Mind-to-mind heteroclinic coordination: Model of sequential episodic memory initiation.

Retrieval of episodic memory is a dynamical process in the large scale brain networks. In social groups, the neural patterns, associated with specific events directly experienced by single members, are encoded, recalled, and shared by all participants. Here, we construct and study the dynamical model for the formation and maintaining of episodic memory in small ensembles of interacting minds. We prove that the unconventional dynamical attractor of this process-the nonsmooth heteroclinic torus-is structurally stable within the Lotka-Volterra-like sets of equations. Dynamics on this torus combines the absence of chaos with asymptotic instability of every separate trajectory; its adequate quantitative characteristics are length-related Lyapunov exponents. Variation of the coupling strength between the participants results in different types of sequential switching between metastable states; we interpret them as stages in formation and modification of the episodic memory.

Modeling rhythmic patterns in the hippocampus.

We investigate different dynamical regimes of the neuronal network in the CA3 area of the hippocampus. The proposed neuronal circuit includes two fast- and two slowly spiking cells which are interconnected by means of dynamical synapses. On the individual level, each neuron is modeled by FitzHugh-Nagumo equations. Three basic rhythmic patterns are observed: the gamma rhythm in which the fast neurons are uniformly spiking, the theta rhythm in which the individual spikes are separated by quiet epochs, and the theta-gamma rhythm with repeated patches of spikes. We analyze the influence of asymmetry of synaptic strengths on the synchronization in the network and demonstrate that strong asymmetry reduces the variety of available dynamical states. The model network exhibits multistability; this results in the occurrence of hysteresis in dependence on the conductances of individual connections. We show that switching between different rhythmic patterns in the network depends on the degree of synchronization between the slow cells.

Non-Markovian approach to globally coupled excitable systems.

We consider stochastic excitable units with three discrete states. Each state is characterized by a waiting time density function. This approach allows for a non-Markovian description of the dynamics of separate excitable units and of ensembles of such units. We discuss the emergence of oscillations in a globally coupled ensemble with excitatory coupling. In the limit of a large ensemble we derive the non-Markovian mean-field equations: nonlinear integral equations for the populations of the three states. We analyze the stability of their steady solutions. Collective oscillations are shown to persist in a large parameter region beyond supercritical and subcritical Hopf bifurcations. We compare the results with simulations of discrete units as well as of coupled FitzHugh-Nagumo systems.

Can the shape of attractor forbid chaotic phase synchronization?

We address the question, which properties of a chaotic oscillator are crucial for its ability/inability to synchronize with external force or other similar oscillators. The decisive role is played by temporal coherency whereas the shape of the attractor is less important. We discuss the role of coordinate-dependent reparameterizations of time which preserve the attractor geometry but greatly influence the coherency. An appropriate reparameterization enables phase synchronization in coupled multiscroll attractors. In contrast, the ability to synchronize phases for nearly isochronous oscillators can be destroyed by a reparameterization which washes out the characteristic time scale.

Coherence resonance near a Hopf bifurcation.

We report on the observation of coherence resonance for a semiconductor laser with short optical feedback close to Hopf bifurcations. Noise-induced self-pulsations are documented by distinct Lorentzian-like features in the power spectrum. The character of coherence is critically related to the type of the bifurcation. In the supercritical case, spectral width and height of the peak are monotonic functions of the noise level. In contrast, for the subcritical bifurcation, the width exhibits a minimum, translating into resonance behavior of the correlation time in the pulsation transients. A theoretical analysis based on the generic model of a self-sustained oscillator demonstrates that these observations are of general nature and are related to the fact that the damping depends qualitatively different on the noise intensity for the subcritical and supercritical case.

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.

Noise-controlled oscillations and their bifurcations in coupled phase oscillators.

We derive in Gaussian approximation dynamical equations for the first two cumulants of the mean field fluctuations in a system of globally coupled stochastic phase oscillators. In these equations the intensity of noise serves as an explicit control parameter. Its variation generates transitions between three dynamical regimes: (i) stationary, (ii) rotatory and (iii) locally oscillatory (breathing). The latter regime has previously not been reported in studies of globally coupled noisy phase oscillators. Our detailed bifurcation analysis is supported by numerical simulations of an ensemble of coupled stochastic phase oscillators. Similar regimes are also found in simulations of globally coupled stochastic FitzHugh-Nagumo elements.

Convective Cahn-Hilliard models: from coarsening to roughening.

In this paper we demonstrate that convective Cahn-Hilliard models, describing phase separation of driven systems (e.g., faceting of growing thermodynamically unstable crystal surfaces), exhibit, with the increase of the driving force, a transition from the usual coarsening regime to a chaotic behavior without coarsening via a pattern-forming state characterized by the formation of various stationary and traveling periodic structures as well as structures with localized oscillations. Relation of this phenomenon to a kinetic roughening of thermodynamically unstable surfaces is discussed.

Phase synchronization in the forced Lorenz system.

We demonstrate that the dynamics of phase synchronization in a chaotic system under weak periodic forcing depends crucially on the distribution of intrinsic characteristic times of this system. Under the external periodic action, the frequency of every unstable periodic orbit is locked to the frequency of the force. In systems which in the autonomous case displays nearly isochronous chaotic rotations, the locking ratio is the same for all periodic orbits; since a typical chaotic orbit wanders between the periodic ones, its phase follows the phase of the force. For the Lorenz attractor with its unbounded times of return onto a Poincaré surface, such state of perfect phase synchronization is inaccessible. Analysis with the help of unstable periodic orbits shows that this state is replaced by another one, which we call "imperfect phase synchronization," and in which we observe alternation of temporal segments, corresponding to different rational values of frequency lockings.