Pattern formation in oscillatory complex networks consisting of excitable nodes

Theta oscillons in behaving rats

Recently discovered constituents of the brain waves-the oscillons-provide high-resolution representation of the extracellular field dynamics. Here we study the most robust, highest-amplitude oscillons that manifest in actively behaving rats and generally correspond to the traditional θ -waves. We show that the resemblances between θ -oscillons and the conventional θ -waves apply to the ballpark characteristics-mean frequencies, amplitudes, and bandwidths. In addition, both hippocampal and cortical oscillons exhibit a number of intricate, behavior-attuned, transient properties that suggest a new vantage point for understanding the θ -rhythms' structure, origins and functions. We demonstrate that oscillons are frequency-modulated waves, with speed-controlled parameters, embedded into a noise background. We also use a basic model of neuronal synchronization to contextualize and to interpret the observed phenomena. In particular, we argue that the synchronicity level in physiological networks is fairly weak and modulated by the animal's locomotion.

Altered patterning of neural activity in a tauopathy mouse model

Alzheimer's disease (AD) is a complex neurodegenerative condition that manifests at multiple levels and involves a spectrum of abnormalities ranging from the cellular to cognitive. Here, we investigate the impact of AD-related tau-pathology on hippocampal circuits in mice engaged in spatial navigation, and study changes of neuronal firing and dynamics of extracellular fields. While most studies are based on analyzing instantaneous or time-averaged characteristics of neuronal activity, we focus on intermediate timescales-spike trains and waveforms of oscillatory potentials, which we consider as single entities. We find that, in healthy mice, spike arrangements and wave patterns (series of crests or troughs) are coupled to the animal's location, speed, and acceleration. In contrast, in tau-mice, neural activity is structurally disarrayed: brainwave cadence is detached from locomotion, spatial selectivity is lost, the spike flow is scrambled. Importantly, these alterations start early and accumulate with age, which exposes progressive disinvolvement the hippocampus circuit in spatial navigation. These features highlight qualitatively different neurodynamics than the ones provided by conventional analyses, and are more salient, thus revealing a new level of the hippocampal circuit disruptions.

Controlled generation of self-sustained oscillations in complex artificial neural networks

Spatially distinct, self-sustained oscillations in artificial neural networks are fundamental to information encoding, storage, and processing in these systems. Here, we develop a method to induce a large variety of self-sustained oscillatory patterns in artificial neural networks and a controlling strategy to switch between different patterns. The basic principle is that, given a complex network, one can find a set of nodes-the minimum feedback vertex set (mFVS), whose removal or inhibition will result in a tree-like network without any loop structure. Reintroducing a few or even a single mFVS node into the tree-like artificial neural network can recover one or a few of the loops and lead to self-sustained oscillation patterns based on these loops. Reactivating various mFVS nodes or their combinations can then generate a large number of distinct neuronal firing patterns with a broad distribution of the oscillation period. When the system is near a critical state, chaos can arise, providing a natural platform for pattern switching with remarkable flexibility. With mFVS guided control, complex networks of artificial neurons can thus be exploited as potential prototypes for local, analog type of processing paradigms.

Discovering hidden layers in quantum graphs

Finding hidden layers in complex networks is an important and a nontrivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multilayer system exist and if so then what is their extent, i.e., how many unknown layers are there. Assuming that the only information available is the time evolution of a wave propagation on a single layer of a network it is indeed possible to uncover that which is hidden by merely observing the dynamics. We present evidence on both synthetic and real-world networks that the frequency spectrum of the wave dynamics can express distinct features in the form of additional frequency peaks. These peaks exhibit dependence on the number of layers taking part in the propagation and thus allowing for the extraction of said number. We show that, in fact, with sufficient observation time, one can fully reconstruct the row-normalized adjacency matrix spectrum. We compare our propositions to a machine learning approach using a wave packet signature method modified for the purposes of multilayer systems.

Spatiotemporal characteristics in systems of diffusively coupled excitable slow-fast FitzHugh-Rinzel dynamical neurons

In this paper, we study an excitable, biophysical system that supports wave propagation of nerve impulses. We consider a slow-fast, FitzHugh-Rinzel neuron model where only the membrane voltage interacts diffusively, giving rise to the formation of spatiotemporal patterns. We focus on local, nonlinear excitations and diverse neural responses in an excitable one- and two-dimensional configuration of diffusively coupled FitzHugh-Rinzel neurons. The study of the emerging spatiotemporal patterns is essential in understanding the working mechanism in different brain areas. We derive analytically the coefficients of the amplitude equations in the vicinity of Hopf bifurcations and characterize various patterns, including spirals exhibiting complex geometric substructures. Furthermore, we derive analytically the condition for the development of antispirals in the neighborhood of the bifurcation point. The emergence of broken target waves can be observed to form spiral-like profiles. The spatial dynamics of the excitable system exhibits two- and multi-arm spirals for small diffusive couplings. Our results reveal a multitude of neural excitabilities and possible conditions for the emergence of spiral-wave formation. Finally, we show that the coupled excitable systems with different firing characteristics participate in a collective behavior that may contribute significantly to irregular neural dynamics.

Heterochromatin Networks: Topology, Dynamics, and Function (a Working Hypothesis)

Open systems can only exist by self-organization as pulsing structures exchanging matter and energy with the outer world. This review is an attempt to reveal the organizational principles of the heterochromatin supra-intra-chromosomal network in terms of nonlinear thermodynamics. The accessibility of the linear information of the genetic code is regulated by constitutive heterochromatin (CHR) creating the positional information in a system of coordinates. These features include scale-free splitting-fusing of CHR with the boundary constraints of the nucleolus and nuclear envelope. The analysis of both the literature and our own data suggests a radial-concentric network as the main structural organization principle of CHR regulating transcriptional pulsing. The dynamic CHR network is likely created together with nucleolus-associated chromatin domains, while the alveoli of this network, including springy splicing speckles, are the pulsing transcription hubs. CHR contributes to this regulation due to the silencing position variegation effect, stickiness, and flexible rigidity determined by the positioning of nucleosomes. The whole system acts in concert with the elastic nuclear actomyosin network which also emerges by self-organization during the transcriptional pulsing process. We hypothesize that the the transcriptional pulsing, in turn, adjusts its frequency/amplitudes specified by topologically associating domains to the replication timing code that determines epigenetic differentiation memory.

Cluster synchronization in networked nonidentical chaotic oscillators

In exploring oscillator synchronization, a general observation is that as the oscillators become nonidentical, e.g., introducing parameter mismatch among the oscillators, the propensity for synchronization will be deteriorated. Yet in realistic systems, parameter mismatch is unavoidable and even worse in some circumstances, the oscillators might follow different types of dynamics. Considering the significance of synchronization to the functioning of many realistic systems, it is natural to ask the following question: Can synchronization be achieved in networked oscillators of clearly different parameters or dynamics? Here, by the model of networked chaotic oscillators, we are able to demonstrate and argue that, despite the presence of parameter mismatch (or different dynamics), stable synchronization can still be achieved on symmetric complex networks. Specifically, we find that when the oscillators are configured on the network in such a way that the symmetric nodes have similar parameters (or follow the same type of dynamics), cluster synchronization can be generated. The stabilities of the cluster synchronization states are analyzed by the method of symmetry-based stability analysis, with the theoretical predictions in good agreement with the numerical results. Our study sheds light on the interplay between symmetry and cluster synchronization in complex networks and give insights into the functionalities of realistic systems where nonidentical nonlinear oscillators are presented and cluster synchronization is crucial.

Winfree loop sustained oscillation in two-dimensional excitable lattices: Prediction and realization

The problem of self-sustained oscillations in excitable complex networks is the central issue under investigation, among which the prediction and the realization of self-sustained oscillations in different kinds of excitable networks are the challenging tasks. In this paper, we extensively investigate the prediction and the realization of a Winfree loop sustained oscillation (WLSO) in two-dimensional (2D) excitable lattices. By analyzing the network structure, the fundamental oscillation source structure (FOSS) of WLSO in a 2D excitable lattice is exposed explicitly. For the suitable combinations of system parameters, the Winfree loop can self-organize on the FOSS to form an oscillation source sustaining the oscillation, and these suitable parameter combinations are predicted by calculating the minimum Winfree loop length and have been further confirmed in numerical simulations. However, the FOSS cannot spontaneously offer the WLSO in 2D excitable lattices in usual cases due to the coupling bidirectionality and the symmetry properties of the lattice. A targeted protection scheme of the oscillation source is proposed by overcoming these two drawbacks. Finally, the WLSO is realized in the 2D excitable lattice successfully.

Reconstruction of dynamic networks with time-delayed interactions in the presence of fast-varying noises

Most complex social, biological and technological systems can be described by dynamic networks. Reconstructing network structures from measurable data is a fundamental problem in almost all interdisciplinary fields. Network nodes interact with each other and those interactions often have diversely distributed time delays. Accurate reconstruction of any targeted interaction to a node requires measured data of all its neighboring nodes together with information on the time delays of interactions from these neighbors. When networks are large, these data are often not available and time-delay factors are deeply hidden. Here we show that fast-varying noise can be of great help in solving these challenging problems. By computing suitable correlations, we can infer the intensity and time delay of any targeted interaction with the data of two related nodes (driving and driven nodes) only while all other nodes in the network are hidden. This method is analytically derived and fully justified by extensive numerical simulations.

Spatiotemporal dynamics in excitable homogeneous random networks composed of periodically self-sustained oscillation

The collective behaviors of networks are often dependent on the network connections and bifurcation parameters, also the local kinetics plays an important role in contributing the consensus of coupled oscillators. In this paper, we systematically investigate the influence of network structures and system parameters on the spatiotemporal dynamics in excitable homogeneous random networks (EHRNs) composed of periodically self-sustained oscillation (PSO). By using the dominant phase-advanced driving (DPAD) method, the one-dimensional (1D) Winfree loop is exposed as the oscillation source supporting the PSO, and the accurate wave propagation pathways from the oscillation source to the whole network are uncovered. Then, an order parameter is introduced to quantitatively study the influence of network structures and system parameters on the spatiotemporal dynamics of PSO in EHRNs. Distinct results induced by the network structures and the system parameters are observed. Importantly, the corresponding mechanisms are revealed. PSO influenced by the network structures are induced not only by the change of average path length (APL) of network, but also by the invasion of 1D Winfree loop from the outside linking nodes. Moreover, PSO influenced by the system parameters are determined by the excitation threshold and the minimum 1D Winfree loop. Finally, we confirmed that the excitation threshold and the minimum 1D Winfree loop determined PSO will degenerate as the system size is expanded.

Sustained dynamics of a weakly excitable system with nonlocal interactions

We investigate a two-dimensional spatially extended system that has a weak sense of excitability, where an excitation wave has a uniform profile and propagates only within a finite range. Using a cellular automaton model of such a weakly excitable system, we show that three kinds of sustained dynamics emerge when nonlocal spatial interactions are provided, where a chain of local wave propagation and nonlocal activation forms an elementary oscillatory cycle. Transition between different oscillation regimes can be understood as different ways of interactions among these cycles. Analytical expressions are given for the oscillation probability near the onset of oscillations.

Minimum Winfree loop determines self-sustained oscillations in excitable Erdös-Rényi random networks

The investigation of self-sustained oscillations in excitable complex networks is very important in understanding various activities in brain systems, among which the exploration of the key determinants of oscillations is a challenging task. In this paper, by investigating the influence of system parameters on self-sustained oscillations in excitable Erdös-Rényi random networks (EERRNs), the minimum Winfree loop (MWL) is revealed to be the key factor in determining the emergence of collective oscillations. Specifically, the one-to-one correspondence between the optimal connection probability (OCP) and the MWL length is exposed. Moreover, many important quantities such as the lower critical connection probability (LCCP), the OCP, and the upper critical connection probability (UCCP) are determined by the MWL. Most importantly, they can be approximately predicted by the network structure analysis, which have been verified in numerical simulations. Our results will be of great importance to help us in understanding the key factors in determining persistent activities in biological systems.

Inferring connectivity in networked dynamical systems: Challenges using Granger causality

Determining the interactions and causal relationships between nodes in an unknown networked dynamical system from measurement data alone is a challenging, contemporary task across the physical, biological, and engineering sciences. Statistical methods, such as the increasingly popular Granger causality, are being broadly applied for data-driven discovery of connectivity in fields from economics to neuroscience. A common version of the algorithm is called pairwise-conditional Granger causality, which we systematically test on data generated from a nonlinear model with known causal network structure. Specifically, we simulate networked systems of Kuramoto oscillators and use the Multivariate Granger Causality Toolbox to discover the underlying coupling structure of the system. We compare the inferred results to the original connectivity for a wide range of parameters such as initial conditions, connection strengths, community structures, and natural frequencies. Our results show a significant systematic disparity between the original and inferred network, unless the true structure is extremely sparse or dense. Specifically, the inferred networks have significant discrepancies in the number of edges and the eigenvalues of the connectivity matrix, demonstrating that they typically generate dynamics which are inconsistent with the ground truth. We provide a detailed account of the dynamics for the Erdős-Rényi network model due to its importance in random graph theory and network science. We conclude that Granger causal methods for inferring network structure are highly suspect and should always be checked against a ground truth model. The results also advocate the need to perform such comparisons with any network inference method since the inferred connectivity results appear to have very little to do with the ground truth system.

The Fundamental Structure and the Reproduction of Spiral Wave in a Two-Dimensional Excitable Lattice

In this paper we have systematically investigated the fundamental structure and the reproduction of spiral wave in a two-dimensional excitable lattice. A periodically rotating spiral wave is introduced as the model to reproduce spiral wave artificially. Interestingly, by using the dominant phase-advanced driving analysis method, the fundamental structure containing the loop structure and the wave propagation paths has been revealed, which can expose the periodically rotating orbit of spiral tip and the charity of spiral wave clearly. Furthermore, the fundamental structure is utilized as the core for artificial spiral wave. Additionally, the appropriate parameter region, in which the artificial spiral wave can be reproduced, is studied. Finally, we discuss the robustness of artificial spiral wave to defects.

Activation process in excitable systems with multiple noise sources: Large number of units

We study the activation process in large assemblies of type II excitable units whose dynamics is influenced by two independent noise terms. The mean-field approach is applied to explicitly demonstrate that the assembly of excitable units can itself exhibit macroscopic excitable behavior. In order to facilitate the comparison between the excitable dynamics of a single unit and an assembly, we introduce three distinct formulations of the assembly activation event. Each formulation treats different aspects of the relevant phenomena, including the thresholdlike behavior and the role of coherence of individual spikes. Statistical properties of the assembly activation process, such as the mean time-to-first pulse and the associated coefficient of variation, are found to be qualitatively analogous for all three formulations, as well as to resemble the results for a single unit. These analogies are shown to derive from the fact that global variables undergo a stochastic bifurcation from the stochastically stable fixed point to continuous oscillations. Local activation processes are analyzed in the light of the competition between the noise-led and the relaxation-driven dynamics. We also briefly report on a system-size antiresonant effect displayed by the mean time-to-first pulse.

Perspective: network-guided pattern formation of neural dynamics

The understanding of neural activity patterns is fundamentally linked to an understanding of how the brain's network architecture shapes dynamical processes. Established approaches rely mostly on deviations of a given network from certain classes of random graphs. Hypotheses about the supposed role of prominent topological features (for instance, the roles of modularity, network motifs or hierarchical network organization) are derived from these deviations. An alternative strategy could be to study deviations of network architectures from regular graphs (rings and lattices) and consider the implications of such deviations for self-organized dynamic patterns on the network. Following this strategy, we draw on the theory of spatio-temporal pattern formation and propose a novel perspective for analysing dynamics on networks, by evaluating how the self-organized dynamics are confined by network architecture to a small set of permissible collective states. In particular, we discuss the role of prominent topological features of brain connectivity, such as hubs, modules and hierarchy, in shaping activity patterns. We illustrate the notion of network-guided pattern formation with numerical simulations and outline how it can facilitate the understanding of neural dynamics.

Emergence of self-sustained oscillations in excitable Erdös-Rényi random networks

We investigate the emergence of self-sustained oscillations in excitable Erdös-Rényi random networks (EERRNs). Interestingly, periodical self-sustained oscillations have been found at a moderate connection probability P. For smaller or larger P, the system evolves into a homogeneous rest state with distinct mechanisms. One-dimensional Winfree loops are discovered as the sources to maintain the oscillations. Moreover, by analyzing these oscillation sources, we propose two criteria to explain the spatiotemporal dynamics obtained in EERRNs. Finally, the two critical connection probabilities for which self-sustained oscillations can emerge are approximately predicted based on these two criteria.

Simulating dynamic plastic continuous neural networks by finite elements

We introduce dynamic plastic continuous neural network (DPCNN), which is comprised of neurons distributed in a nonlinear plastic medium where wire-like connections of neural networks are replaced with the continuous medium. We use finite element method to model the dynamic phenomenon of information processing within the DPCNNs. During the training, instead of weights, the properties of the continuous material at its different locations and some properties of neurons are modified. Input and output can be vectors and/or continuous functions over lines and/or areas. Delay and feedback from neurons to themselves and from outputs occur in the DPCNNs. We model a simple form of the DPCNN where the medium is a rectangular plate of bilinear material, and the neurons continuously fire a signal, which is a function of the horizontal displacement.

Autapse-induced spiral wave in network of neurons under noise

Autapse plays an important role in regulating the electric activity of neuron by feedbacking time-delayed current on the membrane of neuron. Autapses are considered in a local area of regular network of neurons to investigate the development of spatiotemporal pattern, and emergence of spiral wave is observed while it fails to grow up and occupy the network completely. It is found that spiral wave can be induced to occupy more area in the network under optimized noise on the network with periodical or no-flux boundary condition being used. The developed spiral wave with self-sustained property can regulate the collective behaviors of neurons as a pacemaker. To detect the collective behaviors, a statistical factor of synchronization is calculated to investigate the emergence of ordered state in the network. The network keeps ordered state when self-sustained spiral wave is formed under noise and autapse in local area of network, and it independent of the selection of periodical or no-flux boundary condition. The developed stable spiral wave could be helpful for memory due to the distinct self-sustained property.

Structure and dynamics of molecular networks: a novel paradigm of drug discovery: a comprehensive review

Despite considerable progress in genome- and proteome-based high-throughput screening methods and in rational drug design, the increase in approved drugs in the past decade did not match the increase of drug development costs. Network description and analysis not only give a systems-level understanding of drug action and disease complexity, but can also help to improve the efficiency of drug design. We give a comprehensive assessment of the analytical tools of network topology and dynamics. The state-of-the-art use of chemical similarity, protein structure, protein-protein interaction, signaling, genetic interaction and metabolic networks in the discovery of drug targets is summarized. We propose that network targeting follows two basic strategies. The "central hit strategy" selectively targets central nodes/edges of the flexible networks of infectious agents or cancer cells to kill them. The "network influence strategy" works against other diseases, where an efficient reconfiguration of rigid networks needs to be achieved by targeting the neighbors of central nodes/edges. It is shown how network techniques can help in the identification of single-target, edgetic, multi-target and allo-network drug target candidates. We review the recent boom in network methods helping hit identification, lead selection optimizing drug efficacy, as well as minimizing side-effects and drug toxicity. Successful network-based drug development strategies are shown through the examples of infections, cancer, metabolic diseases, neurodegenerative diseases and aging. Summarizing >1200 references we suggest an optimized protocol of network-aided drug development, and provide a list of systems-level hallmarks of drug quality. Finally, we highlight network-related drug development trends helping to achieve these hallmarks by a cohesive, global approach.

The formation mechanism of defects, spiral wave in the network of neurons

A regular network of neurons is constructed by using the Morris-Lecar (ML) neuron with the ion channels being considered, and the potential mechnism of the formation of a spiral wave is investigated in detail. Several spiral waves are initiated by blocking the target wave with artificial defects and/or partial blocking (poisoning) in ion channels. Furthermore, possible conditions for spiral wave formation and the effect of partial channel blocking are discussed completely. Our results are summarized as follows. 1) The emergence of a target wave depends on the transmembrane currents with diversity, which mapped from the external forcing current and this kind of diversity is associated with spatial heterogeneity in the media. 2) Distinct spiral wave could be induced to occupy the network when the target wave is broken by partially blocking the ion channels of a fraction of neurons (local poisoned area), and these generated spiral waves are similar with the spiral waves induced by artificial defects. It is confirmed that partial channel blocking of some neurons in the network could play a similar role in breaking a target wave as do artificial defects; 3) Channel noise and additive Gaussian white noise are also considered, and it is confirmed that spiral waves are also induced in the network in the presence of noise. According to the results mentioned above, we conclude that appropriate poisoning in ion channels of neurons in the network acts as ‘defects’ on the evolution of the spatiotemporal pattern, and accounts for the emergence of a spiral wave in the network of neurons. These results could be helpful to understand the potential cause of the formation and development of spiral waves in the cortex of a neuronal system.

Synchronous patterns in complex systems

When a complex network is slightly desynchronized, a few of the network nodes will be escaping from the uniform synchronization background frequently with a random fashion, leading to the intermittent network synchronization. Here, based on the eigenvectors of the network coupling matrix, we propose a new method which is able to figure out the unstable nodes in the general case of desynchronized complex networks. Moreover, with this method, we are also able to regulate the seemingly random network dynamics into stable and visible synchronous patterns. The efficiency of this method is verified by a variety of network models, including varying the network structures, the node local dynamics, and the desynchronization types. Our studies show that, even for the complex network systems, synchronous patterns can still be identified and characterized.