Unidirectionally Coupled Research Articles

Influence of sinusoidal forcing on the master FitzHugh–Nagumo neuron model and global dynamics of a unidirectionally coupled two-neuron system

In this paper, we investigate a seven-parameter, five-dimensional dynamical system, specifically a unidirectional coupling of two FitzHugh–Nagumo neuron models, with one neuron being sinusoidally driven. This master–slave configuration features neuron N1 as the master, subjected to an external sinusoidal electrical current, and neuron N2 as the slave, interacting with N1 through an electrical force. We report numerical results for three distinct scenarios where N1 operates in (i) periodic, (ii) quasiperiodic, and (iii) chaotic regimes. The primary objective is to explore how the dynamics of the master neuron N1 influence the coupled system’s behavior. To achieve this, we generated cross sections of the seven-dimensional parameter space, known as parameter planes. Our findings reveal that in the periodic regime of N1, the coupled system exhibits period-adding sequences of Arnold tongue-like structures in the parameter planes. Furthermore, regions of multistability can also be identified in these parameter planes of the coupled system. In the quasiperiodic regime, regions of periodic motion are absent, with only regions of quasiperiodic and chaotic dynamics present. In the chaotic regime of N1, the parameter planes display regions of chaos, hyperchaos, and transient hyperchaos.

Mirroring of synchronization in a bi-layer master-slave configuration of Kuramoto oscillators.

The phenomenon of mirroring of synchronization is investigated in dynamically dissimilar, unidirectionally coupled, bi-layer master-slave configuration of globally coupled Kuramoto oscillators. The dynamics of the master layer depends solely on the distribution of the natural frequencies of its oscillators. On the other hand, the slave layer dynamics depends not only on the distribution of the natural frequencies of its oscillators but also on the unidirectional coupling with the master layer. The standard Kuramoto order parameter is used to study synchronization in the individual layers and of the bi-layer network. A transition to a completely mirroring state is observed in the dynamics of the slave layer, as the mirroring coefficient in the unidirectional coupling is increased. We derive analytically and verify numerically the conditions for the slave layer to fully mimic the synchronization properties of the master layer. It is further shown that while the master and slave layers are individually synchronized, the bi-layer network exhibits a state of frustrated synchronization.

Border Collision Bifurcations and Chaos in a Ring of Three Unidirectionally Coupled Nonmonotonic Piecewise Constant Neurons

The bifurcations and chaos in a ring of three nonmonotonic piecewise constant neurons with unidirectional inhibitory coupling are examined. Periodic solutions in the system undergo border collision bifurcations, which are characteristic of piecewise smooth systems. Conditions for the bifurcations of periodic solutions are derived analytically by using a piecewise constant function as the output function of neurons. Multiple periodic solutions are generated through border collision bifurcations and saddle-node bifurcations. Chaotic attractors are generated through border collision bifurcations and degenerate period-doubling bifurcations. The existence of multiple periodic solutions and chaotic attractors is proven to be common in rings of unidirectionally coupled nonmonotonic neurons.

Neuronal heterogeneity modulates phase synchronization between unidirectionally coupled populations with excitation-inhibition balance.

Several experiments and models have highlighted the importance of neuronal heterogeneity in brain dynamics and function. However, how such a cell-to-cell diversity can affect cortical computation, synchronization, and neuronal communication is still under debate. Previous studies have focused on the effect of neuronal heterogeneity in one neuronal population. Here we are specifically interested in the effect of neuronal variability on the phase relations between two populations, which can be related to different cortical communication hypotheses. It has been recently shown that two spiking neuron populations unidirectionally connected in a sender-receiver configuration can exhibit anticipated synchronization (AS), which is characterized by a negative phase lag. This phenomenon has been reported in electrophysiological data of nonhuman primates and human EEG during a visual discrimination cognitive task. In experiments, the unidirectional coupling could be accessed by Granger causality and can be accompanied by either positive or negative phase difference between cortical areas. Here we propose a model of two coupled populations in which the neuronal heterogeneity can determine the dynamical relation between the sender and the receiver and can reproduce phase relations reported in experiments. Depending on the distribution of parameters characterizing the neuronal firing patterns, the system can exhibit both AS and the usual delayed synchronization regime (DS, with positive phase) as well as a zero-lag synchronization regime and phase bistability between AS and DS. Furthermore, we show that our network can present diversity in their phase relations maintaining the excitation-inhibition balance.

Asymptotic Analysis of Bifurcations in Feedforward Networks

A feedforward network is a unidirectionally coupled chain of dynamical systems in which the first cell is coupled to itself, and each successive cell is coupled to the next one. Feedforward networks have gained considerable interest because of their potential to enhance signal amplification and to manipulate the frequency of oscillations. Indeed, it has been shown that the growth rate of the bifurcation undergone by the final cell is much larger than the expected square root growth rate associated with the standard Hopf bifurcation. In this paper, we present a new approach to studying this growth rate phenomenon. We employ a two-time-scale analysis and asymptotic approximations to detect behavior associated with the growth rate phenomenon that has not been previously observed. In particular, we show that the Hopf bifurcation is not the only bifurcation capable of exhibiting this large growth rate behavior. Using asymptotic methods we show that it is not a special property of the Hopf bifurcation that allows for this accelerated growth rate; it is a combination of the unidirectional coupling and the higher-degree nonlinearities that cause this effect. Furthermore, we show that this large growth rate need not persist away from the bifurcation. In fact, the growth rate is asymptotic to the standard square root growth rate as the bifurcation parameter increases.

Intermittent route to generalized synchronization in bidirectionally coupled chaotic oscillators.

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

On the Relationship between the Theory of Cointegration and the Theory of Phase Synchronization

The theory of cointegration has been a leading theory in econometrics with powerful applications to macroeconomics during the last decades. On the other hand, the theory of phase synchronization for weakly coupled complex oscillators has been one of the leading theories in physics for many years with many applications to different areas of science. For example, in neuroscience phase synchronization is regarded as essential for functional coupling of different brain regions. In an abstract sense, both theories describe the dynamic fluctuation around some equilibrium. In this paper, we point out that there exists a very close connection between both theories. Apart from phase jumps, a stochastic version of the Kuramoto equations can be approximated by a cointegrated system of difference equations. As one consequence, the rich theory on statistical inference for cointegrated systems can immediately be applied for statistical inference on phase synchronization based on empirical data. This includes tests for phase synchronization, tests for unidirectional coupling and the identification of the equilibrium from data including phase shifts. We study two examples on a unidirectionally coupled Rössler–Lorenz system and on electrochemical oscillators. The methods from cointegration may also be used to investigate phase synchronization in complex networks. Conversely, there are many interesting results on phase synchronization which may inspire new research on cointegration.

Transition Dynamics of Epileptic Seizures in the Coupled Thalamocortical Network Model

In this paper, based on the two-compartment unidirectionally coupled thalamocortical model network, we investigated the transition dynamics of epileptic seizures, by considering the inhibitory coupling strength from cortical inhibitory interneuronal (IN) population to excitatory pyramidal (PY) neuronal population as the key bifurcation parameter. The results show that in the single compartment thalamocortical model, inner-compartment inhibitory functions of IN can make the system transit from the absence seizures to the tonic oscillations. In the case of two-compartment coupled thalamocortical model network, the inter-compartment inhibitory coupling functions from the first compartment can drive the second compartment to more easily initiate the absence and tonic seizures at the lower inhibitory coupling strengths, respectively. Also, the driven functions can make the amplitudes of these seizures vary irregularly. Detailed investigations reveal that along with the various state transitions, the system consecutively undergoes Hopf bifurcations, fold of cycles bifurcations and torus bifurcations, respectively. In particular, the reinforcing inter-compartment inhibitory coupling function can induce the chaotic dynamics. We highlight the unidirectional coupling functions between two compartments which might give new insights into the propagation and evolution dynamics of epileptic seizures.

The route to synchrony via drum head mode and mixed oscillatory state in star coupled Hindmarsh–Rose neural network

Network of coupled oscillators exhibit different types of spatiotemporal patterns. We report that as the coupling strength increases the unidirectionally coupled Hindmarsh–Rose neuron star network will synchronize. The condition for synchronization has been evaluated using Lyapunov function method. We also discuss the dynamics of the system in the presence of controllers. The control input generate interesting behaviors which consist of clusters of spatially coherent domains depending on the coupling strength. Drum head mode, mixed oscillatory state, desynchrony, and multi cluster states are formed and cluster reduction takes place before settling to complete synchrony. The evolution of a perfectly synchronized state via drum head mode, mixed oscillatory state, and clusters from a desynchronized state is reported for the first time. The parameter values which lead to stable cluster formation is also discussed. Our results suggest that in the presence of controllers the common oscillator in the star network behaves as a driver and generates the transitions and cluster formation acts as a precursor to complete synchrony in Hindmarsh–Rose model with unidirectional star coupling.

Stabilization of metastable dynamical rotating waves in a ring of unidirectionally coupled sigmoidal neurons due to shortcuts

Effects of shortcut connection on metastable dynamical rotating waves in a ring of sigmoidal neurons with unidirectional excitatory coupling are considered. A kinematical equation describing the propagation of wave fronts is derived with a sign function for the output function of neurons. Unstable rotating waves can be stabilized in the presence of an inhibitory shortcut. When a shortcut is excitatory and connects the most distant neurons, the dynamical metastability of rotating waves is lost. The duration of transient rotating waves then increases only linearly with the number of neurons, not exponentially. However, the dynamical metastability of rotating waves remains when a shortcut is local.

Synchronization of strange non-chaotic attractors via unidirectional coupling of quasiperiodically-forced systems

In this paper, we present a numerical investigation on the robust synchronization phenomenon observed in a unidirectionally-coupled quasiperiodically-forced simple nonlinear electronic circuit system exhibiting strange non-chaotic attractors (SNAs) in its dynamics. The SNA obtained in the simple quasiperiodic system is characterized for its SNA behavior. Then, we studied the nature of the synchronized state in unidirectionally coupled SNAs by using the Master-Slave approach. The stability of the synchronized state is studied through the master stability functions (MSF) obtained for coupling different state variables of the drive and response system. The property of robust synchronization is analyzed for one type of coupling of the state variables through phase portraits, conditional lyapunov exponents and the Kaplan-Yorke dimension. The phenomenon of complete synchronization of SNAs via a unidirectional coupling scheme is reported for the first time.

The Dynamics of Small-Sized Ensembles of the Phase-Locked Loops with Unidirectional Couplings

We study collective dynamics of a small-sized chain of the unidirectionally coupled phase-locked loop. The conditions for the synchronous-regime existence are found, the asynchronous selfoscillation regimes and the transitions among them are studied, and the property of inheriting the structure of the parameter space of the chain when a new element is added to it is established.

Analytical treatment for synchronizing chaos through unidirectional coupling and implementation of logic gates

The idea of synchronization can be explicitly demonstrated by both numerical and analytical means on a nonlinear electronic circuit. Also, we introduce a scheme to obtain various logic gate structures, using synchronization of chaotic systems. By a small change in the response parameter of unidirectionally coupled nonlinear systems, one is able to construct various logic behaviours by both numerical and analytical methods.

Efficiently Unidirectional Coupling of Surface Plasmons Based on Asymmetric Cascaded Nanoslits

A compact unidirectional plasmonic coupler based on asymmetric cascaded nanoslits is proposed and theoretically analyzed. The incident photons illuminated on the structure are unidirectionally coupled into the metal–insulator–metal waveguide. The operation principle is clarified and a universal theoretical model based on coupled-mode equations is developed to facilitate structure design. Proof-of-principle demonstrations show that an ultrahigh extinction ratio ER $= 43.8$ dB at a specific wavelength and broadband of 160 nm covering a wide angular range from −20° to 15° with ER ≥ 10 dB are achieved.

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also present multistability shown in the planes of the basin of attractions.

EXPERIMENTAL EVIDENCE FOR VIBRATIONAL RESONANCE AND ENHANCED SIGNAL TRANSMISSION IN CHUA'S CIRCUIT

We consider a single Chua's circuit and a system of a unidirectionally coupled n-Chua's circuits driven by a biharmonic signal with two widely different frequencies ω and Ω, where Ω ≫ ω. We show experimental evidence for vibrational resonance in the single Chua's circuit and undamped signal propagation of a low-frequency signal in the system of n-coupled Chua's circuits where only the first circuit is driven by the biharmonic signal. In the single circuit, we illustrate the mechanism of vibrational resonance and the influence of the biharmonic signal parameters on the resonance. In the n(=75)-coupled Chua's circuits enhanced propagation of low-frequency signal is found to occur for a wide range of values of the amplitude of the high-frequency input signal and coupling parameter. The response amplitude of the ith circuit increases with i and attains a saturation. Moreover, the unidirectional coupling is found to act as a low-pass filter.

Dynamics of three Toda oscillators with nonlinear unidirectional coupling

We study the dynamics of three unidirectionally coupled Toda oscillators with nonlinear coupling function in the form of first three terms of Taylor power series. We analytically investigate how the coupling influence the stability of steady state. Basing on calculation of the first Lyapunov coefficient, we show that destabilization may occur by the sub- or supercritical Andronov-Hopf bifurcation. Born periodic solutions are calculated using path-following as a function of coupling strength and Taylor series coefficients. We present that initially stable or unstable branch of periodic solutions may undergo a sequence of bifurcations including: period doubling, Neimark-Saker and fold.

Coupled skinny baker's maps and the Kaplan–Yorke conjecture

The Kaplan–Yorke conjecture states that for ‘typical’ dynamical systems with a physical measure, the information dimension and the Lyapunov dimension coincide. We explore this conjecture in a neighborhood of a system for which the two dimensions do not coincide because the system consists of two uncoupled subsystems. We are interested in whether coupling ‘typically’ restores the equality of the dimensions. The particular subsystems we consider are skinny baker's maps, and we consider uni-directional coupling. For coupling in one of the possible directions, we prove that the dimensions coincide for a prevalent set of coupling functions, but for coupling in the other direction we show that the dimensions remain unequal for all coupling functions. We conjecture that the dimensions prevalently coincide for bi-directional coupling. On the other hand, we conjecture that the phenomenon we observe for a particular class of systems with uni-directional coupling, where the information and Lyapunov dimensions differ robustly, occurs more generally for many classes of uni-directionally coupled systems (also called skew-product systems) in higher dimensions.

Signal amplification by unidirectional coupling of oscillators

We report our investigation on the input signal amplification in unidirectionally coupled monostable Duffing oscillators in one- and two dimensions with first oscillator alone driven by a weak periodic signal. Applying a perturbation theory, we obtain a set of nonlinear equations for the response amplitude of the coupled oscillators. We identify the conditions for undamped signal propagation with enhanced amplitude through the coupled oscillators. When the number of oscillators increases the response amplitude approaches a limiting value. We determine this limiting value. Also, we analyse the signal amplification in the coupled oscillators in two dimensions with a fraction of oscillators chosen randomly for coupling and forcing.

Synchronization of Boolean networks with time delays

This paper studies analytically synchronization of two deterministic Boolean networks with time delays, which are unidirectionally coupled in the drive-response configuration. Necessary and sufficient conditions of synchronization are established based on the algebraic representations of logical dynamics in terms of the semi-tensor product of matrices. The synchronization conditions show that, for drive-response Boolean networks, different relations between the inherent state delay and the unidirectional coupling delay may result in distinct synchronization phenomena, and both kinds of time delays that ensure synchronization, if existent, are not unique. Examples are worked out to illustrate the present results.