Stability of a neural network model with small-world connections

Functional network characteristics based on EEG of patients in acute ischemic stroke: A pilot study

BACKGROUND

Ischemic stroke is a common type of stroke associated with reorganization of functional network of the brain.

OBJECTIVE

This pilot study aimed to investigate the characteristics of functional brain networks based on EEG in patients with acute ischemic stroke.

METHODS

Seven patients with ischemic stroke within 72 hours of onset and seven healthy controls were enrolled in the study. Dynamic EEG monitoring and clinical information were repeatedly collected within 72 hours (T1), on the 5th day (T2), and on the 7th day (T3) of stroke onset. A directed transfer function was employed to construct functional brain connection patterns. Graph theoretical analysis was performed to evaluate the characteristics of functional brain networks.

RESULTS

First, we found that the brain networks of ischemic stroke patients were quite different from the healthy controls. The clustering coefficient (0.001 <  Threshold <  0.2) in Delta, Theta, and Alpha bands for the patients were significantly lower (P <  0.01) and the shortest path length in all bands (0.001 <  Threshold <  0.2) for the patients were significantly longer (P <  0.01). Moreover, the peaks of the shortest path length for the patients seemed to be higher in all bands with larger thresholds. Secondly, the brain networks for the patients showed a characterized time-variation pattern. The clustering coefficient (0.001 <  Threshold <  0.2) of T1 was higher than that of T2 in alpha band (P <  0.01). The shortest path length (0.001 <  Threshold <  0.2) of T3 was shorter than that of T2 (P <  0.01) in all bands, and the peak of T3 was numerically higher than that of T2 in all bands with narrower thresholds.

CONCLUSION

Functional brain networks in patients with acute ischemic stroke showed impaired global functional integration and decreased efficiency of information transmission compared with healthy subjects. The shortening of the shortest path length during the recovery indicates neural plasticity and reorganization.

Stability and instability of a neuron network with excitatory and inhibitory small-world connections

This study considers a delayed neural network with excitatory and inhibitory shortcuts. The global stability of the trivial equilibrium is investigated based on Lyapunov's direct method and the delay-dependent criteria are obtained. It is shown that both the excitatory and inhibitory shortcuts decrease the stability interval, but a time delay can be employed as a global stabilizer. In addition, we analyze the bounds of the eigenvalues of the adjacent matrix using matrix perturbation theory and then obtain the generalized sufficient conditions for local stability. The possibility of small inhibitory shortcuts is helpful for maintaining stability. The mechanisms of instability, bifurcation modes, and chaos are also investigated. Compared with methods based on mean-field theory, the proposed method can guarantee the stability of the system in most cases with random events. The proposed method is effective for cases where excitatory and inhibitory shortcuts exist simultaneously in the network.

Interplay between excitability type and distributions of neuronal connectivity determines neuronal network synchronization

While the interplay between neuronal excitability properties and global properties of network topology is known to affect network propensity for synchronization, it is not clear how detailed characteristics of these properties affect spatiotemporal pattern formation. Here we study mixed networks, composed of neurons having type I and/or type II phase response curves, with varying distributions of local and random connections and show that not only average network properties, but also the connectivity distribution statistics, significantly affect network synchrony. Namely, we study networks with fixed networkwide properties, but vary the number of random connections that nodes project. We show that varying node excitability (type I vs type II) influences network synchrony most dramatically for systems with long-tailed distributions of the number of random connections per node. This indicates that a cluster of even a few highly rewired cells with a high propensity for synchronization can alter the degree of synchrony in the network as a whole. We show this effect generally on a network of coupled Kuramoto oscillators and investigate the impact of this effect more thoroughly in pulse-coupled networks of biophysical neurons.

Gradient boosting machines, a tutorial

Gradient boosting machines are a family of powerful machine-learning techniques that have shown considerable success in a wide range of practical applications. They are highly customizable to the particular needs of the application, like being learned with respect to different loss functions. This article gives a tutorial introduction into the methodology of gradient boosting methods with a strong focus on machine learning aspects of modeling. A theoretical information is complemented with descriptive examples and illustrations which cover all the stages of the gradient boosting model design. Considerations on handling the model complexity are discussed. Three practical examples of gradient boosting applications are presented and comprehensively analyzed.

Stochastic stability of genetic regulatory networks with a finite set delay characterization

In this paper, the delay-distribution-dependent stability is derived for the stochastic genetic regulatory networks (GRNs) with a finite set delay characterization and interval parameter uncertainties. One important feature of the obtained results here is that the time-varying delays are assumed to be random and the sum of the occurrence probabilities of the delays is assumed to be 1. By employing a new Lyapunov-Krasovskii functional dependent on auxiliary delay parameters which allow the time-varying delays to be not differentiable, less conservative mean-square stochastic stability criteria are obtained. Finally, two examples are given to illustrate the effectiveness and superiority of the derived results.

Graph theoretical model of a sensorimotor connectome in zebrafish

Mapping the detailed connectivity patterns (connectomes) of neural circuits is a central goal of neuroscience. The best quantitative approach to analyzing connectome data is still unclear but graph theory has been used with success. We present a graph theoretical model of the posterior lateral line sensorimotor pathway in zebrafish. The model includes 2,616 neurons and 167,114 synaptic connections. Model neurons represent known cell types in zebrafish larvae, and connections were set stochastically following rules based on biological literature. Thus, our model is a uniquely detailed computational representation of a vertebrate connectome. The connectome has low overall connection density, with 2.45% of all possible connections, a value within the physiological range. We used graph theoretical tools to compare the zebrafish connectome graph to small-world, random and structured random graphs of the same size. For each type of graph, 100 randomly generated instantiations were considered. Degree distribution (the number of connections per neuron) varied more in the zebrafish graph than in same size graphs with less biological detail. There was high local clustering and a short average path length between nodes, implying a small-world structure similar to other neural connectomes and complex networks. The graph was found not to be scale-free, in agreement with some other neural connectomes. An experimental lesion was performed that targeted three model brain neurons, including the Mauthner neuron, known to control fast escape turns. The lesion decreased the number of short paths between sensory and motor neurons analogous to the behavioral effects of the same lesion in zebrafish. This model is expandable and can be used to organize and interpret a growing database of information on the zebrafish connectome.

Fast and robust image segmentation by small-world neural oscillator networks

Inspired by the temporal correlation theory of brain functions, researchers have presented a number of neural oscillator networks to implement visual scene segmentation problems. Recently, it is shown that many biological neural networks are typical small-world networks. In this paper, we propose and investigate two small-world models derived from the well-known LEGION (locally excitatory and globally inhibitory oscillator network) model. To form a small-world network, we add a proper proportion of unidirectional shortcuts (random long-range connections) to the original LEGION model. With local connections and shortcuts, the neural oscillators can not only communicate with neighbors but also exchange phase information with remote partners. Model 1 introduces excitatory shortcuts to enhance the synchronization within an oscillator group representing the same object. Model 2 goes further to replace the global inhibitor with a sparse set of inhibitory shortcuts. Simulation results indicate that the proposed small-world models could achieve synchronization faster than the original LEGION model and are more likely to bind disconnected image regions belonging together. In addition, we argue that these two models are more biologically plausible.

Synchronization of the small-world neuronal network with unreliable synapses

As is well known, synchronization phenomena are ubiquitous in neuronal systems. Recently a lot of work concerning the synchronization of the neuronal network has been accomplished. In these works, the synapses are usually considered reliable, but experimental results show that, in biological neuronal networks, synapses are usually unreliable. In our previous work, we have studied the synchronization of the neuronal network with unreliable synapses; however, we have not paid attention to the effect of topology on the synchronization of the neuronal network. Several recent studies have found that biological neuronal networks have typical properties of small-world networks, characterized by a short path length and high clustering coefficient. In this work, mainly based on the small-world neuronal network (SWNN) with inhibitory neurons, we study the effect of network topology on the synchronization of the neuronal network with unreliable synapses. Together with the network topology, the effects of the GABAergic reversal potential, time delay and noise are also considered. Interestingly, we found a counter-intuitive phenomenon for the SWNN with specific shortcut adding probability, that is, the less reliable the synapses, the better the synchronization performance of the SWNN. We also consider the effects of both local noise and global noise in this work. It is shown that these two different types of noise have distinct effects on the synchronization: one is negative and the other is positive.

Self-sustained irregular activity in 2-D small-world networks of excitatory and inhibitory neurons

In this paper, we study the self-sustained irregular firing activity in 2-D small-world (SW) neural networks consisting of both excitatory and inhibitory neurons by computational modeling. For a proper proportion of unidirectional shortcuts, the stable self-sustained activity with irregular firing states indeed occurs in the considered network. By varying the shortcut density while keeping other system parameters fixed, different levels of irregular firing states, from weakly irregular to Poisson-like and burst firing states, are obtained in 2-D SW neural networks. It is also observed that this activity is sensitive to small perturbations, which might provide a possible mechanism for producing chaos. On the other hand, we find that several other system parameters, such as the network size and refractory period, have significant impact on this activity. Further simulation results show that the 2-D SW neural network can sustain such long-lasting firing behavior by using a smaller number of connections than the random neural network.

Structured information in small-world neural networks

The retrieval abilities of spatially uniform attractor networks can be measured by the global overlap between patterns and neural states. However, we found that nonuniform networks, for instance, small-world networks, can retrieve fragments of patterns (blocks) without performing global retrieval. We propose a way to measure the local retrieval using a parameter that is related to the fluctuation of the block overlaps. Simulation of neural dynamics shows a competition between local and global retrieval. The phase diagram shows a transition from local retrieval to global retrieval when the storage ratio increases and the topology becomes more random. A theoretical approach confirms the simulation results and predicts that the stability of blocks can be improved by dilution.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Topological Determinants of Epileptogenesis in Large-Scale Structural and Functional Models of the Dentate Gyrus Derived From Experimental Data

In temporal lobe epilepsy, changes in synaptic and intrinsic properties occur on a background of altered network architecture resulting from cell loss and axonal sprouting. Although modeling studies using idealized networks indicated the general importance of network topology in epilepsy, it is unknown whether structural changes that actually take place during epileptogenesis result in hyperexcitability. To answer this question, we built a 1:1 scale structural model of the rat dentate gyrus from published in vivo and in vitro cell type–specific connectivity data. This virtual dentate gyrus in control condition displayed globally and locally well connected (“small world”) architecture. The average number of synapses between any two neurons in this network of over one million cells was less than three, similar to that measured for the orders of magnitude smaller C. elegans nervous system. To study how network architecture changes during epileptogenesis, long-distance projecting hilar cells were gradually removed in the structural model, causing massive reductions in the number of total connections. However, as long as even a few hilar cells survived, global connectivity in the network was effectively maintained and, as a result of the spatially restricted sprouting of granule cell axons, local connectivity increased. Simulations of activity in a functional dentate network model, consisting of over 50,000 multicompartmental single-cell models of major glutamatergic and GABAergic cell types, revealed that the survival of even a small fraction of hilar cells was enough to sustain networkwide hyperexcitability. These data indicate new roles for fractionally surviving long-distance projecting hilar cells observed in specimens from epilepsy patients.

Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses

By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness, and global exponential stability of periodic solution for shunting inhibitory cellular neural networks with impulses, dx(ij)dt=-a(ij)x(ij)- summation operator(C(kl)inN(r)(i,j))C(ij) (kl)f(ij)[x(kl)(t)]x(ij)+L(ij)(t), t>0,t not equal t(k); Deltax(ij)(t(k))=x(ij)(t(k) (+))-x(ij)(t(k) (-))=I(k)[x(ij)(t(k))], k=1,2,...] . Furthermore, the numerical simulation shows that our system can occur in many forms of complexities, including periodic oscillation and chaotic strange attractor. To the best of our knowledge, these results have been obtained for the first time. Some researchers have introduced impulses into their models, but analogous results have never been found.

The brainstem reticular formation is a small-world, not scale-free, network

Recently, it has been demonstrated that several complex systems may have simple graph-theoretic characterizations as so-called 'small-world' and 'scale-free' networks. These networks have also been applied to the gross neural connectivity between primate cortical areas and the nervous system of Caenorhabditis elegans. Here, we extend this work to a specific neural circuit of the vertebrate brain--the medial reticular formation (RF) of the brainstem--and, in doing so, we have made three key contributions. First, this work constitutes the first model (and quantitative review) of this important brain structure for over three decades. Second, we have developed the first graph-theoretic analysis of vertebrate brain connectivity at the neural network level. Third, we propose simple metrics to quantitatively assess the extent to which the networks studied are small-world or scale-free. We conclude that the medial RF is configured to create small-world (implying coherent rapid-processing capabilities), but not scale-free, type networks under assumptions which are amenable to quantitative measurement.

Structure of cortical microcircuit theory

Recent experimental and theoretical investigations have made considerable advances in three major areas relating to the structural basis of quantitative cortical microcircuit theory. The first concerns the nature of the cellular units, encompassing the increasingly precise identification and progressively more complete listing of the individual cellular species that constitute the various cortical networks. The second element addresses the problem of heterogeneity, including the demonstration of the importance of cell to cell variability within defined interneuronal populations and the application of the Shannon-Wiener diversity index for the quantitative assessment of the number and relative abundance of interneuronal species. The third component relates to the discovery of basic topological principles underlying the circuit wiring, revealing a surprising order in the architectural design of networks. These new advances deepen our understanding of the computational principles embedded in cortical microcircuits, and they also provide novel opportunities for building realistic models of mammalian cortical microcircuits.