Title: Oscillatory pattern formation in a neural dynamical system governed by a mutual Hamiltonian and gradient vector field structure
Authors: V.V.Gafiychuk (Institute of computer modeling, Krakow University of Technology, 24 Warszawska Street, 31155, Krakow, Poland),
A.K.Prykarpatsky (Department of Applied Mathematics at the AGH University of Science and Technology, 30 Mickiewicz Al. Bl. A4, 30059 Krakow, Poland)

We analyze dynamical systems of general form possessing gradient (symmetric) and Hamiltonian (antisymmetric) flow parts. The relevance of such systems to self-organizing processes is discussed. Coherent structure formation and related gradient flows on matrix Grassmann type manifolds are considered. The corresponding graph model associated with the partition swap neighborhood problem is studied. The criterion for emerging gradient and Hamiltonian flows is established. As an example we consider nonlinear dynamics in a neuron network system described by a simulative vector field. A simple criterion was written in order to establish conditions for the formation of an oscillatory pattern in a model neural system under consideration.

Key words: dynamical system, gradient flow, Hamiltonian flow, self-organization, neural network
PACS: 05.45.-a, 07.05.Mh, 05.65.+b