Control in Inhibitory Genetic Regulatory Network Models

Abstract

The system of two the first order ordinary differential equations arising in the gene regulatory networks theory is studied. The structure of attractors for this system is described for three important behavioral cases: activation, inhibition, mixed activation-inhibition. The geometrical approach combined with the vector field analysis allows treating the problem in full generality. A number of propositions are stated and the proof is geometrical, avoiding complex analytic. Although not all the possible cases are considered, the instructions are given what to do in any particular situation.

On modelling of complex networks

“…We examine the behaviour of solutions and show that two of periodic solutions are stable limit cycles, which attract almost all the trajectories in G. We show what happens if the regulatory matrix changes its structure and the system (1) becomes coupled. Our study has connections with papers [9][10][11][12], where similar problems were studied. The novelty of the current paper is showing how periodic solutions can appear in 3D system and what are the conditions for their existence.…”

On a three-dimensional neural network model

“…For this, the analysis of the phase space is needed. Future states are heavily dependent on attractors of the system, [4], [5]. In this note we will show that the three-dimensional system of the form Eq.…”

Comparative Analysis of Models of Genetic and Neuronal Networks

“…The gene system (2.1) have appeared first in [19] (see also [12]). It was used in [4,7] and in more recent papers [1,2,3,11,13,14,15]. Periodic solutions were in a focus in [5,20].…”

“…Figure13. The projection of the attractor on 3D subspace on (x 2 , x 3 , x 4 ) of system (6.1) with perturbed regulatory matrix (7.2).…”