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The synchronized behaviors of a noisy small-world neuronal network with delay and diversity is numerically studied by calculating a synchronization measure and plotting firing pattern. We show that delay in the information transmission can induce fruitful synchronization transitions, including transition from phase locking to antiphase synchronization, and transition from antiphase synchronization to complete synchronization. Furthermore, the delay-induced complete synchronization can be changed by diversity, which causes the oscillatory-like transition between antiphase synchronization and complete synchronization.

The dynamic scaling behaviors of the Family model and the Etching model on different fractal substrates are studied by means of Monte Carlo simulations, so as to discuss the microscopic mechanisms influencing the dynamic behavior of growth interfaces by changing the structure of the substrates. The Sierpinski arrowhead, crab lattice and dual Sierpinski gasket are employed as the substrates of the growth. These substrates have same fractal dimensions (d f ≈ 1.585), but with different morphologies. It is shown that the structure of the substrates can affect the dynamic scaling properties of the surfaces and interfaces. Although the standard Family-Vicsek scaling is still satisfied in describing the scaling behavior of the growth on fractal substrates, the original continuum equations are invalid. The dynamic behavior of the Family model satisfies the fractional Edwards-Wilkinson equation introduced by Lee and Kim, and the dynamic behavior of the Etching model implies that α + z > 2, which is different from the analytical results of the Kardar-Parisi-Zhang equation. The fractal character of the substrate also affects the lateral behavior of the Etching growth. Interestingly, the same fractal dimensions lead to different scaling exponents. The scaling exponents of the growth models on fractal substrates are determined by not only the fractal dimensions of the substrates, but also the spectral dimensions. Fortunately, it seems that the fractal dimension and the spectral dimension are sufficient to determine the scaling exponents of the growth model on fractal substrates.

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To analyze long-range temporal correlations in surface growth, we study numerically the (1 + 1)-dimensional Kardar-Parisi-Zhang (KPZ) equation driven by temporally correlated noise, and obtain the scaling exponents based on two dierent numerical methods. Our simulations show that the numerical results are in good agreement with the dynamic renormalization group (DRG) predictions, and are also consistent with the simulation results of the ballistic deposition (BD) model.