A Bertrand Duopoly Game with Long-Memory Effects

Reconsidering the Bertrand duopoly game based on the concept of long short-term memory, we construct a fractional-order Bertrand duopoly game by extending the integer-order game to its corresponding fractional-order form. We build such a Bertrand duopoly game, in which both players can make their decisions with long-memory effects. Then, we investigate its Nash equilibria, local stability, and numerical solutions. Using the bifurcation diagram, the phase portrait, time series, and the 0-1 test for chaos, we numerically validate these results and illustrate its complex phenomena, such as bifurcation and chaos

Brain Pharmacokinetics and the Pharmacological Effects on Striatal Neurotransmitter Levels of Pueraria lobata Isoflavonoids in Rat

Isoflavonoids are putatively active components of Pueraria lobata and has been demonstrated prominent neuro-protection effect against cerebrovascular disorders, hypertension or Parkinson's disease (PD). However, the molecular basis for the beneficial effect of Pueraria lobata on nervous systems has not been well revealed. The present study aims to assess striatum exposure to main active isoflavonoids and changes of striatal extracellular neurotransmitters levels in rat brain after intravenous administration of Pueraria lobata isoflavonoids extracts (PLF), to further elucidate its' substantial bases for neuro activities. Fifteen rats were divided into 3 groups (five rats in each group) to receive a dose of PLF at 80 or 160 mg/kg or normal saline (vehicle), respectively. An LC-MS/MS method was employed to determine the concentrations of five main isoflavonoids and multiple neurotransmitters in microdialysate from striatal extracellular fluid (ECF) of the rats. The exposed quantities of puerarin (PU), 3′-methoxypuerarin (MPU), daidzein-8-C-apiosyl-(1-6)-glucoside (DAC), and 3′-hydroxypuerarin (HPU) in striatum were dose-dependent. The content of daidzein (DAZ) was too low to be detected in all dialysate samples through the experiment. Optimal dose PLF (80 mg/kg) promoted DA metabolism and inhibited 5-HT metabolism. No obvious change in the level of GLu was determined. The concentration of GABA presented a temporary decline firstly and then a gradual uptrend followed by a further downtrend. Higher dose (160 mg/kg) PLF could enhance the metabolism of both DA and 5-HT, and lower the extracellular level of GLu, without changing GABA concentrations, which might result in alleviation on excitatory toxicity under conditions, such as ischemia. The results infer that different dose of PLF should be chosen to achieve appropriate neurochemical modulation effects under conditions, such as hypertension or ischemia/stroke. These findings may significantly contribute to a better understanding of the neuroprotective effect of Pueraria lobata and provide new insights into its application toward neuro-degenerative diseases in the future