On trajectories of a system modeling evolution of genetic networks

Abstract

<abstract> <p>A system of ordinary differential equations is considered, which arises in the modeling of genetic networks and artificial neural networks. Any point in phase space corresponds to a state of a network. Trajectories, which start at some initial point, represent future states. Any trajectory tends to an attractor, which can be a stable equilibrium, limit cycle or something else. It is of practical importance to answer the question of whether a trajectory exists which connects two points, or two regions of phase space. Some classical results in the theory of boundary value problems can provide an answer. Some problems cannot be answered and require the elaboration of new approaches. We consider both the classical approach and specific tasks which are related to the features of the system and the modeling object.</p> </abstract>

Comparative Analysis of Models of Gene and Neural Networks

“…Positivity of w ij means the activation of x i by x j , negativity means inhibition, and zero value is for no relation. System (1) was studied, in particular, in the paper [14].…”

“…The graphs of x i (t), i = 1, 2, 3 of the system (1) with the regulatory matrix (4) are shown in Figure 7. Similar systems were considered in papers [14], [19] and [20].…”

A New 3D Chaotic Attractor in Gene Regulatory Network

“…While various sigmoidal functions exist, the one mentioned above is particularly suitable for analysis and visualization [32]. Such systems were considered in [20,[33][34][35].…”

On attractors in systems of ordinary differential equations arising in models of genetic networks

“…Limit cycles can emerge from stable focuses as a result of Andronov -Hopf bifurcation. Attractors were constructed in systems with dimensions higher than three [6], [7]. These attractors are not solutions themselves, but they attract periodic solutions of a system.…”