Synchronous behavior in networks of coupled systems : with applications to neuronal dynamics

Synchronization in networks of interacting dynamical systems is an interesting phenomenon that arises in nature, science and engineering. Examples include the simultaneous flashing of thousands of fireflies, the synchronous firing of action potentials by groups of neurons, cooperative behavior of robots and synchronization of chaotic systems with applications to secure communication. How is it possible that systems in a network synchronize? A key ingredient is that the systems in the network "communicate" information about their state to the systems they are connected to. This exchange of information ultimately results in synchronization of the systems in the network. The question is how the systems in the network should be connected and respond to the received information to achieve synchronization. In other words, which network structures and what kind of coupling functions lead to synchronization of the systems? In addition, since the exchange of information is likely to take some time, can systems in networks show synchronous behavior in presence of time-delays? The first part of this thesis focusses on synchronization of identical systems that interact via diffusive coupling, that is a coupling defined through the weighted difference of the output signals of the systems. The coupling might contain timedelays. In particular, two types of diffusive time-delay coupling are considered: coupling type I is diffusive coupling in which only the transmitted signals contain a time-delay, and coupling type II is diffusive coupling in which every signal is timedelayed. It is proven that networks of diffusive time-delay coupled systems that satisfy a strict semipassivity property have solutions that are ultimately bounded. This means that the solutions of the interconnected systems always enter some compact set in finite time and remain in that set ever after. Moreover, it is proven that nonlinear minimum-phase strictly semipassive systems that interact via diffusive coupling always synchronize provided the interaction is sufficiently strong. If the coupling functions contain time-delays, then these systems synchronize if, in addition to the sufficiently strong interaction, the product of the time-delay and the coupling strength is sufficiently small. Next, the specific role of the topology of the network in relation to synchronization is discussed. First, using symmetries in the network, linear invariant manifolds for networks of the diffusively time-delayed coupled systems are identified. If such a linear invariant manifold is also attracting, then the network possibly shows partial synchronization. Partial synchronization is the phenomenon that some, at least two, systems in the network synchronize with each other but not with every system in the network. It is proven that a linear invariant manifold defined by a symmetry in a network of strictly semipassive systems is attracting if the coupling strength is sufficiently large and the product of the coupling strength and the time-delay is sufficiently small. The network shows partial synchronization if the values of the coupling strength and time-delay for which this manifold is attracting differ from those for which all systems in the network synchronize. Next, for systems that interact via symmetric coupling type II, it is shown that the values of the coupling strength and time-delay for which any network synchronizes can be determined from the structure of that network and the values of the coupling strength and time-delay for which two systems synchronize. In the second part of the thesis the theory presented in the first part is used to explain synchronization in networks of neurons that interact via electrical synapses. In particular, it is proven that four important models for neuronal activity, namely the Hodgkin-Huxley model, the Morris-Lecar model, the Hindmarsh-Rose model and the FitzHugh-Nagumo model, all have the semipassivity property. Since electrical synapses can be modeled by diffusive coupling, and all these neuronal models are nonlinear minimum-phase, synchronization in networks of these neurons happens if the interaction is sufficiently strong and the product of the time-delay and the coupling strength is sufficiently small. Numerical simulations with various networks of Hindmarsh-Rose neurons support this result. In addition to the results of numerical simulations, synchronization and partial synchronization is witnessed in an experimental setup with type II coupled electronic realizations of Hindmarsh-Rose neurons. These experimental results can be fully explained by the theoretical findings that are presented in the first part of the thesis. The thesis continues with a study of a network of pancreatic -cells. There is evidence that these beta-cells are diffusively coupled and that the synchronous bursting activity of the network is related to the secretion of insulin. However, if the network consists of active (oscillatory) beta-cells and inactive (dead) beta-cells, it might happen that, due to the interaction between the active and inactive cells, the activity of the network dies out which results in a inhibition of the insulin secretion. This problem is related to Diabetes Mellitus type 1. Whether the activity dies out or not depends on the number of cells that are active relative to the number of inactive cells. A bifurcation analysis gives estimates of the number of active cells relative to the number of inactive cells for which the network remains active. At last the controlled synchronization problem for all-to-all coupled strictly semipassive systems is considered. In particular, a systematic design procedure is presented which gives (nonlinear) coupling functions that guarantee synchronization of the systems. The coupling functions have the form of a definite integral of a scalar weight function on a interval defined by the outputs of the systems. The advantage of these coupling functions over linear diffusive coupling is that they provide high gain only when necessary, i.e. at those parts of the state space of the network where nonlinearities need to be suppressed. Numerical simulations in networks of Hindmarsh-Rose neurons support the theoretical results

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. [...

Dynamics of Neural Systems: From Intracellular Transport in Neurons to Network Activity

Neurodegenerative diseases such as Alzheimer’s disease (AD) are all results of neurons losing their normal functionality. However, the exact mechanics of neurodegeneration remains obscure. Most of the knowledge about this class of diseases is obtained by studying late stage patients. Therefore, the mechanism proceeding the late stages of such diseases are less understood. Better understanding of respective mechanisms can help developing in early diagnostic tools and techniques to enable more effective treatment methods. Analyzing the dynamics of neural systems can be the key to discover the underlying mechanisms, which lead to neurodegenerative diseases. The dynamics of neural systems can be studied in different scales. At subcellular level, dynamics of axonal transport plays an important role in AD. In particular, anterograde axonal transport conducted by kinesin-1, known conventionally as kinesin, is essential for maintaining functional synapses. The stochastic motion of kinesin in the presence of magnetic nanoparticles is studied. A novel reduced-order-model (ROM) is constructed to simulate the collective dynamics of magnetic nanoparticles that are delivered into cells. The ROM coupled with the kinesin model allows the quantification of the decrease in processivity of kinesin and in its average velocity under external loads caused by chains of magnetic nanoparticles. Changes in the properties of transport induced by perturbations have the potential to decipher normal transport from impaired transport in the state of disease. In single-cell level analysis, Ca2+ transients in ASH neuron of C. elegans model organism is studied in the context of biological conditions such as aging and oxidative stress. A novel mathematical model is established that can describe the unique Ca2+ transients of ASH neuron in C. elegans including its “on” and “off” response. The model provides insight into the mechanism that governs the observed Ca2+ dynamics in ASH neuron. Hence, the proposed mathematical model can be utilized as a tool that offers explanation for changes induced by aging or oxidative stress in the neuron based on the observed Ca2+ dynamics. Network level analysis of neurons does not require methods of extremely high spatial and temporal resolution compared to the analysis in subcellular and cellular level. Yet, malfunction in smaller scales can manifest themselves in dynamics of larger scales. In particular, impairment of synaptic connections and their dynamics can jeopardize the normal functionality of the brain in pathological conditions such as AD. The impact of synaptic deficiencies is investigated on robustness of persistence activity (essential for working memory, which is adversely affected by AD) in excitatory networks with different topologies. Networks with rich-clubs are shown to have higher robustness when their synapses are impaired. Hence, monitoring changes in the properties of the neural network can be utilized as a tool to detect defects in synaptic connections. Moreover, such defects are shown to be more devastating if they occur in synapses of highly active neurons. Impairments of synapses in highly active neurons can be directly linked to subcellular processes such as depletion of synaptic resources. Using stochastic firing rate models, the parameters that govern synaptic dynamics are shown to influence the capability of the model to possess memory. The decrease in the release probability of synaptic vesicles, which can be caused by loss of axonal transport, is shown to have a detrimental effect on memory represented by the firing rate of population models.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145921/1/mirzakh_1.pd