Distinct configurations of synchrony govern the functioning of many natural and technological network systems. For instance, coordinated behavior of oscillatory components determine the coordinated motion of orbiting particle systems, promote successful mating in populations of fireflies, regulate the active power flow in electrical grids, predict global climate change phenomena, dictate the structural development of mother-of-pearl in mollusks, and enable numerous cognitive functions in the brain. Because this rich repertoire of patterns emerges from the properties of the underlying interaction network, being able to control the collective configuration of interdependent units holds a tremendous potential across science and engineering.
To pursue such a tantalizing idea, we set out to develop a rigorous sound framework to optimally control the spatial organization of the network components and their oscillation frequencies with the ultimate goal of achieving desired patterns of synchrony. To quantify the pairwise functional relations between oscillatory units, we defined functional patterns. A functional pattern explicitly encodes the pairwise, local correlations across all of the network oscillators.
Network control to enforce a desired functional pattern from an abnormal or undesired one. In this example, we computed the closest set of coupling strengths and natural frequencies to the original ones that enforce the emergence of the target pattern. Importantly, only a subset of the original parameters has been modified.
When dealing with functional patterns, a few questions arise naturally. Are all functional patterns achievable? Which network configurations allow for the emergence of multiple target functional patterns? And, if a certain functional pattern can be achieved, is it robust to perturbations?
When the oscillators reach an equilibrium, the differences of the oscillator phases become constant, and the network evolves in a phase-locked configuration. In this case, the functional pattern of the network also becomes constant, and is uniquely determined by the phase differences at the equilibrium configuration. In light of this observation, we convert the problem of generating a desired functional pattern into the problem of ensuring a desired phase-locked equilibrium.
Analytical results
In the first part of our work, we focused on addressing these questions from a mathematical point of view. Our analytical results reveal that the interplay between the network structure and the oscillators' natural frequencies dictates whether a desired functional pattern is achievable under several constraints on the type of interconnections that are allowed in the network. Furthermore, we demonstrated that the characterization of which (and how many) compatible patterns simultaneously coexist can be revealed by some algebraic properties of the matrix that describes the network structure. Being able to concurrently assign multiple functional patterns is crucial, for instance, to the investigation and design of memory systems, where different patterns of activity correspond to distinct memories. We also derived analytical results on the stability of functional patterns. That is, whether small deviations of the oscillators phases from the desired configuration lead to vanishing functional perturbations. We demonstrated that the stability of a functional pattern is strictly related to the distance between the oscillators' phases. Finally, by leveraging the aforementioned results, we showed that the problem of adjusting the network weights to generate a desired functional pattern can be cast as a convex optimization problem. The minimization problem presented in the paper is efficiently solvable even for large networks, and may admit multiple minimizers, thus showing that different networks may exhibit the same functional pattern.
In the second part of this work, we applied our methods and theories to an empirically reconstructed brain network and to a model of the New England power distribution network. In the former case, we used a classical oscillator model to map structure to function, and found that local metabolic changes underlie the emergence of functional patterns of recorded neural activity. In the latter case, we used our methods to restore the nominal network power flow after a simulated fault.
Application to brain networks
With respect to the brain network data, our results corroborate the postulate that structural connections in the brain support the intermittent activation of specific functional patterns during rest through regional metabolic changes. Furthermore, we showed that the Kuramoto oscillator model represents an accurate and interpretable mapping between the brain anatomical organization and the functional patterns of frequency-synchronized neural co-fluctuations. Once a good mapping is inferred, it can be used to define a digital generative brain model to replicate how the brain efficiently enacts large-scale integration of information, and to develop personalized intervention schemes for neurological disorders related to abnormal synchronization phenomena.
Replication of empirically recorded functional connectivity in the brain through tuning of the natural frequencies of Kuramoto oscillators. The anatomical brain organization provides the network backbone over which the oscillators evolve. The recorded brain activity time series provide the relative phase differences between co-fluctuating brain regions, and thus define the desired phase differences, which are used to calculate the metabolic change encoded in the oscillators' natural frequencies.
Application to power networks
The second application of our results concerned power delivery networks. Efficient and robust power delivery in electrical grids is crucial for the correct functioning of this critical infrastructure. Modern, reconfigurable power networks are expected to be resilient to distributed faults and malicious cyber-physical attacks while being able to rapidly adapt to varying load demands. In addition, climate change is straining service reliability, as underscored by recent events such as the Texas grid collapse after Winter Storm Uri in February 2021, and the New Orleans blackout after Hurricane Ida in August 2021. Therefore, there exists a dire need to design control methods to efficiently operate these networks and react to unforeseen disruptive events.
We posited that solving the optimization problem presented in our paper to design a local reconfiguration of the network parameters can recover the power distribution before a line fault or provide the smallest parameter changes to steer the load powers to desired values. We showcased the effectiveness of our approach by recovering a desired power distribution in a model of the New England power distribution network after a fault. During a regime of normal operation, we simulated a fault by disconnecting the line between two loads and solved the proposed optimization problem to find the minimum modification of the interconnections that recovers the nominal power distribution.
Fault recovery in the IEEE 39 New England power distribution network through minimal and local intervention. The generator terminal buses illustrate the type of generator (coal, nuclear, hydroelectric). We simulate a fault by disconnecting the transmission line 25 (between loads 13 and 14). The network returns to the initial operative conditions with a localized modification of the neighboring transmission lines' impedances.
All in all, this work presents a simple and mathematically grounded mapping between the structural parameters of arbitrary oscillator networks and their components' functional interactions. The disruptive idea of prescribing patterns in networks of oscillators has been investigated before, yet only partial results had been reported in the literature. Closing the gap in the existing literature, here we demonstrated that the control of patterns of synchrony can be cast as optimal (convex) design and tuning problems. We also investigated the feasibility of such optimizations in the cases of networks whose coupling types are restricted (e.g., either all negative or positive). Finally, our control framework allowed us to prescribe multiple desired equilibria in oscillatory networks, a problem that is relevant in practice and had not been investigated before.
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