Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph

We consider a class of non-linear dynamics on a graph that contains and generalizes various models from network systems and control and study convergence to uniform agreement states using gradient methods. In particular, under the assumption of detailed balance, we provide a method to formulate the governing ODE system in gradient descent form of sum-separable energy functions, which thus represent a class of Lyapunov functions; this class coincides with Csisz\'{a}r's information divergences. Our approach bases on a transformation of the original problem to a mass-preserving transport problem and it reflects a little-noticed general structure result for passive network synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed gradient formulation extends known gradient results in dynamical systems obtained recently by M. Erbar and J. Maas in the context of porous medium equations. Furthermore, we exhibit a novel relationship between inhomogeneous Markov chains and passive non-linear circuits through gradient systems, and show that passivity of resistor elements is equivalent to strict convexity of sum-separable stored energy. Eventually, we discuss our results at the intersection of Markov chains and network systems under sinusoidal coupling

The Kinetic Basis of Morphogenesis

It has been shown recently (Shalygo, 2014) that stationary and dynamic patterns can arise in the proposed one-component model of the analog (continuous state) kinetic automaton, or kinon for short, defined as a reflexive dynamical system with active transport. This paper presents extensions of the model, which increase further its complexity and tunability, and shows that the extended kinon model can produce spatio-temporal patterns pertaining not only to pattern formation but also to morphogenesis in real physical and biological systems. The possible applicability of the model to morphogenetic engineering and swarm robotics is also discussed.Comment: 8 pages. Submitted to the 13th European Conference on Artificial Life (ECAL-2015) on March 10, 2015. Accepted on April 28, 201

Towards Physarum Binary Adders

Plasmodium of \emph{Physarum polycephalum} is a single cell visible by unaided eye. The plasmodium's foraging behaviour is interpreted in terms of computation. Input data is a configuration of nutrients, result of computation is a network of plasmodium's cytoplasmic tubes spanning sources of nutrients. Tsuda et al (2004) experimentally demonstrated that basic logical gates can be implemented in foraging behaviour of the plasmodium. We simplify the original designs of the gates and show --- in computer models --- that the plasmodium is capable for computation of two-input two-output gate $\to$ and three-input two-output $\to$. We assemble the gates in a binary one-bit adder and demonstrate validity of the design using computer simulation.Comment: Biosystems (2010), in press. Please download final version of the paper from the Publishers's sit

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Information theoretical study of cross-talk mediated signal transduction in MAPK pathways

Biochemical networks related to similar functional pathways are often correlated due to cross-talk among the homologous proteins in the different networks. Using a stochastic framework, we address the functional significance of the cross-talk between two pathways. Our theoretical analysis on generic MAPK pathways reveals cross-talk is responsible for developing coordinated fluctuations between the pathways. The extent of correlation evaluated in terms of the information theoretic measure provides directionality to net information propagation. Stochastic time series and scattered plot suggest that the cross-talk generates synchronization within a cell as well as in a cellular population. Depending on the number of input and output, we identify signal integration and signal bifurcation motif that arise due to inter-pathway connectivity in the composite network. Analysis using partial information decomposition quantifies the net synergy in the information propagation through these branched pathways.Comment: Revised version, 17 pages, 5 figure

Excitable Media Seminar

The simulation data presented here, and the conceptual framework developed for their interpretation are, both, in need of substantial refinement and extension. However, granting that they are initial pointers of some merit, and elementary indicators of general principles, several implications follow: the activity patterns of neurons and their assemblies are\ud interdependent with the extracellular milieu in which they are embedded, and to whose time varying composition they contribute. The complexity of this interdependence in the temporal dimension forecloses any time and context invariant relation between what the experimenter may consider stimulus input and its representation in neural activity. Hence, ideas of coding by (quasi)-digital neurons are called in question by the mutual interdependence of neurons and their\ud humoral milieu. Instead, concepts of 'mass action' in the Nervous system gain a new perspective: this time augmented by including the chemical medium surrounding neurons as part of the dynamics of the system as a whole. Accordingly, a meaningful way to describe activity in a neuron assembly would be in terms of a state space in which it can move along an infinite number of trajectories.\u