a clustering state be robust.

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