Chaos in Vallis' asymmetric Lorenz model for El Nino

AbstractWe consider Vallis’ symmetric and asymmetric Lorenz models for El Niño—systems of autonomous ordinary differential equations in 3D—with the usual parameters and, in both cases, by using rigorous numerics, we locate topological horseshoes in iterates of Poincaré return maps. The computer-assisted proofs follow the standard Mischaikow–Mrozek–Zgliczynski approach. The novelty is a dimension reduction method, a direct exploitation of numerical Lorenz-like maps associated to the two components of the Poincaré section

Lyapunov analysis of the spatially discrete-continuous system dynamics

The spatially discrete-continuous dynamical systems, that are composed of a spatially extended medium coupled with a set of lumped elements, are frequently met in different fields, ranging from electronics to multicellular structures in living systems. Due to the natural heterogeneity of such systems, the calculation of Lyapunov exponents for them appears to be a challenging task, since the conventional techniques in this case often become unreliable and inaccurate. The paper suggests an effective approach to calculate Lyapunov exponents for discrete-continuous dynamical systems, which we test in stability analysis of two representative models from different fields. Namely, we consider a mathematical model of a 1D transferred electron device coupled with a lumped resonant circuit, and a phenomenological neuronal model of spreading depolarization, which involves 2D diffusive medium. We demonstrate that the method proposed is able reliably recognize regular, chaotic and hyperchaotic dynamics in the systems under study

Symmetry in Modeling and Analysis of Dynamic Systems

Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries

Systems evolution: the conceptual framework and a formal model

This research addresses to some of the fundamental problems in systems science.T he aim of this study is to: (1) provide a general conceptual framework for systems evolution; (2) develop a formal model for evolving systems based on dynamical systems theory; (3) analyse the evolving behaviour of various systems by using the formal model so far developed. First of all, it is argued that a system, which can be recognized by an observer as a system, is characterised by some emergent properties at a certain level of discourse. These properties are the results of the interactions between the system as components but not reducible to the individual or summative properties of those components. Any system is such an emergent and organized whole, and this whole can be defined and described as an emergent attractor. To maintain the wholeness in a changing environment, an open system may undergo radical changes both in its structure and function. The process of change is what is called of systems evolution. On reviewing the existing theories of self-organization, such as "Theory of Dissipative Structure", "Synergetics", "Hypercycle", "Cellular Automata", "Random Boolean Network" et al., a general conceptual framework for systems evolution has been outlined and it is based on the concept of emergent attractor for open systems. The emphasis is placed on the structural aspect of the process of change. Modem mathematical dynamical systems theory, with the study of nonlinear dynamics as its core, can provide (a) the concept of "attractor" to describe a system as an organized whole; (b) simple geometrical models of complex behaviour, (c) a complete taxonomy of attractors and bifurcation patterns; (d) a mathematical rationale for the explanations of evolutionary processes. Based on this belief, a formal model of evolving systems has been developed by using the language of mathematical dynamical systems theory (DST). Attractors and emergent attractors are formally defined. It is argued that the state of any systems can be described by one of the four fundamental types of attractors ( i. e. point attractor, periodic attractor, quasiperiodic attractor, chaotic attractor) at a certain level. The evolving behaviour of open systems can be analyzed by looking at the loss of structural stability in the systems. For a full analysis of systems evolution, the emphasis is put on the nonlinear inner dynamics which governs evolving systems. In trying to apply this conceptual framework and formal model, the evolving behaviour of various systems at different levels have been discussed. Among them are Benard cells in hydrodynamics, Brusselator in chemical systems, replicator systems in biology (hypercycle), predator-prey-food systems in ecology, and artificial neural networks. The complex dynamical behaviour of these systems, like the existence of various types of attractors and the occurrences of bifurcation when the environment changes, have been discussed. In most of the examples, the results in previous studies are cited directly and they are only re-interpreted by using the conceptual framework and the formal model developed in this research. In the study of artificial neural networks, a simple cellular automata network with only three neurons has been constructed and the activation dynamics has been analysed according to the formal model. Different attractors representing different dynamical behaviour of this network have been identified (point, periodic, quasiperiodic, and chaotic attractor). Similar discussions have been applied to a coupled Wilson-Cowan net. It is believed that the study of systems evolution is one of those attempts to bring systems science out of its primitive stage in which it ought not to be

Dynamics of excitable cells: spike-adding phenomena in action

We study the dynamics of action potentials of some electrically excitable cells: neurons and cardiac muscle cells. Bursting, following a fast–slow dynamics, is the most characteristic behavior of these dynamical systems, and the number of spikes may change due to spike-adding phenomenon. Using analytical and numerical methods we give, by focusing on the paradigmatic 3D Hindmarsh–Rose neuron model, a review of recent results on the global organization of the parameter space of neuron models with bursting regions occurring between saddle-node and homoclinic bifurcations (fold/hom bursting). We provide a generic overview of the different bursting regimes that appear in the parametric phase space of the model and the bifurcations among them. These techniques are applied in two realistic frameworks: insect movement gait changes and the appearance of Early Afterdepolarizations in cardiac dynamics

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

Lineages and molecular heterogeneity in the developing nervous system

Information in the genome unfolds through a dynamic process leading to the molecular and anatomical organization of a physiologically functional organism. The nervous system is the most diverse and intricate architecture generated by this process. It is composed of hundreds of millions of cells of hundreds of different cell types, whose connectivity and interactions are the physiological underpinnings of our capacity to respond to stimuli, our ability to learn and our cognitive capabilities. In this thesis, I explore the formation of tissues in the nervous system during embryonic development. In particular, I focus on changes in molecular composition that lead progenitor cells to generate a complex mix of cell types. The specific aim of this work is to address the lack of complete and systematic knowledge of the heterogeneity of neural tissues and to describe the progression of a cell through different molecular states. To achieve this, I took advantage of the new opportunities offered by single-cell expression profiling technologies to gain a holistic view of a developing tissue. To contextualize the work, I review the relevant literature and conceptual framework. Starting with a historical perspective, I discuss the concept of cell type and how it relates to developmental dynamics and evolution. I then review different aspects of developmental neuroscience, starting with general principles and then focusing on the main areas of interest: the ventral midbrain, the sympathetic nervous system, and postnatal development. Then the technological advances instrumental for this thesis are reviewed, with a focus on analysis methods for single-cell RNA sequencing. Finally, I discuss the relationship between lineages and gene regulation, and I introduce the reader to the idea of a global time derivative of gene expression through traditional systems biology modeling. Then I present the results of three different studies. In paper I, we used single-cell RNA sequencing to describe the cell-type heterogeneity of sympathetic ganglia. We found seven distinct kinds of neurons, where only two had been previously described. Using lineage tracing, we shed light on the developmental origin of the new types. We linked their molecular profile to function and described how they innervate the erector muscles. Paper II describes the embryonic development of the ventral midbrain at the single-cell level. We characterized human and mouse embryonic tissues, identifying cell types and their homologies. We found an uncharacterized heterogeneity among radial glial cells and gained new insight into the timing of dopaminergic neurons specification. Finally, we presented a data-driven strategy to assess the quality of in vitro differentiation protocols. In paper III we addressed the major limitation of studying development with single-cell RNA sequencing: the absence of a temporal dimension. We described an analysis framework that uses the ratio of spliced to unspliced RNA abundance to estimate the time derivative of gene expression. The method was used to predict the future molecular states of cells and to determine their fate bias. In these studies, we produced a rich description of tissue heterogeneity and answered different biological questions. The results were achieved by harnessing the information contained in the data through analysis approaches inspired by developmental or physical principles. In summary, this thesis provides new insight into several aspects of mammalian nervous-system development, and it presents analytical approaches that I predict will inspire future investigation of the developmental dynamics of single-cells

Principles for Making Half-center Oscillators and Rules for Torus Bifurcation in Neuron Models

In this modelling work, we adopted geometric slow-fast dissection and parameter continuation approach to study the following three topics: 1. Principles for making the half-center oscillator, a ubiquitous building block for many rhythm-generating neural networks. 2. Causes of a novel electrical behavior of neurons, amplitude modulation, from the view of dynamical systems; 3. Explanation and predictions for two common types of chaotic dynamics in single neuron model. To make our work as general as possible, we used and built both exemplary biologically plausible Hodgkin-Huxley type neuron models and reduced phenomenological neuron models