In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others We also survey and discuss existing data sets that can be represented as multilayer networks We review attempts to generalize single-layer-network diagnostics to multilayer networks We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks We conclude with a summary and an outlook

Multilayer Networks

A formal theory of symmetries of networks of coupled dynamical
systems, stated in terms of the group of permutations of the nodes that preserve
the network topology, has existed for some time. Global network symmetries
impose strong constraints on the corresponding dynamical systems,
which affect equilibria, periodic states, heteroclinic cycles, and even chaotic
states. In particular, the symmetries of the network can lead to synchrony,
phase relations, resonances, and synchronous or cycling chaos.
Symmetry is a rather restrictive assumption, and a general theory of networks
should be more flexible. A recent generalization of the group-theoretic
notion of symmetry replaces global symmetries by bijections between certain
subsets of the directed edges of the network, the ‘input sets’. Now the symmetry
group becomes a groupoid, which is an algebraic structure that resembles
a group, except that the product of two elements may not be defined. The
groupoid formalism makes it possible to extend group-theoretic methods to
more general networks, and in particular it leads to a complete classification
of ‘robust’ patterns of synchrony in terms of the combinatorial structure of the
network.
Many phenomena that would be nongeneric in an arbitrary dynamical
system can become generic when constrained by a particular network topology.
A network of dynamical systems is not just a dynamical system with
a high-dimensional phase space. It is also equipped with a canonical set of
observables—the states of the individual nodes of the network. Moreover, the
form of the underlying ODE is constrained by the network topology—which
variables occur in which component equations, and how those equations relate
to each other. The result is a rich and new range of phenomena, only a few of
which are yet properly understood.

/pdf/nonlinear-dynamics-of-networks-the-groupoid-formalism-b30bad6eex.pdf

Nonlinear dynamics of networks: the groupoid formalism

Exposure to a variety of toxins and/or infectious agents leads to disease, degeneration and death, often characterised by circumstances in which cells or tissues do not merely die and cease to function but may be more or less entirely obliterated. It is then legitimate to ask the question as to whether, despite the many kinds of agent involved, there may be at least some unifying mechanisms of such cell death and destruction. I summarise the evidence that in a great many cases, one underlying mechanism, providing major stresses of this type, entails continuing and autocatalytic production (based on positive feedback mechanisms) of hydroxyl radicals via Fenton chemistry involving poorly liganded iron, leading to cell death via apoptosis (probably including via pathways induced by changes in the NF-κB system). While every pathway is in some sense connected to every other one, I highlight the literature evidence suggesting that the degenerative effects of many diseases and toxicological insults converge on iron dysregulation. This highlights specifically the role of iron metabolism, and the detailed speciation of iron, in chemical and other toxicology, and has significant implications for the use of iron chelating substances (probably in partnership with appropriate anti-oxidants) as nutritional or therapeutic agents in inhibiting both the progression of these mainly degenerative diseases and the sequelae of both chronic and acute toxin exposure. The complexity of biochemical networks, especially those involving autocatalytic behaviour and positive feedbacks, means that multiple interventions (e.g. of iron chelators plus antioxidants) are likely to prove most effective. A variety of systems biology approaches, that I summarise, can predict both the mechanisms involved in these cell death pathways and the optimal sites of action for nutritional or pharmacological interventions.

/pdf/towards-a-unifying-systems-biology-understanding-of-large-29x7kqlt9i.pdf

Towards a unifying, systems biology understanding of large-scale cellular death and destruction caused by poorly liganded iron: Parkinson’s, Huntington’s, Alzheimer’s, prions, bactericides, chemical toxicology and others as examples

The NF-κB transcription factor was discovered 30 years ago and has since emerged as the master regulator of inflammation and immune homeostasis. It achieves this status by means of the large number of important pro- and antiinflammatory factors under its transcriptional control. NF-κB has a central role in inflammatory diseases such as rheumatoid arthritis, inflammatory bowel disease, and autoimmunity, as well as diseases comprising a significant inflammatory component such as cancer and atherosclerosis. Here, we provide an overview of the studies that form the basis of our understanding of the role of NF-κB subunits and their regulators in controlling inflammation. We also describe the emerging importance of posttranslational modifications of NF-κB in the regulation of inflammation, and highlight the future challenges faced by researchers who aim to target NF-κB transcriptional activity for therapeutic benefit in treating chronic inflammatory diseases.

NF-κB and the Transcriptional Control of Inflammation.

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. The architecture
of a coupled cell network is a graph that indicates how cells are coupled and which cells are
equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that
permits a classification of robust synchrony in terms of network architecture. They also studied
the existence of other robust dynamical patterns using a concept of quotient network. There are
two difficulties with their approach. First, there are examples of networks with robust patterns of
synchrony that are not included in their class of networks; and second, vector fields on the quotient
do not in general lift to vector fields on the original network, thus complicating genericity arguments.
We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in
more than one way, and we show that this approach resolves both difficulties. The theory that we
develop, the “multiarrow formalism,” parallels that of Stewart, Golubitsky, and Pivato. In addition,
we prove that the pattern of synchrony generated by a hyperbolic equilibrium is rigid (the pattern
does not change under small admissible perturbations) if and only if the pattern corresponds to
a balanced equivalence relation. Finally, we use quotient networks to discuss Hopf bifurcation in
homogeneous cell systems with two-color balanced equivalence relations.

/pdf/patterns-of-synchrony-in-coupled-cell-networks-with-multiple-evtt1dbugq.pdf

Patterns of synchrony in coupled cell networks with multiple arrows

Sensitivity analysis of an oscillatory signal transduction pathway

Levelt's Propositions have been a touchstone for experimental and modeling studies of perceptual multistability. We asked whether Levelt's Propositions extend to perceptual multistability involving interocular grouping. To address this question we used split-grating stimuli with complementary halves of the same color. As in previous studies, subjects reported four percepts in alternation: the two stimuli presented to each eye (single-eye percepts), as well as two interocularly grouped, single color percepts (grouped percepts). Most subjects responded to increased color saturation by more frequently reporting a single color image, thus increasing the predominance of grouped percepts (Levelt's Proposition I). In these subjects increased predominance was due to a decrease in the average dominance duration of single-eye percepts, while that of grouped percepts remained largely unaffected. This is in accordance with generalized Levelt's Proposition II which posits that the average dominance duration of the stronger (in this case single-eye) percept is primarily affected by changes in stimulus strength. In accordance with Proposition III, the alternation rate increased as the difference in the strength of the percepts decreased. To explain the mechanism behind these observations, we introduce a hierarchical model consisting of low-level neural populations, each responding to input at a visual hemifield, and higher-level populations representing the percepts. The model exhibits the changes in dominance duration observed in the data, and conforms to all of Levelt's Propositions.

https://www.biorxiv.org/content/biorxiv/early/2016/06/05/057232.full.pdf

Extending Levelt's Propositions to perceptual multistability involving interocular grouping

Background:
Sustained stimulation with 
tumour necrosis factor α (TNFα) induces substantial
oscillations—observed at both the single cell and 
population levels—in the nuclear factor κB (NF-κB) 
system. Although the mechanism has not yet been elucidated fully, a 
core system has been identified consisting of a negative feedback 
loop involving NF-κB (RelA:p50 hetero-dimer) and its 
inhibitor IκBα. Many authors have suggested that 
this core oscillator should couple to other oscillatory pathways.

Results:
First we analyse single-cell data from
experiments in which the NF-κB system is forced by short trains
of strong pulses of TNFα. Power spectra of the ratio of
nuclear to cytoplasmic concentration of NF-κB suggest that the 
cells' responses are entrained by the pulsing 
frequency. Using a recent model of the NF-κB system due to Caroline Horton,
we carried out extensive numerical simulations to analyze the
response frequencies induced by trains of pulses of TNFα stimulation
having a wide range of frequencies and amplitudes.
These studies suggest that for sufficiently 
weak stimulation, various nonlinear resonances should be
observable. To explore further the possibility of probing alternative
feedback mechanisms, we also coupled the model to sinusoidal signals
signals with a wide range of strengths and frequencies. Our results 
show that, at least in simulation, frequencies 
other than those of the forcing and the main NF-κB oscillator
can be excited via sub- and superharmonic resonance, producing
quasiperiodic and even chaotic dynamics. 

Conclusions:
Our numerical results suggest that
the entrainment phenomena observed in pulse-stimulated
experiments is a consequence of the high intensity of the
stimulation. Computational studies based on current models suggest
that resonant interactions between periodoc pulsatile forcing 
and the system's natural frequencies may become evident for 
sufficiently weak stimulation.
Further simulations suggest that the nonlinearities of the
NF-κB feedback oscillator mean that even sinusoidally modulated forcing
can induce a rich variety of nonlinear interactions.

Interactions among oscillatory pathways in the NF-kB signalling.

Ambiguous visual images can generate dynamic and stochastic switches in perceptual interpretation known as perceptual rivalry. Such dynamics have primarily been studied in the context of rivalry between two percepts, but there is growing interest in the neural mechanisms that drive rivalry between more than two percepts. In recent experiments, we showed that split images presented to each eye lead to subjects perceiving four stochastically alternating percepts (Jacot-Guillarmod et al. Vision research, 133, 37–46, 2017): two single eye images and two interocularly grouped images. Here we propose a hierarchical neural network model that exhibits dynamics consistent with our experimental observations. The model consists of two levels, with the first representing monocular activity, and the second representing activity in higher visual areas. The model produces stochastically switching solutions, whose dependence on task parameters is consistent with four generalized Levelt Propositions, and with experiments. Moreover, dynamics restricted to invariant subspaces of the model demonstrate simpler forms of bistable rivalry. Thus, our hierarchical model generalizes past, validated models of binocular rivalry. This neuromechanistic model also allows us to probe the roles of interactions between populations at the network level. Generalized Levelt’s Propositions hold as long as feedback from the higher to lower visual areas is weak, and the adaptation and mutual inhibition at the higher level is not too strong. Our results suggest constraints on the architecture of the visual system and show that complex visual stimuli can be used in perceptual rivalry experiments to develop more detailed mechanistic models of perceptual processing.

/pdf/a-hierarchical-model-of-perceptual-multistability-involving-49th1ezena.pdf

A hierarchical model of perceptual multistability involving interocular grouping.

Statistical analysis of the connectivity of real world networks have revealed interesting featuressuch as community structure, network motif and as on. Such discoveries tempt us to understandthe dynamics of a complex network system by studying those of its subnetworks. This approach isfeasible only if the dynamics of the subnetwork systems can somehow be preserved or partially preservedin the whole system. Most works studied the connectivity structures of networks while very fewconsidered the possibility of translating the dynamics of a subnetwork system to the whole system. Inthis paper, we address this issue by focusing on considering the relations between cycles and fixedpoints of a network system and those of its subnetworks based on Boolean framework. We proved thatat a condition we called agreeable, if X0 is a fixed point of the whole system, then X 0 restricted to thephase-space of one of the subnetwork systems must be a fixed point as well. An equivalent statementon cycles follows from this result. In addition, we discussed the relations between the product of thetransition diagrams (a representation of trajectories) of subnetwork systems and the transition diagramof the whole system.

Yunjiao Wang

Papers

Sensitivity analysis of an oscillatory signal transduction pathway

Extending Levelt's Propositions to perceptual multistability involving interocular grouping

Interactions among oscillatory pathways in the NF-kB signalling.

A hierarchical model of perceptual multistability involving interocular grouping.

Relations between the dynamics of network systems and their subnetworks