Alexander B. Neiman

Nonlinear dynamics of chaotic and stochastic systems : tutorial and modern developments

From the contents: Tutorial * Dynamical Systems * Fluctuations in Dynamic Systems * Synchronization of Periodic Systems * Dynamical Chaos * Routes to Chaos * Synchronization of Chaos * Controlling Chaos * Reconstruction of Dynamical Systems * Stochastic Dynamics * Stochastic Resonance * Synchronization of Stochastic Systems * The Beneficial Role of Noise in Excitable Systems * Noise Induced Transport.

Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons.

We study global bifurcations of the chaotic attractor in a modified Hodgkin-Huxley model of thermally sensitive neurons. The control parameter for this model is the temperature. The chaotic behavior is realized over a wide range of temperatures and is visualized using interspike intervals. We observe an abrupt increase of the interspike intervals in a certain temperature region. We identify this as a homoclinic bifurcation of a saddle-focus fixed point which is embedded in the chaotic attractors. The transition is accompanied by intermittency, which obeys a universal scaling law for the average length of trajectory segments exhibiting only short interspike intervals with the distance from the onset of intermittency. We also present experimental results of interspike interval measurements taken from the crayfish caudal photoreceptor, which qualitatively demonstrate the same bifurcation structure. (c) 2000 American Institute of Physics.

Stochastic resonance in chaotic systems

The phenomenon of stochastic resonance (SR) is investigated for chaotic systems perturbed by white noise and a harmonic force. The bistable discrete map and the Lorenz system are considered as models. It is shown that SR in chaotic systems can be realized via both parameter variation (in the absence of noise) and by variation of the noise intensity with fixed values of the other parameters.

Stochastic resonance in psychophysics and in animal behavior

Abstract. A recent analysis of the energy detector model in sensory psychophysics concluded that stochastic resonance does not occur in a measure of signal detectability (d′), but can occur in a percent-correct measure of performance as an epiphenomenon of nonoptimal criterion placement [Tougaard (2000) Biol Cybern 83: 471–480]. When generalized to signal detection in sensory systems in general, this conclusion is a serious challenge to the idea that stochastic resonance could play a significant role in sensory processing in humans and other animals. It also seems to be inconsistent with recent demonstrations of stochastic resonance in sensory systems of both nonhuman animals and humans using measures of system performance such as signal-to-noise ratio of power spectral densities and percent-correct detections in a two-interval forced-choice paradigm, both closely related to d′. In this paper we address this apparent dilemma by discussing several models of how stochastic resonance can arise in signal detection systems, including especially those that implement a “soft threshold” at the input transform stage. One example involves redefining d′ for energy increments in terms of parameters of the spike-count distribution of FitzHugh–Nagumo neurons. Another involves a Poisson spike generator that receives an exponentially transformed noisy periodic signal. In this case it can be shown that the signal-to-noise ratio of the power spectral density at the signal frequency, which exhibits stochastic resonance, is proportional to d′. Finally, a variant of d′ is shown to exhibit stochastic resonance when calculated directly from the distributions of power spectral densities at the signal frequency resulting from transformation of noise alone and a noisy signal by a sufficiently steep nonlinear response function. All of these examples, and others from the literature, imply that stochastic resonance is more than an epiphenomenon, although significant limitations to the extent to which adding noise can aid detection do exist.

Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems.

We study the stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. In the course of this transition diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed. In order to understand the details and mechanisms of these noise-induced dynamics we consider the thermodynamic limit N-->infinity of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good qualitative agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.

Long-range correlations between letters and sentences in texts

We mapped three long texts to random walks and calculated several correlation measures as Holder exponents, higher-order cumulants and power spectra. By means of computer experiments we have found that shuffling on/or below the sentence level generates strings showing no anomalous diffusion, no higher-order cumulants and no power spectra with 1/fδ-shape. In this way we have shown that the long correlations reflected in these measures are not based on correlations inside sentences but reflect the large-scale structure beyond the sentence level.

Stochastic resonance in two coupled bistable systems

Abstract We consider the collective response of two coupled bistable oscillators driven by independent noise sources to a periodical force. We have found that there exists an optimal value of the coupling strength for which the signal-to-noise ratio of the collective response has its maximal value. The connection of this effect with the phenomenon of stochastic synchronization is established.

Noise-induced impulse pattern modifications at different dynamical period-one situations in a computer model of temperature encoding.

We used a minimal Hodgkin-Huxley type model of cold receptor discharges to examine how noise interferes with the non-linear dynamics of the ionic mechanisms of neuronal stimulus encoding. The model is based on the assumption that spike-generation depends on subthreshold oscillations. With physiologically plausible temperature scaling, it passes through different impulse patterns which, with addition of noise, are in excellent agreement with real experimental data. The interval distributions of purely deterministic simulations, however, exhibit considerable differences compared to the noisy simulations especially at the bifurcations of deterministically period-one discharges. We, therefore, analyzed the effects of noise in different situations of deterministically regular period-one discharges: (1) at high-temperatures near the transition to subthreshold oscillations and to burst discharges, and (2) at low-temperatures close to and more far away from the bifurcations to chaotic dynamics. The data suggest that addition of noise can considerably extend the dynamical behavior of the system with coexistence of different dynamical situations at deterministically fixed parameter constellations. Apart from well-described coexistence of spike-generating and subthreshold oscillations also mixtures of tonic and bursting patterns can be seen and even transitions to unstable period-one orbits seem to appear. The data indicate that cooperative effects between low- and high-dimensional dynamics have to be considered as qualitatively important factors in neuronal encoding.

CONTROLLING STOCHASTIC OSCILLATIONS CLOSE TO A HOPF BIFURCATION BY TIME-DELAYED FEEDBACK

We study the effect of a time-delayed feedback upon a Van der Pol oscillator under the influence of white noise in the regime below the Hopf bifurcation where the deterministic system has a stable fixed point. We show that both the coherence and the frequency of the noise-induced oscillations can be controlled by varying the delay time and the strength of the control force. Approximate analytical expressions for the power spectral density and the coherence properties of the stochastic delay differential equation are developed, and are in good agreement with our numerical simulations. Our analytical results elucidate how the correlation time of the controlled stochastic oscillations can be maximized as a function of delay and feedback strength.

Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells

We employ a Hodgkin-Huxley-type model of basolateral ionic currents in bullfrog saccular hair cells for studying the genesis of spontaneous voltage oscillations and their role in shaping the response of the hair cell to external mechanical stimuli. Consistent with recent experimental reports, we find that the spontaneous dynamics of the model can be categorized using conductance parameters of calcium-activated potassium, inward rectifier potassium, and mechano-electrical transduction (MET) ionic currents. The model is demonstrated for exhibiting a broad spectrum of autonomous rhythmic activity, including periodic and quasi-periodic oscillations with two independent frequencies as well as various regular and chaotic bursting patterns. Complex patterns of spontaneous oscillations in the model emerge at small values of the conductance of Ca2+-activated potassium currents. These patterns are significantly affected by thermal fluctuations of the MET current. We show that self-sustained regular voltage oscillations lead to enhanced and sharply tuned sensitivity of the hair cell to weak mechanical periodic stimuli. While regimes of chaotic oscillations are argued to result in poor tuning to sinusoidal driving, chaotically oscillating cells do provide a high sensitivity to low-frequency variations of external stimuli.