Finite-time synchronization of Markovian neural networks with proportional delays and discontinuous activations

In this paper, finite-time synchronization of neural networks (NNs) with discontinuous activation functions (DAFs), Markovian switching, and proportional delays is studied in the framework of Filippov solution. Since proportional delay is unbounded and different from infinite-time distributed delay and classical finite-time analytical techniques are not applicable anymore, new 1-norm analytical techniques are developed. Controllers with and without the sign function are designed to overcome the effects of the uncertainties induced by Filippov solutions and further synchronize the considered NNs in a finite time. By designing new Lyapunov functionals and using M-matrix method, sufficient conditions are derived to guarantee that the considered NNs realize synchronization in a settling time without introducing any free parameters. It is shown that, though the proportional delay can be unbounded, complete synchronization can still be realized, and the settling time can be explicitly estimated. Moreover, it is discovered that controllers with sign function can reduce the control gains, while controllers without the sign function can overcome chattering phenomenon. Finally, numerical simulations are given to show the effectiveness of theoretical results

Limit set dichotomy and convergence of semiflows defined by cooperative standard CNNs

The paper analyzes some fundamental properties of the solution semiflow of nonsymmetric cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment piecewise-linear (pwl) neuron activation. Two relevant subclasses of SCNNs, corresponding to one-dimensional circular SCNNs with two-sided or single-sided positive interconnections between nearest neighboring neurons only, are considered. For these subclasses it is shown that the associated solution semiflow satisfies the fundamental properties of the CONVERGENCE CRITERION, the NONORDERING OF LIMIT SETS and the LIMIT SET DICHOTOMY, and that this is true although the semiflow is not eventually strongly monotone. As a consequence such CNNs are almost convergent, i.e., almost all solutions converge toward an equilibrium point as time tends to infinity. To the authors' knowledge the paper is the first rigorous investigation on the geometry of limit sets and convergence properties of cooperative SCNNs with a pwl neuron activation. All available convergence results in the literature indeed concern a modified cooperative CNN model where the original pwl activation of the SCNN model is replaced by a continuously differentiable strictly increasing sigmoid function. The main results in the paper are established by conducting a deep analysis of the properties of the omega-limit sets of the solution semiflow defined by the considered subclasses of SCNNs. In doing so the paper exploits and extends some mathematical tools for monotone systems in order that they can be applied to pwl vector fields that govern the dynamics of SCNNs. By using some transformations and referring to specific examples it is also shown that the treatment in the paper can be extended to other subclasses of SCNNs

Modelling the role of polarity and geometry in cell-fate dynamics of mammary organoids

Mammary organoids are three-dimensional structures that are derived from mammary gland cells and can recapitulate the complex architecture and functionality of the mammary gland in vitro. Mammary organoids hold great promise for advancing our understanding of mammary gland biology, breast cancer, and precision medicine. However, phenotypic and genetic instabilities observed in long-term expansion limit their applications to prolonged experiments and large-scale production. A proposed factor driving this organoid-wise heterogeneity is plasticity within mammary epithelial cells, the phenotypic switching of cells. Therefore, we examine the dynamics of key intracellular pathways that govern cell-fate commitment in mammary organoids. Specifically, we explore the influence of local tissue geometry and polarity in cell-cell signalling in stabilising cell-fate determinants using a combination of analytic and numerical multiscale modelling approaches. We introduce interconnected dynamical systems, graph-coupled dynamical systems with input-output representations to describe intercellular signal flow between cells. Exploiting structural properties of the bilayer graphs describing mammary tissue architecture, we derive low-dimensional forms of these models enabling the analytic examination of the interplay of structure and polarity on cell-fate patterning, extending existing methods to include pathway crosstalk and providing rigorous links between low-dimensional and their associated large-scale representations. Supporting the analytic investigations of static spatial domains with cellbased modelling, we provide evidence that sufficiently strong cell-cell signal polarity has the capacity to generate and sustain bilayer laminar patterns of Notch1, a critical cell-fate determinant and inducer of plasticity in mammary epithelial cells. Furthermore, we demonstrate how local tissue curvature can relax the constraints of polarity supporting healthy tissue growth and supporting branching morphologies. Fundamentally, this study highlights the significance of cell signalling polarity as a control mechanism of cell-fate commitment. Thus, the establishment and maintenance of epithelial polarity should be considered in long-term mammary organoid expansion protocol development

A note on the dichotomy of limit sets for cooperative CNNs with delays

The paper considers a class of delayed standard (S) cellular neural networks (CNNs) with non-negative interconnections between distinct neurons and a typical three-segment pwl neuron activation. It is also assumed that such cooperative SCNNs satisfy an irreducibility condition on the interconnection and delayed interconnection matrix. By means of a counterexample it is shown that the solution semiflow associated to such SCNNs in the general case does not satisfy the fundamental property of the omega-limit set dichotomy and is not eventually strongly monotone. The consequences of this result are discussed in the context of the existing methods for addressing convergence of monotone semiflows defined by delayed cooperative dynamical systems