Fluctuation–dissipation of the Kuramoto model on fruit-fly connectomes

We investigate the distance from equilibrium using the Kuramoto model via the degree of fluctuation–dissipation violation as the consequence of different levels of edge weight anisotropies. This is achieved by solving the synchronization equations on the raw, homeostatic weighted and a random inhibitory edge variant of a real full fly (FF) connectome, containing ≃ 1 0 5 neuron cell nodes. We investigate these systems close to their synchronization transition critical points. While the topological(graph) dimension is high: d ≃ 6 the spectral dimensions of the variants, relevant in describing the synchronization behavior, are lower than the upper critical dimension: d s ≃ 2 < d c = 5 , suggesting relevant fluctuation effects and non mean-field scaling behavior. By measuring the auto-correlations and the auto-response functions for small perturbations we calculate the fluctuation–dissipation ratios (FDR) for the different variants of different anisotropy levels of the FF connectome. Numerical evidence is presented that the FDRs follow the level of anisotropy of these non-equilibrium systems in agreement with the expectations, with increasing temporal oscillations. Numerical values for the aging exponents provide confirmation for the predictions of the generalized time-translational-invariance theory. We compare the non-reciprocal connectome results with those on a symmetric Erdős–Rényi random graph of similar size and provide estimates for the Kuramoto model aging behavior. We also present a network analysis of the FF connectome and calculate the level of hierarchy, also related to the anisotropy. Finally, we provide some partial results for the periodically forced Shinomoto–Kuramoto model describing fluctuations in the non-resting state. We provide numerical evidence that fluctuations are maximal in the resting state of the critical brain.