Emergent synchrony in locally coupled neural oscillators

The discovery of long range synchronous oscillations in the visual cortex has triggered much interest in understanding the underlying neural mechanisms and in exploring possible applications of neural oscillations. Many neural models thus proposed end up relying on global connections, leading to the question of whether lateral connections alone can produce remote synchronization. With a formulation different from frequently used phase models, we find that locally coupled neural oscillators can yield global synchrony. The model employs a previously suggested mechanism that the efficacy of the connections is allowed to change on a fast time scale. Based on the known connectivity of the visual cortex, the model outputs closely resemble the experimental findings. Furthermore, we illustrate the potential of locally connected oscillator networks in perceptual grouping and pattern segmentation, which seems missing in globally connected ones.     

Patterns of synchrony in neural networks with spike adaptation

We study the emergence of synchronized burst activity in networks of neurons with spike adaptation. We show that networks of tonically firing adapting excitatory neurons can evolve to a state where the neurons burst in a synchronized manner. The mechanism leading to this burst activity is analyzed in a network of integrate-and-fire neurons with spike adaptation. The dependence of this state on the different network parameters is investigated, and it is shown that this mechanism is robust against inhomogeneities, sparseness of the connectivity, and noise. In networks of two populations, one excitatory and one inhibitory, we show that decreasing the inhibitory feedback can cause the network to switch from a tonically active, asynchronous state to the synchronized bursting state. Finally, we show that the same mechanism also causes synchronized burst activity in networks of more realistic conductance-based model neurons.     

Interactions between two neural populations: A mechanism of chaos and oscillation in neural mass model

Neural mass model developed by Lopes da Silva et al. is able to describe limit cycle behavior in Electroencephalography (EEG) of alpha rhythm and exhibit complex dynamics between cortical areas. In this work, we extend Grimbert and Faugeras's work to study the dynamical behavior caused by interaction of cortical areas. The model is developed with the coupling of two neural populations. We show that various attractors, including equilibrium points, periodic solutions and chaotic strange attractors, could coexist in different ways with different value of the connectivity parameters. The main findings are that: (1) The stable equilibrium points only appear with a small value of the parameter. (2) While the alpha activities always exist for both two populations with proper initial conditions. Interestingly, the coexistence of the multiple alpha-to-epileptic activities implies the multiple coupling ways for these activities in phase. Two neuronal populations with epileptic activities could interact with multiple rhythms depending on their connectivity. (3) For particular interest, chaotic behaviors are identified in four regions divided by the connectivity parameter with the positive maximal Lyapunov exponent. The four types of chaotic attractors have their own structures, but all of them are related to the epileptic activities.     

Dynamics and stability of a delay-locked loop

We discuss the transient behavior of a delay-locked loop which is designed to generate a delay-error signal that is proportional to the difference in the autocorrelation function of the input signal at two points separated by a fixed time2tau_{1}. When the input signal is a sine wave, we present an exact solution which shows that the system is stable and achieves a delay lock with an ambiguity of an integral number of periods. The second input considered is that of a stationary, ergodic, and band-limited Gaussian signal. In this case we present an approximate analysis which predicts that for times long compared to the inverse bandwidth of the random signal that the delay error is log-normally distributed. For this case we develop the almost sure sample stability criterion [1]. When this criterion is met, the system sample solutions are stable with probability one independant of the system amplification. We also develop stability criteria which limit the system amplification for stability of the first and second moments of the time delay.     

Distinguishing causal interactions in neural populations

We describe a theoretical network analysis that can distinguish statistically causal interactions in population neural activity leading to a specific output. We introduce the concept of a causal core to refer to the set of neuronal interactions that are causally significant for the output, as assessed by Granger causality. Because our approach requires extensive knowledge of neuronal connectivity and dynamics, an illustrative example is provided by analysis of Darwin X, a brain-based device that allows precise recording of the activity of neuronal units during behavior. In Darwin X, a simulated neuronal model of the hippocampus and surrounding cortical areas supports learning of a spatial navigation task in a real environment. Analysis of Darwin X reveals that large repertoires of neuronal interactions contain comparatively small causal cores and that these causal cores become smaller during learning, a finding that may reflect the selection of specific causal pathways from diverse neuronal repertoires.     

Dynamics of neuronal populations: eigenfunction theory; some solvable cases

A novel approach to cortical modelling was introduced by Knight and co-workers in 1996. In their presentation cortical dynamics is formulated in terms of interacting populations of neurons, a perspective that is in part motivated by modern cortical imaging. The approach may be regarded as the application of statistical mechanics to neuronal populations, and the simplest exemplar bears a kinship to the Boltzmann equation of kinetic theory. The disarming simplicity of this linear equation hides the complex behaviour it produces. A purpose of this paper is to investigate and reveal its intricacies by treating a series of solvable special cases. In particular we will focus on issues that relate to the spectral analysis of the underlying operators. A fairly thorough treatment is presented for a simple, but still useful example, that has important consequences for more general situations.     

Dynamics of encoding in neuron populations: some general mathematical features

The use of a population dynamics approach promises efficient simulation of large assemblages of neurons. Depending on the issues addressed and the degree of realism incorporated in the simulated neurons, a wide range of different population dynamics formulations can be appropriate. Here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common. This underlying structure serves as a guide toward efficient means of simulation. As an example, we derive the general population firing-rate frequency-response and show how it may be used effectively to address a broad range of interacting-population response and stability problems. A few specific cases will be worked out. A summary of this work appears at the end, before the appendix.     

Dynamics and stability of potential hyper-networked evolutionary games

This paper considers the modeling and convergence of hyper-networked evolutionary games (HNEGs). In an HNEG the network graph is a hypergraph, which allows the fundamental network game to be a multi-player one. Using semi-tensor product of matrices and the fundamental evolutionary equation, the dynamics of an HNEG is obtained and we extend the results about the networked evolutionary games to show whether an HNEG is potential and how to calculate the potential. Then we propose a new strategy updating rule, called the cascading myopic best response adjustment rule (MBRAR), and prove that under the cascading MBRAR the strategies of an HNEG will converge to a pure Nash equilibrium. An example is presented and discussed in detail to demonstrate the theoretical and numerical results.     

Representational accuracy of stochastic neural populations.

Fisher information is used to analyze the accuracy with which a neural population encodes D stimulus features. It turns out that the form of response variability has a major impact on the encoding capacity and therefore plays an important role in the selection of an appropriate neural model. In particular, in the presence of baseline firing, the reconstruction error rapidly increases with D in the case of Poissonian noise but not for additive noise. The existence of limited-range correlations of the type found in cortical tissue yields a saturation of the Fisher information content as a function of the population size only for an additive noise model. We also show that random variability in the correlation coefficient within a neural population, as found empirically, considerably improves the average encoding quality. Finally, the representational accuracy of populations with inhomogeneous tuning properties, either with variability in the tuning widths or fragmented into specialized subpopulations, is superior to the case of identical and radially symmetric tuning curves usually considered in the literature.     

Fisher and Shannon information in finite neural populations

The precision of the neural code is commonly investigated using two families of statistical measures: Shannon mutual information and derived quantities when investigating very small populations of neurons and Fisher information when studying large populations. These statistical tools are no longer the preserve of theorists and are being applied by experimental research groups in the analysis of empirical data. Although the relationship between information-theoretic and Fisher-based measures in the limit of infinite populations is relatively well understood, how these measures compare in finite-size populations has not yet been systematically explored. We aim to close this gap. We are particularly interested in understanding which stimuli are best encoded by a given neuron within a population and how this depends on the chosen measure. We use a novel Monte Carlo approach to compute a stimulus-specific decomposition of the mutual information (the SSI) for populations of up to 256 neurons and show that Fisher information can be used to accurately estimate both mutual information and SSI for populations of the order of 100 neurons, even in the presence of biologically realistic variability, noise correlations, and experimentally relevant integration times. According to both measures, the stimuli that are best encoded are those falling at the flanks of the neuron's tuning curve. In populations of fewer than around 50 neurons, however, Fisher information can be misleading.     

Global asymptotic stability and global exponential stability ofcontinuous-time recurrent neural networks

This paper presents new results on global asymptotic stability (GAS) and global exponential stability (GES) of a general class of continuous-time recurrent neural networks with Lipschitz continuous and monotone nondecreasing activation functions. We first give three sufficient conditions for the GAS of neural networks. These testable sufficient conditions differ from and improve upon existing ones. We then extend an existing GAS result to GES one and also extend the existing GES results to more general cases with less restrictive connection weight matrices and/or partially Lipschitz activation functions     

Dynamics and stability of plane trusses with gusset plates

The effect of gusset plates on free and forced vibration and stability analyses of plane trusses is investigated. The gusset plates are considered to be finite joints possessing mass and rotational flexibility. The bars of the truss are assumed to be elastic Bernoulli-Euler beams with distributed mass. Axial deformation of the bars and the effect of a constant axial force on the bending stiffness are taken into account. On the basis of these assumptions element stiffness matrices are constructed and presented in detail. The general formulation and solution of stability and free and forced vibration problems of trusses is discussed. Examples are presented in detail which demonstrate the effect of the gusset plates on the behavior of trusses under static or dynamic loads.     

Dynamics and stability in optical communication networks: a system theory framework

This paper addresses the problem of dynamics analysis in optical networks from a system control perspective. A general framework for finding the transfer matrix representation of an optical network is developed, based on linear fractional transformations. Under the natural assumption of equal time-delay for all channels in a link, the network transfer matrix is simplified such that channel cross-coupling is evidenced. The optical network stability problem is then reformulated as a robust stability problem and stability conditions are developed by applying μ-analysis.     

Nonlinear one-step-ahead control using neural networks: Control strategy and stability design

A nonlinear one-step-ahead control strategy based on a neural network model is proposed for nonlinear SISO processes. The neural network used for controller design is a feedforward network with external recurrent terms. The training of the neural network model is implemented by using a recursive least-squares (RLS)-based algorithm. Considering the case of the nonlinear processes with time delay, the extension of the mentioned neural control scheme to d-step-ahead predictive neural control is proposed to compensate the influence of the time-delay. Then the stability analysis of the neural-network-based one-step-ahead control system is presented based on Lyapunov theory. From the stability investigation, the stability condition for the neural control system is obtained. The method is illustrated with some simulated examples, including the control of a continuous stirred tank reactor (CSTR).     

Global exponential stability of Hopfield neural networks

In this article, some sufficient criteria are derived for the global exponential stability of the equilibrium of Hopfield neural networks of the form C_i du_i /dt     

Global stability analysis of multitime-scale neural networks

Global asymptotic stability problem is studied for a class of recurrent neural networks with multitime scale. The concerned network involves two coupling terms, i.e., long-term memory and short-term memory, which leads to the difficulty to the dynamics analysis, especially for the case of multiple time varying delays. Some novel stability criteria are proposed on the basis of linear matrix inequality technique for the concerned neural network, which sufficiently consider the inhibitory actions in the different memories. From the viewpoint of biological information, the proposed results obviously improve the existing stability criteria. A numerical example is used to show the effectiveness of the obtained results.     

On the phase reduction and response dynamics of neural oscillator populations

We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli. Recalling and extending results from the literature, we compute phase response curves (PRCs) valid near bifurcations to periodic firing for Hindmarsh-Rose, Hodgkin-Huxley, FitzHugh-Nagumo, and Morris-Lecar models, encompassing the four generic (codimension one) bifurcations. Phase density equations are then used to analyze the role of the bifurcation, and the resulting PRC, in responses to stimuli. In particular, we explore the interplay among stimulus duration, baseline firing frequency, and population-level response patterns. We interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions.     

Critical dynamics study on recurrent neural networks: Globally exponential stability

Critical dynamics research of recurrent neural networks (RNNs) is very meaningful in both theoretical importance and practical significance. Due to the essential difficulty in analysis, there were only a few contributions concerning it. In this paper, we devote to study the critical dynamics behaviors for RNNs with general forms. By exploring some intrinsic features processed naturally by the nonlinear activation mappings of RNNs, and by using matrix measure theory, new criteria are found to ascertain the globally exponential stability of RNNs under the critical conditions. The results obtained here either yield new, or sharpen, extend or unify, to a large extent, most of the existing non-critical conclusions as well as the latest critical results.     

Complete stability in multistable delayed neural networks

We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3n equilibria and 2n positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.     

On the global stability of delayed neural networks

Lyapunov functional methods, combining with some inequality techniques, are employed to study the global asymptotic stability of delayed neural networks. Without assuming Lipschitz conditions on the activation functions, a new sufficient condition is established. Such criteria allows us to include non-Lipschitzian activation functions in the design of delayed neural networks. The result presented here is also discussed from the point of view of its relationship to some previous results.