This note states several results on the exponential functionals of the Brownian motion and their approximations by Markov chains. Starting from M.Yor, such functionals were studied in mathematical finance. At the same time, they play a significant role in different settings: the analysis of diffusions on the class of solvable Lie groups, in particular on the group of (2 X 2) upper triangular matrices, with positive diagonal elements. The discrete random walks cannot properly describe the local structure of diffusion. However, instead of the usual local limit theorem (which is not applicable) its weaker form, namely quasi-local form is given.
According to the currently prevalent theory, hippocampal formation constructs and maintains cognitive spatial maps. Most of the experimental evidence for this theory comes from the studies on navigation in laboratory rats and mice, typically male animals. While these animals exhibit a rich repertoire of behaviors associated with navigation, including locomotion, head movements, whisking, sniffing, raring and scent marking, the contribution of these behavioral patterns to hippocampal activity has not been sufficiently studied. Instead, many publications have considered animal position in space as the single variable that affects the firing of hippocampal place cells and entorhinal grid cells. Here we argue that future work should focus on a more detailed examination of different behaviors exhibited during navigation in order to interpret the cause of spatial tuning in hippocampal neurons. As a step in this direction, we have analyzed data from two datasets, shared online, containing recordings from rats navigating in square and round arenas. Our analyses revealed structured, grid-like navigation patterns, evident from the spatial maps of animal position, velocity and acceleration. Moreover, grid cells available in the datasets exhibited the same spatial periodicity as the navigation parameters. These findings cast doubt on the cognitive-map interpretation of grid cells, since they suggest that neuronal spatial patterns could be caused by behaviors associated with navigation instead of representing a hierarchically high spatial map. Additionally, we speculate that scent marks left by navigating animals could contribute to neuronal responses while rats and mice sniff their environment.
We present a comparative study of several algorithms for an in-plane random walk with a variable step. The goal is to check the efficiency of the algorithm in case where the random walk terminates at some boundary. We recently found that a finite step of the random walk produces a bias in the hitting probability and this bias vanishes in the limit of an infinitesimal step. Therefore, it is important to know how a change in the step size of the random walk influences the performance of simulations. We propose an algorithm with the most effective procedure for the step-length-change protocol.
Nowadays the random search became a widespread and effective tool for solving different complex optimization and adaptation problems. In this work, the problem of an average duration of a random search for one object by another is regarded, depending on various factors on a square field. The problem solution was carried out by holding total experiment with 4 factors and orthogonal plan with 54 lines. Within each line, the initial conditions and the cellular automaton transition rules were simulated and the duration of the search for one object by another was measured. As a result, the regression model of average duration of a random search for an object depending on the four factors considered, specifying the initial positions of two objects, the conditions of their movement and detection is constructed. The most significant factors among the factors considered in the work that determine the average search time are determined. An interpretation is carried out in the problem of random search for an object from the constructed model.The important result of the work is that the qualitative and quantitative influence of initial positions of objects, the size of the lattice and the transition rules on the average duration of search is revealed by means of model obtained. It is shown that the initial neighborhood of objects on the lattice does not guarantee a quick search, if each of them moves. In addition, it is quantitatively estimated how many times the average time of searching for an object can increase or decrease with increasing the speed of the searching object by 1 unit, and also with increasing the field size by 1 unit, with different initial positions of the two objects. The exponential nature of the growth in the number of steps for searching for an object with an increase in the lattice size for other fixed factors is revealed. The conditions for the greatest increase in the average search duration are found: the maximum distance of objects in combination with the immobility of one of them when the field size is changed by 1 unit. (that is, for example, with 4x4 at 5x5) can increase the average search duration in e^1,69≈5,42. The task presented in the work may be relevant from the point of view of application both in the landmark for ensuring the security of the state, and, for example, in the theory of mass service.