Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons

Long-Tailed Characteristic of Spiking Pattern Alternation Induced by Log-Normal Excitatory Synaptic Distribution

Studies of structural connectivity at the synaptic level show that in synaptic connections of the cerebral cortex, the excitatory postsynaptic potential (EPSP) in most synapses exhibits sub-mV values, while a small number of synapses exhibit large EPSPs ( >~1.0 [mV]). This means that the distribution of EPSP fits a log-normal distribution. While not restricting structural connectivity, skewed and long-tailed distributions have been widely observed in neural activities, such as the occurrences of spiking rates and the size of a synchronously spiking population. Many studies have been modeled this long-tailed EPSP neural activity distribution; however, its causal factors remain controversial. This study focused on the long-tailed EPSP distributions and interlateral synaptic connections primarily observed in the cortical network structures, thereby having constructed a spiking neural network consistent with these features. Especially, we constructed two coupled modules of spiking neural networks with excitatory and inhibitory neural populations with a log-normal EPSP distribution. We evaluated the spiking activities for different input frequencies and with/without strong synaptic connections. These coupled modules exhibited intermittent intermodule-alternative behavior, given moderate input frequency and the existence of strong synaptic and intermodule connections. Moreover, the power analysis, multiscale entropy analysis, and surrogate data analysis revealed that the long-tailed EPSP distribution and intermodule connections enhanced the complexity of spiking activity at large temporal scales and induced nonlinear dynamics and neural activity that followed the long-tailed distribution.

FNS allows efficient event-driven spiking neural network simulations based on a neuron model supporting spike latency

Neural modelling tools are increasingly employed to describe, explain, and predict the human brain's behavior. Among them, spiking neural networks (SNNs) make possible the simulation of neural activity at the level of single neurons, but their use is often threatened by the resources needed in terms of processing capabilities and memory. Emerging applications where a low energy burden is required (e.g. implanted neuroprostheses) motivate the exploration of new strategies able to capture the relevant principles of neuronal dynamics in reduced and efficient models. The recent Leaky Integrate-and-Fire with Latency (LIFL) spiking neuron model shows some realistic neuronal features and efficiency at the same time, a combination of characteristics that may result appealing for SNN-based brain modelling. In this paper we introduce FNS, the first LIFL-based SNN framework, which combines spiking/synaptic modelling with the event-driven approach, allowing us to define heterogeneous neuron groups and multi-scale connectivity, with delayed connections and plastic synapses. FNS allows multi-thread, precise simulations, integrating a novel parallelization strategy and a mechanism of periodic dumping. We evaluate the performance of FNS in terms of simulation time and used memory, and compare it with those obtained with neuronal models having a similar neurocomputational profile, implemented in NEST, showing that FNS performs better in both scenarios. FNS can be advantageously used to explore the interaction within and between populations of spiking neurons, even for long time-scales and with a limited hardware configuration.

Impact of perceived distances on international tourism

Worldwide tourism revenues have tripled in the last decade. Yet, there is a gap in our understanding of how distances shape peoples’ travel choices. To understand global tourism patterns we map the flow of tourists around the world onto a complex network and study the impact of two types of distances, geographical and through the World Airline Network, a major infrastructure for tourism. We find that although the World Airline Network serves as infrastructural support for the International Tourism Network, the flow of tourism does not correlate strongly with the extent of flight connections available worldwide. Instead, unidirectional flows appear locally forming communities that shed light on global travelling behaviour since there is only a 15% probability of finding bidirectional tourism between a pair of countries. We find that most tourists travel to neighbouring countries and mainly cover larger distances when there is a direct flight, irrespective of the time it takes. This may be a consequence of one-way cyclic tourism that we uncover by analysing the triangles that are formed by the network of flows in the International Tourism Network.

Emergence of spatially periodic diffusive waves in small-world neuronal networks

It has been observed in experiment that the anatomical structure of neuronal networks in the brain possesses the feature of small-world networks. Yet how the small-world structure affects network dynamics remains to be fully clarified. Here we study the dynamics of a class of small-world networks consisting of pulse-coupled integrate-and-fire (I&F) neurons. Under stochastic Poisson drive, we find that the activity of the entire network resembles diffusive waves. To understand its underlying mechanism, we analyze the simplified regular-lattice network consisting of firing-rate-based neurons as an approximation to the original I&F small-world network. We demonstrate both analytically and numerically that, with strongly coupled connections, in the absence of noise, the activity of the firing-rate-based regular-lattice network spatially forms a static grating pattern that corresponds to the spatial distribution of the firing rate observed in the I&F small-world neuronal network. We further show that the spatial grating pattern with different phases comprise the continuous attractor of both the I&F small-world and firing-rate-based regular-lattice network dynamics. In the presence of input noise, the activity of both networks is perturbed along the continuous attractor, which gives rise to the diffusive waves. Our numerical simulations and theoretical analysis may potentially provide insights into the understanding of the generation of wave patterns observed in cortical networks.

Temporal-specific complexity of spiking patterns in spontaneous activity induced by a dual complex network structure

Temporal fluctuation of neural activity in the brain has an important function in optimal information processing. Spontaneous activity is a source of such fluctuation. The distribution of excitatory postsynaptic potentials (EPSPs) between cortical pyramidal neurons can follow a log-normal distribution. Recent studies have shown that networks connected by weak synapses exhibit characteristics of a random network, whereas networks connected by strong synapses have small-world characteristics of small path lengths and large cluster coefficients. To investigate the relationship between temporal complexity spontaneous activity and structural network duality in synaptic connections, we executed a simulation study using the leaky integrate-and-fire spiking neural network with log-normal synaptic weight distribution for the EPSPs and duality of synaptic connectivity, depending on synaptic weight. We conducted multiscale entropy analysis of the temporal spiking activity. Our simulation demonstrated that, when strong synaptic connections approach a small-world network, specific spiking patterns arise during irregular spatio-temporal spiking activity, and the complexity at the large temporal scale (i.e., slow frequency) is enhanced. Moreover, we confirmed through a surrogate data analysis that slow temporal dynamics reflect a deterministic process in the spiking neural networks. This modelling approach may improve the understanding of the spatio-temporal complex neural activity in the brain.

Effect of inhibitory spike-timing-dependent plasticity on fast sparsely synchronized rhythms in a small-world neuronal network

We consider the Watts-Strogatz small-world network (SWN) consisting of inhibitory fast spiking Izhikevich interneurons. This inhibitory neuronal population has adaptive dynamic synaptic strengths governed by the inhibitory spike-timing-dependent plasticity (iSTDP). In previous works without iSTDP, fast sparsely synchronized rhythms, associated with diverse cognitive functions, were found to appear in a range of large noise intensities for fixed strong synaptic inhibition strengths. Here, we investigate the effect of iSTDP on fast sparse synchronization (FSS) by varying the noise intensity D. We employ an asymmetric anti-Hebbian time window for the iSTDP update rule [which is in contrast to the Hebbian time window for the excitatory STDP (eSTDP)]. Depending on values of D, population-averaged values of saturated synaptic inhibition strengths are potentiated [long-term potentiation (LTP)] or depressed [long-term depression (LTD)] in comparison with the initial mean value, and dispersions from the mean values of LTP/LTD are much increased when compared with the initial dispersion, independently of D. In most cases of LTD where the effect of mean LTD is dominant in comparison with the effect of dispersion, good synchronization (with higher spiking measure) is found to get better via LTD, while bad synchronization (with lower spiking measure) is found to get worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to that in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD). Emergences of LTD and LTP of synaptic inhibition strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, we also investigate the effects of network architecture on FSS by changing the rewiring probability p of the SWN in the presence of iSTDP.

Circuit variability interacts with excitatory-inhibitory diversity of interneurons to regulate network encoding capacity

Local interneurons (LNs) in the Drosophila olfactory system exhibit neuronal diversity and variability, yet it is still unknown how these features impact information encoding capacity and reliability in a complex LN network. We employed two strategies to construct a diverse excitatory-inhibitory neural network beginning with a ring network structure and then introduced distinct types of inhibitory interneurons and circuit variability to the simulated network. The continuity of activity within the node ensemble (oscillation pattern) was used as a readout to describe the temporal dynamics of network activity. We found that inhibitory interneurons enhance the encoding capacity by protecting the network from extremely short activation periods when the network wiring complexity is very high. In addition, distinct types of interneurons have differential effects on encoding capacity and reliability. Circuit variability may enhance the encoding reliability, with or without compromising encoding capacity. Therefore, we have described how circuit variability of interneurons may interact with excitatory-inhibitory diversity to enhance the encoding capacity and distinguishability of neural networks. In this work, we evaluate the effects of different types and degrees of connection diversity on a ring model, which may simulate interneuron networks in the Drosophila olfactory system or other biological systems.

From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity

It is clear that the topological structure of a neural network somehow determines the activity of the neurons within it. In the present work, we ask to what extent it is possible to examine the structural features of a network and learn something about its activity? Specifically, we consider how the centrality (the importance of a node in a network) of a neuron correlates with its firing rate. To investigate, we apply an array of centrality measures, including In-Degree, Closeness, Betweenness, Eigenvector, Katz, PageRank, Hyperlink-Induced Topic Search (HITS) and NeuronRank to Leaky-Integrate and Fire neural networks with different connectivity schemes. We find that Katz centrality is the best predictor of firing rate given the network structure, with almost perfect correlation in all cases studied, which include purely excitatory and excitatory–inhibitory networks, with either homogeneous connections or a small-world structure. We identify the properties of a network which will cause this correlation to hold. We argue that the reason Katz centrality correlates so highly with neuronal activity compared to other centrality measures is because it nicely captures disinhibition in neural networks. In addition, we argue that these theoretical findings are applicable to neuroscientists who apply centrality measures to functional brain networks, as well as offer a neurophysiological justification to high level cognitive models which use certain centrality measures.

A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design

This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is its multiple attractors caused by different initial values. With the change of parameters, the system experiences mono-stability, bi-stability, mono-periodicity, bi-periodicity, one strange attractor, and two coexisting strange attractors. The complex dynamic behaviors of the system are revealed by analyzing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the system. Using the new chaotic system, an S-Box is developed for cryptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies.

Stochastic spike synchronization in a small-world neural network with spike-timing-dependent plasticity

We consider the Watts-Strogatz small-world network (SWN) consisting of subthreshold neurons which exhibit noise-induced spikings. This neuronal network has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). In previous works without STDP, stochastic spike synchronization (SSS) between noise-induced spikings of subthreshold neurons was found to occur in a range of intermediate noise intensities. Here, we investigate the effect of additive STDP on the SSS by varying the noise intensity. Occurrence of a "Matthew" effect in synaptic plasticity is found due to a positive feedback process. As a result, good synchronization gets better via long-term potentiation of synaptic strengths, while bad synchronization gets worse via long-term depression. Emergences of long-term potentiation and long-term depression of synaptic strengths are intensively investigated via microscopic studies based on the pair-correlations between the pre- and the post-synaptic IISRs (instantaneous individual spike rates) as well as the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, the effects of multiplicative STDP (which depends on states) on the SSS are studied and discussed in comparison with the case of additive STDP (independent of states). These effects of STDP on the SSS in the SWN are also compared with those in the regular lattice and the random graph.

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.

Emergence of ultrafast sparsely synchronized rhythms and their responses to external stimuli in an inhomogeneous small-world complex neuronal network

We consider an inhomogeneous small-world network (SWN) composed of inhibitory short-range (SR) and long-range (LR) interneurons, and investigate the effect of network architecture on emergence of synchronized brain rhythms by varying the fraction of LR interneurons p_long. The betweenness centralities of the LR and SR interneurons (characterizing the potentiality in controlling communication between other interneurons) are distinctly different. Hence, in view of the betweenness, SWNs we consider are inhomogeneous, unlike the "canonical" Watts-Strogatz SWN with nearly the same betweenness centralities. For small p_long, the load of communication traffic is much concentrated on a few LR interneurons. However, as p_long is increased, the number of LR connections (coming from LR interneurons) increases, and then the load of communication traffic is less concentrated on LR interneurons, which leads to better efficiency of global communication between interneurons. Sparsely synchronized rhythms are thus found to emerge when passing a small critical value p_long^(c)(≃0.16). The population frequency of the sparsely synchronized rhythm is ultrafast (higher than 100 Hz), while the mean firing rate of individual interneurons is much lower (∼30 Hz) due to stochastic and intermittent neural discharges. These dynamical behaviors in the inhomogeneous SWN are also compared with those in the homogeneous Watts-Strogatz SWN, in connection with their network topologies. Particularly, we note that the main difference between the two types of SWNs lies in the distribution of betweenness centralities. Unlike the case of the Watts-Strogatz SWN, dynamical responses to external stimuli vary depending on the type of stimulated interneurons in the inhomogeneous SWN. We consider two cases of external time-periodic stimuli applied to sub-populations of the LR and SR interneurons, respectively. Dynamical responses (such as synchronization suppression and enhancement) to these two cases of stimuli are studied and discussed in relation to the betweenness centralities of stimulated interneurons, representing the effectiveness for transfer of stimulation effect in the whole network.

Autaptic Connections Shift Network Excitability and Bursting

We examine the role of structural autapses, when a neuron synapses onto itself, in driving network-wide bursting behavior. Using a simple spiking model of neuronal activity, we study how autaptic connections affect activity patterns, and evaluate if controllability significantly affects changes in bursting from autaptic connections. Adding more autaptic connections to excitatory neurons increased the number of spiking events and the number of network-wide bursts. We observed excitatory synapses contributed more to bursting behavior than inhibitory synapses. We evaluated if neurons with high average controllability, predicted to push the network into easily achievable states, affected bursting behavior differently than neurons with high modal controllability, thought to influence the network into difficult to reach states. Results show autaptic connections to excitatory neurons with high average controllability led to higher burst frequencies than adding the same number of self-looping connections to neurons with high modal controllability. The number of autapses required to induce bursting was lowered by adding autapses to high degree excitatory neurons. These results suggest a role of autaptic connections in controlling network-wide bursts in diverse cortical and subcortical regions of mammalian brain. Moreover, they open up new avenues for the study of dynamic neurophysiological correlates of structural controllability.

The shaping of intrinsic membrane potential oscillations: positive/negative feedback, ionic resonance/amplification, nonlinearities and time scales

The generation of intrinsic subthreshold (membrane potential) oscillations (STOs) in neuronal models requires the interaction between two processes: a relatively fast positive feedback that favors changes in voltage and a slower negative feedback that opposes these changes. These are provided by the so-called resonant and amplifying gating variables associated to the participating ionic currents. We investigate both the biophysical and dynamic mechanisms of generation of STOs and how their attributes (frequency and amplitude) depend on the model parameters for biophysical (conductance-based) models having qualitatively different types of resonant currents (activating and inactivating) and an amplifying current. Combinations of the same types of ionic currents (same models) in different parameter regimes give rise to different types of nonlinearities in the voltage equation: quasi-linear, parabolic-like and cubic-like. On the other hand, combinations of different types of ionic currents (different models) may give rise to the same type of nonlinearities. We examine how the attributes of the resulting STOs depend on the combined effect of these resonant and amplifying ionic processes, operating at different effective time scales, and the various types of nonlinearities. We find that, while some STO properties and attribute dependencies on the model parameters are determined by the specific combinations of ionic currents (biophysical properties), and are different for models with different such combinations, others are determined by the type of nonlinearities and are common for models with different types of ionic currents. Our results highlight the richness of STO behavior in single cells as the result of the various ways in which resonant and amplifying currents interact and affect the generation and termination of STOs as control parameters change. We make predictions that can be tested experimentally and are expected to contribute to the understanding of how rhythmic activity in neuronal networks emerge from the interplay of the intrinsic properties of the participating neurons and the network connectivity.

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.

Effect of intermodular connection on fast sparse synchronization in clustered small-world neural networks

We consider a clustered network with small-world subnetworks of inhibitory fast spiking interneurons and investigate the effect of intermodular connection on the emergence of fast sparsely synchronized rhythms by varying both the intermodular coupling strength J(inter) and the average number of intermodular links per interneuron M(syn)(inter). In contrast to the case of nonclustered networks, two kinds of sparsely synchronized states such as modular and global synchronization are found. For the case of modular sparse synchronization, the population behavior reveals the modular structure, because the intramodular dynamics of subnetworks make some mismatching. On the other hand, in the case of global sparse synchronization, the population behavior is globally identical, independently of the cluster structure, because the intramodular dynamics of subnetworks make perfect matching. We introduce a realistic cross-correlation modularity measure, representing the matching degree between the instantaneous subpopulation spike rates of the subnetworks, and examine whether the sparse synchronization is global or modular. Depending on its magnitude, the intermodular coupling strength J(inter) seems to play "dual" roles for the pacing between spikes in each subnetwork. For large J(inter), due to strong inhibition it plays a destructive role to "spoil" the pacing between spikes, while for small J(inter) it plays a constructive role to "favor" the pacing between spikes. Through competition between the constructive and the destructive roles of J(inter), there exists an intermediate optimal J(inter) at which the pacing degree between spikes becomes maximal. In contrast, the average number of intermodular links per interneuron M(syn)(inter) seems to play a role just to favor the pacing between spikes. With increasing M(syn)(inter), the pacing degree between spikes increases monotonically thanks to the increase in the degree of effectiveness of global communication between spikes. Furthermore, we employ the realistic sub- and whole-population order parameters, based on the instantaneous sub- and whole-population spike rates, to determine the threshold values for the synchronization-unsynchronization transition in the sub- and whole populations, and the degrees of global and modular sparse synchronization are also measured in terms of the realistic sub- and whole-population statistical-mechanical spiking measures defined by considering both the occupation and the pacing degrees of spikes. It is expected that our results could have implications for the role of the brain plasticity in some functional behaviors associated with population synchronization.

Global segregation of cortical activity and metastable dynamics

Cortical activity exhibits persistent metastable dynamics. Assemblies of neurons transiently couple (integrate) and decouple (segregate) at multiple spatiotemporal scales; both integration and segregation are required to support metastability. Integration of distant brain regions can be achieved through long range excitatory projections, but the mechanism supporting long range segregation is not clear. We argue that the thalamocortical matrix connections, which project diffusely from the thalamus to the cortex and have long been thought to support cortical gain control, play an equally-important role in cortical segregation. We present a computational model of the diffuse thalamocortical loop, called the competitive cross-coupling (CXC) spiking network. Simulations of the model show how different levels of tonic input from the brainstem to the thalamus could control dynamical complexity in the cortex, directing transitions between sleep, wakefulness and high attention or vigilance. The model also explains how mutually-exclusive activity could arise across large portions of the cortex, such as between the default-mode and task-positive networks. It is robust to noise but does not require noise to autonomously generate metastability. We conclude that the long range segregation observed in brain activity and required for global metastable dynamics could be provided by the thalamocortical matrix, and is strongly modulated by brainstem input to the thalamus.

Self-organized criticality in cortical assemblies occurs in concurrent scale-free and small-world networks

The spontaneous activity of cortical networks is characterized by the emergence of different dynamic states. Although several attempts were accomplished to understand the origin of these dynamics, the underlying factors continue to be elusive. In this work, we specifically investigated the interplay between network topology and spontaneous dynamics within the framework of self-organized criticality (SOC). The obtained results support the hypothesis that the emergence of critical states occurs in specific complex network topologies. By combining multi-electrode recordings of spontaneous activity of in vitro cortical assemblies with theoretical models, we demonstrate that different 'connectivity rules' drive the network towards different dynamic states. In particular, scale-free architectures with different degree of small-worldness account better for the variability observed in experimental data, giving rise to different dynamic states. Moreover, in relationship with the balance between excitation and inhibition and percentage of inhibitory hubs, the simulated cortical networks fall in a critical regime.

Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons

We are interested in noise-induced firings of subthreshold neurons which may be used for encoding environmental stimuli. Noise-induced population synchronization was previously studied only for the case of global coupling, unlike the case of subthreshold spiking neurons. Hence, we investigate the effect of complex network architecture on noise-induced synchronization in an inhibitory population of subthreshold bursting Hindmarsh-Rose neurons. For modeling complex synaptic connectivity, we consider the Watts-Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization. Thus, noise-induced burst synchronization (synchrony on the slow bursting time scale) and spike synchronization (synchrony on the fast spike time scale) are found to appear in a synchronized region of the [Formula: see text]-[Formula: see text] plane ([Formula: see text]: synaptic inhibition strength and [Formula: see text]: noise intensity). As the rewiring probability [Formula: see text] is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the [Formula: see text]-[Formula: see text] plane, while the region of the burst synchronization decreases slowly. We separate the slow bursting and the fast spiking time scales via frequency filtering, and characterize the noise-induced burst and spike synchronizations by employing realistic order parameters and statistical-mechanical measures introduced in our recent work. Thus, the bursting and spiking thresholds for the burst and spike synchronization transitions are determined in terms of the bursting and spiking order parameters, respectively. Furthermore, we also measure the degrees of burst and spike synchronizations in terms of the statistical-mechanical bursting and spiking measures, respectively.

Synchronization, non-linear dynamics and low-frequency fluctuations: analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

In this paper, the topographical relationship between functional connectivity (intended as inter-regional synchronization), spectral and non-linear dynamical properties across cortical areas of the healthy human brain is considered. Based upon functional MRI acquisitions of spontaneous activity during wakeful idleness, node degree maps are determined by thresholding the temporal correlation coefficient among all voxel pairs. In addition, for individual voxel time-series, the relative amplitude of low-frequency fluctuations and the correlation dimension (D2), determined with respect to Fourier amplitude and value distribution matched surrogate data, are measured. Across cortical areas, high node degree is associated with a shift towards lower frequency activity and, compared to surrogate data, clearer saturation to a lower correlation dimension, suggesting presence of non-linear structure. An attempt to recapitulate this relationship in a network of single-transistor oscillators is made, based on a diffusive ring (n = 90) with added long-distance links defining four extended hub regions. Similarly to the brain data, it is found that oscillators in the hub regions generate signals with larger low-frequency cycle amplitude fluctuations and clearer saturation to a lower correlation dimension compared to surrogates. The effect emerges more markedly close to criticality. The homology observed between the two systems despite profound differences in scale, coupling mechanism and dynamics appears noteworthy. These experimental results motivate further investigation into the heterogeneity of cortical non-linear dynamics in relation to connectivity and underline the ability for small networks of single-transistor oscillators to recreate collective phenomena arising in much more complex biological systems, potentially representing a future platform for modelling disease-related changes.

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.

Abrupt and gradual transitions between low and hyperexcited firing frequencies in neuronal models with fast synaptic excitation: a comparative study

Hyperexcitability of neuronal networks is one of the hallmarks of epileptic brain seizure generation, and results from a net imbalance between excitation and inhibition that promotes excessive abnormal firing frequencies. The transition between low and high firing frequencies as the levels of recurrent AMPA excitation change can occur either gradually or abruptly. We used modeling, numerical simulations, and dynamical systems tools to investigate the biophysical and dynamic mechanisms that underlie these two identified modes of transition in recurrently connected neurons via AMPA excitation. We compare our results and demonstrate that these two modes of transition are qualitatively different and can be linked to different intrinsic properties of the participating neurons.

Metastability and chimera states in modular delay and pulse-coupled oscillator networks

Modular networks of delay-coupled and pulse-coupled oscillators are presented, which display both transient (metastable) synchronization dynamics and the formation of a large number of "chimera" states characterized by coexistent synchronized and desynchronized subsystems. We consider networks based on both community and small-world topologies. It is shown through simulation that the metastable behaviour of the system is dependent in all cases on connection delay, and a critical region is found that maximizes indices of both metastability and the prevalence of chimera states. We show dependence of phase coherence in synchronous oscillation on the level and strength of external connectivity between communities, and demonstrate that synchronization dynamics are dependent on the modular structure of the network. The long-term behaviour of the system is considered and the relevance of the model briefly discussed with emphasis on biological and neurobiological systems.

Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol oscillators near a codimension-two bifurcation point

Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincaré map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves.

Interactions of excitatory and inhibitory feedback topologies in facilitating pattern separation and retrieval

Within the brain, the interplay between connectivity patterns of neurons and their spatiotemporal dynamics is believed to be intricately linked to the bases of behavior, such as the process of storing, consolidating, and retrieving memory traces. Memory is believed to be stored in the synaptic patterns of anatomical circuitry in the form of increased connectivity densities within subpopulations of neurons. At the same time, memory recall is thought to correspond to activation of discrete areas of the brain corresponding to those memories. Such regional subpopulations can selectively activate during memory recall or retrieval, signifying the process of accessing a single memory or concept. It has been shown previously that recovery of single memory activity patterns is mediated by global neuromodulation signifying transition into different cognitive states such as sleep or awake exploration. We examine how underlying topology can affect memory awake activation and sleep reactivation when such memories share increasing proportions of neurons. The results show that while single memory activation is diminished with increased overlap, pattern separation can be recovered by offsetting excitatory associations between two memories with targeted and heterogeneous inhibitory feedback. Such findings point to the importance of excitatory-to-inhibitory current balance at both the global and local levels in the context of memory retrieval and replay, and highlight the role of network topology in memory management processes.

Self-sustaining oscillations in complex networks of excitable elements

Random networks of symmetrically coupled, excitable elements can self-organize into coherently oscillating states if the networks contain loops (indeed loops are abundant in random networks) and if the initial conditions are sufficiently random. In the oscillating state, signals propagate in a single direction and one or a few network loops are selected as driving loops in which the excitation periodically circulates. We analyze the mechanism, describe the oscillating states, identify the pacemaker loops, and explain key features of their distribution.

Neurophysiological and computational principles of cortical rhythms in cognition

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Self-sustained irregular activity in 2-D small-world networks of excitatory and inhibitory neurons

In this paper, we study the self-sustained irregular firing activity in 2-D small-world (SW) neural networks consisting of both excitatory and inhibitory neurons by computational modeling. For a proper proportion of unidirectional shortcuts, the stable self-sustained activity with irregular firing states indeed occurs in the considered network. By varying the shortcut density while keeping other system parameters fixed, different levels of irregular firing states, from weakly irregular to Poisson-like and burst firing states, are obtained in 2-D SW neural networks. It is also observed that this activity is sensitive to small perturbations, which might provide a possible mechanism for producing chaos. On the other hand, we find that several other system parameters, such as the network size and refractory period, have significant impact on this activity. Further simulation results show that the 2-D SW neural network can sustain such long-lasting firing behavior by using a smaller number of connections than the random neural network.

Propagation failures, breathing fronts, and nonannihilation collisions in the ferroin-bromate-pyrocatechol system

The emergence of propagating pulses was investigated with the photosensitive ferroin-bromate-pyrocatechol reaction in capillary tubes, in which various interesting spatiotemporal behaviors such as propagation failure, breathing fronts, and transitions between propagating pulses and fronts have been observed. Rather than a mutual annihilation, the collision of a propagating pulse and a growing front forces the front to recede gradually. A phase diagram in the pyrocatechol-bromate concentration space shows that the pulse instabilities take place throughout the conditions at which the system generates wave activities, suggesting that the presence of coupled autocatalytic feedbacks may facilitate the onset of pulse instabilities.

Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrate-and-fire neurons

We investigate numerically the collective dynamical behavior of pulse-coupled nonleaky integrate-and-fire neurons that are arranged on a two-dimensional small-world network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Besides a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulentlike patterns, which-depending on network parameters-interfere with the global collective firing of the neurons. We attribute the various network dynamics to distinct regimes in the parameter space. For the same network parameters different network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spatiotemporal dynamics of biological networks.

Dynamical complexity in small-world networks of spiking neurons

A computer model is described which is used to assess the dynamical complexity of a class of networks of spiking neurons with small-world properties. Networks are constructed by forming an initially segregated set of highly intraconnected clusters and then applying a probabilistic rewiring method reminiscent of the Watts-Strogatz procedure to make intercluster connections. Causal density, which counts the number of independent significant interactions among a system's components, is used to assess dynamical complexity. This measure was chosen because it employs lagged observations, and is therefore more sensitive to temporally smeared evidence of segregation and integration than its alternatives. The results broadly support the hypothesis that small-world topology promotes dynamical complexity, but reveal a narrow parameter range within which this occurs for the network topology under investigation, and suggest an inverse correlation with phase synchrony inside this range.

Narrow-band oscillations in probabilistic cellular automata

Dynamical properties of neural populations are studied using probabilistic cellular automata. Previous work demonstrated the emergence of critical behavior as the function of system noise and density of long-range axonal connections. Finite-size scaling theory identified critical properties, which were consistent with properties of a weak Ising universality class. The present work extends the studies to neural populations with excitatory and inhibitory interactions. It is shown that the populations can exhibit narrow-band oscillations when confined to a range of inhibition levels, with clear boundaries marking the parameter region of prominent oscillations. Phase diagrams have been constructed to characterize unimodal, bimodal, and quadromodal oscillatory states. The significance of these findings is discussed in the context of large-scale narrow-band oscillations in neural tissues, as observed in electroencephalographic and magnetoencephalographic measurements.

Introduction: optimization in networks

The recent surge in the network modeling of complex systems has set the stage for a new era in the study of fundamental and applied aspects of optimization in collective behavior. This Focus Issue presents an extended view of the state of the art in this field and includes articles from a large variety of domains in which optimization manifests itself, including physical, biological, social, and technological networked systems.