Higher-order interactions and heteroclinic network dynamics

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.

The proposed project will create mathematical knowledge, that is relevant to understand, design, and control real-world networks of interacting dynamical systems, and thus creates societal impact in the long term. Specifically, the potential beneficiaries of this research are
(*) mathematicians working in dynamical systems theory and network science,
(*) scientists studying coupled oscillatory processes, including neuroscientists,
(*) the research staff on this project,
(*) STEM students and the general public.

First, the proposed research will have an impact on researchers in the mathematical sciences. This includes primarily mathematicians in the dynamical systems community: The anticipated results on network dynamical systems with continuous symmetries will drive new developments in the dynamical systems community, both theoretical and applied. Moreover, impact will extend to the network science community: Networks with higher-order interactions - as considered in the proposed project - have only recently caught attention in the network science community. Hence, communicating the results to the network science community will lead to new perspectives and collaborations between the dynamical systems and the network science communities. Importantly, the proposed project will bridge dynamical systems and network science through a planned workshop.

Second, the proposed research will generate impact beyond the mathematical sciences. The theoretical results will link to neuroscience modelling. Through a workshop to bring together mathematicians and (theoretical and clinical) neuroscientists, we will explore new directions how to exploit dynamical insights to work towards new treatment approaches for neurological disease. This includes Parkinson's disease, which affects 127,000 people in the UK according to the NHS. In the longer term, the proposed research thus has the potential to increase the quality of life in the UK.

Third, the researchers funded by the project will benefit. Delivering the project will be a crucial step for the PI to establish his own research group and further develop his skills needed as a future leader in mathematical sciences in the UK. The project will also generate impact on the career of the PDRA who will be involved in the project. Apart from the mathematical skills that will be developed during the project, the PDRA will have the opportunity to develop organizational, leadership, and writing skills that will be crucial for her/his future career.

Finally, through teaching and outreach activities, the project will impact not only students but also the general public. These activities will stimulate interest in the mathematical challenges that underlie the functioning of networks of interacting dynamical systems we take for granted every day. Importantly, communicating these challenges and the solution approaches that are part of the proposed project will inspire a future generation of mathematicians and scientists.