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Controlled synchronization in regular delay-coupled networks of Hindmarsh-Rose neurons
... The Lyapunov function method and the speed-gradient method have also been effectively utilized in designing and analyzing control algorithms for synchronization and chaos control problems involving Hindmarsh-Rose models and their networks (Plotnikov, 2021;Semenov et al., 2022). ...
A version of the speed-gradient evolution models for systems obeying the maximum information entropy principle developed by H. Haken in his book of 1988 is proposed in this article. An explicit relation specifying system dynamics for general linear constraints is established. Two versions of the human brain entropy detailed balance-breaking model are proposed. In addition, the contours of a new scientific field called cybernetical neuroscience dedicated to the control of neural systems have been outlined.
We study synchronization in delay-coupled neural networks of heterogeneous nodes. It is well known that heterogeneities in the nodes hinder synchronization when becoming too large. We show that an adaptive tuning of the overall coupling strength can be used to counteract the effect of the heterogeneity. Our adaptive controller is demonstrated on ring networks of FitzHugh–Nagumo systems which are paradigmatic for excitable dynamics but can also — depending on the system parameters — exhibit self-sustained periodic firing. We show that the adaptively tuned time-delayed coupling enables synchronization even if parameter heterogeneities are so large that excitable nodes coexist with oscillatory ones.
We consider networks of delay-coupled Stuart-Landau oscillators. In these
systems, the coupling phase has been found to be a crucial control parameter.
By proper choice of this parameter one can switch between different synchronous
oscillatory states of the network. Applying the speed-gradient method, we
derive an adaptive algorithm for an automatic adjustment of the coupling phase
such that a desired state can be selected from an otherwise multistable regime.
We propose goal functions based on both the difference of the oscillators and a
generalized order parameter and demonstrate that the speed-gradient method
allows one to find appropriate coupling phases with which different states of
synchronization, e.g., in-phase oscillation, splay or various cluster states,
can be selected.
Homoclinic bifurcations of both equilibria and periodic orbits are argued to be critical for understanding the dynamics of the Hindmarsh–Rose model in particular, as well as of some square-wave bursting models of neurons of the Hodgkin–Huxley type. They explain very well various transitions between the tonic spiking and bursting oscillations in the model. We present the approach that allows for constructing Poincaré return mapping via the averaging technique. We show that a modified model can exhibit the blue sky bifurcation, as well as, a bistability of the coexisting tonic spiking and bursting activities. A new technique for localizing a slow motion manifold and periodic orbits on it is also presented.
Following the discovery of context-dependent synchronization of oscillatory neuronal responses in the visual system, the role of neural synchrony in cortical networks has been expanded to provide a general mechanism for the coordination of distributed neural activity patterns. In the current paper, we present an update of the status of this hypothesis through summarizing recent results from our laboratory that suggest important new insights regarding the mechanisms, function and relevance of this phenomenon. In the first part, we present recent results derived from animal experiments and mathematical simulations that provide novel explanations and mechanisms for zero and nero-zero phase lag synchronization. In the second part, we shall discuss the role of neural synchrony for expectancy during perceptual organization and its role in conscious experience. This will be followed by evidence that indicates that in addition to supporting conscious cognition, neural synchrony is abnormal in major brain disorders, such as schizophrenia and autism spectrum disorders. We conclude this paper with suggestions for further research as well as with critical issues that need to be addressed in future studies.
First recognized in 1665 by Christiaan Huygens, synchronization phenomena are abundant in science, nature, engineering and social life. Systems as diverse as clocks, singing crickets, cardiac pacemakers, firing neurons and applauding audiences exhibit a tendency to operate in synchrony. These phenomena are universal and can be understood within a common framework based on modern nonlinear dynamics. The first half of this book describes synchronization without formulae, and is based on qualitative intuitive ideas. The main effects are illustrated with experimental examples and figures, and the historical development is outlined. The remainder of the book presents the main effects of synchronization in a rigorous and systematic manner, describing classical results on synchronization of periodic oscillators, and recent developments in chaotic systems, large ensembles, and oscillatory media. This comprehensive book will be of interest to a broad audience, from graduate students to specialist researchers in physics, applied mathematics, engineering and natural sciences.
The book summarizes the state-of-the-art of research on control of self-organizing nonlinear systems with contributions from leading international experts in the field. The first focus concerns recent methodological developments including control of networks and of noisy and time-delayed systems. As a second focus, the book features emerging concepts of application including control of quantum systems, soft condensed matter, and biological systems. Special topics reflecting the active research in the field are the analysis and control of chimera states in classical networks and in quantum systems, the mathematical treatment of multiscale systems, the control of colloidal and quantum transport, the control of epidemics and of neural network dynamics.
Explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition.
In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology.
Dynamical Systems in Neuroscience presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties. The book introduces dynamical systems, starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines.
Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum—or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience.
An additional chapter on synchronization, with more advanced material, can be found at the author's website, www.izhikevich.com.
The problem is considered of asymptotic synchronization by states in networks of identical linear agents in the application
of the consensual output feedback. For the networks with fixed topology and without delay in the information transmission,
on the basis of the passification theorem and the Agaev-Chebotarev theorem, the possibility is established of the provision
of synchronization (consensus) of strong feedback under the assumption of the strict passification of agents and the existence
of the incoming spanning tree in the information graph. In contrast to the known works, in which only the problems with the
number of controls equal to the number of variables of the state of agents are investigated, in this work a substantially
more complex case is considered, where the number of controls is less than the number of variables of the state, namely: the
control is scalar. The results are illustrated by the example for the ring-shaped network of four dual integrators.
Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.
In Thailand, male Pteroptyx malaccae fireflies, congregated in trees, flash in rhythmic synchrony with a period of about 560 ± 6 msec (at 28° C). Photometric
and cinematographic records indicate that the range of flash coincidence is of the order of ± 20 msec. This interval is considerably
shorter than the minimum eye-lantern response latency and suggests that the Pteroptyx synchrony is regulated by central nervous feedback from preceding activity cycles, as in the human "sense of rhythm," rather
than by direct contemporaneous response to the flashes of other individuals. Observations on the development of synchrony
among Thai fireflies indoors, the results of experiments on phase-shifting in the American Photinus pyralis and comparisons with synchronization between crickets and between human beings are compatible with the suggestion.
The Hodgkin-Huxley model1 of the nerve impulse consists of
four coupled nonlinear differential equations, six functions and seven
constants. Because of the complexity of these equations and the
necessity for numerical solution, it is difficult to use them in
simulations of interactions in small neural networks. Thus, it would be
useful to have a second-order differential equation which predicted
correctly properties such as the frequency-current relationship.
Fitzhugh2 introduced a second-order model of the nerve
impulse, but his equations predict an action potential duration which is
similar to the inter-spike interval3 and they do not give a
reasonable frequency-current relationship. To develop a second-order
model having few parameters but which does not have these disadvantages,
we have generalized the second-order Fitzhugh equations2, and
based the form of the functions in the new equations on voltage-clamp
data obtained from a snail neurone. We report here an unexpected
property of the resulting equations-the x and y null clines in the phase
plane lie close together when the phase point is on the recovery side of
the phase plane. The resulting slow movement along the phase path gives
a long inter-spike interval, a property not shown clearly by previous
models2,4. The model also predicts the linearity of the
frequency-current relationship, and may be useful for studying detailed
interactions in networks containing small numbers of neurones.
On self-synchronization and controlled synchronization
- I Blekhman
- A Fradkov
- H Nijmeijer
- A Y Pogromsky
I. Blekhman, A. Fradkov, H.Nijmeijer, and A. Y. Pogromsky, "On selfsynchronization and controlled synchronization," Systems & Control
Letters, vol. 31, no. 5, 1997.