Persistence and boundedness in a two-species chemotaxis-competition system with singular sensitivity and indirect signal production.

This paper deals with a two-species chemotaxis-competition system involving singular sensitivity and indirect signal production: $ \begin{equation*} \begin{cases} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi_1\nabla\cdot(\frac{u}{z^{k}}\nabla z)+\mu_1 u(1-u-a_1v), &x\in\Omega,\ t>0,\\ v_{t} = \nabla\cdot(D(v)\nabla v)-\chi_2\nabla\cdot(\frac{v}{z^{k}}\nabla z)+\mu_2 v(1-v-a_2 u), &x\in\Omega,\ t>0,\\ w_{t} = \Delta w-w+u+v,&x\in\Omega,\ t>0,\\ z_{t} = \Delta z-z+w,&x\in\Omega,\ t>0,\\ \end{cases} \end{equation*} $ where $ \Omega\subset R^{n} $ is a convex smooth bounded domain with homogeneous Neumann boundary conditions. The diffusion functions $ D(u), D(v) $ are assumed to fulfill $ D(u)\geq(u+1)^{\theta_1} $ and $ D(v)\geq(v+1)^{\theta_2} $ with $ \theta_1, \theta_2 > 0 $, respectively. The parameters are $ k\in (0, \frac{1}{2})\cup (\frac{1}{2}, 1] $, $ \chi_ {i} > 0, (i = 1, 2) $.

After-effects of repetitive anodal transcranial direct current stimulation on learning and memory in a rat model of Alzheimer's disease.

Repetitive anodal transcranial direct current stimulation (tDCS) in a rat model of Alzheimer's disease (AD) has been shown to have distinct neuroprotective effects. Moreover, the effects of anodal tDCS not only occur during the stimulation but also persist after the stimulation has ended (after-effects). Here, the duration of the after-effects induced by repetitive anodal tDCS was investigated based on our previous studies.