Predicting the evolution of multispecies ecological systems is an intriguing problem. A sufficiently complex model with the necessary predicting power requires solutions that are structurally stable. Small variations of the system parameters should not qualitatively perturb its solutions. When one is interested in just asymptotic results of evolution (as time goes to infinity), then the problem has a straightforward mathematical image involving simple attractors (fixed points or limit cycles) of a dynamical system. However, for an accurate prediction of evolution, the analysis of transient solutions is critical. In this paper, in the framework of the traditional Lotka-Volterra model (generalized in some sense), we show that the transient solution representing multispecies sequential competition can be reproducible and predictable with high probability.

Recent results of imaging technologies and non-linear dynamics make possible to relate the structure and dynamics of functional brain networks to different mental tasks and to build theoretical models for the description and prediction of cognitive activity. Such models are non-linear dynamical descriptions of the interaction of the core components—brain modes—participating in a specific mental function. The dynamical images of different mental processes depend on their temporal features. The dynamics of many cognitive functions are transient. They are often observed as a chain of sequentially changing metastable states. A stable heteroclinic channel (SHC) consisting of a chain of saddles—metastable states—connected by unstable separatrices is a mathematical image for robust transients. In this paper we focus on hierarchical chunking dynamics that can represent several forms of transient cognitive activity. Chunking is a dynamical phenomenon that nature uses to perform information processing of long sequences by dividing them in shorter information items. Chunking, for example, makes more efficient the use of short-term memory by breaking up long strings of information (like in language where one can see the separation of a novel on chapters, paragraphs, sentences, and finally words). Chunking is important in many processes of perception, learning, and cognition in humans and animals. Based on anatomical information about the hierarchical organization of functional brain networks, we propose a cognitive network architecture that hierarchically chunks and super-chunks switching sequences of metastable states produced by winnerless competitive heteroclinic dynamics.

Understanding and predicting the behavior of complex multiagent systems like brain or ecological food net requires new approaches and paradigms. Traditional analyses based on just asymptotic results of behavior as time goes to infinity, or on straightforward mathematical images that can accommodate only fixed points or limit cycles do not tell much about these systems. To obtain sensible dynamical models of natural phenomena, such as the reproducible order observed in ecological, cognitive or behavioral experiments, one cannot afford to neglect the transient dynamics of the underlying complex network. In disclosing such dynamical mechanisms, the focus of interest must be on reproducible or, even, structurally stable transients. In this tutorial, we formulate the Winnerless Competition (WLC) principle that induces robust transient dynamics in open complex networks. The main point of WLC principle is the transformation of the acquired information into ensemble (spatio)-temporal output via intrinsic transient dynamics of the network. Such encoding provides a reproducible transient response, whose geometrical image in phase space is a stable heteroclinic sequence. We compile a diverse list of natural phenomena which can be rigorously modeled by the WLC. Together with the experimental and numerical results of the networks with different levels of complexity, we evaluate the robustness and reproducibility of the WLC dynamics and discuss the advantages of future possible application of the discussed approach.

No abstract

Attentional networks that integrate many cortical and subcortical elements dynamically control mental processes to focus on specific events and make a decision. The resources of attentional processing are finite. Nevertheless, we often face situations in which it is necessary to simultaneously process several modalities, for example, to switch attention between players in a soccer field. Here we use a global brain mode description to build a model of attentional control dynamics. This model is based on sequential information processing stability conditions that are realized through nonsymmetric inhibition in cortical circuits. In particular, we analyze the dynamics of attentional switching and focus in the case of parallel processing of three interacting mental modalities. Using an excitatory-inhibitory network, we investigate how the bifurcations between different attentional control strategies depend on the stimuli and analyze the relationship between the time of attention focus and the strength of the stimuli. We discuss the interplay between attention and decision-making: in this context, a decision-making process is a controllable bifurcation of the attention strategy. We also suggest the dynamical evaluation of attentional resources in neural sequence processing.