Pattern Formation over Multigraphs

Abstract

Two of the most common pattern formation mechanisms are Turing-patterning in reaction-diffusion systems and lateral inhibition of neighboring cells. In this paper, we introduce a broad dynamical model of interconnected modules to study the emergence of patterns, with the above mentioned two mechanisms as special cases. Our results do not restrict the number of modules or their complexity, allow multiple layers of communication channels with possibly different interconnection structure, and do not assume symmetric connections between two connected modules. Leveraging only the static input/output properties of the subsystems and the spectral properties of the interconnection matrices, we characterize the stability of the homogeneous fixed points as well as sufficient conditions for the emergence of spatially non-homogeneous patterns. To obtain these results, we rely on properties of the graphs together with tools from monotone systems theory. As application examples, we consider patterning in neural networks, in reaction-diffusion systems, and contagion processes over random graphs.

Modelling polarity-driven laminar patterns in bilayer tissues with mixed signalling mechanisms

“…We have previously analysed the role of polarity in laminar pattern formation using interconnected methods for a single juxtracrine signalling mechanism [34]. In this study, we generalise and extend these results to include multiple signalling mechanisms of any type using a multilayer graph approach as defined in [32]. Namely, we explore the interplay of multilayer network topology and edge weights in laminar pattern formation in bilayer tissues using dynamical systems of generic competitive kinetics.…”

“…These quotient systems were then used to provide pattern existence conditions for prescribed cellular patterns for interconnected juxtacrine models. These methods of pattern predictions were later extended in [31] and [32] to simultaneously couple diffusive and non-diffusive signalling mechanisms within the interconnected dynamical systems framework using directed multilayer graphs, namely graphs with unidirectional edges connected to cells with multiple-input and output signals. However, the influence of edge weights with on pattern existence and convergence in these multi-channel interconnected systems is yet to be investigated.…”

“…Provided the intracellular proteins regulated by the prescribed GRN react monotonically to intercellular stimuli, then global dynamics become predictable in a closed-loop of cells and facilitates the introduction of control theoretic tools for pattern stability [30]. Although, the restriction to bipartite connectivity graphs was imposed in [29] and [32] as a sufficient measure to preserve the monotonic behaviour of lateral-inhibition models in the large-scale forms, these restrictions limit the biological applications. Such conditions can be relaxed when seeking pattern existence in quotient systems as demonstrated in [30] but how such behaviour translates to the large-scale counterpart is not fully understood.…”

“…Initially, we present conditions for the existence, uniqueness and instability of a homogeneous steady state for large-scale multi-input-multi-output (MIMO) dynamical system which extends the conditions of [32] to yield analytically applicable statements for low-spatial order intracellular GRNs. Thereafter we use methods of multilayer graph partitioning to derive polarity conditions for the existence of laminar patterning in large-scale systems.…”

“…Thereafter we use methods of multilayer graph partitioning to derive polarity conditions for the existence of laminar patterning in large-scale systems. Critically, we demonstrate the graph commutativity requirements imposed in [32] for simultaneous diagonalisation can be relaxed when seeking patterns of only two states, allowing a broader range of quotient connectivities to be explored.…”