A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta–Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.

Regulation of retinal ganglion cell production by Sonic hedgehog

Previous work has shown that production of retinal ganglion cells is in part regulated by inhibitory factors secreted by ganglion cell themselves; however, the identities of these molecules are not known. Recent studies have demonstrated that the signaling molecule Sonic hedgehog (Shh) secreted by differentiated retinal ganglion cells is required to promote the progression of ganglion cell differentiation wave front and to induce its own expression. We present evidence that Shh signals play a role to negatively regulate ganglion cell genesis behind the differentiation wave front. Higher levels of Shh expression are detected behind the wave front as ganglion cells accumulate, while the Patched 1 receptor of Shh is expressed in adjacent retinal progenitor cells. Retroviral-mediated overexpression of Shh results in reduced ganglion cell proportions in vivo and in vitro. Conversely, inhibiting endogenous Shh activity by anti-Shh antibodies leads to an increased production of ganglion cells. Shh signals modulate ganglion cell production within the normal period of ganglion cell genesis in vitro without significantly affecting cell proliferation or cell death. Moreover, Shh signaling affects progenitor cell specification towards the ganglion cell fate during or soon after their last mitotic cycle. Thus, Shh derived from differentiated ganglion cells serves as a negative regulator behind the differentiation wave front to control ganglion cell genesis from the competent progenitor pool. Based on these results and other recent findings, we propose that Shh signals secreted by early-differentiated retinal neurons play dual roles at distinct concentration thresholds to orchestrate the progression of retinal neurogenic wave and the emergence of new neurons.