Exponential sensitivity of noise-driven switching in genetic networks

Cells are known to utilize biochemical noise to probabilistically switch between distinct gene expression states. We demonstrate that such noise-driven switching is dominated by tails of probability distributions and is therefore exponentially sensitive to changes in physiological parameters such as transcription and translation rates. However, provided mRNA lifetimes are short, switching can still be accurately simulated using protein-only models of gene expression. Exponential sensitivity limits the robustness of noise-driven switching, suggesting cells may use other mechanisms in order to switch reliably.

The Construction of Arbitrary Stable Dynamics in Non-Linear Neural Networks

In this paper, two methods for constructing systems of ordinary differential equations realizing any fixed finite set of equilibria in any fixed finite dimension are introduced; no spurious equilibria are possible for either method. By using the first method, one can construct a system with the fewest number of equilibria, given a fixed set of attractors. Using a strict Lyapunov function for each of these differential equations, a large class of systems with the same set of equilibria is constructed. A method of fitting these nonlinear systems to trajectories is proposed. In addition, a general method which will produce an arbitrary number of periodic orbits of shapes of arbitrary complexity is also discussed. A more general second method is given to construct a differential equation which converges to a fixed given finite set of equilibria. This technique is much more general in that it allows this set of equilibria to have any of a large class of indices which are consistent with the Morse Inequalities. It is clear that this class is not universal, because there is a large class of additional vector fields with convergent dynamics which cannot be constructed by the above method. The easiest way to see this is to enumerate the set of Morse indices which can be obtained by the above method and compare this class with the class of Morse indices of arbitrary differential equations with convergent dynamics. The former set of indices are a proper subclass of the latter, therefore, the above construction cannot be universal. In general, it is a difficult open problem to construct a specific example of a differential equation with a given fixed set of equilibria, permissible Morse indices, and permissible connections between stable and unstable manifolds. A strict Lyapunov function is given for this second case as well. This strict Lyapunov function as above enables construction of a large class of examples consistent with these more complicated dynamics and indices. The determination of all the basins of attraction in the general case for these systems is also difficult and open.

Simulations of X and Y Retinal Ganglion Cell Behavior with a Nonlinear Push-pull Model of Spatiotemporal Retinal Processing

This article describes a nonlinear model of neural processing in the vertebrate retina, comprising model photoreceptors, model push-pull bipolar cells, and model ganglion cells. Previous analyses and simulations have shown that with a choice of parameters that mimics beta cells, the model exhibits X-like linear spatial summation (null response to contrast-reversed gratings) in spite of photoreceptor nonlinearities; on the other hand, a choice of parameters that mimics alpha cells leads to Y-like frequency doubling. This article extends the previous work by showing that the model can replicate qualitatively many of the original findings on X and Y cells with a fixed choice of parameters. The results generally support the hypothesis that X and Y cells can be seen as functional variants of a single neural circuit. The model also suggests that both depolarizing and hyperpolarizing bipolar cells converge onto both ON and OFF ganglion cell types. The push-pull connectivity enables ganglion cells to remain sensitive to deviations about the mean output level of nonlinear photoreceptors. These and other properties of the push-pull model are discussed in the general context of retinal processing of spatiotemporal luminance patterns.