Abstract:
Excavations (area of 50 x 50 cm and 20 cm depth) at about 15 m from the cave entrance revealed various vertebrate fauna. As individual numbers the mammals and the frogs predominated as bone remains. All other taxa were with low percent of occurrence. The trogloxenic species dominated than the troglophilic. Considering the cave characteristics and the taxonomical identity of the bones we proposed two main ways of bone accumulation in this cave in recent times.

Abstract:
For a class of coupled limit cycle oscillators, we give a condition on a linear coupling operator that is necessary and sufficient for exponential stability of the synchronous solution. We show that with certain modifications our method of analysis applies to networks with partial, time-dependent, and nonlinear coupling schemes, as well as to ensembles of local systems with nonperiodic attractors. We also study robustness of synchrony to noise. To this end, we analytically estimate the degree of coherence of the network oscillations in the presence of noise. Our estimate of coherence highlights the main ingredients of stochastic stability of the synchronous regime. In particular, it quantifies the contribution of the network topology. The estimate of coherence for the randomly perturbed network can be used as means for analytic inference of degree of stability of the synchronous solution of the unperturbed deterministic network. Furthermore, we show that in large networks, the effects of noise on the dynamics of each oscillator can be effectively controlled by varying the strength of coupling, which provides a powerful mechanism of denoising. This suggests that the organization of oscillators in a coupled network may play an important role in maintaining robust oscillations in random environment. The analysis is complemented with the results of numerical simulations of a neuronal network. PACS: 05.45.Xt, 05.40.Ca Keywords: synchronization, coupled oscillators, denoising, robustness to noise, compartmental model

Abstract:
A unified approach for analyzing synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented with a discussion of numerical simulations of a compartmental model of a neuron.

Abstract:
We use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking continuum limit for certain nonlocally coupled networks and to extend this method to cover many complex networks, for which it has not been applied before. Specifically, for dynamical networks on convergent sequences of simple and weighted graphs, we prove convergence of solutions of the initial-value problems for discrete models to those of the limiting continuous equations. In addition, for sequences of simple graphs converging to {0, 1}-valued graphons, it is shown that the convergence rate depends on the fractal dimension of the boundary of the support of the graph limit. These results are then used to study the regions of continuity of chimera states and the attractors of the nonlocal Kuramoto equation on certain multipartite graphs. Furthermore, the analytical tools developed in this work are used in the rigorous justification of the continuum limit for networks on random graphs that we undertake in a companion paper (Medvedev, 2013). As a by-product of the analysis of the continuum limit on deterministic and random graphs, we identify the link between this problem and the convergence analysis of several classical numerical schemes: the collocation, Galerkin, and Monte-Carlo methods. Therefore, our results can be used to characterize convergence of these approximate methods of solving initial-value problems for nonlinear evolution equations with nonlocal interactions.

Abstract:
For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in [9] justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.

Abstract:
A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and those whose topology and strength of connections may vary in time. The graph theoretic interpretation of the analytical results is emphasized. We show how the spectral properties, such as algebraic connectivity and total effective resistance, as well as the geometric properties, such the dimension and the structure of the cycle subspace of the underlying graph, shape stability of the corresponding CPs. In addition, we explore certain implications of the spectral graph theory to CP design. In particular, we point out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly. Similarly, we highlight the benefits of using random versus regular network topologies for CP design. We illustrate these observations with numerical examples and refer to the relevant graph-theoretic results. Keywords: consensus protocol, dynamical network, synchronization, robustness to noise, algebraic connectivity, effective resistance, expander, random graph

Abstract:
We study statistical properties of the irregular bursting arising in a class of neuronal models close to the transition from spiking to bursting. Prior to the transition to bursting, the systems in this class develop chaotic attractors, which generate irregular spiking. The chaotic spiking gives rise to irregular bursting. The duration of bursts near the transition can be very long. We describe the statistics of the number of spikes and the interspike interval distributions within one burst as functions of the distance from criticality.

Abstract:
The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is analyzed in this work. When the number of oscillators in the network goes to infinity, the model acquires a family of steady state solutions of degree q, called q-twisted states. We show that this class of solutions plays an important role in the formation of spatial patterns in the Kuramoto model on SW graphs. In particular, the analysis of q-twisted elucidates the role of long-range random connections in shaping the attractors in this model. We develop two complementary approaches for studying q-twisted states in the coupled oscillator model on SW graphs: the linear stability analysis and the numerical continuation. The former approach shows that long-range random connections in the SW graphs promote synchronization and yields the estimate of the synchronization rate as a function of the SW randomization parameter. The continuation shows that the increase of the long-range connections results in patterns consisting of one or several plateaus separated by sharp interfaces. These results elucidate the pattern formation mechanisms in nonlocally coupled dynamical systems on random graphs.

Abstract:
In the article, on the basis of systematization of presentations about enterprise stability of the native and western scientists, more detailed definition of the given category is offered. With the help of the authors of economic-mathematical model, we have described the correlation stability and its main factors

Abstract:
We consider a model of a square-wave bursting neuron residing in the regime of tonic spiking. Upon introduction of small stochastic forcing, the model generates irregular bursting. The statistical properties of the emergent bursting patterns are studied in the present work. In particular, we identify two principal statistical regimes associated with the noise-induced bursting. In the first case, (type I) bursting oscillations are created mainly due to the fluctuations in the fast subsystem. In the alternative scenario, type II bursting, the random perturbations in the slow dynamics play a dominant role. We propose two classes of randomly perturbed slow-fast systems that realize type I and type II scenarios. For these models, we derive the Poincare maps. The analysis of the linearized Poincare maps of the randomly perturbed systems explains the distributions of the number of spikes within one burst and reveals their dependence on the small and control parameters present in the models. The mathematical analysis of the model problems is complemented by the numerical experiments with a generic Hodgkin-Huxley type model of a bursting neuron.