The aim of this talk is to analyze stochastic lattice equations driven by a non-trivial multiplicative fractional Brownian motion (fBm) with Hurst parameter in (1/2,1). We will obtain the existence of a unique solution for the model, that will rely on a fixed point argument, based on nice estimates satisfied by the stochastic integral with an fBm as integrator. Further, we will focus on investigating the long time behavior of the solution, proving that when zero is a solution of the model and the initial condition belongs to a neighborhood of zero, then the corresponding solution of the lattice equation is exponentially stable with some exponential rate.
Cognitive theory has decomposed human mental abilities into cognitive (sub-) systems and cognitive neuroscience succeeded in disclosing a host of relationships between cognitive systems and specific structures of the human brain. However, an explanation of why specific functions are located in specific places had still been missing, along with a neurobiological model that makes concrete the neuronal circuits that carry thoughts and meaning. Brain theory now offers an avenue towards explaining brain-mind relationships and to spell out cognition in terms of neuron circuits in a neuromechanistic sense. Central to this endeavor is the theoretical construct of an elementary functional neuronal unit above the level of individual neurons and below that of whole brain areas and systems: the distributed neuronal assembly (DNA) or thought circuit (TC). I will argued that DNA/TC theory of cognition offers an integrated explanatory perspective on brain mechanisms underlying a range of cognitive processes, including perception, action, language, attention, memory, decision, and conceptual thought (1,2). DNAs are proposed to carry all of these functions and their inner structure (e.g., core and halo subcomponents) along with their functional activation dynamics (e.g., ignition and reverberation processes) explain crucial questions about cortical localization of cognitive function, including the question why memory and decisions draw on frontoparietal ‘multi-demand’ areas although memory formation is normally driven by information in the senses and in the motor system. We suggest that the ability of building DNAs/TCs spread-out over different cortical areas is the key mechanism for a range of specifically human sensorimotor, linguistic and conceptual capacities and that the cell assembly mechanism of overlap reduction is crucial for differentiating a rich vocabulary of actions, symbols and concepts (3).1 Pulvermüller, F. (2013). How neurons make meaning: Brain mechanisms for embodied and abstract-symbolic semantics. Trends in Cognitive Sciences, 17(9), 458-470.2 Pulvermüller, F., & Fadiga, L. (2010). Active perception: Sensorimotor circuits as a cortical basis for language. Nature Reviews Neuroscience, 11(5), 351-360.3 Pulvermüller, F., Garagnani, M., & Wennekers, T. (2014). Thinking in circuits: Towards neurobiological explanation in cognitive neuroscience. Biological Cybernetics, in press.
In this talk an ODE system modeling oscillatory patterns of mood alternations in manic-depression, also known as bipolar disorder, is analyzed. The model is four-dimensional, contains many parameters of different orders of magnitude, and non-polynomial nonlinearities. This poses several challenges and the analysis of the model must be based on identifying and using hierarchies of local approximations based on various –- hidden -– forms of time scale separation.
I will explain some concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method in combination with standard techniques from dynamical systems theory which I use to understand the geometry of self-sustained (non-classical relaxation) oscillations.
Many evolutionary and self-organization pressures can be characterized information theoretically not only because it's an approximation useful in designing biologically-inspired systems, but also because numerous optimal structures evolve/self-organize in nature when information dynamics approach critical points. The talk will focus on information dynamics of computation within spatio-temporal systems in terms of three fundamental operations: information storage, transfer, and modification, quantifying these operations on a local scale in space and time. The methods will be exemplified in different contexts, including swarms, computational neuroscience, and random Boolean networks. In addition, we shall discuss a relation between Fisher information and phase transitions / order parameters, drawing from both thermodynamics and statistical estimation theory, as well as a thermodynamic interpretation of transfer entropy.
Consider a stochastic differential equation on a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behaviour of a small transversal perturbation of order $\epsilon$. An average principle is shown to hold such that the component transversal to the leaves converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to invariant measures on the leaves, as $\epsilon$ goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system.
Models from e.g. fluid dynamics or systems biology often can be described by ordinary differential equations. While the long-term behavior of those models as time tends to infinity is quite well understood, the theory of transient solution behavior is still in its infancy. We present some results as well as open questions.
I will discuss an eigenvalue perturbation formula for transfer operators of (open) dynamical systems and how this formula is related to exponential hitting time distributions and extreme value theory for processes generated by chaotic dynamical systems.
The aim of this talk is to present briefly some "user-friendly" methods and techniques (mainly, frequency-domain approaches) for the analysis and control of the dynamical systems in presence of delays. The presentation is as simple as possible, focusing on the main intuitive ideas to develop theoretical results, and their potential use in practical applications. Single and multiple delays will be both considered. Classical control schemes (as, for example, the well-known PI and PID controllers, or Smith predictors) will be also revisited.
The talk ends with the analysis of some control schemes used in the motion synchronization in shared virtual environments.
In this talk I am going to present a study on the influence of noise on oscillations in multiple time scale systems. We start with a brief introduction to fast-slow dynamical systems. We introduce the normal form of a folded node singularity in three dimensions with two slow variables and one fast variable. In this context, we encounter canard (="duck") orbits that produce local oscillations. The influence of noise on this singularity will be explored. Our results are expected to form a main building block to understand mixed-mode oscillations that appear in many different applications.
Vovk's randomness criterion characterizes sequences that are random relative to two distinct computable probability measures. The uniqueness of the criterion lies in the fact that, unlike the standard criterion based on the likelihood ratio test, it is expressed in terms of a geometrical quantity, the Hellinger distance, on the space of probability measures. In this talk, I generalize the randomness criteria to a wider class of geometrical quantities, the alpha-divergences with -1
The impact of spatial disorder [1] and time-dependent noise [2] on diffusion in chaotic dynamical systems is studied. As an example, we consider deterministic random walks in a one-dimensional periodic array of scatterers modeled by a parameter-dependent coupled chaotic map. In computer simulations we find a crossover from deterministic to stochastic diffusion under variation of the perturbation strength related to different asymptotic laws for the diffusion coefficient. Typical signatures of this scenario are multiple suppression and enhancement of normal diffusion. These results are explained by simple theoretical approximations showing that the oscillations emerge as a direct consequence of the unperturbed deterministic diffusion coefficient, which is known to be a fractal function of control parameters [3].
Problems of complex behavior of random dynamical systems is investigated based on numerically observed noise-induced phenomena in Belousov-Zhabotinsky map (BZ map) and modified Lasota-Mackey map with presence of noise. We found that (i) both noise-induced chaos and noise-induced order robustly coexist, and that (ii) asymptotic periodicity of density is varied by noise amplitude. Applications to time series analysis are also discussed.
Evolutionary dynamics is based on mutation, selection and random drift. When the success of a certain type depends on others, game-theoretic approaches are more appropriate than optimization arguments.
Traditionally, evolutionary game theory considers infinitely large populations, where stochastic effects can be neglected. Only recently, stochastic processes have been applied to model evolutionary game dynamics in finite populations analytically. In this context, analytical results are obtained that can be very different from the usual results of the replicator dynamics.
After a general introduction to evolutionary game theory, I will discuss these stochastic approaches. It will be shown that the connection to the traditional approaches based on the replicator dynamics can be obtained by an approximation for large populations. As applications of evolutionary game dynamics, I will discuss mechanisms for the evolution of cooperation and for the emergence of punishment.
The talk motivates an integral-differential equation, which generalizes the reaction-diffusion equation, the Swift-Hohenberg equation and the Kuramoto-Sivashinsky equation. In addition, it is shown how this generalization allows for the treatment of finite propagation speeds in these systems. A more detailed discussion of the Cattaneo-equation for diffusion systems illustrates the importance of the proposed equation.