Synchronization in Networks of Hindmarsh-Rose Neurons

Synchronization is deemed to play an important role in information processing in many neuronal systems. In this work, using a well known technique due to Pecora and Carroll, we investigate the existence of a synchronous state and the bifurcation diagram of a network of synaptically coupled neurons described by the Hindmarsh-Rose model. Through the analysis of the bifurcation diagram, the different dynamics of the possible synchronous states are evidenced. Furthermore, the influence of the topology on the synchronization properties of the network is shown through an exampl

Weak connections form an infinite number of patterns in the brain

This work is supporting in part by NSFC (61172070), Innovation Research Team of Shaanxi Province (2013KCT-04), Key Program of Nature science Foundation of Shaanxi Province (20162DJC-01) and EPSRC (EP/I032606/1).Peer reviewedPublisher PD

Neuron models of the generic bifurcation type:network analysis and data modeling

Minimal nonlinear dynamic neuron models of the generic bifurcation type may provide the middle way between the detailed models favored by experimentalists and the simplified threshold and rate model of computational neuroscientists. This thesis investigates to which extent generic bifurcation type models grasp the essential dynamical features that may turn out play a role in cooperative neural behavior. The thesis considers two neuron models, of increasing complexity, and one model of synaptic interactions. The FitzHugh-Nagumo model is a simple two-dimensional model capable only of spiking behavior, and the Hindmarsh-Rose model is a three-dimensional model capable of more complex dynamics such as bursting and chaos. The model for synaptic interactions is a memory-less nonlinear function, known as fast threshold modulation (FTM). By means of a combination of nonlinear system theory and bifurcation analysis the dynamical features of the two models are extracted. The most important feature of the FitzHugh-Nagumo model is its dynamic threshold: the spike threshold does not only depend on the absolute value, but also on the amplitude of changes in the membrane potential. Part of the very complex, intriguing bifurcation structure of the Hindmarsh-Rose model is revealed. By considering basic networks of FTM-coupled FitzHugh-Nagumo (spiking) or Hindmarsh-Rose (bursting) neurons, two main cooperative phenomena, synchronization and coincidence detections, are addressed. In both cases it is illustrated that pulse coupling in combination with the intrinsic dynamics of the models provides robustness. In large scale networks of FTM-coupled bursting neurons, the stability of complete synchrony is independent from the network topology and depends only on the number of inputs to each neuron. The analytical results are obtained under very restrictive and biologically implausible hypotheses, but simulations show that the theoretical predictions hold in more realistic cases as well. Finally, the realism of the models is put to a test by identification of their parameters from in vitro measurements. The identification problem is addressed by resorting to standard techniques combined with heuristics based on the results of the reported mathematical analysis and on a priori knowledge from neuroscience. The FitzHugh-Nagumo model is only able to model pyramidal neurons and even then performs worse than simple threshold models; it should be used only when the advantages of the more realistic threshold mechanism are prevalent. The Hindmarsh-Rose model can model much of the diversity of neocortical neurons; it can be used as a model in the study of heterogeneous networks and as a realistic model of a pyramidal neuron

Synchronization in dynamical networks:synchronizability, neural network models and EEG analysis

Complex dynamical networks are ubiquitous in many fields of science from engineering to biology, physics, and sociology. Collective behavior, and in particular synchronization,) is one of the most interesting consequences of interaction of dynamical systems over complex networks. In this thesis we study some aspects of synchronization in dynamical networks. The first section of the study discuses the problem of synchronizability in dynamical networks. Although synchronizability, i.e. the ease by which interacting dynamical systems can synchronize their activity, has been frequently used in research studies, there is no single interpretation for that. Here we give some possible interpretations of synchronizability and investigate to what extent they coincide. We show that in unweighted dynamical networks different interpretations of synchronizability do not lie in the same line, in general. However, in networks with high degrees of synchronization properties, the networks with properly assigned weights for the links or the ones with well-performed link rewirings, the different interpretations of synchronizability go hand in hand. We also show that networks with nonidentical diffusive connections whose weights are assigned using the connection-graph-stability method are better synchronizable compared to networks with identical diffusive couplings. Furthermore, we give an algorithm based on node and edge betweenness centrality measures to enhance the synchronizability of dynamical networks. The algorithm is tested on some artificially constructed dynamical networks as well as on some real-world networks from different disciplines. In the second section we study the synchronization phenomenon in networks of Hindmarsh-Rose neurons. First, the complete synchronization of Hindmarsh-Rose neurons over Newman-Watts networks is investigated. By numerically solving the differential equations of the dynamical network as well as using the master-stability-function method we determine the synchronizing coupling strength for diffusively coupled Hindmarsh-Rose neurons. We also consider clustered networks with dense intra-cluster connections and sparse inter-cluster links. In such networks, the synchronizability is more influenced by the inter-cluster links than intra-cluster connections. We also consider the case where the neurons are coupled through both electrical and chemical connections and obtain the synchronizing coupling strength using numerical calculations. We investigate the behavior of interacting locally synchronized gamma oscillations. We construct a network of minimal number of neurons producing synchronized gamma oscillations. By simulating giant networks of this minimal module we study the dependence of the spike synchrony on some parameters of the network such as the probability and strength of excitatory/inhibitory couplings, parameter mismatch, correlation of thalamic input and transmission time-delay. In the third section of the thesis we study the interdependencies within the time series obtained through electroencephalography (EEG) and give the EEG specific maps for patients suffering from schizophrenia or Alzheimer's disease. Capturing the collective coherent spatiotemporal activity of neuronal populations measured by high density EEG is addressed using measures estimating the synchronization within multivariate time series. Our EEG power analysis on schizophrenic patients, which is based on a new parametrization of the multichannel EEG, shows a relative increase of power in alpha rhythm over the anterior brain regions against its reduction over posterior regions. The correlations of these patterns with the clinical picture of schizophrenia as well as discriminating of the schizophrenia patients from normal control subjects supports the concept of hypofrontality in schizophrenia and renders the alpha rhythm as a sensitive marker of it. By applying a multivariate synchronization estimator, called S-estimator, we reveal the whole-head synchronization topography in schizophrenia. Our finding shows bilaterally increased synchronization over temporal brain regions and decreased synchronization over the postcentral/parietal brain regions. The topography is stable over the course of several months as well as over all conventional EEG frequency bands. Moreover, it correlates with the severity of the illness characterized by positive and negative syndrome scales. We also reveal the EEG features specific to early Alzheimer's disease by applying multivariate phase synchronization method. Our analyses result in a specific map characterized by a decrease in the values of phase synchronization over the fronto-temporal and an increase over temporo-parieto-occipital region predominantly of the left hemisphere. These abnormalities in the synchronization maps correlate with the clinical scores associated to the patients and are able to discriminate patients from normal control subjects with high precision

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

Network complexity and synchronous behavior an experimental approach

We discuss synchronization in networks of Hindmarsh-Rose neurons that are inter-connected via gap junctions, also known as electrical synapses. We present theoretical results for interactions without time-delay. These results are supported by experiments with a setup consisting of sixteen electronic equivalents of the Hindmarsh-Rose neuron. We show experimental results of networks where time-delay on the interaction is taken into account. We discuss in particular the influence of the network topology on the synchronization