Neuronal morphology is an essential element for brain activity and function. We take advantage of current availability of brain-wide neuron digital reconstructions of the Pyramidal cells from a mouse brain, and analyze several emergent features of brain-wide neuronal morphology. We observe that axonal trees are self-affine while dendritic trees are self-similar. We also show that tree size appear to be random, independent of the number of dendrites within single neurons. Moreover, we consider inhomogeneous branching model which stochastically generates rooted 3-Cayley trees for the brain-wide neuron topology. Based on estimated order-dependent branching probability from actual axonal and dendritic trees, our inhomogeneous model quantitatively captures a number of topological features including size and shape of both axons and dendrites. This sheds lights on a universal mechanism behind the topological formation of brain-wide axonal and dendritic trees.

This work is devoted to the investigation of the most probable transition path for stochastic dynamical systems driven by either symmetric α-stable Lévy motion (0 < α < 1) or Brownian motion. For stochastic dynamical systems with Brownian motion, minimizing an action functional is a general method to determine the most probable transition path. We have developed a method based on path integrals to obtain the most probable transition path of stochastic dynamical systems with symmetric α-stable Lévy motion or Brownian motion, and the most probable path can be characterized by a deterministic dynamical system.

The minimal number of straight line segments required to construct a polygonal presentation of the knot K in the cubic lattice is called the lattice stick number of the knot K, denoted by . It is known that if the crossing number of K, , satisfies , and the main result of this paper is to improve this to if . Furthermore, we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers.

This work is devoted to the investigation of the most probable transition time between metastable states for stochastic dynamical systems with non-vanishing Brownian noise. Instead of minimizing the Onsager–Machlup action functional, we examine the maximum probability that the solution process of the system stays in a neighbourhood (or a tube) of a transition path, in order to characterize the most probable transition path. We first establish the exponential decay lower bound and a power law decay upper bound for the maximum of this probability. Based on these estimates, we further derive the lower and upper bounds for the most probable transition time, under suitable conditions. Finally, we illustrate our results in simple stochastic dynamical systems, and highlight the relation with some relevant works.

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