Phase and amplitude dynamics of coupled oscillator systems on complex networks.

We investigated locking behaviors of coupled limit-cycle oscillators with phase and amplitude dynamics. We focused on how the dynamics are affected by inhomogeneous coupling strength and by angular and radial shifts in coupling functions. We performed mean-field analyses of oscillator systems with inhomogeneous coupling strength, testing Gaussian, power-law, and brain-like degree distributions. Even for oscillators with identical intrinsic frequencies and intrinsic amplitudes, we found that the coupling strength distribution and the coupling function generated a wide repertoire of phase and amplitude dynamics. These included fully and partially locked states in which high-degree or low-degree nodes would phase-lead the network. The mean-field analytical findings were confirmed via numerical simulations. The results suggest that, in oscillator systems in which individual nodes can independently vary their amplitude over time, qualitatively different dynamics can be produced via shifts in the coupling strength distribution and the coupling form. Of particular relevance to information flows in oscillator networks, changes in the non-specific drive to individual nodes can make high-degree nodes phase-lag or phase-lead the rest of the network.

Various synchronous states due to coupling strength inhomogeneity and coupling functions in systems of coupled identical oscillators.

We study the effects of coupling strength inhomogeneity and coupling functions on locking behaviors of coupled identical oscillators, some of which are relatively weakly coupled to others while some are relatively strongly coupled. Through the stability analysis and numerical simulations, we show that several categories of fully locked or partially locked states can emerge and obtain the conditions for these categories. In this system with coupling strength inhomogeneity, locked and drifting subpopulations are determined by the coupling strength distribution and the shape of the coupling functions. Even the strongly coupled oscillators can drift while weakly coupled oscillators can be locked. The simulation results with Gaussian and power-law distributions for coupling strengths are compared and discussed.

Mechanisms of hysteresis in human brain networks during transitions of consciousness and unconsciousness: Theoretical principles and empirical evidence.

Hysteresis, the discrepancy in forward and reverse pathways of state transitions, is observed during changing levels of consciousness. Identifying the underlying mechanism of hysteresis phenomena in the brain will enhance the ability to understand, monitor, and control state transitions related to consciousness. We hypothesized that hysteresis in brain networks shares the same underlying mechanism of hysteresis as other biological and non-biological networks. In particular, we hypothesized that the principle of explosive synchronization, which can mediate abrupt state transitions, would be critical to explaining hysteresis in the brain during conscious state transitions. We analyzed high-density electroencephalogram (EEG) that was acquired in healthy human volunteers during conscious state transitions induced by the general anesthetics sevoflurane or ketamine. We developed a novel method to monitor the temporal evolution of EEG networks in a parameter space, which consists of the strength and topography of EEG-based networks. Furthermore, we studied conditions of explosive synchronization in anatomically informed human brain network models. We identified hysteresis in the trajectory of functional brain networks during state transitions. The model study and empirical data analysis explained various hysteresis phenomena during the loss and recovery of consciousness in a principled way: (1) more potent anesthetics induce a larger hysteresis; (2) a larger range of EEG frequencies facilitates transitions into unconsciousness and impedes the return of consciousness; (3) hysteresis of connectivity is larger than that of EEG power; and (4) the structure and strength of functional brain networks reconfigure differently during the loss vs. recovery of consciousness. We conclude that the hysteresis phenomena observed during the loss and recovery of consciousness are generic network features. Furthermore, the state transitions are grounded in the same principle as state transitions in complex non-biological networks, especially during perturbation. These findings suggest the possibility of predicting and modulating hysteresis of conscious state transitions in large-scale brain networks.

Structure Shapes Dynamics and Directionality in Diverse Brain Networks: Mathematical Principles and Empirical Confirmation in Three Species.

Identifying how spatially distributed information becomes integrated in the brain is essential to understanding higher cognitive functions. Previous computational and empirical studies suggest a significant influence of brain network structure on brain network function. However, there have been few analytical approaches to explain the role of network structure in shaping regional activities and directionality patterns. In this study, analytical methods are applied to a coupled oscillator model implemented in inhomogeneous networks. We first derive a mathematical principle that explains the emergence of directionality from the underlying brain network structure. We then apply the analytical methods to the anatomical brain networks of human, macaque, and mouse, successfully predicting simulation and empirical electroencephalographic data. The results demonstrate that the global directionality patterns in resting state brain networks can be predicted solely by their unique network structures. This study forms a foundation for a more comprehensive understanding of how neural information is directed and integrated in complex brain networks.

General relationship of global topology, local dynamics, and directionality in large-scale brain networks.

The balance of global integration and functional specialization is a critical feature of efficient brain networks, but the relationship of global topology, local node dynamics and information flow across networks has yet to be identified. One critical step in elucidating this relationship is the identification of governing principles underlying the directionality of interactions between nodes. Here, we demonstrate such principles through analytical solutions based on the phase lead/lag relationships of general oscillator models in networks. We confirm analytical results with computational simulations using general model networks and anatomical brain networks, as well as high-density electroencephalography collected from humans in the conscious and anesthetized states. Analytical, computational, and empirical results demonstrate that network nodes with more connections (i.e., higher degrees) have larger amplitudes and are directional targets (phase lag) rather than sources (phase lead). The relationship of node degree and directionality therefore appears to be a fundamental property of networks, with direct applicability to brain function. These results provide a foundation for a principled understanding of information transfer across networks and also demonstrate that changes in directionality patterns across states of human consciousness are driven by alterations of brain network topology.

Model-based simulation and prediction of an antiviral strategy against influenza A infection.

There is a strong need to develop novel strategies in using antiviral agents to efficiently treat influenza infections. Thus, we constructed a rule-based mathematical model that reflects the complicated interactions of the host immunity and viral life cycle and analyzed the key controlling steps of influenza infections. The main characteristics of the pandemic and seasonal influenza strains were estimated using parameter values derived from cells infected with Influenza A/California/04/2009 and Influenza A/NewCaledonia/20/99, respectively. The quantitative dynamics of the infected host cells revealed a more aggressive progression of the pandemic strain than the seasonal strain. The perturbation of each parameter in the model was then tested for its effects on viral production. In both the seasonal and pandemic strains, the inhibition of the viral release (kC), the reinforcement of viral attachment (kV), and an increased transition rate of infected cells into activated cells (kI) exhibited significant suppression effects on the viral production; however, these inhibitory effects were only observed when the numerical perturbations were performed at the early stages of the infection. In contrast, combinatorial perturbations of both the inhibition of viral release and either the reinforcement of the activation of infected cells or the viral attachment exhibited a significant reduction in the viral production even at a later stage of infection. These results suggest that, in addition to blocking the viral release, a combination therapy that also enhances either the viral attachment or the transition of the infected cells might provide an alternative for effectively controlling progressed influenza infection.

Core-periphery disparity in fractal behavior of complex networks.

We show that there is a disparity in fractal scaling behavior of the core and peripheral parts of empirical small-world scale-free networks. We decompose the network into a core and a periphery and measure the fractal dimension of each part separately using the box-counting method. We find that the core of small-world scale-free networks has a nonfractal structure, whereas the periphery exhibits either fractal or nonfractal scaling. The fractal dimension of the periphery is found to coincide with one for the whole network.