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This paper aims to investigate the stochastic response of the van der Pol (VDP) oscillator with two kinds of fractional derivatives under Gaussian white noise excitation. First, the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique. Then, the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution. Finally, the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator. The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order, the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator. An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary.

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(E)-2-(1-hydroxyl-4-oxocyclohexyl) ethyl caffeine (HOEC), a naturally caffeic ester isolated from Incarvillea mairei, has been reported to possess anti-inflammatory activity by targeting 5-lipoxygenase. However, its other potential activities have yet to be explored. In this study, we measured antioxidant activity of HOEC using the DPPH free radical-scavenging assay. Then, we exposed rat pheochromocytoma (PC12) cells to hydrogen peroxide (H2O2)-induced damage and investigated the antioxidant activity of HOEC. Cell viability, lactate dehydrogenase (LDH) release, cellular morphology, Hoechst 33342 fluorescent staining, and apoptosis of the PC12 cells were assessed after treatment with 0.3-10 μM HOEC for 2 h and exposure to 600 μM H2O2. Additionally, glutathione reductase (GR), superoxide dismutase (SOD), lipid peroxidation malondialdehyde (MDA), and intracellular reactive oxygen species (ROS) accumulation were assayed after the PC12 cells were exposed to H2O2. To investigate mechanism, apoptosis-related protein were evaluated, including cleaved caspase 3/7, cleaved PARP, Bcl-2, Bcl-XL, and cytochrome c. The results showed that HOEC possessed potent antioxidant activity and pre-treatment with HOEC prior to H2O2 exposure significantly increased cell viability, reduced the release of LDH, ameliorated changes in cell morphology, and inhibited apoptosis. Further, HOEC did the following: reduced intracellular accumulation of ROS and MDA; rescued loss of SOD and GR activities; inhibited activated caspase-3 and caspase-7, cleaved PARP, and cytochrome c release; up-regulated the antiapoptosis-related protein Bcl-2 and Bcl-XL; and down-regulated the apoptosis-related proteins Bax and Bad. These findings suggested that HOEC may be a therapeutic agent for treating oxidative stress-derived neurodegenerative disorders.