Synchronization of bursting neurons: what matters in the network topology

We study the influence of coupling strength and network topology on synchronization behavior in pulse-coupled networks of bursting Hindmarsh-Rose neurons. Surprisingly, we find that the stability of the completely synchronous state in such networks only depends on the number of signals each neuron receives, independent of all other details of the network topology. This is in contrast with linearly coupled bursting neurons where complete synchrony strongly depends on the network structure and number of cells. Through analysis and numerics, we show that the onset of synchrony in a network with any coupling topology admitting complete synchronization is ensured by one single condition.     

Uniqueness of the Ground State in the Feshbach Renormalization Analysis

In the operator theoretic renormalization analysis introduced by Bach, Fr?hlich, and Sigal, we prove uniqueness of the ground state.     

Wegner Estimate for Random Magnetic Laplacian on {\mathbb{Z}^{2}}

We consider a two-dimensional magnetic Schr?dinger operator on a square lattice with a spatially stationary random magnetic field. We prove Anderson localization near the spectral edges. We use a new approach to establish a Wegner estimate that does not rely on the monotonicity of the energy on the random parameters.     

A Vanishing Theorem for Operators in Fock Space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space, we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.     

The Independence on Boundary Conditions for the Thermodynamic Limit of Charged Systems

We study systems containing electrons and nuclei. Based on the fact that the Thermodynamic limit exists for systems with Dirichlet boundary conditions, we prove that the same limit is obtained if one imposes other boundary conditions such as Neumann, periodic, or elastic boundary conditions. The result is proven for all limiting sequences of domains which are obtained by scaling a bounded open set, with smooth boundary, except for isolated edges and corners. Work partially supported by EU grant HPRN-CT-2002-00277.     

Modeling phase synchronization in systems of two and three coupled oscillators

We determine regions of synchronization of two and three globally coupled oscillators and describe the main mechanisms and bifurcations through which the synchronization of systems is lost.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 311–327, July–September, 2004.