Higher-order interactions and heteroclinic network dynamics

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Networks of coupled dynamical nodes are ubiquitous in science and technology and influence many parts of our everyday lives. Indeed, ecological networks of interacting species, neurons in the brain and coupled rotors as a model for a power grid are examples of networks of coupled oscillatory nodes. Such networks can give rise to a wide range of collective dynamics - the joint dynamics of the coupled nodes - such as synchrony. Crucially, the network function or dysfunction often depends on the collective dynamics. For example, neurological diseases, such as Parkinson's disease, have long been associated with excessive neural synchronization.The collective dynamics of a network, that is, whether the nodes synchronize or show other dynamical behaviour, depends on the network structure and interactions. The network structure determines whether a node influences other nodes. The network interactions determine how a node influences other nodes. In many real-world networks, the network interactions include "higher-order" coupling, for example, the influence of one node onto another may depend on the state of a third node. However, such interactions are often omitted in commonly studied networks.The proposed project will elucidate the collective dynamics of coupled oscillator networks. The main question we address here is how network structure and interactions - with a particular focus on higher-order interactions - shape the collective dynamics. We will investigate objects called heteroclinic structures and elucidate how they organize the dynamics for interactions that are relevant for real-world networks.The project will yield new results in dynamical systems theory and their application. Moreover, we will investigate how the results can lead to new ways to control dynamics. Insights into how network structure and interactions shape the dynamics can be employed to understand what part of the network one has to tune to get oscillators to synchronize (or not). Since Parkinson's disease has been associated with excessive synchrony, this could eventually lead to new ways to tune network parameter to restore healthy brain functionality.