Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique

Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.

Sex, ducks, and rock "n" roll: Mathematical model of sexual response

In this paper, we derive and analyze a mathematical model of a sexual response. As a starting point, we discuss two studies that proposed a connection between a sexual response cycle and a cusp catastrophe and explain why that connection is incorrect but suggests an analogy with excitable systems. This then serves as a basis for derivation of a phenomenological mathematical model of a sexual response, in which the variables represent levels of physiological and psychological arousal. Bifurcation analysis is performed to identify stability properties of the model's steady state, and numerical simulations are performed to illustrate different types of behavior that can be observed in the model. Solutions corresponding to the dynamics associated with the Masters-Johnson sexual response cycle are represented by "canard"-like trajectories that follow an unstable slow manifold before making a large excursion in the phase space. We also consider a stochastic version of the model, for which spectrum, variance, and coherence of stochastic oscillations around a deterministically stable steady state are found analytically, and confidence regions are computed. Large deviation theory is used to explore the possibility of stochastic escape from the neighborhood of the deterministically stable steady state, and the methods of an action plot and quasi-potential are employed to compute most probable escape paths. We discuss implications of the results for facilitating better quantitative understanding of the dynamics of a human sexual response and for improving clinical practice.

Turing instability mechanism of short-memory formation in multilayer FitzHugh-Nagumo network

Introduction

The study of brain function has been favored by scientists, but the mechanism of short-term memory formation has yet to be precise.

Research problem

Since the formation of short-term memories depends on neuronal activity, we try to explain the mechanism from the neuron level in this paper.

Research contents and methods

Due to the modular structures of the brain, we analyze the pattern properties of the FitzHugh-Nagumo model (FHN) on a multilayer network (coupled by a random network). The conditions of short-term memory formation in the multilayer FHN model are obtained. Then the time delay is introduced to more closely match patterns of brain activity. The properties of periodic solutions are obtained by the central manifold theorem.

Conclusion

When the diffusion coeffcient, noise intensity np, and network connection probability p reach a specific range, the brain forms a relatively vague memory. It is found that network and time delay can induce complex cluster dynamics. And the synchrony increases with the increase of p. That is, short-term memory becomes clearer.

Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information

The problem of synthesis of stochastic sensitivity for equilibrium modes in nonlinear randomly forced dynamical systems with incomplete information is considered. We construct a feedback regulator that uses noisy data on some system state coordinates. For parameters of the regulator providing assigned stochastic sensitivity, a quadratic matrix equation is derived. Attainability of the assigned stochastic sensitivity is reduced to the solvability of this equation. We suggest a constructive algorithm for solving this quadratic matrix equation. These general theoretical results are used to solve the problem of stabilizing equilibrium modes of nonlinear stochastic oscillators under conditions of incomplete information. Details of our approach are illustrated on the example of a van der Pol oscillator.

Anomalous climate dynamics induced by multiplicative and additive noises

Anomalous behavior of a nonlinear climate-vegetation model governed by the multiplicative and additive noises is revealed on the basis of stochastic sensitivity analysis. A specific feature of this model is the bistability with the coexistence of "snowball" equilibrium and "warm" attractor in the form of equilibrium or cycle. It is found that multiplicative and additive noises shift probabilistic distribution in opposite directions. The multiplicative noise introduced into the death rate of vegetation changes the dispersion of random states and their localization in the phase diagram. This type of noise cools down the system and is responsible for its transition to the snowball state. On the contrary, the additive noise warms up the climate with increasing noise intensity. A cumulative effect of multiplicative and additive noises occurs under their simultaneous influence. This effect determining the evolutionary behavior of a climate-vegetation system depends on the ratio of intensities of these noises.

Dynamic behaviors of the FitzHugh-Nagumo neuron model with state-dependent impulsive effects

In present work, in order to reproduce spiking and bursting behavior of real neurons, a new hybrid biological neuron model is established and analyzed by combining the FitzHugh-Nagumo (FHN) neuron model, the threshold for spike initiation and the state-dependent impulsive effects (impulse resetting process). Firstly, we construct Poincaré mappings under different conditions by means of geometric analysis, and then obtain some sufficient criteria for the existence and stability of order-1 or order-2 periodic solution to the impulsive neuron model by finding the fixed point of Poincaré mapping and some geometric analysis techniques. Numerical simulations are given to illustrate and verify our theoretical results. The bifurcation diagrams are presented to describe the phenomena of period-doubling route to chaos, which implies that the dynamic behavior of the neuron model become more complex due to impulsive effects. Furthermore, the correctness and effectiveness of the proposed FitzHugh-Nagumo neuron model with state-dependent impulsive effects are verified by circuit simulation. Finally, the conclusions of this paper are analyzed and summarized, and the effects of random factors on the electrophysiological activities of neuron are discussed by numerical simulation.

Noise-induced spiking-bursting transition in the neuron model with the blue sky catastrophe

We study a special variant of the noise-induced transition between spiking and bursting regimes associated with the blue sky catastrophe bifurcation in the Hindmarsh-Rose neuron model. We show that in the parameter region close to the bifurcation value, where the only attractor of the system is the limit cycle of tonic spiking type, noise can transform the spiking oscillatory regime to the bursting one. This phenomenon is studied by means of power spectral density and interspike intervals statistics. We show that noise shifts the bifurcation value, so that bursting activity can be observed for a wider parameter range. Moreover, we reveal that the stochastic spiking-bursting transitions in this system are accompanied by the change in sign of the Lyapunov exponent. We perform a detailed quantitative analysis of these phenomena with an approach that uses a concept of the stochastic sensitivity function, the confidence domains method, and Mahalanobis metrics.

Degeneracy in the robust expression of spectral selectivity, subthreshold oscillations, and intrinsic excitability of entorhinal stellate cells

Biological heterogeneities are ubiquitous and play critical roles in the emergence of physiology at multiple scales. Although neurons in layer II (LII) of the medial entorhinal cortex (MEC) express heterogeneities in channel properties, the impact of such heterogeneities on the robustness of their cellular-scale physiology has not been assessed. Here, we performed a 55-parameter stochastic search spanning nine voltage- or calcium-activated channels to assess the impact of channel heterogeneities on the concomitant emergence of 10 in vitro electrophysiological characteristics of LII stellate cells (SCs). We generated 150,000 models and found a heterogeneous subpopulation of 449 valid models to robustly match all electrophysiological signatures. We employed this heterogeneous population to demonstrate the emergence of cellular-scale degeneracy in SCs, whereby disparate parametric combinations expressing weak pairwise correlations resulted in similar models. We then assessed the impact of virtually knocking out each channel from all valid models and demonstrate that the mapping between channels and measurements was many-to-many, a critical requirement for the expression of degeneracy. Finally, we quantitatively predict that the spike-triggered average of SCs should be endowed with theta-frequency spectral selectivity and coincidence detection capabilities in the fast gamma-band. We postulate this fast gamma-band coincidence detection as an instance of cellular-scale-efficient coding, whereby SC response characteristics match the dominant oscillatory signals in LII MEC. The heterogeneous population of valid SC models built here unveils the robust emergence of cellular-scale physiology despite significant channel heterogeneities, and forms an efficacious substrate for evaluating the impact of biological heterogeneities on entorhinal network function.

NEW & NOTEWORTHY We assessed the impact of heterogeneities in channel properties on the robustness of cellular-scale physiology of medial entorhinal cortical stellate neurons. We demonstrate that neuronal models with disparate channel combinations were endowed with similar physiological characteristics, as a consequence of the many-to-many mapping between channel properties and the physiological characteristics that they modulate. We predict that the spike-triggered average of stellate cells should be endowed with theta-frequency spectral selectivity and fast gamma-band coincidence detection capabilities.

Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis

A phenomenon of the noise-induced oscillatory multistability in glycolysis is studied. As a basic deterministic skeleton, we consider the two-dimensional Higgins model. The noise-induced generation of mixed-mode stochastic oscillations is studied in various parametric zones. Probabilistic mechanisms of the stochastic excitability of equilibria and noise-induced splitting of randomly forced cycles are analysed by the stochastic sensitivity function technique. A parametric zone of supersensitive Canard-type cycles is localized and studied in detail. It is shown that the generation of mixed-mode stochastic oscillations is accompanied by the noise-induced transitions from order to chaos.

Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model

We study the phenomenon of noise-induced torus bursting on the base of the three-dimensional Hindmarsh-Rose neuron model forced by additive noise. We show that in the parametric zone close to the Neimark-Sacker bifurcation, where the deterministic system exhibits rapid tonic spiking oscillations, random disturbances can turn tonic spiking into bursting, which is characterized by the formation of a peculiar dynamical structure resembling that of a torus. This phenomenon is confirmed by the changes in dispersion of random trajectories as well as the power spectral density and interspike intervals statistics. In particular, we show that as noise increases, the system undergoes P and D bifurcations, transitioning from order to chaos. We ultimately characterize the transition from stochastic (tonic) spiking to bursting by stochastic sensitivity functions.

Activation process in excitable systems with multiple noise sources: One and two interacting units

We consider the coaction of two distinct noise sources on the activation process of a single excitable unit and two interacting excitable units, which are mathematically described by the Fitzhugh-Nagumo equations. We determine the most probable activation paths around which the corresponding stochastic trajectories are clustered. The key point lies in introducing appropriate boundary conditions that are relevant for a class II excitable unit, which can be immediately generalized also to scenarios involving two coupled units. We analyze the effects of the two noise sources on the statistical features of the activation process, in particular demonstrating how these are modified due to the linear or nonlinear form of interactions. Universal properties of the activation process are qualitatively discussed in the light of a stochastic bifurcation that underlies the transition from a stochastically stable fixed point to continuous oscillations.

Stochastic sensitivity analysis of noise-induced suppression of firing and giant variability of spiking in a Hodgkin-Huxley neuron model

We study the stochastic dynamics of a Hodgkin-Huxley neuron model in a regime of coexistent stable equilibrium and a limit cycle. In this regime, noise may suppress periodic firing by switching the neuron randomly to a quiescent state. We show that at a critical value of the injected current, the mean firing rate depends weakly on noise intensity, while the neuron exhibits giant variability of the interspike intervals and spike count. To reveal the dynamical origin of this noise-induced effect, we develop the stochastic sensitivity analysis and use the Mahalanobis metric for this four-dimensional stochastic dynamical system. We show that the critical point of giant variability corresponds to the matching of the Mahalanobis distances from attractors (stable equilibrium and limit cycle) to a three-dimensional surface separating their basins of attraction.

Stochastic sensitivity analysis of the noise-induced excitability in a model of a hair bundle

We study effect of weak noise on the dynamics of a hair bundle model near the excitability threshold and near a subcritical Hopf bifurcation. We analyze numerically noise-induced structural changes in the probability density and the power spectral density of the model. In particular, we show that weak noise can induce oscillations with two distinct frequencies in both excitable and limit-cycle regimes. We then applied a recently developed technique of stochastic sensitivity functions which allows us to estimate threshold values of noise intensity corresponding to these transitions.

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system.

Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect

We study a stochastically forced predator-prey model with Allee effect. In the deterministic case, this model exhibits non-trivial stable equilibrium or limit cycle corresponding to the coexistence of both species. Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of the dispersion of random states in stochastic attractors. Our method allows to construct confidence domains and estimate the threshold value of the intensity for noise generating a transition from the coexistence to the extinction.