Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Harnessing Intelligence from Brain Cells In Vitro

Harnessing intelligence from brain cells in vitro requires a multidisciplinary approach integrating wetware, hardware, and software. Wetware comprises the in vitro brain cells themselves, where differentiation from induced pluripotent stem cells offers ethical scalability; hardware typically involves a life support system and a setup to record the activity from and deliver stimulation to the brain cells; and software is required to control the hardware and process the signals coming from and going to the brain cells. This review provides a broad summary of the foundational technologies underpinning these components, along with outlining the importance of technology integration. Of particular importance is that this new technology offers the ability to extend beyond traditional methods that assess primarily the survival and spontaneous activity of neural cultures. Instead, the focus returns to the core function of neural tissue: the neurocomputational ability to process information and respond accordingly. Therefore, this review also covers work that, despite the relatively early state of current technology, has provided novel and meaningful understandings in the field of neuroscience along with opening exciting avenues for future research.

Response of neuronal populations to phase-locked stimulation: model-based predictions and validation

Modulation of neuronal oscillations holds promise for the treatment of neurological disorders. Nonetheless, conventional stimulation in a continuous open-loop manner can lead to side effects and suboptimal efficiency. Closed-loop strategies such as phase-locked stimulation aim to address these shortcomings by offering a more targeted modulation. While theories have been developed to understand the neural response to stimulation, their predictions have not been thoroughly tested using experimental data. Using a mechanistic coupled oscillator model, we elaborate on two key predictions describing the response to stimulation as a function of the phase and amplitude of ongoing neural activity. To investigate these predictions, we analyze electrocorticogram recordings from a previously conducted study in Parkinsonian rats, and extract the corresponding phase and response curves. We demonstrate that the amplitude response to stimulation is strongly correlated to the derivative of the phase response ([Formula: see text] > 0.8) in all animals except one, thereby validating a key model prediction. The second prediction postulates that the stimulation becomes ineffective when the network synchrony is high, a trend that appeared missing in the data. Our analysis explains this discrepancy by showing that the neural populations in Parkinsonian rats did not reach the level of synchrony for which the theory would predict ineffective stimulation. Our results highlight the potential of fine-tuning stimulation paradigms informed by mathematical models that consider both the ongoing phase and amplitude of the targeted neural oscillation.Significance Statement This study validates a mathematical model of coupled oscillators in predicting the response of neural activity to stimulation for the first time. Our findings also offer further insights beyond this validation. For instance, the demonstrated correlation between phase response and amplitude response is indeed a key theoretical concept within a subset of mathematical models. This prediction can bring about clinical implications in terms of predictive power for manipulation of neural activity. Additionally, while phase dependence in modulation has been previously studied, we propose a general framework for studying amplitude dependence as well. Lastly, our study reconciles the seemingly contradictory views of pathologic hypersynchrony and theoretical low synchrony in Parkinson's disease.

Co-evolutionary control of a class of coupled mixed-feedback systems

Oscillatory behavior is ubiquitous in many natural and engineered systems, often emerging through self-regulating mechanisms. In this paper, we address the challenge of stabilizing a desired oscillatory pattern in a networked system where neither the internal dynamics nor the interconnections can be changed. To achieve this, we propose two distinct control strategies. The first requires the full knowledge of the system generating the desired oscillatory pattern, while the second only needs local error information. In addition, the controllers are implemented as co-evolutionary, or adaptive, rules of some edges in an extended plant-controller network. We validate our approach in several insightful scenarios, including synchronization and systems with time-varying network structures.

Mean-field analysis of synaptic alterations underlying deficient cortical gamma oscillations in schizophrenia

Deficient gamma oscillations in the prefrontal cortex (PFC) of individuals with schizophrenia (SZ) are proposed to arise from alterations in the excitatory drive to fast-spiking interneurons (E → I) and in the inhibitory drive from these interneurons to excitatory neurons (I → E). Consistent with this idea, prior postmortem studies showed lower levels of molecular and structural markers for the strength of E → I and I → E synapses and also greater variability in E → I synaptic strength in PFC of SZ. Moreover, simulating these alterations in a network of quadratic integrate-and-fire (QIF) neurons revealed a synergistic effect of their interactions on reducing gamma power. In this study, we aimed to investigate the dynamical nature of this synergistic interaction at macroscopic level by deriving a mean-field description of the QIF model network that consists of all-to-all connected excitatory neurons and fast-spiking interneurons. Through a series of numerical simulations and bifurcation analyses, findings from our mean-field model showed that the macroscopic dynamics of gamma oscillations are synergistically disrupted by the interactions among lower strength of E → I and I → E synapses and greater variability in E → I synaptic strength. Furthermore, the two-dimensional bifurcation analyses showed that this synergistic interaction is primarily driven by the shift in Hopf bifurcation due to lower E → I synaptic strength. Together, these simulations predict the nature of dynamical mechanisms by which multiple synaptic alterations interact to robustly reduce PFC gamma power in SZ, and highlight the utility of mean-field model to study macroscopic neural dynamics and their alterations in the illness.

Morphological and Electrical Properties of Proteinoid-Actin Networks

Proteinoids, or thermal proteins, are produced by heating amino acids. Proteinoids form hollow microspheres in water. The microspheres produce oscillation of electrical potential. Actin is a filament-forming protein responsible for communication, information processing and decision making in eukaryotic cells. We synthesize randomly organized networks of proteinoid microspheres spanned by actin filaments and study their morphology and electrical potential oscillatory dynamics. We analyze proteinoid-actin networks' responses to electrical stimulation. The signals come from logistic maps, the Lorenz attractor, the Rossler oscillator, and the FitzHugh-Nagumo system. We show how the networks attenuated the signals produced by these models. We demonstrate that emergent logical patterns derived from oscillatory behavior of proteinoid-actin networks show characteristics of Boolean logic gates, providing evidence for the computational ability to combine different components through architectural changes in the dynamic interface. Our experimental laboratory study paves a base for generation of proto-neural networks and implementation of neuromorphic computation with them.

Direct and Retrograde Wave Propagation in Unidirectionally Coupled Wilson-Cowan Oscillators

Some biological systems exhibit both direct and retrograde propagating signal waves despite unidirectional coupling. To explain this phenomenon, we study a chain of unidirectionally coupled Wilson-Cowan oscillators. Surprisingly, we find that changes in the homogeneous global input to the chain suffice to reverse the wave propagation direction. To obtain insights, we analyze the frequencies and bifurcations of the limit cycle solutions of the chain as a function of the global input. Specifically, we determine that the directionality of wave propagation is controlled by differences in the intrinsic frequencies of oscillators caused by the differential proximity of the oscillators to a homoclinic bifurcation.

From chimeras to extensive chaos in networks of heterogeneous Kuramoto oscillator populations

Populations of coupled oscillators can exhibit a wide range of complex dynamical behavior, from complete synchronization to chimera and chaotic states. We can, thus, expect complex dynamics to arise in networks of such populations. Here, we analyze the dynamics of networks of populations of heterogeneous mean-field coupled Kuramoto-Sakaguchi oscillators and show that the instability that leads to chimera states in a simple two-population model also leads to extensive chaos in large networks of coupled populations. Formally, the system consists of a complex network of oscillator populations whose mesoscopic behavior evolves according to the Ott-Antonsen equations. By considering identical parameters across populations, the system contains a manifold of homogeneous solutions where all populations behave identically. Stability analysis of these homogeneous states provided by the master stability function formalism shows that non-trivial dynamics might emerge on a wide region of the parameter space for arbitrary network topologies. As examples, we first revisit the two-population case and provide a complete bifurcation diagram. Then, we investigate the emergent dynamics in large ring and Erdös-Rényi networks. In both cases, transverse instabilities lead to extensive space-time chaos, i.e., irregular regimes whose complexity scales linearly with the system size. Our work provides a unified analytical framework to understand the emergent dynamics of networks of oscillator populations, from chimera states to robust high-dimensional chaos.

The functional role of oscillatory dynamics in neocortical circuits: A computational perspective

The dynamics of neuronal systems are characterized by hallmark features such as oscillations and synchrony. However, it has remained unclear whether these characteristics are epiphenomena or are exploited for computation. Due to the challenge of selectively interfering with oscillatory network dynamics in neuronal systems, we simulated recurrent networks of damped harmonic oscillators in which oscillatory activity is enforced in each node, a choice well supported by experimental findings. When trained on standard pattern recognition tasks, these harmonic oscillator recurrent networks (HORNs) outperformed nonoscillatory architectures with respect to learning speed, noise tolerance, and parameter efficiency. HORNs also reproduced a many characteristic features of neuronal systems, such as the cerebral cortex and the hippocampus. In trained HORNs, stimulus-induced interference patterns holistically represent the result of comparing sensory evidence with priors stored in recurrent connection weights, and learning-induced weight changes are compatible with Hebbian principles. Implementing additional features characteristic of natural networks, such as heterogeneous oscillation frequencies, inhomogeneous conduction delays, and network modularity, further enhanced HORN performance without requiring additional parameters. Taken together, our model allows us to give plausible a posteriori explanations for features of natural networks whose computational role has remained elusive. We conclude that neuronal systems are likely to exploit the unique dynamics of recurrent oscillator networks whose computational superiority critically depends on the oscillatory patterning of their nodal dynamics. Implementing the proposed computational principles in analog hardware is expected to enable the design of highly energy-efficient and self-adapting devices that could ideally complement existing digital technologies.

Continuum limit of the adaptive Kuramoto model

We investigate the dynamics of the adaptive Kuramoto model with slow adaptation in the continuum limit, N→∞. This model is distinguished by dense multistability, where multiple states coexist for the same system parameters. The underlying cause of this multistability is that some oscillators can lock at different phases or switch between locking and drifting depending on their initial conditions. We identify new states, such as two-cluster states. To simplify the analysis, we introduce an approximate reduction of the model via row-averaging of the coupling matrix. We derive a self-consistency equation for the reduced model and present a stability diagram illustrating the effects of positive and negative adaptation. Our theoretical findings are validated through numerical simulations of a large finite system. Comparisons of previous work highlight the significant influence of adaptation on synchronization behavior.

Stochastic Kuramoto oscillators with inertia and higher-order interactions

The impact of noise in coupled oscillators with pairwise interactions has been extensively explored. Here, we study stochastic second-order coupled Kuramoto oscillators with higher-order interactions and show that as noise strength increases, the critical points associated with synchronization transitions shift toward higher coupling values. By employing the perturbation analysis, we obtain an expression for the forward critical point as a function of inertia and noise strength. Further, for overdamped systems, we show that as noise strength increases, the first-order transition switches to second-order even for higher-order couplings. We include a discussion on the nature of critical points obtained through Ott-Antonsen ansatz.

Revolutionizing treatment for disorders of consciousness: a multidisciplinary review of advancements in deep brain stimulation

Among the existing research on the treatment of disorders of consciousness (DOC), deep brain stimulation (DBS) offers a highly promising therapeutic approach. This comprehensive review documents the historical development of DBS and its role in the treatment of DOC, tracing its progression from an experimental therapy to a detailed modulation approach based on the mesocircuit model hypothesis. The mesocircuit model hypothesis suggests that DOC arises from disruptions in a critical network of brain regions, providing a framework for refining DBS targets. We also discuss the multimodal approaches for assessing patients with DOC, encompassing clinical behavioral scales, electrophysiological assessment, and neuroimaging techniques methods. During the evolution of DOC therapy, the segmentation of central nuclei, the recording of single-neurons, and the analysis of local field potentials have emerged as favorable technical factors that enhance the efficacy of DBS treatment. Advances in computational models have also facilitated a deeper exploration of the neural dynamics associated with DOC, linking neuron-level dynamics with macroscopic behavioral changes. Despite showing promising outcomes, challenges remain in patient selection, precise target localization, and the determination of optimal stimulation parameters. Future research should focus on conducting large-scale controlled studies to delve into the pathophysiological mechanisms of DOC. It is imperative to further elucidate the precise modulatory effects of DBS on thalamo-cortical and cortico-cortical functional connectivity networks. Ultimately, by optimizing neuromodulation strategies, we aim to substantially enhance therapeutic outcomes and greatly expedite the process of consciousness recovery in patients.

Detailed-level modelling of influence spreading on complex networks

The progress in high-performance computing makes it increasingly possible to build detailed models to investigate spreading processes on complex networks. However, current studies have been lacking detailed computational methods to describe spreading processes in large complex networks. To fill this gap we present a new modelling approach for analysing influence spreading via individual nodes and links on various network structures. The proposed influence-spreading model uses a probability matrix to capture the spreading probability from one node to another in the network. This approach enables analysing network characteristics in a number of applications and spreading processes using metrics that are consistent with the quantities used to model the network structures. In addition, this study combines sub-models and offers a comprehensive look at different applications and metrics previously discussed in cases of social networks, community detection, and epidemic spreading. Here, we also note that the centrality measures based on the probability matrix are used to identify the most significant nodes in the network. Furthermore, the model can be expanded to include additional properties, such as introducing individual breakthrough probabilities for the nodes and specific temporal distributions for the links.

Co-evolutionary dynamics for two adaptively coupled Theta neurons

Natural and technological networks exhibit dynamics that can lead to complex cooperative behaviors, such as synchronization in coupled oscillators and rhythmic activity in neuronal networks. Understanding these collective dynamics is crucial for deciphering a range of phenomena from brain activity to power grid stability. Recent interest in co-evolutionary networks has highlighted the intricate interplay between dynamics on and of the network with mixed time scales. Here, we explore the collective behavior of excitable oscillators in a simple network of two Theta neurons with adaptive coupling without self-interaction. Through a combination of bifurcation analysis and numerical simulations, we seek to understand how the level of adaptivity in the coupling strength, a, influences the dynamics. We first investigate the dynamics possible in the non-adaptive limit; our bifurcation analysis reveals stability regions of quiescence and spiking behaviors, where the spiking frequencies mode-lock in a variety of configurations. Second, as we increase the adaptivity a, we observe a widening of the associated Arnol'd tongues, which may overlap and give room for multi-stable configurations. For larger adaptivity, the mode-locked regions may further undergo a period-doubling cascade into chaos. Our findings contribute to the mathematical theory of adaptive networks and offer insights into the potential mechanisms underlying neuronal communication and synchronization.

Analyzing top-down visual attention in the context of gamma oscillations: a layer- dependent network-of- networks approach

Top-down visual attention is a fundamental cognitive process that allows individuals to selectively attend to salient visual stimuli in the environment. Recent empirical findings have revealed that gamma oscillations participate in the modulation of visual attention. However, computational studies face challenges when analyzing the attentional process in the context of gamma oscillation due to the unstable nature of gamma oscillations and the complexity induced by the layered fashion in the visual cortex. In this study, we propose a layer-dependent network-of-networks approach to analyze such attention with gamma oscillations. The model is validated by reproducing empirical findings on orientation preference and the enhancement of neuronal response due to top-down attention. We perform parameter plane analysis to classify neuronal responses into several patterns and find that the neuronal response to sensory and attention signals was modulated by the heterogeneity of the neuronal population. Furthermore, we revealed a counter-intuitive scenario that the excitatory populations in layer 2/3 and layer 5 exhibit opposite responses to the attentional input. By modification of the original model, we confirmed layer 6 plays an indispensable role in such cases. Our findings uncover the layer-dependent dynamics in the cortical processing of visual attention and open up new possibilities for further research on layer-dependent properties in the cerebral cortex.

Evoked resonant neural activity long-term dynamics can be reproduced by a computational model with vesicle depletion

Subthalamic deep brain stimulation (DBS) robustly generates high-frequency oscillations known as evoked resonant neural activity (ERNA). Recently the importance of ERNA has been demonstrated through its ability to predict the optimal DBS contact in the subthalamic nucleus in patients with Parkinson's disease. However, the underlying mechanisms of ERNA are not well understood, and previous modelling efforts have not managed to reproduce the wealth of published data describing the dynamics of ERNA. Here, we aim to present a minimal model capable of reproducing the characteristics of the slow ERNA dynamics published to date. We make biophysically-motivated modifications to the Kuramoto model and fit its parameters to the slow dynamics of ERNA obtained from data. Our results demonstrate that it is possible to reproduce the slow dynamics of ERNA (over hundreds of seconds) with a single neuronal population, and, crucially, with vesicle depletion as one of the key mechanisms behind the ERNA frequency decay in our model. We further validate the proposed model against experimental data from Parkinson's disease patients, where it captures the variations in ERNA frequency and amplitude in response to variable stimulation frequency, amplitude, and to stimulation pulse bursting. We provide a series of predictions from the model that could be the subject of future studies for further validation.

Model-agnostic neural mean field with a data-driven transfer function

As one of the most complex systems known to science, modeling brain behavior and function is both fascinating and extremely difficult. Empirical data is increasingly available from ex vivo human brain organoids and surgical samples, as well as in vivo animal models, so the problem of modeling the behavior of large-scale neuronal systems is more relevant than ever. The statistical physics concept of a mean-field model offers a tractable way to bridge the gap between single-neuron and population-level descriptions of neuronal activity, by modeling the behavior of a single representative neuron and extending this to the population. However, existing neural mean-field methods typically either take the limit of small interaction sizes, or are applicable only to the specific neuron models for which they were derived. This paper derives a mean-field model by fitting a transfer function called Refractory SoftPlus, which is simple yet applicable to a broad variety of neuron types. The transfer function is fitted numerically to simulated spike time data, and is entirely agnostic to the underlying neuronal dynamics. The resulting mean-field model predicts the response of a network of randomly connected neurons to a time-varying external stimulus with a high degree of accuracy. Furthermore, it enables an accurate approximate bifurcation analysis as a function of the level of recurrent input. This model does not assume large presynaptic rates or small postsynaptic potential size, allowing mean-field models to be developed even for populations with large interaction terms.

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

Pulse Shape and Voltage-Dependent Synchronization in Spiking Neuron Networks

Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic insights into how neurons self-organize to produce coherent collective behavior. These networks use simple spiking neuron models, such as the θ-neuron or the quadratic integrate-and-fire (QIF) neuron, that replicate the essential features of real neural dynamics. Interactions between neurons are modeled with infinitely narrow pulses, or spikes, rather than the more complex dynamics of real synapses. To make these networks biologically more plausible, it has been proposed that they must also account for the finite width of the pulses, which can have a significant impact on the network dynamics. However, the derivation and interpretation of these pulses are contradictory, and the impact of the pulse shape on the network dynamics is largely unexplored. Here, I take a comprehensive approach to pulse coupling in networks of QIF and θ-neurons. I argue that narrow pulses activate voltage-dependent synaptic conductances and show how to implement them in QIF neurons such that their effect can last through the phase after the spike. Using an exact low-dimensional description for networks of globally coupled spiking neurons, I prove for instantaneous interactions that collective oscillations emerge due to an effective coupling through the mean voltage. I analyze the impact of the pulse shape by means of a family of smooth pulse functions with arbitrary finite width and symmetric or asymmetric shapes. For symmetric pulses, the resulting voltage coupling is not very effective in synchronizing neurons, but pulses that are slightly skewed to the phase after the spike readily generate collective oscillations. The results unveil a voltage-dependent spike synchronization mechanism at the heart of emergent collective behavior, which is facilitated by pulses of finite width and complementary to traditional synaptic transmission in spiking neuron networks.

Criticality and partial synchronization analysis in Wilson-Cowan and Jansen-Rit neural mass models

Synchronization is a phenomenon observed in neuronal networks involved in diverse brain activities. Neural mass models such as Wilson-Cowan (WC) and Jansen-Rit (JR) manifest synchronized states. Despite extensive research on these models over the past several decades, their potential of manifesting second-order phase transitions (SOPT) and criticality has not been sufficiently acknowledged. In this study, two networks of coupled WC and JR nodes with small-world topologies were constructed and Kuramoto order parameter (KOP) was used to quantify the amount of synchronization. In addition, we investigated the presence of SOPT using the synchronization coefficient of variation. Both networks reached high synchrony by changing the coupling weight between their nodes. Moreover, they exhibited abrupt changes in the synchronization at certain values of the control parameter not necessarily related to a phase transition. While SOPT was observed only in JR model, neither WC nor JR model showed power-law behavior. Our study further investigated the global synchronization phenomenon that is known to exist in pathological brain states, such as seizure. JR model showed global synchronization, while WC model seemed to be more suitable in producing partially synchronized patterns.

On the higher-order smallest ring-star network of Chialvo neurons under diffusive couplings

Network dynamical systems with higher-order interactions are a current trending topic, pervasive in many applied fields. However, our focus in this work is neurodynamics. We numerically study the dynamics of the smallest higher-order network of neurons arranged in a ring-star topology. The dynamics of each node in this network is governed by the Chialvo neuron map, and they interact via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through multi-body interactions. We deploy the higher-order coupling strength as the primary bifurcation parameter. We start by analyzing our model using standard tools from dynamical systems theory: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of disparate chaotic attractors. We also observe an interesting route to chaos from a fixed point via period-doubling and the appearance of cyclic quasiperiodic closed invariant curves. Furthermore, we numerically observe the existence of codimension-1 bifurcation points: saddle-node, period-doubling, and Neimark-Sacker. We also qualitatively study the typical phase portraits of the system, and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure, respectively. Finally, we study the synchronization behavior among the neurons using the cross correlation coefficient and the Kuramoto order parameter. We conjecture that unfolding these patterns and behaviors of the network model will help us identify different states of the nervous system, further aiding us in dealing with various neural diseases and nervous disorders.

Strong delayed negative feedback

In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit.

Classification of musical intervals by spiking neural networks: Perfect student in solfége classes

We investigate a spike activity of a network of excitable FitzHugh-Nagumo neurons, which is under constant two-frequency auditory signals. The neurons are supplemented with linear frequency filters and nonlinear input signal converters. We show that it is possible to configure the network to recognize a specific frequency ratio (musical interval) by selecting the parameters of the neurons, input filters, and coupling between neurons. A set of appropriately configured subnetworks with different topologies and coupling strengths can serve as a classifier for musical intervals. We have found that the selective properties of the classifier are due to the presence of a specific topology of coupling between the neurons of the network.

Synchronization dynamics of phase oscillators on power grid models

We investigate topological and spectral properties of models of European and US-American power grids and of paradigmatic network models as well as their implications for the synchronization dynamics of phase oscillators with heterogeneous natural frequencies. We employ the complex-valued order parameter-a widely used indicator for phase ordering-to assess the synchronization dynamics and observe the order parameter to exhibit either constant or periodic or non-periodic, possibly chaotic temporal evolutions for a given coupling strength but depending on initial conditions and the systems' disorder. Interestingly, both topological and spectral characteristics of the power grids point to a diminished capability of these networks to support a temporarily stable synchronization dynamics. We find non-trivial commonalities between the synchronization dynamics of oscillators on seemingly opposing topologies.

Global density equations for interacting particle systems with stochastic resetting: From overdamped Brownian motion to phase synchronization

A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic "agents." Kinetic methods for reducing the complexity of such systems typically involve the derivation of nonlinear partial differential equations for the corresponding global densities. In recent years, there has been considerable interest in the mean field limit of interacting particle systems with long-range interactions. Two major examples are interacting Brownian particles in the overdamped regime and the Kuramoto model of coupled phase oscillators. In this paper, we analyze these systems in the presence of local or global stochastic resetting, where the position or phase of each particle independently or simultaneously resets to its original value at a random sequence of times generated by a Poisson process. In each case, we derive the Dean-Kawasaki (DK) equation describing hydrodynamic fluctuations of the global density and then use a mean field ansatz to obtain the corresponding nonlinear McKean-Vlasov (MV) equation in the thermodynamic limit. In particular, we show how the MV equation for global resetting is driven by a Poisson noise process, reflecting the fact that resetting is common to all of the particles and, thus, induces correlations that cannot be eliminated by taking a mean field limit. We then investigate the effects of local and global resetting on nonequilibrium stationary solutions of the macroscopic dynamics and, in the case of the Kuramoto model, the reduced dynamics on the Ott-Antonsen manifold.

Preserved Auditory Steady State Response and Envelope-Following Response in Severe Brainstem Dysfunction Highlight the Need for Cross-Checking

OBJECTIVES

Commercially available auditory steady state response (ASSR) systems are widely used to obtain hearing thresholds in the pediatric population objectively. Children are often examined during natural or induced sleep so that the recorded ASSRs are of subcortical origin, the inferior colliculus being often designated as the main ASSR contributor in these conditions. This report presents data from a battery of auditory neurophysiological objective tests obtained in 3 cases of severe brainstem dysfunction in sleeping children. In addition to ASSRs, envelope-following response (EFR) recordings designed to distinguish peripheral (cochlear nerve) from central (brainstem) were recorded to document the effect of brainstem dysfunction on the two types of phase-locked responses.

DESIGN

Results obtained in the 3 children with severe brainstem dysfunctions were compared with those of age-matched controls. The cases were identified as posterior fossa tumor, undiagnosed (UD), and Pelizaeus-Merzbacher-Like Disease. The standard audiological objective tests comprised tympanograms, distortion product otoacoustic emissions, click-evoked auditory brainstem responses (ABRs), and ASSRs. EFRs were recorded using horizontal (EFR-H) and vertical (EFR-V) channels and a stimulus phase rotation technique allowing isolation of the EFR waveforms in the time domain to obtain direct latency measurements.

RESULTS

The brainstem dysfunctions of the 3 children were revealed as abnormal (weak, absent, or delayed) ABRs central waves with a normal wave I. In addition, they all presented a summating and cochlear microphonic potential in their ABRs, coupled with a normal wave I, which implies normal cochlear and cochlear nerve function. EFR-H and EFR-V waveforms were identified in the two cases in whom they were recorded. The EFR-Hs onset latencies, response durations, and phase-locking values did not differ from their respective age-matched control values, indicating normal cochlear nerve EFRs. In contrast, the EFR-V phase-locking value and onset latency varied from their control values. Both patients had abnormal but identifiable and significantly phase-locked brainstem EFRs, even in a case with severely distorted ABR central waves. ASSR objective audiograms were recorded in two cases. They showed normal or slightly elevated (explained by a slight transmission loss) thresholds that do not yield any clue about their brainstem dysfunction, revealing the method's lack of sensitivity to severe brainstem dysfunction.

CONCLUSIONS

The present study, performed on 3 sleeping children with severe brainstem dysfunction but normal cochlear responses (cochlear microphonic potential, summating potential, and ABR wave I), revealed the differential sensitivity of three auditory electrophysiological techniques. Estimated thresholds obtained by standard ASSR recordings (cases UD and Pelizaeus-Merzbacher-Like Disease) provided no clue to the brainstem dysfunction clearly revealed by the click-evoked ABR. EFR recordings (cases posterior fossa tumor and UD) showed preserved central responses with abnormal latencies and low phase-locking values, whereas the peripheral EFR attributed to the cochlear nerve was normal. The one case (UD) for which the three techniques could be performed confirms this sensitivity gradient, emphasizing the need for applying the Cross-Check Principle by avoiding resorting to ASSR recording alone. The entirely normal EFR-H recordings observed in two cases further strengthen the hypothesis of its cochlear nerve origin in sleeping children.

Mean-field analysis of synaptic alterations underlying deficient cortical gamma oscillations in schizophrenia

Deficient gamma oscillations in the prefrontal cortex (PFC) of individuals with schizophrenia (SZ) are proposed to arise from alterations in the excitatory drive to fast-spiking interneurons ( E → I ) and in the inhibitory drive from these interneurons to excitatory neurons ( I → E ) . Consistent with this idea, prior postmortem studies showed lower levels of molecular and structural markers for the strength of E → I and I → E synapses and also greater variability in E → I synaptic strength in PFC of SZ. Moreover, simulating these alterations in a network of quadratic integrate-and-fire (QIF) neurons revealed a synergistic effect of their interactions on reducing gamma power. In this study, we aimed to investigate the dynamical nature of this synergistic interaction at macroscopic level by deriving a mean-field description of the QIF model network that consists of all-to-all connected excitatory neurons and fast-spiking interneurons. Through a series of numerical simulations and bifurcation analyses, findings from our mean-field model showed that the macroscopic dynamics of gamma oscillations are synergistically disrupted by the interactions among lower strength of E → I and I → E synapses and greater variability in E → I synaptic strength. Furthermore, the two-dimensional bifurcation analyses showed that this synergistic interaction is primarily driven by the shift in Hopf bifurcation due to lower E → I synaptic strength. Together, these simulations predict the nature of dynamical mechanisms by which multiple synaptic alterations interact to robustly reduce PFC gamma power in SZ, and highlight the utility of mean-field model to study macroscopic neural dynamics and their alterations in the illness.

Neuronal travelling waves explain rotational dynamics in experimental datasets and modelling

Spatiotemporal properties of neuronal population activity in cortical motor areas have been subjects of experimental and theoretical investigations, generating numerous interpretations regarding mechanisms for preparing and executing limb movements. Two competing models, representational and dynamical, strive to explain the relationship between movement parameters and neuronal activity. A dynamical model uses the jPCA method that holistically characterizes oscillatory activity in neuron populations by maximizing the data rotational dynamics. Different rotational dynamics interpretations revealed by the jPCA approach have been proposed. Yet, the nature of such dynamics remains poorly understood. We comprehensively analyzed several neuronal-population datasets and found rotational dynamics consistently accounted for by a traveling wave pattern. For quantifying rotation strength, we developed a complex-valued measure, the gyration number. Additionally, we identified parameters influencing rotation extent in the data. Our findings suggest that rotational dynamics and traveling waves are typically the same phenomena, so reevaluation of the previous interpretations where they were considered separate entities is needed.

Model-Agnostic Neural Mean Field With The Refractory SoftPlus Transfer Function

Due to the complexity of neuronal networks and the nonlinear dynamics of individual neurons, it is challenging to develop a systems-level model which is accurate enough to be useful yet tractable enough to apply. Mean-field models which extrapolate from single-neuron descriptions to large-scale models can be derived from the neuron's transfer function, which gives its firing rate as a function of its synaptic input. However, analytically derived transfer functions are applicable only to the neurons and noise models from which they were originally derived. In recent work, approximate transfer functions have been empirically derived by fitting a sigmoidal curve, which imposes a maximum firing rate and applies only in the diffusion limit, restricting applications. In this paper, we propose an approximate transfer function called Refractory SoftPlus, which is simple yet applicable to a broad variety of neuron types. Refractory SoftPlus activation functions allow the derivation of simple empirically approximated mean-field models using simulation results, which enables prediction of the response of a network of randomly connected neurons to a time-varying external stimulus with a high degree of accuracy. These models also support an accurate approximate bifurcation analysis as a function of the level of recurrent input. Finally, the model works without assuming large presynaptic rates or small postsynaptic potential size, allowing mean-field models to be developed even for populations with large interaction terms.

Integrability of a Globally Coupled Complex Riccati Array: Quadratic Integrate-and-Fire Neurons, Phase Oscillators, and All in Between

We present an exact dimensionality reduction for dynamics of an arbitrary array of globally coupled complex-valued Riccati equations. It generalizes the Watanabe-Strogatz theory [Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70, 2391 (1993).PRLTAO0031-900710.1103/PhysRevLett.70.2391] for sinusoidally coupled phase oscillators and seamlessly includes quadratic integrate-and-fire neurons as the real-valued special case. This simple formulation reshapes our understanding of a broad class of coupled systems-including a particular class of phase-amplitude oscillators-which newly fall under the category of integrable systems. Precise and rigorous analysis of complex Riccati arrays is now within reach, paving a way to a deeper understanding of emergent behavior of collective dynamics in coupled systems.

Determining bifurcations to explosive synchronization for networks of coupled oscillators with higher-order interactions

We determine bifurcations from gradual to explosive synchronization in coupled oscillator networks with higher-order coupling using self-consistency analysis. We obtain analytic bifurcation values for generic symmetric natural frequency distributions. We show that nonsynchronized, drifting, oscillators are non-negligible and play a crucial role in bifurcation. As such, the entire natural frequency distribution must be accounted for, rather than just the shape at the center. We verify our results for Lorentzian- and Gaussian-distributed natural frequencies.

Chimeras and traveling waves in ensembles of Kuramoto oscillators off the Poisson manifold

We examine how perturbations off the Poisson manifold affect chimeras and traveling waves (TWs) in Kuramoto models with two sub-populations. Our numerical study is based on simulations on invariant manifolds, which contain von Mises probability distributions. Our study demonstrates that chimeras and TWs off the Poisson manifold always "breathe", and the effect of breathing is more pronounced further from the Poisson manifold. On the other side, TWs arising in similar models on the sphere always breathe moderately, no matter if the dynamics take place near the Poisson manifold or far away from it.

Modelling cortical network dynamics

We have investigated the theoretical constraints of the interactions between coupled cortical columns. Each cortical column consists of a set of neural populations where each population is modelled as a neural mass. The existence of semi-stable states within a cortical column is dependent on the type of interaction between the neuronal populations, i.e., the form of the synaptic kernels. Current-to-current coupling has been shown, in contrast to potential-to-current coupling, to create semi-stable states within a cortical column. The interaction between semi-stable states of the cortical columns is studied where we derive the dynamics for the collected activity. For small excitations the dynamics follow the Kuramoto model; however, in contrast to previous work we derive coupled equations between phase and amplitude dynamics with the possibility of defining connectivity as a stationary and dynamic variable. The turbulent flow of phase dynamics which occurs in networks of Kuramoto oscillators would indicate turbulent changes in dynamic connectivity for coupled cortical columns which is something that has been recorded in epileptic seizures. We used the results we derived to estimate a seizure propagation model which allowed for inversions using the Laplace assumption (Dynamic Causal Modelling). The seizure propagation model was trialed on simulated data, and future work will investigate the estimation of the connectivity matrix from empirical data. This model can be used to predict changes in seizure evolution after virtual changes in the connectivity network, something that could be of clinical use when applied to epilepsy surgical cases.

Validating an algebraic approach to characterizing resonator networks

Resonator networks are ubiquitous in natural and engineered systems, such as solid-state materials, electrical circuits, quantum processors, and even neural tissue. To understand and manipulate these networks it is essential to characterize their building blocks, which include the mechanical analogs of mass, elasticity, damping, and coupling of each resonator element. While these mechanical parameters are typically obtained from response spectra using least-squares fitting, this approach requires a priori knowledge of all parameters and is susceptible to large error due to convergence to local minima. Here we validate an alternative algebraic means to characterize resonator networks with no or minimal a priori knowledge. Our approach recasts the equations of motion of the network into a linear homogeneous algebraic equation and solves the equation with a set of discrete measured network response vectors. For validation, we employ our approach on noisy simulated data from a single resonator and a coupled resonator pair, and we characterize the accuracy of the recovered parameters using high-dimension factorial simulations. Generally, we find that the error is inversely proportional to the signal-to-noise ratio, that measurements at two frequencies are sufficient to recover all parameters, and that sampling near the resonant peaks is optimal. Our simple, powerful tool will enable future efforts to ascertain network properties and control resonator networks in diverse physical domains.

Synaptic dependence of dynamic regimes when coupling neural populations

In this article we focus on the study of the collective dynamics of neural networks. The analysis of two recent models of coupled "next-generation" neural mass models allows us to observe different global mean dynamics of large neural populations. These models describe the mean dynamics of all-to-all coupled networks of quadratic integrate-and-fire spiking neurons. In addition, one of these models considers the influence of the synaptic adaptation mechanism on the macroscopic dynamics. We show how both models are related through a parameter and we study the evolution of the dynamics when switching from one model to the other by varying that parameter. Interestingly, we have detected three main dynamical regimes in the coupled models: Rössler-type (funnel type), bursting-type, and spiking-like (oscillator-type) dynamics. This result opens the question of which regime is the most suitable for realistic simulations of large neural networks and shows the possibility of the emergence of chaotic collective dynamics when synaptic adaptation is very weak.

Complex localization mechanisms in networks of coupled oscillators: Two case studies

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

Stable chimera states: A geometric singular perturbation approach

Over the past decades, chimera states have attracted considerable attention given their unexpected symmetry-breaking spatiotemporal nature and simultaneously exhibiting synchronous and incoherent behaviors under specific conditions. Despite relevant precursory results of such unforeseen states for diverse physical and topological configurations, there remain structures and mechanisms yet to be unveiled. In this work, using mean-field techniques, we analyze a multilayer network composed of two populations of heterogeneous Kuramoto phase oscillators with coevolutive coupling strengths. Moreover, we employ the geometric singular perturbation theory through the inclusion of a time-scale separation between the dynamics of the network elements and the adaptive coupling strength connecting them, gaining a better insight into the behavior of the system from a fast-slow dynamics perspective. Consequently, we derive the necessary and sufficient condition to produce stable chimera states when considering a coevolutionary intercoupling strength. Additionally, under the aforementioned constraint and with a suitable adaptive law election, it is possible to generate intriguing patterns, such as persistent breathing chimera states. Thereafter, we analyze the geometric properties of the mean-field system with a coevolutionary intracoupling strength and demonstrate the production of stable chimera states. Next, we give arguments for the presence of such patterns in the associated network under specific conditions. Finally, relaxation oscillations and canard cycles, seemingly related to breathing chimeras, are numerically produced under identified conditions due to the geometry of our system.

Delta-alpha cross-frequency coupling for different brain regions

Neural interactions occur on different levels and scales. It is of particular importance to understand how they are distributed among different neuroanatomical and physiological relevant brain regions. We investigated neural cross-frequency couplings between different brain regions according to the Desikan-Killiany brain parcellation. The adaptive dynamic Bayesian inference method was applied to EEG measurements of healthy resting subjects in order to reconstruct the coupling functions. It was found that even after averaging over all subjects, the mean coupling function showed a characteristic waveform, confirming the direct influence of the delta-phase on the alpha-phase dynamics in certain brain regions and that the shape of the coupling function changes for different regions. While the averaged coupling function within a region was of similar form, the region-averaged coupling function was averaged out, which implies that there is a common dependence within separate regions across the subjects. It was also found that for certain regions the influence of delta on alpha oscillations is more pronounced and that oscillations that influence other are more evenly distributed across brain regions than the influenced oscillations. When presenting the information on brain lobes, it was shown that the influence of delta emanating from the brain as a whole is greatest on the alpha oscillations of the cingulate frontal lobe, and at the same time the influence of delta from the cingulate parietal brain lobe is greatest on the alpha oscillations of the whole brain.

Nonlinear slow-timescale mechanisms in synaptic plasticity

Learning and memory rely on synapses changing their strengths in response to neural activity. However, there is a substantial gap between the timescales of neural electrical dynamics (1-100 ms) and organism behaviour during learning (seconds-minutes). What mechanisms bridge this timescale gap? What are the implications for theories of brain learning? Here I first cover experimental evidence for slow-timescale factors in plasticity induction. Then I review possible underlying cellular and synaptic mechanisms, and insights from recent computational models that incorporate such slow-timescale variables. I conclude that future progress in understanding brain learning across timescales will require both experimental and computational modelling studies that map out the nonlinearities implemented by both fast and slow plasticity mechanisms at synapses, and crucially, their joint interactions.

Modular architecture facilitates noise-driven control of synchrony in neuronal networks

High-level information processing in the mammalian cortex requires both segregated processing in specialized circuits and integration across multiple circuits. One possible way to implement these seemingly opposing demands is by flexibly switching between states with different levels of synchrony. However, the mechanisms behind the control of complex synchronization patterns in neuronal networks remain elusive. Here, we use precision neuroengineering to manipulate and stimulate networks of cortical neurons in vitro, in combination with an in silico model of spiking neurons and a mesoscopic model of stochastically coupled modules to show that (i) a modular architecture enhances the sensitivity of the network to noise delivered as external asynchronous stimulation and that (ii) the persistent depletion of synaptic resources in stimulated neurons is the underlying mechanism for this effect. Together, our results demonstrate that the inherent dynamical state in structured networks of excitable units is determined by both its modular architecture and the properties of the external inputs.

Multi-modal and multi-model interrogation of large-scale functional brain networks

Existing whole-brain models are generally tailored to the modelling of a particular data modality (e.g., fMRI or MEG/EEG). We propose that despite the differing aspects of neural activity each modality captures, they originate from shared network dynamics. Building on the universal principles of self-organising delay-coupled nonlinear systems, we aim to link distinct features of brain activity - captured across modalities - to the dynamics unfolding on a macroscopic structural connectome. To jointly predict connectivity, spatiotemporal and transient features of distinct signal modalities, we consider two large-scale models - the Stuart Landau and Wilson and Cowan models - which generate short-lived 40 Hz oscillations with varying levels of realism. To this end, we measure features of functional connectivity and metastable oscillatory modes (MOMs) in fMRI and MEG signals - and compare them against simulated data. We show that both models can represent MEG functional connectivity (FC), functional connectivity dynamics (FCD) and generate MOMs to a comparable degree. This is achieved by adjusting the global coupling and mean conduction time delay and, in the WC model, through the inclusion of balance between excitation and inhibition. For both models, the omission of delays dramatically decreased the performance. For fMRI, the SL model performed worse for FCD and MOMs, highlighting the importance of balanced dynamics for the emergence of spatiotemporal and transient patterns of ultra-slow dynamics. Notably, optimal working points varied across modalities and no model was able to achieve a correlation with empirical FC higher than 0.4 across modalities for the same set of parameters. Nonetheless, both displayed the emergence of FC patterns that extended beyond the constraints of the anatomical structure. Finally, we show that both models can generate MOMs with empirical-like properties such as size (number of brain regions engaging in a mode) and duration (continuous time interval during which a mode appears). Our results demonstrate the emergence of static and dynamic properties of neural activity at different timescales from networks of delay-coupled oscillators at 40 Hz. Given the higher dependence of simulated FC on the underlying structural connectivity, we suggest that mesoscale heterogeneities in neural circuitry may be critical for the emergence of parallel cross-modal functional networks and should be accounted for in future modelling endeavours.

Mean-Field Approximations With Adaptive Coupling for Networks With Spike-Timing-Dependent Plasticity

Understanding the effect of spike-timing-dependent plasticity (STDP) is key to elucidating how neural networks change over long timescales and to design interventions aimed at modulating such networks in neurological disorders. However, progress is restricted by the significant computational cost associated with simulating neural network models with STDP and by the lack of low-dimensional description that could provide analytical insights. Phase-difference-dependent plasticity (PDDP) rules approximate STDP in phase oscillator networks, which prescribe synaptic changes based on phase differences of neuron pairs rather than differences in spike timing. Here we construct mean-field approximations for phase oscillator networks with STDP to describe part of the phase space for this very high-dimensional system. We first show that single-harmonic PDDP rules can approximate a simple form of symmetric STDP, while multiharmonic rules are required to accurately approximate causal STDP. We then derive exact expressions for the evolution of the average PDDP coupling weight in terms of network synchrony. For adaptive networks of Kuramoto oscillators that form clusters, we formulate a family of low-dimensional descriptions based on the mean-field dynamics of each cluster and average coupling weights between and within clusters. Finally, we show that such a two-cluster mean-field model can be fitted to synthetic data to provide a low-dimensional approximation of a full adaptive network with symmetric STDP. Our framework represents a step toward a low-dimensional description of adaptive networks with STDP, and could for example inform the development of new therapies aimed at maximizing the long-lasting effects of brain stimulation.

Influence and influenceability: global directionality in directed complex networks

Knowing which nodes are influential in a complex network and whether the network can be influenced by a small subset of nodes is a key part of network analysis. However, many traditional measures of importance focus on node level information without considering the global network architecture. We use the method of trophic analysis to study directed networks and show that both 'influence' and 'influenceability' in directed networks depend on the hierarchical structure and the global directionality, as measured by the trophic levels and trophic coherence, respectively. We show that in directed networks trophic hierarchy can explain: the nodes that can reach the most others; where the eigenvector centrality localizes; which nodes shape the behaviour in opinion or oscillator dynamics; and which strategies will be successful in generalized rock-paper-scissors games. We show, moreover, that these phenomena are mediated by the global directionality. We also highlight other structural properties of real networks related to influenceability, such as the pseudospectra, which depend on trophic coherence. These results apply to any directed network and the principles highlighted-that node hierarchy is essential for understanding network influence, mediated by global directionality-are applicable to many real-world dynamics.

Synchronization transitions in Kuramoto networks with higher-mode interaction

Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes. We describe the multistability induced by the presence of a second coupling mode. By extending the collective coordinate approach, we describe the emergence of various states observed in the transition from incoherence to coherence. Remarkably, our analysis suggests that, in essence, the two-mode coupling gives rise to states characterized by two independent but interacting groups of oscillators. We believe that these findings will stimulate future research on dynamical systems, including complex interaction functions beyond the Kuramoto-type.

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches.

Exponentially Long Transient Time to Synchronization of Coupled Chaotic Circle Maps in Dense Random Networks

We study the transition to synchronization in large, dense networks of chaotic circle maps, where an exact solution of the mean-field dynamics in the infinite network and all-to-all coupling limit is known. In dense networks of finite size and link probability of smaller than one, the incoherent state is meta-stable for coupling strengths that are larger than the mean-field critical coupling. We observe chaotic transients with exponentially distributed escape times and study the scaling behavior of the mean time to synchronization.

Heteroclinic switching between chimeras in a ring of six oscillator populations

In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.

Neural Cross-Frequency Coupling Functions in Sleep

The human brain presents a heavily connected complex system. From a relatively fixed anatomy, it can enable a vast repertoire of functions. One important brain function is the process of natural sleep, which alters consciousness and voluntary muscle activity. On neural level, these alterations are accompanied by changes of the brain connectivity. In order to reveal the changes of connectivity associated with sleep, we present a methodological framework for reconstruction and assessment of functional interaction mechanisms. By analyzing EEG (electroencephalogram) recordings from human whole night sleep, first, we applied a time-frequency wavelet transform to study the existence and strength of brainwave oscillations. Then we applied a dynamical Bayesian inference on the phase dynamics in the presence of noise. With this method we reconstructed the cross-frequency coupling functions, which revealed the mechanism of how the interactions occur and manifest. We focus our analysis on the delta-alpha coupling function and observe how this cross-frequency coupling changes during the different sleep stages. The results demonstrated that the delta-alpha coupling function was increasing gradually from Awake to NREM3 (non-rapid eye movement), but only during NREM2 and NREM3 deep sleep it was significant in respect of surrogate data testing. The analysis on the spatially distributed connections showed that this significance is strong only for within the single electrode region and in the front-to-back direction. The presented methodological framework is for the whole-night sleep recordings, but it also carries general implications for other global neural states.

Chaotic chimera attractors in a triangular network of identical oscillators

A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)1539-375510.1103/PhysRevE.82.016216]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.

Complex dynamics in adaptive phase oscillator networks

Networks of coupled dynamical units give rise to collective dynamics such as the synchronization of oscillators or neurons in the brain. The ability of the network to adapt coupling strengths between units in accordance with their activity arises naturally in a variety of contexts, including neural plasticity in the brain, and adds an additional layer of complexity: the dynamics on the nodes influence the dynamics of the network and vice versa. We study a minimal model of Kuramoto phase oscillators including a general adaptive learning rule with three parameters (strength of adaptivity, adaptivity offset, adaptivity shift), mimicking learning paradigms based on spike-time-dependent plasticity. Importantly, the strength of adaptivity allows to tune the system away from the limit of the classical Kuramoto model, corresponding to stationary coupling strengths and no adaptation and, thus, to systematically study the impact of adaptivity on the collective dynamics. We carry out a detailed bifurcation analysis for the minimal model consisting of N=2 oscillators. The non-adaptive Kuramoto model exhibits very simple dynamic behavior, drift, or frequency-locking; but once the strength of adaptivity exceeds a critical threshold non-trivial bifurcation structures unravel: A symmetric adaptation rule results in multi-stability and bifurcation scenarios, and an asymmetric adaptation rule generates even more intriguing and rich dynamics, including a period-doubling cascade to chaos as well as oscillations displaying features of both librations and rotations simultaneously. Generally, adaptation improves the synchronizability of the oscillators. Finally, we also numerically investigate a larger system consisting of N=50 oscillators and compare the resulting dynamics with the case of N=2 oscillators.