Phase response function for oscillators with strong forcing or coupling

Phase response curve (PRC) is an extremely useful tool for studying the response of oscillatory systems, e.g. neurons, to sparse or weak stimulation. Here we develop a framework for studying the response to a series of pulses which are frequent or/and strong so that the standard PRC fails. We show that in this case, the phase shift caused by each pulse depends on the history of several previous pulses. We call the corresponding function which measures this shift the phase response function (PRF). As a result of the introduction of the PRF, a variety of oscillatory systems with pulse interaction, such as neural systems, can be reduced to phase systems. The main assumption of the classical PRC model, i.e. that the effect of the stimulus vanishes before the next one arrives, is no longer a restriction in our approach. However, as a result of the phase reduction, the system acquires memory, which is not just a technical nuisance but an intrinsic property relevant to strong stimulation. We illustrate the PRF approach by its application to various systems, such as Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF allows predicting the dynamics of forced and coupled oscillators even when the PRC fails

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback

Multistable jittering in oscillators with pulsatile delayed feedback

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in the last years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. In the bifurcation point numerous regimes with non-equal interspike intervals emerge simultaneously. We show that this bifurcation is triggered by the steepness of the oscillator's phase resetting curve and that the number of the emerging, so-called "jittering" regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the "multi-jitter" bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit

Embedding the dynamics of a single delay system into a feed-forward ring

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type

Hierarchical frequency clusters in adaptive networks of phase oscillators

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 29, 103134 (2019) and may be found at https://doi.org/10.1063/1.5097835.Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency multiclusters by investigating the effect of the time scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multiclusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.DFG, 411803875, Dynamik gekoppelter Systeme mit Zeitverzögerungen und deren AnwendungenDFG, 308748074, Komplexe dynamische Netzwerke: Effekte von heterogenen, adaptiven und zeitverzögerten Kopplunge

Sociocultural-Interdeterminist Dialogical Perspective of Intercultural Mutual Understanding Comprehension Deepening

Abstract Sociocultural-interdeterminist dialogical approach focuses attention on the caused interdeteministic character of mutual relations of situational, personal and activity determinants of the behaviour, staticized in concrete historical cultural context. For example, a change in a situational context leads to changes both in the person, and in his or her activity. Achievement of changes in the person and his or her activity assumes creation of the special conditions promoting to their actualization, etc. For understanding of the nature of progress in culture and science the theoretical construct "cultural-scientific tradition" is offered, allowing to trace changes in common cultural and scientific worldview throughout the human history. Consideration of personality problems is carried out in a three-dimensional continuum conscious-unconscious-existential, allowing to capture all variety of displays of psychological phenomenology. Proceeding from understanding of culture as orientational and normative structure of behaviour and as communicative matrix by means of which the behaviour is interpreted, integrated, coordinated and authorized (R. Priest), efficiency of use of the conceptual device of the epistemic approach of Michel Foucault and cultural framing of E. Goffman to understanding of intercultural interaction specificity is proved. The epistemic differences of European (graphic), Chinese, and classic Arabic languages are demonstrated. Intercultural competence is considered in aspect of ability of formation of shared meanings and experiences on the basis of mastering by knowledge about originalities of language, values and norms, experiences and behavioral algorithms of each other. The basis of such mutual understanding creates the dialogue that assumes unconditional acceptance of another based on tolerance and pluralism, the joint extension of the horizons directed on formation and development of coordinated and mutually endured values and senses. As unit of the analysis of intercultural interaction the evaluation of a sharedness of meanings and experiences is offered. Results of empirical research of the given approach on the example of formation of interpersonal mutual understanding of the Belarus and Chinese students are presented

Dialogical Interdetermination in Psychological Phenomenology of Education: an Example of Teachers’ Professional Deformation

Introduction: a distinctive feature of modern psychological knowledge is an extreme degree of disintegration manifested in an infinite array of publications describing local fragments of the studied reality outside the context of integrity. Simultaneously, development of knowledge without its metatheoretical interpretation gives it a sporadic character and, consequently, restricts the optimal solutions. The author’s attempt to solve this urgent problem is presented in the framework of sociocultural-interdeterminist dialogical metatheory of integration of psychological knowledge. Methodological foundations with substantial characterisation of metatheory are described. A research objective is to present innovative and heuristic potential of the meta-approach demonstration illustrated through the teacher’s psychological anti-deforming model. Materials and Methods: the methodological basis of research is presented by the sociocultural interdeterminist dialogical approach to education phenomenology analysis which innovative potential is illustrated by the example of teacher’s personality deformation. System analysis, comparative method, systematisation and conceptualisation of scientific ideas, classification and typifications, research object and subject modeling were used during the study. Results: the foundations and innovative potential of the sociocultural-interdeterminist dialogical meta-approach to social phenomenology analysis are given a thorough account. The teacher’s personality professional deformation main criteria are given (authoritativeness, rigidity, self-perception non-criticality, role expansionism and pedagogical indifference), the personality deformation operational definition is formulated. The concept of psychological interdeterminants of professional deformation is introduced, the process of interdetermination of personality’s deformation phenomenon is revealed, that is: interdependence of per¬sonal, environmental and activity constituents. As an environmental component the eco-cultural dialogical educational environment with its structural components is described: favorable social-psychological climate, teachers work satisfaction, social support, responsibility, awareness, open type of organisational culture in academic institution and democratic management style. One of the research result is the teacher’s personality anti-deforming psychological interdeterminants model. Discussion and Conclusions: sociocultural-interdeterminist dialogical meta-approach implies the consideration of social phenomenology in the context of the interdependence of cultural, personal, and environmental interdeterminants. This approach suggests the phenomenon under study has a new quality. Therefore, achievement of required results, including educational practice, presupposes the formation of personal prerequisites that fosters new quality. Respectively, teacher’s professional deformation prevention presupposes enhancement of their psychological culture level, diversity of their vital activity spheres, modelling of components of the eco-cultural dialogical environment in educational establishment

Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays

Coupling delays may cause drastic changes in the dynamics of oscillatory networks. In the present paper we investigate how coupling delays alter synchronization processes in networks of all-to-all coupled pulse oscillators. We derive an analytic criterion for the stability of synchrony and study the synchronization areas in the space of the delay and coupling strength. Specific attention is paid to the scenario of destabilization on the borders of the synchronization area. We show that in bifurcation points the system possesses homoclinic loops, which give rise to complex long- or quasi-periodic solutions. These newly born solutions are characterized by a synchronous group, from which an oscillator periodically escapes, laps one period, and rejoins. We call such a dynamical regime “phase slip patterns”

Multistable jittering in oscillators with pulsatile delayed feedback

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show that the number of the emerging, so-called jittering regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the multi-jitter bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit

Embedding the dynamics of a single delay system into a feed-forward ring

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type