Island myriads in periodic potentials.

A phenomenon of emergence of stability islands in phase space is reported for two periodic potentials with tiling symmetries, one square and the other hexagonal, inspired by bidimensional Hamiltonian models of optical lattices. The structures found, here termed as island myriads, resemble web-tori with notable fractality and arise at energy levels reaching that of unstable equilibria. In general, the myriad is an arrangement of concentric island chains with properties relying on the translational and rotational symmetries of the potential functions. In the square system, orbits within the myriad come in isochronous pairs and can have different periodic closure, either returning to their initial position or jumping to identical sites in neighbor cells of the lattice, therefore impacting transport properties. As seen when compared to a more generic case, i.e., the rectangular lattice, the breaking of square symmetry disrupts the myriad even for small deviations from its equilateral configuration. For the hexagonal case, the myriad was found but in attenuated form, mostly due to extra instabilities in the potential surface that prevent the stabilization of orbits forming the chains.

Adaptive exponential integrate-and-fire model with fractal extension.

The description of neuronal activity has been of great importance in neuroscience. In this field, mathematical models are useful to describe the electrophysical behavior of neurons. One successful model used for this purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is composed of two ordinary differential equations. Usually, this model is considered in the standard formulation, i.e., with integer order derivatives. In this work, we propose and study the fractal extension of Adex model, which in simple terms corresponds to replacing the integer derivative by non-integer. As non-integer operators, we choose the fractal derivatives. We explore the effects of equal and different orders of fractal derivatives in the firing patterns and mean frequency of the neuron described by the Adex model. Previous results suggest that fractal derivatives can provide a more realistic representation due to the fact that the standard operators are generalized. Our findings show that the fractal order influences the inter-spike intervals and changes the mean firing frequency. In addition, the firing patterns depend not only on the neuronal parameters but also on the order of respective fractal operators. As our main conclusion, the fractal order below the unit value increases the influence of the adaptation mechanism in the spike firing patterns.

Impact of periodic vaccination in SEIRS seasonal model.

We study three different strategies of vaccination in an SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) seasonal forced model, which are (i) continuous vaccination; (ii) periodic short-time localized vaccination, and (iii) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating intensive immunization at inflection of the transmissivity curve. Therefore, a vaccination rate of 44.4% of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years, and show the relevance of the immunization campaign annual reaching for controlling the infection spreading. Regarding the dynamical behavior of the model, our simulations show that chaotic and periodic solutions as well as bi-stable regions depend on the vaccination parameters range.

Multistability and chaos in SEIRS epidemic model with a periodic time-dependent transmission rate.

In this work, we study the dynamics of a susceptible-exposed-infectious-recovered-susceptible epidemic model with a periodic time-dependent transmission rate. Emphasizing the influence of the seasonality frequency on the system dynamics, we analyze the largest Lyapunov exponent along parameter planes finding large chaotic regions. Furthermore, in some ranges, there are shrimp-like periodic structures. We highlight the system multistability, identifying the coexistence of periodic orbits for the same parameter values, with the infections maximum distinguishing by up one order of magnitude, depending only on the initial conditions. In this case, the basins of attraction have self-similarity. Parametric configurations, for which both periodic and non-periodic orbits occur, cover 13.20% of the evaluated range. We also identified the coexistence of periodic and chaotic attractors with different maxima of infectious cases, where the periodic scenario peak reaches approximately 50% higher than the chaotic one.

Nontwist field line mapping in a tokamak with ergodic magnetic limiter.

For tokamaks with uniform magnetic shear, Martin and Taylor have proposed a symplectic map which has been used to describe the magnetic field lines at the plasma edge perturbed by an ergodic magnetic limiter. We propose an analytical magnetic field line map, based on the Martin-Taylor map, for a tokamak with arbitrary safety factor profile. With the inclusion of a nonmonotonic profile, we obtain a nontwist map which presents the characteristic properties of degenerate systems, such as the twin islands scenario, shearless curve, and separatrix reconnection. We estimate the width of the islands and describe their changes of shape for large values of the limiter current. From our numerical simulations about the shearless curve, we show that its position and aspect depend on the control parameters.

The Roles of Potassium and Calcium Currents in the Bistable Firing Transition.

Healthy brains display a wide range of firing patterns, from synchronized oscillations during slow-wave sleep to desynchronized firing during movement. These physiological activities coexist with periods of pathological hyperactivity in the epileptic brain, where neurons can fire in synchronized bursts. Most cortical neurons are pyramidal regular spiking (RS) cells with frequency adaptation and do not exhibit bursts in current-clamp experiments (in vitro). In this work, we investigate the transition mechanism of spike-to-burst patterns due to slow potassium and calcium currents, considering a conductance-based model of a cortical RS cell. The joint influence of potassium and calcium ion channels on high synchronous patterns is investigated for different synaptic couplings (gsyn) and external current inputs (I). Our results suggest that slow potassium currents play an important role in the emergence of high-synchronous activities, as well as in the spike-to-burst firing pattern transitions. This transition is related to the bistable dynamics of the neuronal network, where physiological asynchronous states coexist with pathological burst synchronization. The hysteresis curve of the coefficient of variation of the inter-spike interval demonstrates that a burst can be initiated by firing states with neuronal synchronization. Furthermore, we notice that high-threshold (IL) and low-threshold (IT) ion channels play a role in increasing and decreasing the parameter conditions (gsyn and I) in which bistable dynamics occur, respectively. For high values of IL conductance, a synchronous burst appears when neurons are weakly coupled and receive more external input. On the other hand, when the conductance IT increases, higher coupling and lower I are necessary to produce burst synchronization. In light of our results, we suggest that channel subtype-specific pharmacological interactions can be useful to induce transitions from pathological high bursting states to healthy states.

The Role of Potassium and Calcium Currents in the Bistable Firing Transition.

Healthy brains display a wide range of firing patterns, from synchronized oscillations during slowwave sleep to desynchronized firing during movement. These physiological activities coexist with periods of pathological hyperactivity in the epileptic brain, where neurons can fire in synchronized bursts. Most cortical neurons are pyramidal regular spiking cells (RS) with frequency adaptation and do not exhibit bursts in current-clamp experiments ( in vitro ). In this work, we investigate the transition mechanism of spike-to-burst patterns due to slow potassium and calcium currents, considering a conductance-based model of a cortical RS cell. The joint influence of potassium and calcium ion channels on high synchronous patterns is investigated for different synaptic couplings ( g syn ) and external current inputs ( I ). Our results suggest that slow potassium currents play an important role in the emergence of high-synchronous activities, as well as in the spike-to-burst firing pattern transitions. This transition is related to bistable dynamics of the neuronal network, where physiological asynchronous states coexist with pathological burst synchronization. The hysteresis curve of the coefficient of variation of the inter-spike interval demonstrates that a burst can be initiated by firing states with neuronal synchronization. Furthermore, we notice that high-threshold ( I L ) and low-threshold ( I T ) ion channels play a role in increasing and decreasing the parameter conditions ( g syn and I ) in which bistable dynamics occur, respectively. For high values of I L conductance, a synchronous burst appears when neurons are weakly coupled and receive more external input. On the other hand, when the conductance I T increases, higher coupling and lower I are necessary to produce burst synchronization. In light of our results, we suggest that channel subtype-specific pharmacological interactions can be useful to induce transitions from pathological high bursting states to healthy states.

Stickiness and recurrence plots: An entropy-based approach.

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels.

Finite-time recurrence analysis of chaotic trajectories in Hamiltonian systems.

In this work, we show that a finite-time recurrence analysis of different chaotic trajectories in two-dimensional non-linear Hamiltonian systems provides useful prior knowledge of their dynamical behavior. By defining an ensemble of initial conditions, evolving them until a given maximum iteration time, and computing the recurrence rate of each orbit, it is possible to find particular trajectories that widely differ from the average behavior. We show that orbits with high recurrence rates are the ones that experience stickiness, being dynamically trapped in specific regions of the phase space. We analyze three different non-linear maps and present our numerical observations considering particular features in each of them. We propose the described approach as a method to visually illustrate and characterize regions in phase space with distinct dynamical behaviors.

Suppression of chaotic bursting synchronization in clustered scale-free networks by an external feedback signal.

Oscillatory activities in the brain, detected by electroencephalograms, have identified synchronization patterns. These synchronized activities in neurons are related to cognitive processes. Additionally, experimental research studies on neuronal rhythms have shown synchronous oscillations in brain disorders. Mathematical modeling of networks has been used to mimic these neuronal synchronizations. Actually, networks with scale-free properties were identified in some regions of the cortex. In this work, to investigate these brain synchronizations, we focus on neuronal synchronization in a network with coupled scale-free networks. The networks are connected according to a topological organization in the structural cortical regions of the human brain. The neuronal dynamic is given by the Rulkov model, which is a two-dimensional iterated map. The Rulkov neuron can generate quiescence, tonic spiking, and bursting. Depending on the parameters, we identify synchronous behavior among the neurons in the clustered networks. In this work, we aim to suppress the neuronal burst synchronization by the application of an external perturbation as a function of the mean-field of membrane potential. We found that the method we used to suppress synchronization presents better results when compared to the time-delayed feedback method when applied to the same model of the neuronal network.

Emergence of Neuronal Synchronisation in Coupled Areas.

One of the most fundamental questions in the field of neuroscience is the emergence of synchronous behaviour in the brain, such as phase, anti-phase, and shift-phase synchronisation. In this work, we investigate how the connectivity between brain areas can influence the phase angle and the neuronal synchronisation. To do this, we consider brain areas connected by means of excitatory and inhibitory synapses, in which the neuron dynamics is given by the adaptive exponential integrate-and-fire model. Our simulations suggest that excitatory and inhibitory connections from one area to another play a crucial role in the emergence of these types of synchronisation. Thus, in the case of unidirectional interaction, we observe that the phase angles of the neurons in the receiver area depend on the excitatory and inhibitory synapses which arrive from the sender area. Moreover, when the neurons in the sender area are synchronised, the phase angle variability of the receiver area can be reduced for some conductance values between the areas. For bidirectional interactions, we find that phase and anti-phase synchronisation can emerge due to excitatory and inhibitory connections. We also verify, for a strong inhibitory-to-excitatory interaction, the existence of silent neuronal activities, namely a large number of excitatory neurons that remain in silence for a long time.

Curry-Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems.

The routes to chaos play an important role in predictions about the transitions from regular to irregular behavior in nonlinear dynamical systems, such as electrical oscillators, chemical reactions, biomedical rhythms, and nonlinear wave coupling. Of special interest are dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. If the latter satisfies the so-called twist property, the corresponding dissipative version can be called a "dissipative twist system." Transitions to chaos in these systems are well established; for instance, the Curry-Yorke route describes the transition from a quasiperiodic attractor on torus to chaos passing by a chaotic banded attractor. In this paper, we study the transitions from an attractor on torus to chaotic motion in dissipative nontwist systems. We choose the dissipative standard nontwist map, which is a non-conservative version of the standard nontwist map. In our simulations, we observe the same transition to chaos that happens in twist systems, known as a soft one, where the quasiperiodic attractor becomes wrinkled and then chaotic through the Curry-Yorke route. By the Lyapunov exponent, we study the nature of the orbits for a different set of parameters, and we observe that quasiperiodic motion and periodic and chaotic behavior are possible in the system. We observe that they can coexist in the phase space, implying in multistability. The different coexistence scenarios were studied by the basin entropy and by the boundary basin entropy.

Dynamics of epidemics: Impact of easing restrictions and control of infection spread.

During an infectious disease outbreak, mathematical models and computational simulations are essential tools to characterize the epidemic dynamics and aid in design public health policies. Using these tools, we provide an overview of the possible scenarios for the COVID-19 pandemic in the phase of easing restrictions used to reopen the economy and society. To investigate the dynamics of this outbreak, we consider a deterministic compartmental model (SEIR model) with an additional parameter to simulate the restrictions. In general, as a consequence of easing restrictions, we obtain scenarios characterized by high spikes of infections indicating significant acceleration of the spreading disease. Finally, we show how such undesirable scenarios could be avoided by a control strategy of successive partial easing restrictions, namely, we tailor a successive sequence of the additional parameter to prevent spikes in phases of low rate of transmissibility.

Influence of Autapses on Synchronization in Neural Networks With Chemical Synapses.

A great deal of research has been devoted on the investigation of neural dynamics in various network topologies. However, only a few studies have focused on the influence of autapses, synapses from a neuron onto itself via closed loops, on neural synchronization. Here, we build a random network with adaptive exponential integrate-and-fire neurons coupled with chemical synapses, equipped with autapses, to study the effect of the latter on synchronous behavior. We consider time delay in the conductance of the pre-synaptic neuron for excitatory and inhibitory connections. Interestingly, in neural networks consisting of both excitatory and inhibitory neurons, we uncover that synchronous behavior depends on their synapse type. Our results provide evidence on the synchronous and desynchronous activities that emerge in random neural networks with chemical, inhibitory and excitatory synapses where neurons are equipped with autapses.

Influence of Delayed Conductance on Neuronal Synchronization.

In the brain, the excitation-inhibition balance prevents abnormal synchronous behavior. However, known synaptic conductance intensity can be insufficient to account for the undesired synchronization. Due to this fact, we consider time delay in excitatory and inhibitory conductances and study its effect on the neuronal synchronization. In this work, we build a neuronal network composed of adaptive integrate-and-fire neurons coupled by means of delayed conductances. We observe that the time delay in the excitatory and inhibitory conductivities can alter both the state of the collective behavior (synchronous or desynchronous) and its type (spike or burst). For the weak coupling regime, we find that synchronization appears associated with neurons behaving with extremes highest and lowest mean firing frequency, in contrast to when desynchronization is present when neurons do not exhibit extreme values for the firing frequency. Synchronization can also be characterized by neurons presenting either the highest or the lowest levels in the mean synaptic current. For the strong coupling, synchronous burst activities can occur for delays in the inhibitory conductivity. For approximately equal-length delays in the excitatory and inhibitory conductances, desynchronous spikes activities are identified for both weak and strong coupling regimes. Therefore, our results show that not only the conductance intensity, but also short delays in the inhibitory conductance are relevant to avoid abnormal neuronal synchronization.

Ratchet current in nontwist Hamiltonian systems.

Non-monotonic area-preserving maps violate the twist condition locally in phase space, giving rise to shearless invariant barriers surrounded by twin island chains in these regions of phase space. For the extended standard nontwist map, with two resonant perturbations with distinct wave numbers, we investigate the presence of such barriers and their associated island chains and compare our results with those that have been reported for the standard nontwist map with only one perturbation. Furthermore, we determine in the control parameter space the existence of the shearless barrier and the influence of the additional wave number on this condition. We show that only for odd second wave numbers are the twin island chains symmetrical. Moreover, for even wave numbers, the lack of symmetry between the chains of twin islands generates a ratchet effect that implies a directed transport in the phase space.

Basin of attraction for chimera states in a network of Rössler oscillators.

Chimera states are spatiotemporal patterns in which coherent and incoherent dynamics coexist simultaneously. These patterns were observed in both locally and nonlocally coupled oscillators. We study the existence of chimera states in networks of coupled Rössler oscillators. The Rössler oscillator can exhibit periodic or chaotic behavior depending on the control parameters. In this work, we show that the existence of coherent, incoherent, and chimera states depends not only on the coupling strength, but also on the initial state of the network. The initial states can belong to complex basins of attraction that are not homogeneously distributed. Due to this fact, we characterize the basins by means of the uncertainty exponent and basin stability. In our simulations, we find basin boundaries with smooth, fractal, and riddled structures.

State-dependent vulnerability of synchronization.

A state-dependent vulnerability of synchronization is shown to exist in a complex network composed of numerically simulated electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the current state of its trajectory. We address such state dependence by systematically perturbing the synchronized system at states equally distributed along its trajectory. We find the states at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive states by defining a safety index between them. Finally, we demonstrate that the observed vulnerability is due to the existence of an unstable chaotic set in the system's state space.

Dynamical thermalization in time-dependent billiards.

Numerical experiments of the statistical evolution of an ensemble of noninteracting particles in a time-dependent billiard with inelastic collisions reveals the existence of three statistical regimes for the evolution of the speed ensemble, namely, diffusion plateau, normal growth/exponential decay, and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Furthermore, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations, the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for an intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for "thermalization," where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space toward a stationary distribution function with a kind of "reservoir temperature" determined by the boundary oscillation amplitude and the restitution coefficient.

Spike-burst chimera states in an adaptive exponential integrate-and-fire neuronal network.

Chimera states are spatiotemporal patterns in which coherence and incoherence coexist. We observe the coexistence of synchronous (coherent) and desynchronous (incoherent) domains in a neuronal network. The network is composed of coupled adaptive exponential integrate-and-fire neurons that are connected by means of chemical synapses. In our neuronal network, the chimera states exhibit spatial structures both with spike and burst activities. Furthermore, those desynchronized domains not only have either spike or burst activity, but we show that the structures switch between spikes and bursts as the time evolves. Moreover, we verify the existence of multicluster chimera states.