Synchronization of electrically coupled resonate-and-fire neurons

Electrical coupling between neurons is broadly present across brain areas and is typically assumed to synchronize network activity. However, intrinsic properties of the coupled cells can complicate this simple picture. Many cell types with strong electrical coupling have been shown to exhibit resonant properties, and the subthreshold fluctuations arising from resonance are transmitted through electrical synapses in addition to action potentials. Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons, a hybrid neuron model with linear subthreshold dynamics and discrete post-spike reset. We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuity in the dynamics. We find that both spikes and resonant subthreshold fluctuations can jointly promote synchronization. The subthreshold contribution is strongest when the voltage exhibits a significant post-spike elevation in voltage, or plateau. Additionally, we show that the geometry of trajectories approaching the spiking threshold causes a "reset-induced shear" effect that can oppose synchrony in the presence of network asymmetry, despite having no effect on the phase-locking of symmetrically coupled pairs

On the r-matrix structure of the hyperbolic BC(n) Sutherland model

Working in a symplectic reduction framework, we construct a dynamical r-matrix for the classical hyperbolic BC(n) Sutherland model with three independent coupling constants. We also examine the Lax representation of the dynamics and its equivalence with the Hamiltonian equation of motion.Comment: 20 page

Lie Algebroid Yang Mills with Matter Fields

Lie algebroid Yang-Mills theories are a generalization of Yang-Mills gauge theories, replacing the structural Lie algebra by a Lie algebroid E. In this note we relax the conditions on the fiber metric of E for gauge invariance of the action functional. Coupling to scalar fields requires possibly nonlinear representations of Lie algebroids. In all cases, gauge invariance is seen to lead to a condition of covariant constancy on the respective fiber metric in question with respect to an appropriate Lie algebroid connection. The presentation is kept in part explicit so as to be accessible also to a less mathematically oriented audience.Comment: 24 pages, accepted for publication in J. Geom. Phy

Monopole and Dyon Spectra in N=2 SYM with Higher Rank Gauge Groups

We derive parts of the monopole and dyon spectra for N=2 super-Yang--Mills theories in four dimensions with gauge groups G of rank r>1 and matter multiplets. Special emphasis is put on G=SU(3) and those matter contents that yield perturbatively finite theories. There is no direct interpretation of the soliton spectra in terms of naive selfduality under strong--weak coupling and exchange of electric and magnetic charges. We argue that, in general, the standard procedure of finding the dyon spectrum will not give results that support a conventional selfduality hypothesis --- the SU(2) theory with four fundamental hypermultiplets seems to be an exception. Possible interpretations of the results are discussed.Comment: 24 pages, plain tex, 6 figures. uuencoded compressed tar file. Three references adde