Emergent hypernetworks in weakly coupled oscillators

Abstract

Networks of weakly coupled oscillators had a profound impact on our understanding of complex systems. Studies on model reconstruction from data have shown prevalent contributions from hypernetworks with triplet and higher interactions among oscillators, in spite that such models were originally defined as oscillator networks with pairwise interactions. Here, we show that hypernetworks can spontaneously emerge even in the presence of pairwise albeit nonlinear coupling given certain triplet frequency resonance conditions. The results are demonstrated in experiments with electrochemical oscillators and in simulations with integrate-and-fire neurons. By developing a comprehensive theory, we uncover the mechanism for emergent hypernetworks by identifying appearing and forbidden frequency resonant conditions. Furthermore, it is shown that microscopic linear (difference) coupling among units results in coupled mean fields, which have sufficient nonlinearity to facilitate hypernetworks. Our findings shed light on the apparent abundance of hypernetworks and provide a constructive way to predict and engineer their emergence.

Fig. 1. Emergent hypernetworks in an electrochemical network experiment.

a Experimental setup. b Schematic illustration of the electrochemical experiment with the nonlinear feedback. The blue, orange, yellow, and green lines represent the elements 1 to 4, respectively. The electrode potential signals (E_k) of the four (nearly) isolated electrodes are nonlinearly modulated and fed back with a delay τ to the corresponding circuit potential (V_k), which drives the metal dissolution. (The delay is implemented by storing the past data in the memory of the computer.) c Representation of the in a ring network topology used in the experiment. d Electrode potential time series. e Filtered and fitted (dark red line) instantaneous frequency using LASSO for hypernetwork reconstruction corresponding from top to bottom to oscillators 1 to 4, respectively. f Experimental recovery of the phase interactions given by a hypernetwork.

Fig. 2. Emergent higher-order interactions from the original ring network.

Coupling functions appearing in Eq. (12) of node 1. Colors correspond to signs in the phase combination with blue standing for positive and orange for negative. a Resonant interaction term appearing as ${}^2{G}_1^{23}$. b Resonant interaction term appearing as ${}^2{G}_1^{43}$. Finally, c is a nonresonant term and d ${}^2{G}_1^{24}$ is a forbidden term (it does not appear). These new interaction terms can be predicted from the combinatorics of the original network and coupling function.

Fig. 3. Normal form theory explains the experimental results.

We show the time series of the slow phase ϕ₁ and ϕ₂ from experimental data (solid) and the prediction of the emergent hypernetwork (dashed) capturing higher-order interactions. The vector field describing the phase interaction is obtained from first principles. The coefficients of the vector field are obtained by least-square minimization.

Fig. 4. Interacting subpopulations lead to higher order interaction of mean-fields.

a The original network of coupled subpopulations (with four distinct colours, namely, red, yellow, blue and orange). Oscillators are interacting by an internal coupling constant μ and inter-subpopulations coupling constant α. b Higher order phase interaction of the mean-fields represented with the same colors as in a (red, yellow, blue and orange). Applying our approach we uncover that the phase interaction between the mean-fields is described by a hypernetwork. c The mean-field slow phase variables φ₁ (green) and φ₂ (purple) were computed from the data collected from the simulations of mean fields on the associated network. The dashed curve is the simulation of the vector field of the slow phases φ_1,2 reconstructed from data using the Lasso method.