Humboldt-Universität zu Berlin - Mathematisch-Naturwissen­schaft­liche Fakultät - International Research Training Group 1740


Project A3: Clusters of oscillatory states on networks with local and global coupling

Research Team: R. Köberle (USP), E. Macau (INPE), R. Medrano (UNIFESP), M. Rosenblum (UP), M. Zaks (UP)

Outline: Dynamics of a network results from the interplay of its topology, on-site behavior of each unit and the kind of interaction between the units. Depending on the way in which the interaction influences the difference between the instantaneous states of units, we distinguish the attracting interactions and the repulsive ones. Historically, most attention has been paid to the former kind: it benefits different forms of adjustment between all oscillating subsystems. In our project we intend to consider networks where interaction is repulsive, as well as those in which both kinds of interaction are enabled. Repulsive interaction, in contrast, disfavors close states and tries to bring the units apart. Since in structured networks the latter tendency is bounded (i.e. three units cannot all be in counterphase to each other), a kind of balanced regime is commonly established, often in the form of spatially disjoint clusters. We intend to investigate the onset and stability of such states in different setups.

Research Topic: For regular networks with repulsive local coupling we will derive the conditions under which the sponta- neous onset of a certain form of global coupling takes place: the uniform synchronous state is replaced by spatially disjoined clusters, spreading over the network. In the clustered oscil- latory state, each unit is connected to all clusters in the network and receives from them the same input. For a particular class of coupling functions, a kind of integrability arises: at fixed parameter values the system has a continuous attracting family of periodic states. We will study stability of these oscillations, as well as more complicated quasiperiodic and chaotic regimes. Starting with phase oscillators and clusters of equal size, we will generalize the approach to higher-dimensional units and arbitrary cluster sizes, and consider effects caused by minor deviations from regularity in the coupling pattern. Next, we will consider the case where the ensemble consists of two or several groups, which contribute differently to the mean field and/or respond differently to the mean field forcing. We will investigate how the interplay between attractive and repulsive coupling affects the transition between synchro- nous and cluster states, including fuzzy clusters. Again, starting with phase oscillators, we will proceed with the analysis of neuronal models with excitatory and inhibitory connections.