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An important problem in systems neuroscience is to understand how information is communicated among brain regions, and it has been proposed that communication is mediated by neuronal oscillations, such as rhythms in the gamma band. We sought to investigate this idea by using a network model with two components, a source (sending) and a target (receiving) component, both built to resemble local populations in the cerebral cortex. To measure the effectiveness of communication, we used population-level correlations in spike times between the source and target. We found that after correcting for a response time that is independent of initial conditions, spike-time correlations between the source and target are significant, due in large measure to the alignment of firing events in their gamma rhythms. But, we also found that regular oscillations cannot produce the results observed in our model simulations of cortical neurons. Surprisingly, it is the irregularity of gamma rhythms, the absence of internal clocks, together with the malleability of these rhythms and their tendency to align with external pulses — features that are known to be present in gamma rhythms in the real cortex — that produced the results observed. These findings and the mechanistic explanations we offered are our primary results. Our secondary result is a mathematical relationship between correlations and the sizes of the samples used for their calculation. As improving technology enables recording simultaneously from increasing numbers of neurons, this relationship could be useful for interpreting results from experimental recordings. [ABSTRACT FROM AUTHOR]

This article presents an idealized mathematical model of the cerebral cortex, focusing on the dynamical interaction of neurons. The author proposes a network architecture more consistent with neuroanatomy than in previous studies, borrows ideas from nonequilibrium statistical mechanics and calls attention to the fact that the brain is a large and complex dynamical system. The ideas proposed are illustrated with a realistic model of the visual cortex. [ABSTRACT FROM AUTHOR]

Infectious diseases are among the most prominent threats to mankind. When preventive health care cannot be provided, a viable means of disease control is the isolation of individuals who may be infected. To study the impact of isolation, we propose a system of delay differential equations and offer our model analysis based on the geometric theory of semi-flows. Calibrating the response to an outbreak in terms of the fraction of infectious individuals isolated and the speed with which this is done, we deduce the minimum response required to curb an incipient outbreak, and predict the ensuing endemic state should the infection continue to spread. [ABSTRACT FROM AUTHOR]

This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations. Rigorous results on existence and uniqueness of nonequilibrium steady states are proved. These network models are then compared to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models are investigated and explained by correlations in spiking, or partial synchronization, working in concert with "nonlinearities" in the time evolution of membrane potentials. The use of simple random walks and their first passage times to simulate fluctuations in neuronal membrane potentials and interspike times is also considered. [ABSTRACT FROM AUTHOR]

To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth. [ABSTRACT FROM AUTHOR]

We prove the absolute continuity of stable foliations for mappings of Banach spaces satisfying conditions consistent with time- t maps of certain classes of dissipative PDEs. This property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space; it is also used in proofs of ergodicity. Absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory. On Banach spaces, the absence of nice geometric properties poses some additional difficulties. [ABSTRACT FROM AUTHOR]

We consider $$C^2$$ Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs. [ABSTRACT FROM AUTHOR]

We review some developments that are direct outgrowths of, or closely related to, the idea of SRB measures as introduced by Sinai, Ruelle and Bowen in the 1970s. These new directions of research include the emergence of strange attractors in periodically forced dynamical systems, random attractors in systems defined by stochastic differential equations, SRB measures for infinite dimensional systems including those defined by large classes of dissipative PDEs, quasi-static distributions for slowly varying time-dependent systems, and surviving distributions in leaky dynamical systems. [ABSTRACT FROM AUTHOR]

This paper is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their 'local environments' rather than directly with one another. The physical domain of the system can be a bounded region of $$\mathbb R^d$$ for any $$d \ge 1$$ . We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. In the Hamiltonian context, this would mean that at any location x in the domain, local marginal distributions of NESS tend to a probability with density $$\frac{1}{Z} e^{-\beta (x) H}$$ , permitting one to define the local temperature at x to be $$\beta (x)^{-1}$$ . We prove also that in the infinite volume limit, the mean energy profile of NESS satisfies Laplace's equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at x to that of joint hitting distributions of certain random walks starting from x, and prove that the walks in question become increasingly independent as system size tends to infinity. [ABSTRACT FROM AUTHOR]

This numerical study documents and analyzes emergent spiking behavior in local neuronal populations. Emphasis is given to a phenomenon we call clustering, by which we refer to a tendency of random groups of neurons large and small to spontaneously coordinate their spiking activity in some fashion. Using a sparsely connected network of integrate-and-fire neurons, we demonstrate that spike clustering occurs ubiquitously in both high firing and low firing regimes. As a practical tool for quantifying such spike patterns, we propose a simple scheme with two parameters, one setting the temporal scale and the other the amount of deviation from the mean to be regarded as significant. Viewing population activity as a sequence of events, meaning relatively brief durations of elevated spiking, separated by inter-event times, we observe that background activity tends to give rise to extremely broad distributions of event sizes and inter-event times, while driving a system imposes a certain regularity on its inter-event times, producing a rhythm consistent with broad-band gamma oscillations. We note also that event sizes and inter-event times decorrelate very quickly. Dynamical analyses supported by numerical evidence are offered. [ABSTRACT FROM AUTHOR]

This paper proposes that the network dynamics of the mammalian visual cortex are highly structured and strongly shaped by temporally localized barrages of excitatory and inhibitory firing we call 'multiple-firing events' (MFEs). Our proposal is based on careful study of a network of spiking neurons built to reflect the coarse physiology of a small patch of layer 2/3 of V1. When appropriately benchmarked this network is capable of reproducing the qualitative features of a range of phenomena observed in the real visual cortex, including spontaneous background patterns, orientation-specific responses, surround suppression and gamma-band oscillations. Detailed investigation into the relevant regimes reveals causal relationships among dynamical events driven by a strong competition between the excitatory and inhibitory populations. It suggests that along with firing rates, MFE characteristics can be a powerful signature of a regime. Testable predictions based on model observations and dynamical analysis are proposed. [ABSTRACT FROM AUTHOR]

We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems. [ABSTRACT FROM AUTHOR]

This paper contains rigorous results for a simple stochastic model of heat conduction similar to the KMP (Knipnis-Marchiori-Presutti) model but with possibly energy-dependent interaction rates. We prove the existence and uniqueness of nonequilibrium steady states, their relation to Lebesgue measure, and exponential convergence to steady states from suitable initial conditions. [ABSTRACT FROM AUTHOR]

Randomly connected networks of neurons driven by Poisson inputs are often assumed to produce 'homogeneous' dynamics, characterized by largely independent firing and approximable by diffusion processes. At the same time, it is well known that such networks can fire synchronously. Between these two much studied scenarios lies a vastly complex dynamical landscape that is relatively unexplored. In this paper, we discuss a phenomenon which commonly manifests in these intermediate regimes, namely brief spurts of spiking activity which we call multiple firing events (MFE). These events do not depend on structured network architecture nor on structured input; they are an emergent property of the system. We came upon them in an earlier modeling paper, in which we discovered, through a careful benchmarking process, that MFEs are the single most important dynamical mechanism behind many of the V1 phenomena we were able to replicate. In this paper we explain in a simpler setting how MFEs come about, as well as their potential dynamic consequences. Although the mechanism underlying MFEs cannot easily be captured by current population dynamics models, this phenomena should not be ignored during analysis; there is a growing body of evidence that such collaborative activity may be a key towards unlocking the possible functional properties of many neuronal networks. [ABSTRACT FROM AUTHOR]

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are 'sufficiently weak', an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of 'sticky' phase-space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience. [ABSTRACT FROM AUTHOR]

We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to $\frac{1}{2} \sqrt{T_{L}T_{R}}$ where T and T are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier's Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach. [ABSTRACT FROM AUTHOR]

We consider smooth (not necessarily invertible) maps of Hilbert spaces preserving ergodic Borel probability measures, and prove the existence of hyperbolic periodic orbits and horseshoes in the absence of zero Lyapunov exponents. These results extend Katok's work on diffeomorphisms of compact manifolds to infinite dimensions, with potential applications to some classes of periodically forced PDEs. [ABSTRACT FROM AUTHOR]

We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model. [ABSTRACT FROM AUTHOR]

We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network’s initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced. [ABSTRACT FROM AUTHOR]

We present the results of a detailed study of energy correlations at steady state for a 1-D model of coupled energy and matter transport. Our aim is to discover—via theoretical arguments, conjectures, and numerical simulations—how spatial covariances scale with system size, their relations to local thermodynamic quantities, and the randomizing effects of heat baths. Among our findings are that short-range covariances respond quadratically to local temperature gradients, and long-range covariances decay linearly with macroscopic distance. These findings are consistent with exact results for the simple exclusion and KMP models. [ABSTRACT FROM AUTHOR]

We consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general (nonlinear) dissipative parabolic PDEs. Results on invariant measures and their exponential mixing properties are proved, and applications to 2D Navier–Stokes systems are included. [ABSTRACT FROM AUTHOR]

We prove that simple mechanical systems, when subjected to external periodic forcing, can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied. In particular, the existence of global strange attractors with fully stochastic properties is proved for a class of second order ODEs. [ABSTRACT FROM AUTHOR]

We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy. [ABSTRACT FROM AUTHOR]

The setting of this paper consists of a map making “nice” returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises naturally in differentiable dynamical systems with some expanding or hyperbolic properties. [ABSTRACT FROM AUTHOR]

In neuroscience, computational modeling is an effective way to gain insight into cortical mechanisms, yet the construction and analysis of large-scale network models—not to mention the extraction of underlying principles—are themselves challenging tasks, due to the absence of suitable analytical tools and the prohibitive costs of systematic numerical exploration of high-dimensional parameter spaces. In this paper, we propose a data-driven approach assisted by deep neural networks (DNN). The idea is to first discover certain input-output relations, and then to leverage this information and the superior computation speeds of the well-trained DNN to guide parameter searches and to deduce theoretical understanding. To illustrate this novel approach, we used as a test case a medium-size network of integrate-and-fire neurons intended to model local cortical circuits. With the help of an accurate yet extremely efficient DNN surrogate, we revealed the statistics of model responses, providing a detailed picture of model behavior. The information obtained is both general and of a fundamental nature, with direct application to neuroscience. Our results suggest that the methodology proposed can be scaled up to larger and more complex biological networks when used in conjunction with other techniques of biological modeling. [ABSTRACT FROM AUTHOR]

For centuries isolation has been the main control strategy of unforeseen epidemic outbreaks. When implemented in full and without delay, isolation is very effective. However, flawless implementation is seldom feasible in practice. We present an epidemic model called SIQ with an isolation protocol, focusing on the consequences of delays and incomplete identification of infected hosts. The continuum limit of this model is a system of Delay Differential Equations, the analysis of which reveals clearly the dependence of epidemic evolution on model parameters including disease reproductive number, isolation probability, speed of identification of infected hosts and recovery rates. Our model offers estimates on minimum response capabilities needed to curb outbreaks, and predictions of endemic states when containment fails. Critical response capability is expressed explicitly in terms of parameters that are easy to obtain, to assist in the evaluation of funding priorities involving preparedness and epidemics management. [ABSTRACT FROM AUTHOR]