27 May 2021
Abstract: By designing feedback to control populations of oscillators, synchronisation in neurons causing epileptic seizures can be broken up. Oscillators (e.g. neurons) can be coupled on different levels. The most basic level is through links between pairs of oscillators. These pairwise links fail to explain phenomena such as peer pressure. The nonpairwise ‘links’ make such phenomena possible. Even though the effects of these nonpairwise interactions have been observed, described and modelled in a wide range of oscillatory systems, controlling nonpairwise interactions in arbitrary systems has remained a mainly unexplored area. We generalize synchronisation engineering to control nonpairwise interactions in arbitrary systems. As a first step, we design nonlinear time-delayed feedback that introduces bifurcations away from splay configuration into arbitrary systems. Controlling nonpairwise interactions might advance the design of minimum-power stimuli for the treatment of epilepsy.
Session Abstract: Phase reduction is a powerful technique for the analysis and modelling of oscillatory dynamics in various applications, including power grids and biological systems, such as neural systems. A phase reduction describes the dynamics of weakly coupled limit-cycle oscillators in terms of their phase, a single variable for each oscillator. The effect of coupling between nodes in the network on the evolution of these phase variables is then captured by a single, phase response function, which can be inferred from data, or can be computed for given dynamical models. A barrier to the application of phase reduction approaches is that obtaining them is not straightforward, particularly when the oscillators cannot be isolated. Additionally, phase reductions are formally defined for weak coupling, that is, when the strength of coupling between units is small compared to the intrinsic oscillatory dynamics of each node. When the weak assumption does not hold, higher-order phase reduction techniques, or inclusion of additional dynamic variables, are often necessary to extend the validity of phase reduction. This minisymposium will highlight recent advances in the applicability of phase reduction techniques, covering extensions to phase-amplitude coordinates, efficient approaches for phase response function inference, and approaches for understanding and controlling higher order network interactions.
Talk at the SIAM Conference on Applications of Dynamical Systems 2021 (DS21).