Adaptive myelination causes slow oscillations in recurrent neural loops.

The brain is known to be plastic, i.e., capable of changing and reorganizing as it develops and accumulates experience. Recently, a novel form of brain plasticity was described which is activity-dependent myelination of nerve fibers. Since the speed of propagation of action potentials along axons depends significantly on their degree of myelination, this process leads to adaptive change of axonal delays depending on the neural activity. To understand the possible influence of the adaptive delays on the behavior of neural networks, we consider a simple setup, a neuronal oscillator with delayed feedback. We show that introducing the delay plasticity into this circuit can lead to the occurrence of slow oscillations which are impossible with a constant delay.

Adaptation rules inducing synchronization of heterogeneous Kuramoto oscillator network with triadic couplings.

A class of adaptation functions is found for which a synchronous mode with different number of phase clusters exists in a network of phase oscillators with triadic couplings. This mode is implemented in a fairly wide range of initial conditions and the maximum number of phase clusters is four. The joint influence of coupling strength and adaptation parameters on synchronization in the network has been studied. The desynchronization transition under variation of the adaptation parameter occurs abruptly and begins with the highest-frequency oscillator, spreading hierarchically to all other elements.

Internal dynamics of recurrent neural networks trained to generate complex spatiotemporal patterns.

How complex patterns generated by neural systems are represented in individual neuronal activity is an essential problem in computational neuroscience as well as machine learning communities. Here, based on recurrent neural networks in the form of feedback reservoir computers, we show microscopic features resulting in generating spatiotemporal patterns including multicluster and chimera states. We show the effect of individual neural trajectories as well as whole-network activity distributions on exhibiting particular regimes. In addition, we address the question how trained output weights contribute to the autonomous multidimensional dynamics.

Transient Phase Clusters in a Two-Population Network of Kuramoto Oscillators with Heterogeneous Adaptive Interaction.

Adaptive interactions are an important property of many real-word network systems. A feature of such networks is the change in their connectivity depending on the current states of the interacting elements. In this work, we study the question of how the heterogeneous character of adaptive couplings influences the emergence of new scenarios in the collective behavior of networks. Within the framework of a two-population network of coupled phase oscillators, we analyze the role of various factors of heterogeneous interaction, such as the rules of coupling adaptation and the rate of their change in the formation of various types of coherent behavior of the network. We show that various schemes of heterogeneous adaptation lead to the formation of transient phase clusters of various types.

Multitask computation through dynamics in recurrent spiking neural networks.

In this work, inspired by cognitive neuroscience experiments, we propose recurrent spiking neural networks trained to perform multiple target tasks. These models are designed by considering neurocognitive activity as computational processes through dynamics. Trained by input-output examples, these spiking neural networks are reverse engineered to find the dynamic mechanisms that are fundamental to their performance. We show that considering multitasking and spiking within one system provides insightful ideas on the principles of neural computation.

Disordered quenching in arrays of coupled Bautin oscillators.

In this work, we study the phenomenon of disordered quenching in arrays of coupled Bautin oscillators, which are the normal form for bifurcation in the vicinity of the equilibrium point when the first Lyapunov coefficient vanishes and the second one is nonzero. For particular parameter values, the Bautin oscillator is in a bistable regime with two attractors-the equilibrium and the limit cycle-whose basins are separated by the unstable limit cycle. We consider arrays of coupled Bautin oscillators and study how they become quenched with increasing coupling strength. We analytically show the existence and stability of the dynamical regimes with amplitude disorder in a ring of coupled Bautin oscillators with identical natural frequencies. Next, we numerically provide evidence that disordered oscillation quenching holds for rings as well as chains with nonidentical natural frequencies and study the characteristics of this effect.

Emergence and synchronization of a reversible core in a system of forced adaptively coupled Kuramoto oscillators.

We report on the phenomenon of the emergence of mixed dynamics in a system of two adaptively coupled phase oscillators under the action of a harmonic external force. We show that in the case of mixed dynamics, oscillations in forward and reverse time become similar, especially at some specific frequencies of the external force. We demonstrate that the mixed dynamics prevents forced synchronization of a chaotic attractor. We also show that if an external force is applied to a reversible core formed in an autonomous case, the fractal dimension of the reversible core decreases. In addition, with increasing amplitude of the external force, the average distance between the chaotic attractor and the chaotic repeller on the global Poincaré secant decreases almost to zero. Therefore, at the maximum intersection, we see a trajectory belonging approximately to a reversible core in the numerical simulation.

The third type of chaos in a system of two adaptively coupled phase oscillators.

We study a new type of attractor, the so-called reversible core, which is a mathematical image of mixed dynamics, in a strongly dissipative time-irreversible system of two adaptively coupled phase oscillators. The existence of mixed dynamics in this system was proved in our previous article [A. A. Emelianova and V. I. Nekorkin, Chaos 29, 111102 (2019)]. In this paper, we attempt to identify the dynamic mechanisms underlying the existence of mixed dynamics. We give the region of the existence of mixed dynamics on the parameter plane and demonstrate in what way, when a type of attractor changes, its main characteristics, such as its fractal dimension and the sum of Lyapunov exponents, transform. We demonstrate that when mixed dynamics appear in the system, the average frequencies of the oscillations in forward and reverse time begin to almost coincide, and its spectra gradually approach each other with an increase in the parameter responsible for the presence of mixed dynamics.

Collective dynamics of rate neurons for supervised learning in a reservoir computing system.

In this paper, we study collective dynamics of the network of rate neurons which constitute a central element of a reservoir computing system. The main objective of the paper is to identify the dynamic behaviors inside the reservoir underlying the performance of basic machine learning tasks, such as generating patterns with specified characteristics. We build a reservoir computing system which includes a reservoir-a network of interacting rate neurons-and an output element that generates a target signal. We study individual activities of interacting rate neurons, while implementing the task and analyze the impact of the dynamic parameter-a time constant-on the quality of implementation.

Itinerant chimeras in an adaptive network of pulse-coupled oscillators.

In a network of pulse-coupled oscillators with adaptive coupling, we discover a dynamical regime which we call an "itinerant chimera." Similarly as in classical chimera states, the network splits into two domains, the coherent and the incoherent. The drastic difference is that the composition of the domains is volatile, i.e., the oscillators demonstrate spontaneous switching between the domains. This process can be seen as traveling of the oscillators from one domain to another or as traveling of the chimera core across the network. We explore the basic features of the itinerant chimeras, such as the mean and the variance of the core size, and the oscillators lifetime within the core. We also study the scaling behavior of the system and show that the observed regime is not a finite-size effect but a key feature of the collective dynamics which persists even in large networks.

Transient chaos in the Lorenz-type map with periodic forcing.

We consider a case study of perturbing a system with a boundary crisis of a chaotic attractor by periodic forcing. In the static case, the system exhibits persistent chaos below the critical value of the control parameter but transient chaos above the critical value. We discuss what happens to the system and particularly to the transient chaotic dynamics if the control parameter periodically oscillates. We find a non-exponential decaying behavior of the survival probability function, study the impact of the forcing frequency and amplitude on the escape rate, analyze the phase-space image of the observed dynamics, and investigate the influence of initial conditions.

Hierarchical transitions in multiplex adaptive networks of oscillatory units.

In this work, we consider two-layer multiplex networks of coupled Stuart-Landau oscillators. The first layer contains oscillators with amplitude heterogeneity and all-to-all adaptive links, while the second layer contains identical oscillators all-to-all coupled by links with constant weights. The links between different layers are adaptive and organized in a one-to-one manner. We study the evolution of one-layer and two-layer networks depending on intra- and interlayer coupling strengths and show hierarchical transitions between oscillatory and quenched regimes.

Transient sequences in a hypernetwork generated by an adaptive network of spiking neurons.

We propose a model of an adaptive network of spiking neurons that gives rise to a hypernetwork of its dynamic states at the upper level of description. Left to itself, the network exhibits a sequence of transient clustering which relates to a traffic in the hypernetwork in the form of a random walk. Receiving inputs the system is able to generate reproducible sequences corresponding to stimulus-specific paths in the hypernetwork. We illustrate these basic notions by a simple network of discrete-time spiking neurons together with its FPGA realization and analyse their properties.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Mean-field dynamics of a population of stochastic map neurons.

We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis. It is demonstrated that the mean-field model can qualitatively account for stability and bifurcations of the exact system, capturing all the generic forms of collective behavior, including macroscopic excitability, subthreshold oscillations, periodic or chaotic spiking, and chaotic bursting dynamics. Apart from qualitative analogies, we find a substantial quantitative agreement between the exact and the approximate system, as reflected in matching of the parameter domains admitting the different dynamical regimes, as well as the characteristic properties of the associated time series. The effective model is further shown to reproduce with sufficient accuracy the phase response curves of the exact system and the assembly's response to external stimulation of finite amplitude and duration.

Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale.

In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.

Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators.

The impact of connectivity and individual dynamics on the basin stability of the burst synchronization regime in small-world networks consisting of chaotic slow-fast oscillators is studied. It is shown that there are rewiring probabilities corresponding to the largest basin stabilities, which uncovers a reason for finding small-world topologies in real neuronal networks. The impact of coupling density and strength as well as the nodal parameters of relaxation or excitability are studied. Dynamic mechanisms are uncovered that most strongly influence basin stability of the burst synchronization regime.

Artificial Electrical Morris-Lecar Neuron.

In this paper, an experimental electronic neuron based on a complete Morris-Lecar model is presented, which is able to become an experimental unit tool to study collective association of coupled neurons. The circuit design is given according to the ionic currents of this model. The experimental results are compared with the theoretical prediction, leading to a good agreement between them, which therefore validate the circuit. The use of some parts of the circuit is also possible for other neurons models, namely for those based on ionic currents.

Cross-frequency synchronization of oscillators with time-delayed coupling.

We carry out theoretical and experimental studies of cross-frequency synchronization of two pulse oscillators with time-delayed coupling. In the theoretical part of the paper we utilize the concept of phase resetting curves and analyze the system dynamics in the case of weak coupling. We construct a Poincaré map and obtain the synchronization zones in the parameter space for m:n synchronization. To challenge the theoretical results we designed an electronic circuit implementing the coupled oscillators and studied its dynamics experimentally. We show that the developed theory predicts dynamical properties of the realistic system, including location of the synchronization zones and bifurcations inside them.

Modular networks with delayed coupling: synchronization and frequency control.

We study the collective dynamics of modular networks consisting of map-based neurons which generate irregular spike sequences. Three types of intramodule topology are considered: a random Erdös-Rényi network, a small-world Watts-Strogatz network, and a scale-free Barabási-Albert network. The interaction between the neurons of different modules is organized by relatively sparse connections with time delay. For all the types of the network topology considered, we found that with increasing delay two regimes of module synchronization alternate with each other: inphase and antiphase. At the same time, the average rate of collective oscillations decreases within each of the time-delay intervals corresponding to a particular synchronization regime. A dual role of the time delay is thus established: controlling a synchronization mode and degree and controlling an average network frequency. Furthermore, we investigate the influence on the modular synchronization by other parameters: the strength of intermodule coupling and the individual firing rate.

Dense neuron clustering explains connectivity statistics in cortical microcircuits.

Local cortical circuits appear highly non-random, but the underlying connectivity rule remains elusive. Here, we analyze experimental data observed in layer 5 of rat neocortex and suggest a model for connectivity from which emerge essential observed non-random features of both wiring and weighting. These features include lognormal distributions of synaptic connection strength, anatomical clustering, and strong correlations between clustering and connection strength. Our model predicts that cortical microcircuits contain large groups of densely connected neurons which we call clusters. We show that such a cluster contains about one fifth of all excitatory neurons of a circuit which are very densely connected with stronger than average synapses. We demonstrate that such clustering plays an important role in the network dynamics, namely, it creates bistable neural spiking in small cortical circuits. Furthermore, introducing local clustering in large-scale networks leads to the emergence of various patterns of persistent local activity in an ongoing network activity. Thus, our results may bridge a gap between anatomical structure and persistent activity observed during working memory and other cognitive processes.