Narrow Results

Asynchronous and Synchronous languages have been in use for the specification of reactive systems. One of the main distinguishing features of these two classes lies in the way nondeterminism is used for the specification of programs. From this viewpoint, we analyze CSP (a typical asynchronous language) and ESTEREL (a synchronous language). The synchronous language Esterel is based on the notions of determinism, input nondeterminacy and parallelism whereas CSP is built on the notions of nondeterminism, concurrency and distribution. The main objectives of the study are to assess:

• The role of nondeterminism in the specification of the behaviour and realization of programs: A clear distinction between local and global nondeterminism enables us to distinguish between implementational nondeterminism and environmental/input nondeterminism. The results in this direction would enable one to achieve observable determinism where the implementational choices can be hidden and thus, analyze the program behaviour with reference to the real environmental nondeterminism in the specification. This leads to a proper refinement of specifications and aids in deriving distributed implementations of finite state transition systems that are not necessarily deterministic.

• The implementability of asynchronous languages through synchronous languages. The implementability of asynchronous languages in synchronous languages not only provides a realistic implementation but also provides higher level abstractions (such as multi-process interactions) for reactive specifications using features such as broadcast, interrupts, exception handling mechanisms etc.

Epilepsy is a neurological disorder that affects approximately one percent of the world population. Anti-epileptic drugs are ineffective in 25~30% of cases. Electrical stimulation to control seizures may be an additive therapy. We applied noninvasive transcutaneous focal electrical stimulation (TFES) via concentric ring electrodes on the scalp of rats after inducing seizures with pentylenetetrazole. We found a significant increase in synchrony within the beta-gamma bands during seizures and that TFES significantly reduced the synchrony of the beta-gamma activity and increased synchrony in the delta band.

In this study, we present a novel low-frequency electrical stimulation paradigm for mesial temporal lobe epilepsy (MTLE). The paradigm utilizes the hippocampal commissure as a unique stimulation target to simultaneously influence large portions of the bilateral hippocampal network. When applied to an acute rat model of MTLE, animals that received stimulation exhibited an 88% reduction in the signal power of the bilateral epileptiform activity relative to the control group. In addition, the stimulation entrained the hippocampal network's spontaneous epileptiform activity and disrupted its bilateral synchrony.

Brain function has been proposed to arise as a result of the coordinated activity between distributed brain areas. An important issue in the study of brain activity is the characterization of the synchrony among these areas and the resulting complexity of the system. However, the variety of ways to define and, hence, measure brain synchrony and complexity has sometimes led to inconsistent results. Here, we study the relationship between synchrony and commonly used complexity estimators of electroencephalogram (EEG) activity and we explore how simulated lesions in anatomically based cortical networks would affect key functional measures of activity. We explored this question using different types of neural network lesions while the brain dynamics was modeled with a time-delayed set of 66 Kuramoto oscillators. Each oscillator modeled a region of the cortex (node), and the connectivity and spatial location between different areas informed the creation of a network structure (edges). Each type of lesion consisted on successive lesions of nodes or edges during the simulation of the neural dynamics. For each type of lesion, we measured the synchrony among oscillators and three complexity estimators (Higuchi’s Fractal Dimension, Sample Entropy and Lempel-Ziv Complexity) of the simulated EEGs. We found a general negative correlation between EEG complexity metrics and synchrony but Sample Entropy and Lempel-Ziv showed a positive correlation with synchrony when the edges of the network were deleted. This suggests an intricate relationship between synchrony of the system and its estimated complexity. Hence, complexity seems to depend on the multiple states of interaction between the oscillators of the system. Our results can contribute to the interpretation of the functional meaning of EEG complexity.

In this paper, we use a reduced two-compartment neuron model to investigate the interaction between extracellular subthreshold electric field and synchrony in small world networks. It is observed that network synchronization is closely related to the strength of electric field and geometric properties of the two-compartment model. Specifically, increasing the electric field induces a gradual improvement in network synchrony, while increasing the geometric factor results in an abrupt decrease in synchronization of network. In addition, increasing electric field can make the network become synchronous from asynchronous when the geometric parameter is set to a given value. Furthermore, it is demonstrated that network synchrony can also be affected by the firing frequency and dynamical bifurcation feature of single neuron. These results highlight the effect of weak field on network synchrony from the view of biophysical model, which may contribute to further understanding the effect of electric field on network activity.

Many electrophysiological experiments have shown that epileptic seizures often originate from the synchronous activities of abnormally excitable neurons. The dynamic process of epilepsy is very complex, and characterized by a seemingly rapid and dramatic birth of new oscillations, essentially leading to a propagation and amplification of the original aberrant activity. It is very difficult to thoroughly understand the mechanism from a theoretical standpoint, however some special work can prove helpful. Here we present a theoretical framework to investigate chaos and complexity in the synchrony of excitable neurons in an effort to study the collective oscillations within a neural network. As endogenous rhythms, oscillations arise because most cellular processes contain feedback. The Chay model of excitable neurons is chosen because the model describes the abnormal process, where spiking can be transformed into bursting via bifurcation.

In our study, the Chay model is regarded as an abnormal oscillator and coupled via a resistor representing the effect of gap junctions (electrical synapses). In this paper, we present some models developed from the original Chay model, for the synchrony of two cells and a 2D neural network. Lyapunov exponent and phase portrait are utilized to evaluate the chaotic dynamics. Finally, approximate entropy is utilized to measure its complexity. Our results show that the synchrony of abnormal oscillations can occur when the coupling strength of the gap junction is sufficiently large. It is also found that the concentration of Ca²⁺ ions does not synchronize. In the 2D network, approximate entropies of different oscillations with strong coupling strength are greater than those with weak coupling strength. It is indicated that synchronous neurons have greater ability to produce new oscillations than asynchronous ones. This work shows that nonlinear analytical methods may prove useful in elucidating the mechanisms of pathologic conditions, where new oscillations are born and propagated, such as in epilepsy.

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Let Γ be a permutation group acting on the set of cells. We define the group network , whose architecture is entirely determined by the group orbits of Γ. We prove that if Γ has the "balanced extension property" then every balanced coloring of is a fixed-point coloring relative to the automorphism group of the group network. This theorem applies in particular when Γ is cyclic or dihedral, acting on cells as the symmetries of a regular polygon, and in these cases the automorphism group is Γ itself. In general, however, the automorphism group may be larger than Γ. Several examples of this phenomenon are discussed, including the finite simple group of order 168 in its permutation representation of degree 7. More dramatically, for some choices of Γ there exist balanced colorings of that are not fixed-point colorings. For example, there exists an exotic balanced 2-coloring when Γ is the symmetry group of the two-dimensional square lattice. This coloring is doubly periodic, and its reduction modulo 8 leads to a finite group with similar properties. Although these patterns do not arise from fixed-point spaces, we provide a group-theoretic explanation of their balance property in terms of a sublattice of index two.

Network architecture can lead to robust synchrony in coupled maps and to codimension one bifurcations from synchronous fixed-points at which the associated Jacobian is nilpotent.

We discuss the codimension one synchrony-breaking period-doubling bifurcations for three-cell coupled maps. Interesting phenomena occur for all these coupled maps — a branch of period-2 points with amplitude growing as |λ|^⅙ for coupled networks of feed-forward type, as well as multiple (two) branches of period-2 points with amplitude growing as |λ|^½ for coupled networks of feed-forward type.

We also discuss how some results related to patterns of synchrony that are valid for coupled vector fields are also valid for coupled maps.

Noise-induced synchronization is an unexpected phenomenon which we show here to have biological relevance for multitrophic ecological systems. We focus on two uncoupled Hastings–Powell systems having oscillatory dynamics. Despite the fact the systems are uncoupled, it is shown that they can nevertheless synchronize by applying common Gaussian noise forcing to a suitably chosen variable in both systems. Complete synchronization is studied in the case of identical oscillators, and intermittent synchronization is analyzed when the oscillators are nonidentical. We discuss how this unusual synchronization phenomenon has features akin to Ecology's well-known "Moran effect."

In this paper, we study the combined effect of dynamic chemical and electrical synapses in time-delay-induced phase-transition to synchrony in coupled bursting neurons. Time-delay in coupled nonlinear oscillators or in a network of coupled nonlinear oscillators has been found to be responsible for striking dynamical behaviors such as phase-flip-transitions. These phenomena lead to synchrony or out of synchrony in different oscillators of the system. Here, we show that synaptic parameters, more precisely the neurotransmitters binding time constant influences the phase-flip-transitions of the system. We discuss how the system goes to the phase-flip-transitions when both electrical and dynamic chemical synapses are taken into account. The fourth-order Hindmarsh–Rose neuronal oscillator is considered here for the study of these transitions. A discussion on the importance of these results in brain researches is given, particularly to understand the collective dynamics of bursting neurons.

Hopf bifurcation in two groups of Kuramoto's phase oscillators with delay-coupled interactions is investigated on the Ott–Antonsen's manifold. We find that the reduced delay differential system undergoes Hopf bifurcations when the coupling strength between two groups exceeds some critical values. With the increasing of time delay, stability switches are observed which leads to the synchrony switches for the Kuramoto system. The direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by deriving the normal forms on the center manifold. With respect to the Kuramoto system, simulations are performed to support our analytic results.

Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorings occur in infinite planar lattices, both square and hexagonal, with various short-range couplings. In some cases, a balanced two-coloring remains balanced when colors are swapped along suitable diagonals, giving rise to uncountably many distinct exotic colorings. We explain this phenomenon in terms of iterated orbit colorings, in which the quotient of the lattice by an orbit coloring has extra symmetries, allowing new orbit colorings on the quotient, which then lift back to the lattice. We apply the same construction to several other exotic lattice colorings. Two appendices discuss how to modify the notion of balance for networks with diffusive coupling, and how to formalize the differential equations in infinitely many variables that arise for lattices.

Patterns of synchrony in networks of coupled dynamical systems can be represented as colorings of the nodes, in which nodes of the same color are synchronous. Balanced colorings, where nodes of the same color have color-isomorphic input sets, correspond to dynamically invariant subspaces, which can have a significant effect on the typical bifurcations of network dynamical systems. Orbit colorings for subgroups of the automorphism (symmetry) group of the network are always balanced, although the converse is false. We compute the automorphism groups of all doubly periodic quotient networks of the square lattice with nearest-neighbor coupling, and classify the “exotic” cases where this quotient network has extra automorphisms not induced by automorphisms of the square lattice. These comprise five isolated exceptions and two infinite families with wreath product symmetry. We also comment briefly on implications for bifurcations to doubly periodic patterns in square lattice models.

Pattern formation, dynamics and bifurcations for lattice models are strongly influenced by the symmetry of the lattice. However, network structure introduces additional constraints, which sometimes affect the resulting behavior. We compute the automorphism groups of all doubly periodic quotient networks of the hexagonal lattice with nearest-neighbor coupling, with emphasis on “exotic” cases where this quotient network has extra automorphisms not induced by automorphisms of the square lattice. These cases comprise three isolated networks and two infinite families with wreath product structure. We briefly discuss the implications for pattern formation, dynamics and bifurcations. This paper is a sequel to a similar analysis of the square lattice and uses similar methods.

Symmetries in the external world constrain the evolution of neuronal circuits that allow organisms to sense the environment and act within it. Many small “modular” circuits can be viewed as approximate discretizations of the relevant symmetries, relating their forms to the functions they perform. The recent development of a formal theory of dynamics and bifurcations of networks of coupled differential equations permits the analysis of some aspects of network behavior without invoking specific model equations or numerical simulations. We review basic features of this theory, compare it to equivariant dynamics, and examine the subtle effects of symmetry when combined with network structure. We illustrate the relation between form and function through examples drawn from neurobiology, including locomotion, peristalsis, visual perception, balance, hearing, location detection, decision-making, and the connectome of the nematode Caenorhabditis elegans.

Robust synchrony in network dynamics is governed by balanced colorings and the corresponding quotient network, also formalized in terms of graph fibrations. Dynamics and bifurcations are constrained — often in surprising ways — by the associated synchrony subspaces, which are invariant under all admissible ordinary differential equations (ODEs). The class of admissible ODEs is determined by a groupoid, whose objects are the input sets of nodes and whose morphisms are input isomorphisms between those sets. We define the coloring subgroupoid corresponding to a coloring, leading to groupoid interpretations of colorings and quotient networks. The first half of the paper is mainly tutorial. The second half, which is new, characterizes the structure of the network groupoid and proves that the groupoid of the quotient network is the quotient of the network groupoid by a normal subgroupoid of transition elements.