Oscillatory neurocomputing with ring attractors: a network architecture for mapping locations in space onto patterns of neural synchrony

Theories of neural coding seek to explain how states of the world are mapped onto states of the brain. Here, we compare how an animal''s location in space can be encoded by two different kinds of brain states: population vectors stored by patterns of neural firing rates, versus synchronization vectors stored by patterns of synchrony among neural oscillators. It has previously been shown that a population code stored by spatially tuned ‘grid cells’ can exhibit desirable properties such as high storage capacity and strong fault tolerance; here it is shown that similar properties are attainable with a synchronization code stored by rhythmically bursting ‘theta cells’ that lack spatial tuning. Simulations of a ring attractor network composed from theta cells suggest how a synchronization code might be implemented using fewer neurons and synapses than a population code with similar storage capacity. It is conjectured that reciprocal connections between grid and theta cells might control phase noise to correct two kinds of errors that can arise in the code: path integration and teleportation errors. Based upon these analyses, it is proposed that a primary function of spatially tuned neurons might be to couple the phases of neural oscillators in a manner that allows them to encode spatial locations as patterns of neural synchrony.     

Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.     

Fast Synchronization of Ultradian Oscillators Controlled by Delta-Notch Signaling with Cis-Inhibition

While it is known that a large fraction of vertebrate genes are under the control of a gene regulatory network (GRN) forming a clock with circadian periodicity, shorter period oscillatory genes like the Hairy-enhancer-of split (Hes) genes are discussed mostly in connection with the embryonic process of somitogenesis. They form the core of the somitogenesis-clock, which orchestrates the periodic separation of somites from the presomitic mesoderm (PSM). The formation of sharp boundaries between the blocks of many cells works only when the oscillators in the cells forming the boundary are synchronized. It has been shown experimentally that Delta-Notch (D/N) signaling is responsible for this synchronization. This process has to happen rather fast as a cell experiences at most five oscillations from its ‘birth’ to its incorporation into a somite. Computer simulations describing synchronized oscillators with classical modes of D/N-interaction have difficulties to achieve synchronization in an appropriate time. One approach to solving this problem of modeling fast synchronization in the PSM was the consideration of cell movements. Here we show that fast synchronization of Hes-type oscillators can be achieved without cell movements by including D/N cis-inhibition, wherein the mutual interaction of DELTA and NOTCH in the same cell leads to a titration of ligand against receptor so that only one sort of molecule prevails. Consequently, the symmetry between sender and receiver is partially broken and one cell becomes preferentially sender or receiver at a given moment, which leads to faster entrainment of oscillators. Although not yet confirmed by experiment, the proposed mechanism of enhanced synchronization of mesenchymal cells in the PSM would be a new distinct developmental mechanism employing D/N cis-inhibition. Consequently, the way in which Delta-Notch signaling was modeled so far should be carefully reconsidered.     

Resting-State Temporal Synchronization Networks Emerge from Connectivity Topology and Heterogeneity

Spatial patterns of coherent activity across different brain areas have been identified during the resting-state fluctuations of the brain. However, recent studies indicate that resting-state activity is not stationary, but shows complex temporal dynamics. We were interested in the spatiotemporal dynamics of the phase interactions among resting-state fMRI BOLD signals from human subjects. We found that the global phase synchrony of the BOLD signals evolves on a characteristic ultra-slow (<0.01Hz) time scale, and that its temporal variations reflect the transient formation and dissolution of multiple communities of synchronized brain regions. Synchronized communities reoccurred intermittently in time and across scanning sessions. We found that the synchronization communities relate to previously defined functional networks known to be engaged in sensory-motor or cognitive function, called resting-state networks (RSNs), including the default mode network, the somato-motor network, the visual network, the auditory network, the cognitive control networks, the self-referential network, and combinations of these and other RSNs. We studied the mechanism originating the observed spatiotemporal synchronization dynamics by using a network model of phase oscillators connected through the brain’s anatomical connectivity estimated using diffusion imaging human data. The model consistently approximates the temporal and spatial synchronization patterns of the empirical data, and reveals that multiple clusters that transiently synchronize and desynchronize emerge from the complex topology of anatomical connections, provided that oscillators are heterogeneous.     

Node vulnerability under finite perturbations in complex networks

A measure to quantify vulnerability under perturbations (attacks, failures, large fluctuations) in ensembles (networks) of coupled dynamical systems is proposed. Rather than addressing the issue of how the network properties change upon removal of elements of the graph (the strategy followed by most of the existing methods for studying the vulnerability of a network based on its topology), here a dynamical definition of vulnerability is introduced, referring to the robustness of a collective dynamical state to perturbing events occurring over a fixed topology. In particular, we study how the collective (synchronized) dynamics of a network of chaotic units is disrupted under the action of a finite size perturbation on one of its nodes. Illustrative examples are provided for three systems of identical chaotic oscillators coupled according to three distinct well-known network topologies. A quantitative comparison between the obtained vulnerability rankings and the classical connectivity/centrality rankings is made that yields conclusive results. Possible applications of the proposed strategy and conclusions are also discussed.     

Using Pareto optimality to explore the topology and dynamics of the human connectome

Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain''s topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an ‘economical’ small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.     

Synchronization of coupled biological oscillators under spatially heterogeneous environmental forcing

Spatially heterogeneous intensities of environmental signals are common in nature, being caused, e.g., by rugged or curved surfaces leading to varying angles of incidence and intensities. In this work, we perform numerical studies of one-dimensional arrays of coupled phase oscillators driven by a periodic signal with spatially heterogeneous amplitude, considering both random and gradual amplitude distributions of the driving. We compare the effects of global and next-neighbor interactions, respectively, on the mutual and forced synchronization in the array. Weak global coupling leads to full mutual synchronization for all studied driving configurations. The degree of external synchronization follows a majority rule, depending on the number of externally entrained oscillators in the uncoupled case. The effects of next-neighbor coupling depend on the spatial distribution of the driving amplitude. For random distributions, local interactions show the same qualitative effects as global coupling. In contrast, for gradual distributions and large driving heterogeneities, next-neighbor coupling is detrimental to both mutual and external synchronization. We discuss these observations with respect to fundamental aspects of heterogeneity and variability of dynamical systems, as well as the intercellular synchronization of circadian oscillators.     

Socially synchronized circadian oscillators

Daily rhythms of physiology and behaviour are governed by an endogenous timekeeping mechanism (a circadian ‘clock’). The alternation of environmental light and darkness synchronizes (entrains) these rhythms to the natural day–night cycle, and underlying mechanisms have been investigated using singly housed animals in the laboratory. But, most species ordinarily would not live out their lives in such seclusion; in their natural habitats, they interact with other individuals, and some live in colonies with highly developed social structures requiring temporal synchronization. Social cues may thus be critical to the adaptive function of the circadian system, but elucidating their role and the responsible mechanisms has proven elusive. Here, we highlight three model systems that are now being applied to understanding the biology of socially synchronized circadian oscillators: the fruitfly, with its powerful array of molecular genetic tools; the honeybee, with its complex natural society and clear division of labour; and, at a different level of biological organization, the rodent suprachiasmatic nucleus, site of the brain''s circadian clock, with its network of mutually coupled single-cell oscillators. Analyses at the ‘group’ level of circadian organization will likely generate a more complex, but ultimately more comprehensive, view of clocks and rhythms and their contribution to fitness in nature.     

Input-Dependent Frequency Modulation of Cortical Gamma Oscillations Shapes Spatial Synchronization and Enables Phase Coding

Fine-scale temporal organization of cortical activity in the gamma range (∼25–80Hz) may play a significant role in information processing, for example by neural grouping (‘binding’) and phase coding. Recent experimental studies have shown that the precise frequency of gamma oscillations varies with input drive (e.g. visual contrast) and that it can differ among nearby cortical locations. This has challenged theories assuming widespread gamma synchronization at a fixed common frequency. In the present study, we investigated which principles govern gamma synchronization in the presence of input-dependent frequency modulations and whether they are detrimental for meaningful input-dependent gamma-mediated temporal organization. To this aim, we constructed a biophysically realistic excitatory-inhibitory network able to express different oscillation frequencies at nearby spatial locations. Similarly to cortical networks, the model was topographically organized with spatially local connectivity and spatially-varying input drive. We analyzed gamma synchronization with respect to phase-locking, phase-relations and frequency differences, and quantified the stimulus-related information represented by gamma phase and frequency. By stepwise simplification of our models, we found that the gamma-mediated temporal organization could be reduced to basic synchronization principles of weakly coupled oscillators, where input drive determines the intrinsic (natural) frequency of oscillators. The gamma phase-locking, the precise phase relation and the emergent (measurable) frequencies were determined by two principal factors: the detuning (intrinsic frequency difference, i.e. local input difference) and the coupling strength. In addition to frequency coding, gamma phase contained complementary stimulus information. Crucially, the phase code reflected input differences, but not the absolute input level. This property of relative input-to-phase conversion, contrasting with latency codes or slower oscillation phase codes, may resolve conflicting experimental observations on gamma phase coding. Our modeling results offer clear testable experimental predictions. We conclude that input-dependency of gamma frequencies could be essential rather than detrimental for meaningful gamma-mediated temporal organization of cortical activity.     

A Canonical Circuit for Generating Phase-Amplitude Coupling

‘Phase amplitude coupling’ (PAC) in oscillatory neural activity describes a phenomenon whereby the amplitude of higher frequency activity is modulated by the phase of lower frequency activity. Such coupled oscillatory activity – also referred to as ‘cross-frequency coupling’ or ‘nested rhythms’ – has been shown to occur in a number of brain regions and at behaviorally relevant time points during cognitive tasks; this suggests functional relevance, but the circuit mechanisms of PAC generation remain unclear. In this paper we present a model of a canonical circuit for generating PAC activity, showing how interconnected excitatory and inhibitory neural populations can be periodically shifted in to and out of oscillatory firing patterns by afferent drive, hence generating higher frequency oscillations phase-locked to a lower frequency, oscillating input signal. Since many brain regions contain mutually connected excitatory-inhibitory populations receiving oscillatory input, the simplicity of the mechanism generating PAC in such networks may explain the ubiquity of PAC across diverse neural systems and behaviors. Analytic treatment of this circuit as a nonlinear dynamical system demonstrates how connection strengths and inputs to the populations can be varied in order to change the extent and nature of PAC activity, importantly which phase of the lower frequency rhythm the higher frequency activity is locked to. Consequently, this model can inform attempts to associate distinct types of PAC with different network topologies and physiologies in real data.     

Dynamic Network Drivers of Seizure Generation,Propagation and Termination in Human Neocortical Epilepsy

The epileptic network is characterized by pathologic, seizure-generating ‘foci’ embedded in a web of structural and functional connections. Clinically, seizure foci are considered optimal targets for surgery. However, poor surgical outcome suggests a complex relationship between foci and the surrounding network that drives seizure dynamics. We developed a novel technique to objectively track seizure states from dynamic functional networks constructed from intracranial recordings. Each dynamical state captures unique patterns of network connections that indicate synchronized and desynchronized hubs of neural populations. Our approach suggests that seizures are generated when synchronous relationships near foci work in tandem with rapidly changing desynchronous relationships from the surrounding epileptic network. As seizures progress, topographical and geometrical changes in network connectivity strengthen and tighten synchronous connectivity near foci—a mechanism that may aid seizure termination. Collectively, our observations implicate distributed cortical structures in seizure generation, propagation and termination, and may have practical significance in determining which circuits to modulate with implantable devices.     

Nodes Having a Major Influence to Break Cooperation Define a Novel Centrality Measure: Game Centrality

Cooperation played a significant role in the self-organization and evolution of living organisms. Both network topology and the initial position of cooperators heavily affect the cooperation of social dilemma games. We developed a novel simulation program package, called ‘NetworGame’, which is able to simulate any type of social dilemma games on any model, or real world networks with any assignment of initial cooperation or defection strategies to network nodes. The ability of initially defecting single nodes to break overall cooperation was called as ‘game centrality’. The efficiency of this measure was verified on well-known social networks, and was extended to ‘protein games’, i.e. the simulation of cooperation between proteins, or their amino acids. Hubs and in particular, party hubs of yeast protein-protein interaction networks had a large influence to convert the cooperation of other nodes to defection. Simulations on methionyl-tRNA synthetase protein structure network indicated an increased influence of nodes belonging to intra-protein signaling pathways on breaking cooperation. The efficiency of single, initially defecting nodes to convert the cooperation of other nodes to defection in social dilemma games may be an important measure to predict the importance of nodes in the integration and regulation of complex systems. Game centrality may help to design more efficient interventions to cellular networks (in forms of drugs), to ecosystems and social networks.     

Application of Hierarchical Dissociated Neural Network in Closed-Loop Hybrid System Integrating Biological and Mechanical Intelligence

Neural networks are considered the origin of intelligence in organisms. In this paper, a new design of an intelligent system merging biological intelligence with artificial intelligence was created. It was based on a neural controller bidirectionally connected to an actual mobile robot to implement a novel vehicle. Two types of experimental preparations were utilized as the neural controller including ‘random’ and ‘4Q’ (cultured neurons artificially divided into four interconnected parts) neural network. Compared to the random cultures, the ‘4Q’ cultures presented absolutely different activities, and the robot controlled by the ‘4Q’ network presented better capabilities in search tasks. Our results showed that neural cultures could be successfully employed to control an artificial agent; the robot performed better and better with the stimulus because of the short-term plasticity. A new framework is provided to investigate the bidirectional biological-artificial interface and develop new strategies for a future intelligent system using these simplified model systems.     

Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems

In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.     

Fundamental structural units of the Escherichia coli nucleoid revealed by atomic force microscopy

A small container of several to a few hundred µm³ (i.e. bacterial cells and eukaryotic nuclei) contains extremely long genomic DNA (i.e. mm and m long, respectively) in a highly organized fashion. To understand how such genomic architecture could be achieved, Escherichia coli nucleoids were subjected to structural analyses under atomic force microscopy, and found to change their structure dynamically during cell growth, i.e. the nucleoid structure in the stationary phase was more tightly compacted than in the log phase. However, in both log and stationary phases, a fundamental fibrous structure with a diameter of ~80 nm was found. In addition to this ‘80 nm fiber’, a thinner ‘40 nm fiber’ and a higher order ‘loop’ structure were identified in the log phase nucleoid. In the later growth phases, the nucleoid turned into a ‘coral reef structure’ that also possessed the 80 nm fiber units, and, finally, into a ‘tightly compacted nucleoid’ that was stable in a mild lysis buffer. Mutant analysis demonstrated that these tight compactions of the nucleoid required a protein, Dps. From these results and previously available information, we propose a structural model of the E.coli nucleoid.     

Failure of Delayed Feedback Deep Brain Stimulation for Intermittent Pathological Synchronization in Parkinson’s Disease

Suppression of excessively synchronous beta-band oscillatory activity in the brain is believed to suppress hypokinetic motor symptoms of Parkinson’s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize the synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson’s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronous dynamics used to study the delayed feedback stimulation. This study explores the action of delayed feedback stimulation on partially synchronized oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces experimentally observed fine temporal structure of the synchronous dynamics. When the parameters of our model are such that the synchrony is unphysiologically strong, the feedback exerts a desynchronizing action. However, when the network is tuned to reproduce the highly variable temporal patterns observed experimentally, the same kind of delayed feedback may actually increase the synchrony. As network parameters are changed from the range which produces complete synchrony to those favoring less synchronous dynamics, desynchronizing delayed feedback may gradually turn into synchronizing stimulation. This suggests that delayed feedback DBS in Parkinson’s disease may boost rather than suppress synchronization and is unlikely to be clinically successful. The study also indicates that delayed feedback stimulation may not necessarily exhibit a desynchronization effect when acting on a physiologically realistic partially synchronous dynamics, and provides an example of how to estimate the stimulation effect.     

Transient resetting: a novel mechanism for synchrony and its biological examples

The study of synchronization in biological systems is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. In this paper, by using simple dynamical systems theory, we present a novel mechanism, named transient resetting, for the synchronization of uncoupled biological oscillators with stimuli. This mechanism not only can unify and extend many existing results on (deterministic and stochastic) stimulus-induced synchrony, but also may actually play an important role in biological rhythms. We argue that transient resetting is a possible mechanism for the synchronization in many biological organisms, which might also be further used in the medical therapy of rhythmic disorders. Examples of the synchronization of neural and circadian oscillators as well as a chaotic neuron model are presented to verify our hypothesis.