Nonlinear dynamic elements with noisy sinusoidal forcing: Enhancing response via nonlinear coupling

Realistic modeling of mesoscopic ephaptic coupling in the human brain

Several decades of research suggest that weak electric fields may influence neural processing, including those induced by neuronal activity and proposed as a substrate for a potential new cellular communication system, i.e., ephaptic transmission. Here we aim to model mesoscopic ephaptic activity in the human brain and explore its trajectory during aging by characterizing the electric field generated by cortical dipoles using realistic finite element modeling. Extrapolating from electrophysiological measurements, we first observe that modeled endogenous field magnitudes are comparable to those in measurements of weak but functionally relevant self-generated fields and to those produced by noninvasive transcranial brain stimulation, and therefore possibly able to modulate neuronal activity. Then, to evaluate the role of these fields in the human cortex in large MRI databases, we adapt an interaction approximation that considers the relative orientation of neuron and field to estimate the membrane potential perturbation in pyramidal cells. We use this approximation to define a simplified metric (EMOD1) that weights dipole coupling as a function of distance and relative orientation between emitter and receiver and evaluate it in a sample of 401 realistic human brain models from healthy subjects aged 16-83. Results reveal that ephaptic coupling, in the simplified mesoscopic modeling approach used here, significantly decreases with age, with higher involvement of sensorimotor regions and medial brain structures. This study suggests that by providing the means for fast and direct interaction between neurons, ephaptic modulation may contribute to the complexity of human function for cognition and behavior, and its modification across the lifespan and in response to pathology.

Amphibian sacculus and the forced Kuramoto model with intrinsic noise and frequency dispersion

The amphibian sacculus (AS) is an end organ that specializes in the detection of low-frequency auditory and vestibular signals. In this paper, we propose a model for the AS in the form of an array of phase oscillators with long-range coupling, subject to a steady load that suppresses spontaneous oscillations. The array is exposed to significant levels of frequency dispersion and intrinsic noise. We show that such an array can be a sensitive and robust subthreshold detector of low-frequency stimuli, though without significant frequency selectivity. The effects of intrinsic noise and frequency dispersion are contrasted. Intermediate levels of intrinsic noise greatly enhance the sensitivity through stochastic resonance. Frequency dispersion, on the other hand, only degrades detection sensitivity. However, frequency dispersion can play a useful role in terms of the suppression of spontaneous activity. As a model for the AS, the array parameters are such that the system is poised near a saddle-node bifurcation on an invariant circle. However, by a change of array parameters, the same system also can be poised near an emergent Andronov-Hopf bifurcation and thereby function as a frequency-selective detector.

A neuron model of stochastic resonance using rectangular pulse trains

Stochastic resonance (SR) is the enhanced representation of a weak input signal by the addition of an optimal level of broadband noise to a nonlinear (threshold) system. Since its discovery in the 1980s the domain of input signals shown to be applicable to SR has greatly expanded, from strictly periodic inputs to now nearly any aperiodic forcing function. The perturbations (noise) used to generate SR have also expanded, from white noise to now colored noise or vibrational forcing. This study demonstrates that a new class of perturbations can achieve SR, namely, series of stochastically generated biphasic pulse trains. Using these pulse trains as 'noise' we show that a Hodgkin Huxley model neuron exhibits SR behavior when detecting weak input signals. This result is of particular interest to neuroscience because nearly all artificial neural stimulation is implemented with square current or voltage pulses rather than broadband noise, and this new method may facilitate the translation of the performance gains achievable through SR to neural prosthetics.

What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology

Stochastic resonance is said to be observed when increases in levels of unpredictable fluctuations--e.g., random noise--cause an increase in a metric of the quality of signal transmission or detection performance, rather than a decrease. This counterintuitive effect relies on system nonlinearities and on some parameter ranges being "suboptimal". Stochastic resonance has been observed, quantified, and described in a plethora of physical and biological systems, including neurons. Being a topic of widespread multidisciplinary interest, the definition of stochastic resonance has evolved significantly over the last decade or so, leading to a number of debates, misunderstandings, and controversies. Perhaps the most important debate is whether the brain has evolved to utilize random noise in vivo, as part of the "neural code". Surprisingly, this debate has been for the most part ignored by neuroscientists, despite much indirect evidence of a positive role for noise in the brain. We explore some of the reasons for this and argue why it would be more surprising if the brain did not exploit randomness provided by noise--via stochastic resonance or otherwise--than if it did. We also challenge neuroscientists and biologists, both computational and experimental, to embrace a very broad definition of stochastic resonance in terms of signal-processing "noise benefits", and to devise experiments aimed at verifying that random variability can play a functional role in the brain, nervous system, or other areas of biology.

Stochastic resonance in the Fermi-Pasta-Ulam chain

We consider a damped beta-Fermi-Pasta-Ulam chain, driven at one boundary subjected to stochastic noise. It is shown that, for a fixed driving amplitude and frequency, increasing the noise intensity, the system's energy resonantly responds to the modulating frequency of the forcing signal. Multiple peaks appear in the signal-to-noise ratio, signaling the phenomenon of stochastic resonance. The presence of multiple peaks is explained by the existence of many stable and metastable states that are found when solving this boundary value problem for a semicontinuum approximation of the model. Stochastic resonance is shown to be generated by transitions between these states.

Noise sensitivity of phase-synchronization time in stochastic resonance: theory and experiment

Recent numerical and heuristic arguments have revealed that the average phase-synchronization time between the input and the output associated with stochastic resonance is highly sensitive to noise variation. In particular there is evidence that this average time exhibits a cusplike behavior as the noise strength varies through the optimal value. Here we present an explicit formula for the average phase-synchronization time in terms of the phase diffusion coefficient and the average frequency difference between the input and the output signals. We also provide experimental evidence for the cusplike behavior by using a bistable microelectronic-circuit system.

Noise-enhanced Fisher information in parallel arrays of sensors with saturation

This paper investigates stochastic resonance in parallel arrays of uncoupled saturating devices. The Fisher information is used to demonstrate the possibility of noise improved parameter estimation for arbitrary parametric signals. Especially, it is shown that improvement by noise always occurs in these arrays, for any configuration of the input signal, even in optimal configuration. The results contribute to establish stochastic resonance in parallel uncoupled arrays as a general mechanism of enhancement by noise, which can occur in wide classes of nonlinearities and for various information processing tasks. It can supplement other mechanisms of stochastic resonance that take place in isolated nonlinearities but generally in restricted configurations of the input signal.

Adaptive Stochastic Resonance in Noisy Neurons Based on Mutual Information

Noise can improve how memoryless neurons process signals and maximize their throughput information. Such favorable use of noise is the so-called "stochastic resonance" or SR effect at the level of threshold neurons and continuous neurons. This paper presents theoretical and simulation evidence that 1) lone noisy threshold and continuous neurons exhibit the SR effect in terms of the mutual information between random input and output sequences, 2) a new statistically robust learning law can find this entropy-optimal noise level, and 3) the adaptive SR effect is robust against highly impulsive noise with infinite variance. Histograms estimate the relevant probability density functions at each learning iteration. A theorem shows that almost all noise probability density functions produce some SR effect in threshold neurons even if the noise is impulsive and has infinite variance. The optimal noise level in threshold neurons also behaves nonlinearly as the input signal amplitude increases. Simulations further show that the SR effect persists for several sigmoidal neurons and for Gaussian radial-basis-function neurons.

Ergodicity of spike trains: when does trial averaging make sense?

Neuronal information processing is often studied on the basis of spiking patterns. The relevant statistics such as firing rates calculated with the peri-stimulus time histogram are obtained by averaging spiking patterns over many experimental runs. However, animals should respond to one experimental stimulation in real situations, and what is available to the brain is not the trial statistics but the population statistics. Consequently, physiological ergodicity, namely, the consistency between trial averaging and population averaging, is implicitly assumed in the data analyses, although it does not trivially hold true. In this letter, we investigate how characteristics of noisy neural network models, such as single neuron properties, external stimuli, and synaptic inputs, affect the statistics of firing patterns. In particular, we show that how high membrane potential sensitivity to input fluctuations, inability of neurons to remember past inputs, external stimuli with large variability and temporally separated peaks, and relatively few contributions of synaptic inputs result in spike trains that are reproducible over many trials. The reproducibility of spike trains and synchronous firing are contrasted and related to the ergodicity issue. Several numerical calculations with neural network examples are carried out to support the theoretical results.

Noise and coupling affect signal detection and bursting in a simulated physiological neural network

Signal detection in the CNS relies on a complex interaction between the numerous synaptic inputs to the detecting cells. Two effects, stochastic resonance (SR) and coherence resonance (CR) have been shown to affect signal detection in arrays of basic neuronal models. Here, an array of simulated hippocampal CA1 neurons was used to test the hypothesis that physiological noise and electrical coupling can interact to modulate signal detection in the CA1 region of the hippocampus. The array was tested using varying levels of coupling and noise with different input signals. Detection of a subthreshold signal in the network improved as the number of detecting cells increased and as coupling was increased as predicted by previous studies in SR; however, the response depended greatly on the noise characteristics present and varied from SR predictions at times. Careful evaluation of noise characteristics may be necessary to form conclusions about the role of SR in complex systems such as physiological neurons. The coupled array fired synchronous, periodic bursts when presented with noise alone. The synchrony of this firing changed as a function of noise and coupling as predicted by CR. The firing was very similar to certain models of epileptiform activity, leading to a discussion of CR as a possible simple model of epilepsy. A single neuron was unable to recruit its neighbors to a periodic signal unless the signal was very close to the synchronous bursting frequency. These findings, when viewed in comparison with physiological parameters in the hippocampus, suggest that both SR and CR can have significant effects on signal processing in vivo.

Stochastic synchronization in globally coupled phase oscillators

Cooperative effects of periodic force and noise in globally coupled systems are studied using a nonlinear diffusion equation for the number density. The amplitude of the order parameter oscillation is enhanced in an intermediate range of noise strength for a globally coupled bistable system, and the order parameter oscillation is entrained to the external periodic force in an intermediate range of noise strength. These enhancement phenomena of the response of the order parameter in the deterministic equations are interpreted as stochastic resonance and stochastic synchronization in globally coupled systems.

Experimental observation of stochastic resonance in a linear electronic array

We report the experimental observation of array-enhanced stochastic resonance, spatiotemporal synchronization, and noise-enhanced propagation in a simple coupled linear array of bistable electronic triggers. In addition, we highlight an analogy between charge density wave (CDW) like conductivity and spatiotemporal synchronization in stochastic resonances, several aspects of which are supported by the experimental evidence presented here. This may prove to be important in the understanding of nonlinear conductivity in CDW solids.

Monostable array-enhanced stochastic resonance

We present a simple nonlinear system that exhibits multiple distinct stochastic resonances. By adjusting the noise and coupling of an array of underdamped, monostable oscillators, we modify the array's natural frequencies so that the spectral response of a typical oscillator in an array of N oscillators exhibits N-1 different stochastic resonances. Such families of resonances may elucidate and facilitate a variety of noise-mediated cooperative phenomena, such as noise-enhanced propagation, in a broad class of similar nonlinear systems.

Information transmission in parallel threshold arrays: suprathreshold stochastic resonance

The information transmitted through a parallel summing array of noisy threshold elements with a common threshold is considered. In particular, using theoretical and numerical analysis, a recently reported [N. G. Stocks, Phys. Rev. Lett. 84, 2310 (2000)] form of stochastic resonance, termed suprathreshold stochastic resonance (SSR), is studied in detail. SSR is observed to occur in arrays with two or more elements and, unlike stochastic resonance (SR) in a single element, gives rise to noise-induced information gains that occur independent of the setting of the threshold or the size of the signal. The transmitted information is maximized when all thresholds are set to coincide with the signal mean. In this situation, and for large arrays, the noise can enhance performance up to approximately half the theoretical noiseless channel capacity. The theory is tested against digital simulation.

Effects of correlated and independent noise on signal processing in neuronal systems

Stochastic resonance has recently received considerable attention demonstrating that noise can play a constructive role in signal processing. We investigate the effects of input noise on sensory processing via numerical simulation when they are independent of each other or spatially correlated in a globally coupled neuronal network. The network exhibits a coherent behavior in the absence of stimulation. Such ongoing activity has a remarkable influence on neuronal responses to stimuli. In the presence of a subthreshold periodic signal, the activity averaged over neurons can convey precise information about the stimulus in the case of independent noise. On the other hand, when the noise is correlated among the neurons, the average response is nearly as noisy and variable as the responses of the individual neurons. Thus, the spatially correlated noise diminishes the beneficial effects of pooling, although it can evoke synchronous firings of neurons. These suggest that response variability in cortical activity may be closely related to the correlation in input noise.

Chaos-nonchaos phase transitions induced by external noise in ensembles of nonlinearly coupled oscillators

Nonlinear dynamical behaviors of ensembles of nonlinearly coupled oscillators subjected to external noise are studied on the basis of nonlinear Fokker-Planck equations. The effects of two kinds of noise, the Langevin noise and the noise introduced in the coupling strength, are investigated, and phase transitions involving chaos-nonchaos bifurcations are found to occur as the noise level is changed. An H theorem is proposed for the nonlinear Fokker-Planck equation to ensure stability of the Gaussian type solution that is approached for large times.

General measures for signal-noise separation in nonlinear dynamical systems

We propose the straight phi divergences from statistics and information theory (IT) as a set of separation indices between signal and noise in stochastic nonlinear dynamical systems (SNDS). The straight phi divergences provide a more informative alternative to the signal-to-noise ratio (SNR) and have the advantage of being applicable to virtually any kind of stochastic system. Moreover, straight phi divergences are intimately connected to various fundamental limits in IT. Using the properties of straight phi divergences, we show that the classical stochastic resonance (SR) curve can be interpreted as the performance of a nonoptimal, or mismatched, detector applied to the output of a SNDS. Indeed, for a prototype double-well system with forcing in the form of white Gaussian noise plus a possible embedded signal, the whole information loss can be attributed to this mismatch; an optimal detection procedure (for the signal) gives the same performance when based on the output as when based on the input of the system. More generally, it follows that, when characterizing signal-noise separation (or system performance) of SNDS in terms of criteria that do not correspond to IT limits, the choice of criterion can be crucial. The indicated figure of merit will then not be universal and will be relevant only to some family of applications, such as the classical (narrow-band SNR) SR criterion, which is relevant for narrow-band post processing. We illustrate the theory using simple SNDS excited by both wide- and narrow-band signals; however, we stress that the results are applicable to a much larger class of signals and systems.

Nonlinear stochastic resonance: the saga of anomalous output-input gain

We reconsider stochastic resonance (SR) for an overdamped bistable dynamics driven by a harmonic force and Gaussian noise from the viewpoint of the gain behavior, i.e., the signal-to-noise ratio (SNR) at the output divided by that at the input. The primary issue addressed in this work is whether a gain exceeding unity can occur for this archetypal SR model, for subthreshold signals that are beyond the regime of validity of linear response theory: in contrast to nondynamical threshold systems, we find that the nonlinear gain in this conventional SR system exceeds unity only for suprathreshold signals, where SR for the spectral amplification and/or the SNR no longer occurs. Moreover, the gain assumes, at weak to moderate noise strengths, rather small (minimal) values for near-threshold signal amplitudes. The SNR gain generically exhibits a distinctive nonmonotonic behavior versus both the signal amplitude at fixed noise intensity and the noise intensity at fixed signal amplitude. We also test the validity of linear response theory; this approximation is strongly violated for weak noise. At strong noise, however, its validity regime extends well into the large driving regime above threshold. The prominent role of physically realistic noise color is studied for exponentially correlated Gaussian noise of constant intensity scaling and also for constant variance scaling; the latter produces a characteristic, resonancelike gain behavior. The gain for this typical SR setup is further contrasted with the gain behavior for a "soft" potential model.

Stochastic resonance enhanced by dichotomic noise in a bistable system

We study linear responses of a stochastic bistable system driven by dichotomic noise to a weak periodic signal. We show that the effect of stochastic resonance can be greatly enhanced in comparison with the conventional case when dichotomic forcing is absent, that is, both the signal-to-noise ratio and the spectral power amplification reach much greater values than in the standard stochastic resonance setup.

Colored-noise-induced chaotic array synchronization

The effect of a time-correlated Gaussian noise on one-dimensional arrays consisting of diffusively coupled chaotic cells is analyzed. A resonance effect between the time scale of the chaotic attractor and the colored Gaussian noise has been found. As well, depending on the number of cells, coupling, and noise strength, an improvement of the synchronization or a poor synchronization between cells within the array can occur for some values of the time correlation. These nonlinear cooperative effects are studied in terms of a linear stability analysis around the uniform synchronized behavior.

Response of coupled noisy excitable systems to weak stimulation

It is known that coupling can enhance the response of noisy bistable devices to weak periodic modulation. This work examines whether a similar phenomenon occurs in the active rotator model for excitable systems. We study the dynamics of assemblies of weakly periodically modulated active rotators. The addition of noise to these brings about a number of behaviors that have no counterpart in networks of bistable systems. The analysis of the dynamics of the solution of the Fokker-Planck equation of active rotator networks shows that these new behaviors are similar to generic responses of periodically forced autonomous oscillators. This is because noise alone, in the absence of other inputs, can regularize the dynamics of single active rotators through coherence resonance, and lead to regular synchronous activity at the level of networks. We argue that similar phenomena take place in a broad class of excitable systems.

Aperiodic stochastic resonance in chaotic maps

It is shown by means of numerical simulations that aperiodic stochastic resonance occurs in chaotic one-dimensional maps with various kinds of intermittency. The effect appears in the absence of external noise, as the system control parameter is varied. In the case of input signals slowly varying in time the analytic treatment, using the adiabatic approximation based on the expressions for the mean laminar phase duration, yields the input-output covariance function comparable with numerical results. (c) 1998 American Institute of Physics.

Effect of syncytium structure of receptor systems on stochastic resonance induced by chaotic potential fluctuation

To study a role of syncytium structure of sensory receptor systems in the detection of weak signals through stochastic resonance, we present a model of a receptor system with syncytium structure in which receptor cells are interconnected by gap junctions. The apical membrane of each cell includes two kinds of ion channels whose gating processes are described by the deterministic model. The membrane potential of each cell fluctuates chaotically or periodically, depending on the dynamical state of collective channel gating. The chaotic fluctuation of membrane potential acts as internal noise for the stochastic resonance. The detection ability of the system increases as the electric conductance between adjacent cells generated by the gap junction increases. This effect of gap junctions arises mainly from the fact that the synchronization of chaotic fluctuation of membrane potential between the receptor cells is strengthened as the density of gap junctions is increased.

Stochastic resonance in mammalian neuronal networks

We present stochastic resonance observed in the dynamics of neuronal networks from mammalian brain. Both sinusoidal signals and random noise were superimposed into an applied electric field. As the amplitude of the noise component was increased, an optimization (increase then decrease) in the signal-to-noise ratio of the network response to the sinusoidal signal was observed. The relationship between the measures used to characterize the dynamics is discussed. Finally, a computational model of these neuronal networks that includes the neuronal interactions with the electric field is presented to illustrate the physics behind the essential features of the experiment. (c) 1998 American Institute of Physics.

Stochastic resonance in coupled nonlinear dynamic elements

We investigate the response of a linear chain of diffusively coupled diode resonators under the influence of thermal noise. We also examine the connection between spatiotemporal stochastic resonance and the presence of kink-antikink pairs in the array. The interplay of nucleation rates and kink speeds is briefly addressed. The experimental results are supplemented with simulations on a coupled map lattice. We furthermore present analytical results for the synchronization and signal processing properties of a Phi(4) field theory and explore the effects of various forms of nonlinear coupling. (c) 1998 American Institute of Physics.