Oleg V. Maslennikov

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.

Dynamic boundary crisis in the Lorenz-type map.

Effects of the slowly varying control parameters on bifurcations of the equilibria and limit cycles have been previously studied in detail. In this paper, the concept of dynamic bifurcations is extended to chaotic phenomena. We consider this problem for a Lorenz-type map. As the control parameter passes through a critical value, the dynamic boundary crisis of a chaotic attractor takes place. We discover and analyze the effects of delayed exit from the chaotic region and non-exponential decay of the number of surviving trajectory points. The property of the delay increase with increasing rate of the control parameter change has also been demonstrated and explained.

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.

Map-Based Approach to Problems of Spiking Neural Network Dynamics

Mathematical modeling of phenomena in living systems by using discrete-time systems has a long history. In particular, in the 1940s N. Wiener and A. Rosenblueth developed a cellular automaton system for modeling the propagation of excitation pulses in the cardiac tissue. Cellular automata are regular lattices of elements (cells), each having a finite number of specific states. These states are updated synchronously at discrete time moments, according to some fixed rule. Recently, a new class of discrete-time systems has aroused considerable interest for studying cooperative processes in large-scale neural networks: systems of the coupled nonlinear maps. The state of a map varies at discrete time moments as a cellular automaton, but unlike the latter, takes continuous values. Map-based models hold certain advantages over continuous-time models, i.e. differential equation systems. For example, for reproducing oscillatory properties in continuous-time systems one needs at least two dimensions, and at least three dimensions for chaotic behavior. In discrete time both types of dynamics can be described even in a one-dimensional map. Such a benefit is especially important when modeling complex activity regimes of individual neurons as well as large-scale neural circuits composed of various structural units interacting with each other. For example, to simulate in continuous time the regime of chaotic spike-bursting oscillations, one of the key neural behaviors, one needs to have at least a three-dimensional system of nonlinear differential equations. On the other hand, there are discrete-time two-dimensional systems [1, 2] adequately reproducing this oscillatory activity as well as many other dynamical regimes. For example, the model of Chialvo [3] allows to simulate, among other things, the so-called normal and supernormal excitability. The model of Rulkov [4, 5] has several modifications, one of which is adjusted to simulate different spiking and bursting oscillatory regimes, the other can generate the so-called sub-threshold oscillations, i.e., small-amplitude oscillations below a threshold of excitability. Here we propose the authors’ model [6–9] of neural activity in the form of a two-dimensional map and describe some activity modes which it can reproduce. We show that the model is fairly universal and generates many regimes of neuronal electrical activity. Next, we present the results of modeling the dynamics of a complex neural structure, the olivo-cerebellar system (OCS) of vertebrates, using our basic discrete-time model of neural activity.

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’.

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.

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 assemblys response to external stimulation of finite amplitude and duration.

Synchronization and cluster sequences in modular and evolving map-based neural networks

The impact of modularity and time delay for spiking neural networks is considered in the first part of this report. We show for different complex topologies that time delay controls regimes of inter-module synchronization as well as the oscillatory rate of modules. In the second part of the work we study a paradigmatic model of the evolving neural network whose topology is influenced by nodal dynamics which results in generating different cluster sequences. We show the conditions under which the sequences are robust to small perturbations of initial conditions, parameter detuning, and noise, while at the same are selective to information stimuli.

Synchronization and Control in Modular Networks of Spiking Neurons

In this paper, we consider the dynamics of two types of modular neural networks. The first network consists of two modules of non-interacting neurons while each neuron inhibits all the neurons of an opposite module. We explain the mechanism for emergence of anti-phase group bursts in the network and showed that the collective behavior underlies a regular response of the system to external pulse stimulation. The networks of the second type contain modules with complex topology which are connected by relatively sparse excitatory delayed coupling. We found a dual role of the inter-module coupling delay in the collective network dynamics. First, with increasing time delay, in-phase and anti-phase regimes, where individual spikes form rhythmic modular burst-like oscillations, alternate with each other. Second, the average frequency of the collective oscillations in each of these regimes decreases with increasing inter-module coupling delay.

Spatio-Temporal Patterns in a Large-Scale Discrete-Time Neuron Network

The formation of spatio-temporal patterns is one of the most important forms of collective electrical activity of neural networks. Such forms of activity have been detected experimentally in different neural structures, including the structures in visual [4] and somatosensory [11] cortex, in the temporal lobe [7], in the inferior olives [5], etc. Modeling of the network structure and dynamics can be a possible way of identifying mechanisms of the pattern appearance and disappearance in such large-scale systems.