Explosive Behaviors on Coupled Fractional-Order System

Fractional derivatives provide a prominent platform for various chemical and physical system with memory and hereditary properties, while most of the previous differential systems used to describe dynamic phenomena including oscillation quenching are integer order. Here, effects of fractional derivative on the transition process from oscillatory state to stationary state are illustrated for the first time on mean-filed coupled oscillators. It is found the fractional derivative could induce the emergence of a first-order discrete transition with hysteresis between oscillatory and stationary state. However, if the fractional derivative is smaller than the critical value, the transition will be invertible. Besides, the theoretical conditions for the steady state are calculated via Lyapunov indirect method which probe that, the backward transition point is unrelated to mean-field density. Our result is a step forward in enlightening the control mechanism of explosive phenomenon, which is of great importance to highlight the function of fractional-order derivative in the emergence of collective behaviors on coupled nonlinear model.

Spontaneous Fluctuations in Oscillatory Brain State Cause Differences in Transcranial Magnetic Stimulation Effects Within and Between Individuals

Transcranial magnetic stimulation (TMS) can cause measurable effects on neural activity and behavioral performance in healthy volunteers. In addition, TMS is increasingly used in clinical practice for treating various neuropsychiatric disorders. Unfortunately, TMS-induced effects show large intra- and inter-subject variability, hindering its reliability, and efficacy. One possible source of this variability may be the spontaneous fluctuations of neuronal oscillations. We present recent studies using multimodal TMS including TMS-EMG (electromyography), TMS-tACS (transcranial alternating current stimulation), and concurrent TMS-EEG-fMRI (electroencephalography, functional magnetic resonance imaging), to evaluate how individual oscillatory brain state affects TMS signal propagation within targeted networks. We demonstrate how the spontaneous oscillatory state at the time of TMS influences both immediate and longer-lasting TMS effects. These findings indicate that at least part of the variability in TMS efficacy may be attributable to the current practice of ignoring (spontaneous) oscillatory fluctuations during TMS. Ignoring this state-dependent spread of activity may cause great individual variability which so far is poorly understood and has proven impossible to control. We therefore also compare two technical solutions to directly account for oscillatory state during TMS, namely, to use (a) tACS to externally control these oscillatory states and then apply TMS at the optimal (controlled) brain state, or (b) oscillatory state-triggered TMS (closed-loop TMS). The described multimodal TMS approaches are paramount for establishing more robust TMS effects, and to allow enhanced control over the individual outcome of TMS interventions aimed at modulating information flow in the brain to achieve desirable changes in cognition, mood, and behavior.

Metacommunity stability and persistence for predation turnoff in selective patches

Predation as an important trophic interaction of ecological communities controls the large-scale patterns of species distribution, population abundance and community structure. Numerous studies address that predation can mediate diversity and regulate the ecological community and food web stability through changes in the behaviour, morphology, development, and abundance of prey. Since predation has large effects on persistence and diversity, the local loss or removal of predation in a community can trigger a cascade of extinctions. In ecological theory, the effect of predation removal has been well studied in foodwebs, but it remains unclear in the case of a spatially distributed community connected by dispersal. In this study, the interaction between local and spatial processes is taken into account, we present how a predation turnoff in selective patches affects the stability and persistence of a metacommunity. Using a simple predator-prey metacommunity with a diffusive dispersal, we show the impact of predation on synchronized, asynchronized and source-sink dynamics. Our results reveal that predation turnoff in very few patches alters a perfectly synchronized oscillatory state into multicluster states consisting of various patterns. In a source-sink behaviour, predation turnoff in a source patch reduces the number of sink patches and changes the clusters. In general, predation turnoff in a finite number of patches increases the number of clusters through asynchronized (inhomogeneous) states, whereas predation turnoff in a larger number of patches can lead to the complete extinction of predators. Typically, there exists a critical number of patches below which the predation turnoff results in asynchronized states and above that predation turnoff leads to a synchronized state in prey population with complete extinction of predators. Further, our results identify the network configurations that exhibit a unique number of clusters. Moreover, prey density from the patches where predation is absent goes to a saturating state near the carrying capacity. Thus, this study stresses that predation turnoff in selective patches acts as a stabilizing mechanism that can promote metacommunity persistence.

Low-Reynolds-number, biflagellated Quincke swimmers with multiple forms of motion

In the limit of zero Reynolds number (Re), swimmers propel themselves exploiting a series of nonreciprocal body motions. For an artificial swimmer, a proper selection of the power source is required to drive its motion, in cooperation with its geometric and mechanical properties. Although various external fields (magnetic, acoustic, optical, etc.) have been introduced, electric fields are rarely utilized to actuate such swimmers experimentally in unbounded space. Here we use uniform and static electric fields to demonstrate locomotion of a biflagellated sphere at low Re via Quincke rotation. These Quincke swimmers exhibit three different forms of motion, including a self-oscillatory state due to elastohydrodynamic–electrohydrodynamic interactions. Each form of motion follows a distinct trajectory in space. Our experiments and numerical results demonstrate a method to generate, and potentially control, the locomotion of artificial flagellated swimmers.

Identifying the drivers of stochastic Dansgaard–Oeschger scale variability in a GCM

<p>The driver of the Dansgaard-Oeschger (DO) events remains uncertain, in part because many models do not show similar behaviour of a climate system tipped into a DO oscillatory state. Here we present results from glacial simulations of the HadCM3 GCM that show stochastic DO-scale variability. This is driven by variations in AMOC strength in response to North Atlantic salinity fluctuations. This represents a salt oscillator, driven by the salinity gradient between the subtropical gyre and Nordic seas. We give a mechanistic explanation of the feedbacks that drive this oscillator, particular the interplay between surface fluxes and advection. We identify that the key trigger that pushes the system into this oscillatory mode is the height of the North American ice sheet, which alters atmospheric and subsequently ocean circulation patterns. Our results highlight that glacial background conditions and ice sheet height act to push the system past a tipping point and into an oscillatory state on a timescale comparable to the DO events.</p>

Mathematical modeling of Erk activity waves in regenerating zebrafish scales

Erk signaling regulates cellular decisions in many biological contexts. Recently, we have reported a series of Erk activity traveling waves that coordinate regeneration of osteoblast tissue in zebrafish scales. These waves originate from a central source region, propagate as expanding rings, and impart cell growth, thus controlling tissue morphogenesis. Here, we present a minimal reaction-diffusion model for Erk activity waves. The model considers three components: Erk, a diffusible Erk-activator, and an Erk-inhibitor. Erk stimulates both its activator and inhibitor, forming a positive and negative feedback loop, respectively. Our model shows that this system can be excitable and propagate Erk activity waves. Waves originate from a pulsatile source which is modeled by adding a localized basal production of the activator that switches the source region from an excitable to an oscillatory state. As Erk activity periodically rises in the source, it can trigger an excitable wave which travels across the entire tissue. Analysis of the model finds that positive feedback controls the properties of the traveling wavefront and that negative feedback controls the duration of Erk activity peak and the period of Erk activity waves. The geometrical properties of the waves facilitate constraints on the effective diffusivity of the activator, indicating that waves are an efficient mechanism to transfer growth factor signaling rapidly across a large tissue.Significance statementSignaling waves represent a possible mechanism of spatiotemporal organization of multicellular tissues. We have recently shown that waves of activity of the kinase Erk control osteoblast regeneration in adult zebrafish scales. Here, we present a detailed characterization of a mathematical model of these signaling waves. We show that a source region poised in an oscillatory state can broadcast traveling waves of Erk activity in the surrounding excitable tissue. The dynamics of the source control the number and frequency of waves. Geometrical arguments support the notion that excitable Erk waves are an effective mechanism to transport growth factor signaling across a large regenerating tissue.

Pavlovian eyeblink conditioning is severely impaired in tottering mice

Cacna1a encodes the pore-forming α1A subunit of CaV2.1 voltage-dependent calcium channels, which regulate neuronal excitability and synaptic transmission. Purkinje cells in the cortex of cerebellum abundantly express these CaV2.1 channels. Here, we show that homozygous tottering (tg) mice, which carry a loss-of-function Cacna1a mutation, exhibit severely impaired learning in Pavlovian eyeblink conditioning, which is a cerebellar dependent learning task. Performance of reflexive eyeblinks is unaffected in tg mice. Transient seizure activity in tg mice further corrupted the amplitude of eyeblink CRs. Our results indicate that normal calcium homeostasis is imperative for cerebellar learning and that the oscillatory state of the brain can affect the expression thereof.

Effect of time delay on the synchronization of excitatory-inhibitory neural networks

We study a model for a network of synaptically coupled, excitable neurons to identify the role of coupling delays in generating different network behaviors. The network consists of two distinct populations, each of which contains one excitatory-inhibitory neuron pair. The two pairs are coupled via delayed synaptic coupling between the excitatory neurons, while each inhibitory neuron is connected only to the corresponding excitatory neuron in the same population. We show that multiple equilibria can exist depending on the strength of the excitatory coupling between the populations. We conduct linear stability analysis of the equilibria and derive necessary conditions for delay-induced Hopf bifurcation. We show that these can induce two qualitatively different phase-locked behaviors, with the type of behavior determined by the sizes of the coupling delays. Numerical bifurcation analysis and simulations supplement and confirm our analytical results. Our work shows that the resting equilibrium point is unaffected by the coupling, thus the network exhibits bistability between a rest state and an oscillatory state. This may help understand how rhythms spontaneously arise neuronal networks.

Taming of the Hopf bifurcation in a driven El Niño model

In this paper, we consider the well-known Vallis model for El Niño driven by an external excitation. The bifurcation studies on the driven Vallis model are conducted with different control parameters. Then we discuss about the taming of the Hopf bifurcation by varying the driving function. We could note that the system changes its state from stable steady state to oscillatory state and vice versa which is achieved by changing the driving function. Finally, two parameter bifurcation plots are derived to show that impact of the driving function on the system bifurcation points.

The Effect of a Non-Local Fractional Operator in an Asymmetrical Glucose-Insulin Regulatory System: Analysis, Synchronization and Electronic Implementation

For studying biological conditions with higher precision, the memory characteristics defined by the fractional-order versions of living dynamical systems have been pointed out as a meaningful approach. Therefore, we analyze the dynamics of a glucose-insulin regulatory system by applying a non-local fractional operator in order to represent the memory of the underlying system, and whose state-variables define the population densities of insulin, glucose, and β-cells, respectively. We focus mainly on four parameters that are associated with different disorders (type 1 and type 2 diabetes mellitus, hypoglycemia, and hyperinsulinemia) to determine their observation ranges as a relation to the fractional-order. Like many preceding works in biosystems, the resulting analysis showed chaotic behaviors related to the fractional-order and system parameters. Subsequently, we propose an active control scheme for forcing the chaotic regime (an illness) to follow a periodic oscillatory state, i.e., a disorder-free equilibrium. Finally, we also present the electronic realization of the fractional glucose-insulin regulatory model to prove the conceptual findings.