Hopf Bifurcation in Mean Field Explains Critical Avalanches in Excitation-Inhibition Balanced Neuronal Networks: A Mechanism for Multiscale Variability

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.

A general description of criticality in neural network models

Recent experimental observations have supported the hypothesis that the cerebral cortex operates in a dynamical regime near criticality, where the neuronal network exhibits a mixture of ordered and disordered patterns. However, A comprehensive study of how criticality emerges and how to reproduce it is still lacking. In this study, we investigate coupled networks with conductance-based neurons and illustrate the co-existence of different spiking patterns, including asynchronous irregular (AI) firing and synchronous regular (SR) state, along with a scale-invariant neuronal avalanche phenomenon (criticality). We show that fast-acting synaptic coupling can evoke neuronal avalanches in the mean-dominated regime but has little effect in the fluctuation-dominated regime. In a narrow region of parameter space, the network exhibits avalanche dynamics with power-law avalanche size and duration distributions. We conclude that three stages which may be responsible for reproducing the synchronized bursting: mean-dominated subthreshold dynamics, fast-initiating a spike event, and time-delayed inhibitory cancellation. Remarkably, we illustrate the mechanisms underlying critical avalanches in the presence of noise, which can be explained as a stochastic crossing state around the Hopf bifurcation under the mean-dominated regime. Moreover, we apply the ensemble Kalman filter to determine and track effective connections for the neuronal network. The method is validated on noisy synthetic BOLD signals and could exactly reproduce the corresponding critical network activity. Our results provide a special perspective to understand and model the criticality, which can be useful for large-scale modeling and computation of brain dynamics.

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.

Excitation-Inhibition Balance, Neural Criticality, and Activities in Neuronal Circuits

Neural activities in local circuits exhibit complex and multilevel dynamic features. Individual neurons spike irregularly, which is believed to originate from receiving balanced amounts of excitatory and inhibitory inputs, known as the excitation-inhibition balance. The spatial-temporal cascades of clustered neuronal spikes occur in variable sizes and durations, manifested as neural avalanches with scale-free features. These may be explained by the neural criticality hypothesis, which posits that neural systems operate around the transition between distinct dynamic states. Here, we summarize the experimental evidence for and the underlying theory of excitation-inhibition balance and neural criticality. Furthermore, we review recent studies of excitatory-inhibitory networks with synaptic kinetics as a simple solution to reconcile these two apparently distinct theories in a single circuit model. This provides a more unified understanding of multilevel neural activities in local circuits, from spontaneous to stimulus-response dynamics.

Multilevel and multifaceted brain response features in spiking, ERP and ERD: experimental observation and simultaneous generation in a neuronal network model with excitation-inhibition balance

Brain as a dynamic system responds to stimulations with specific patterns affected by its inherent ongoing dynamics. The patterns are manifested across different levels of organization-from spiking activity of neurons to collective oscillations in local field potential (LFP) and electroencephalogram (EEG). The multilevel and multifaceted response activities show patterns seemingly distinct and non-comparable from each other, but they should be coherently related because they are generated from the same underlying neural dynamic system. A coherent understanding of the interrelationships between different levels/aspects of activity features is important for understanding the complex brain functions. Here, based on analysis of data from human EEG, monkey LFP and neuronal spiking, we demonstrated that the brain response activities from different levels of neural system are highly coherent: the external stimulus simultaneously generated event-related potentials, event-related desynchronization, and variation in neuronal spiking activities that precisely match with each other in the temporal unfolding. Based on a biologically plausible but generic network of conductance-based integrate-and-fire excitatory and inhibitory neurons with dense connections, we showed that the multiple key features can be simultaneously produced at critical dynamical regimes supported by excitation-inhibition (E-I) balance. The elucidation of the inherent coherency of various neural response activities and demonstration of a simple dynamical neural circuit system having the ability to simultaneously produce multiple features suggest the plausibility of understanding high-level brain function and cognition from elementary and generic neuronal dynamics.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-022-09889-w.

Quasiuniversal scaling in mouse-brain neuronal activity stems from edge-of-instability critical dynamics

The brain is in a state of perpetual reverberant neural activity, even in the absence of specific tasks or stimuli. Shedding light on the origin and functional significance of such a dynamical state is essential to understanding how the brain transmits, processes, and stores information. An inspiring, albeit controversial, conjecture proposes that some statistical characteristics of empirically observed neuronal activity can be understood by assuming that brain networks operate in a dynamical regime with features, including the emergence of scale invariance, resembling those seen typically near phase transitions. Here, we present a data-driven analysis based on simultaneous high-throughput recordings of the activity of thousands of individual neurons in various regions of the mouse brain. To analyze these data, we construct a unified theoretical framework that synergistically combines a phenomenological renormalization group approach and techniques that infer the general dynamical state of a neural population, while designing complementary tools. This strategy allows us to uncover strong signatures of scale invariance that are "quasiuniversal" across brain regions and experiments, revealing that all the analyzed areas operate, to a greater or lesser extent, near the edge of instability.

Criticality in reservoir computer of coupled phase oscillators

Accumulating evidence shows that the cerebral cortex is operating near a critical state featured by power-law size distribution of neural avalanche activities, yet evidence of this critical state in artificial neural networks mimicking the cerebral cortex is still lacking. Here we design an artificial neural network of coupled phase oscillators and, by the technique of reservoir computing in machine learning, train it for predicting chaos. It is found that when the machine is properly trained, oscillators in the reservoir are synchronized into clusters whose sizes follow a power-law distribution. This feature, however, is absent when the machine is poorly trained. Additionally, it is found that despite the synchronization degree of the original network, once properly trained, the reservoir network is always developed to the same critical state, exemplifying the "attractor" nature of this state in machine learning. The generality of the results is verified in different reservoir models and by different target systems, and it is found that the scaling exponent of the distribution is independent of the reservoir details and the bifurcation parameters of the target system, but is modified when the dynamics of the target system is changed to a different type. The findings shed light on the nature of machine learning, and are helpful to the design of high-performance machines in physical systems.

Sandpile cascades on oscillator networks: The BTW model meets Kuramoto

Cascading failures abound in complex systems and the Bak-Tang-Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a "cascade of larger cascades," which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.

Rational designing of oscillatory rhythmicity for memory rescue in plasticity-impaired learning networks

In the brain, oscillatory strength embedded in network rhythmicity is important for processing experiences, and this process is disrupted in certain psychiatric disorders. The use of rhythmic network stimuli can change these oscillations and has shown promise in terms of improving cognitive function, although the underlying mechanisms are poorly understood. Here, we combine a two-layer learning model, with experiments involving genetically modified mice, that provides precise control of experience-driven oscillations by manipulating long-term potentiation of excitatory synapses onto inhibitory interneurons (LTP_E→I). We find that, in the absence of LTP_E→I, impaired network dynamics and memory are rescued by activating inhibitory neurons to augment the power in theta and gamma frequencies, which prevents network overexcitation with less inhibitory rebound. In contrast, increasing either theta or gamma power alone was less effective. Thus, inducing network changes at dual frequencies is involved in memory encoding, indicating a potentially feasible strategy for optimizing network-stimulating therapies.

Neuronal Population Transitions Across a Quiescent-to-Active Frontier and Bifurcation

The mechanistic understanding of why neuronal population activity hovers on criticality remains unresolved despite the availability of experimental results. Without a coherent mathematical framework, the presence of power-law scaling is not straightforward to reconcile with findings implying epileptiform activity. Although multiple pictures have been proposed to relate the power-law scaling of avalanche statistics to phase transitions, the existence of a phase boundary in parameter space is until now an assumption. Herein, a framework based on differential inclusions, which departs from approaches constructed from differential equations, is shown to offer an adequate consolidation of evidences apparently connected to criticality and those linked to hyperexcitability. Through this framework, the phase boundary is elucidated in a parameter space spanned by variables representing levels of excitation and inhibition in a neuronal network. The interpretation of neuronal populations based on this approach offers insights on the role of pharmacological and endocrinal signaling in the homeostatic regulation of neuronal population activity.

Criticality enhances the multilevel reliability of stimulus responses in cortical neural networks

Cortical neural networks exhibit high internal variability in spontaneous dynamic activities and they can robustly and reliably respond to external stimuli with multilevel features–from microscopic irregular spiking of neurons to macroscopic oscillatory local field potential. A comprehensive study integrating these multilevel features in spontaneous and stimulus–evoked dynamics with seemingly distinct mechanisms is still lacking. Here, we study the stimulus–response dynamics of biologically plausible excitation–inhibition (E–I) balanced networks. We confirm that networks around critical synchronous transition states can maintain strong internal variability but are sensitive to external stimuli. In this dynamical region, applying a stimulus to the network can reduce the trial-to-trial variability and shift the network oscillatory frequency while preserving the dynamical criticality. These multilevel features widely observed in different experiments cannot simultaneously occur in non-critical dynamical states. Furthermore, the dynamical mechanisms underlying these multilevel features are revealed using a semi-analytical mean-field theory that derives the macroscopic network field equations from the microscopic neuronal networks, enabling the analysis by nonlinear dynamics theory and linear noise approximation. The generic dynamical principle revealed here contributes to a more integrative understanding of neural systems and brain functions and incorporates multimodal and multilevel experimental observations. The E–I balanced neural network in combination with the effective mean-field theory can serve as a mechanistic modeling framework to study the multilevel neural dynamics underlying neural information and cognitive processes.

The complexity and variability of brain dynamical activity range from neuronal spiking and neural avalanches to oscillatory local field potentials of local neural circuits in both spontaneous and stimulus-evoked states. Such multilevel variable brain dynamics are functionally and behaviorally relevant and are principal components of the underlying circuit organization. To more comprehensively clarify their neural mechanisms, we use a bottom-up approach to study the stimulus–response dynamics of neural circuits. Our model assumes the following key biologically plausible components: excitation–inhibition (E–I) neuronal interaction and chemical synaptic coupling. We show that the circuits with E–I balance have a special dynamic sub-region, the critical region. Circuits around this region could account for the emergence of multilevel brain response patterns, both ongoing and stimulus-induced, observed in different experiments, including the reduction of trial-to-trial variability, effective modulation of gamma frequency, and preservation of criticality in the presence of a stimulus. We further analyze the corresponding nonlinear dynamical principles using a novel and highly generalizable semi-analytical mean-field theory. Our computational and theoretical studies explain the cross-level brain dynamical organization of spontaneous and evoked states in a more integrative manner.

Less is more: Wiring-economical modular networks support self-sustained firing-economical neural avalanches for efficient processing

The brain network is notably cost-efficient, while the fundamental physical and dynamic mechanisms underlying its economical optimization in network structure and activity have not been determined. In this study, we investigate the intricate cost-efficient interplay between structure and dynamics in biologically plausible spatial modular neuronal network models. We observe that critical avalanche states from excitation-inhibition balance under modular network topology with less wiring cost can also achieve lower costs in firing but with strongly enhanced response sensitivity to stimuli. We derive mean-field equations that govern the macroscopic network dynamics through a novel approximate theory. The mechanism of low firing cost and stronger response in the form of critical avalanches is explained as a proximity to a Hopf bifurcation of the modules when increasing their connection density. Our work reveals the generic mechanism underlying the cost-efficient modular organization and critical dynamics widely observed in neural systems, providing insights into brain-inspired efficient computational designs.

Long range temporal correlations (LRTCs) in MEG-data during emerging psychosis: Relationship to symptoms, medication-status and clinical trajectory

Long-Range Temporal Correlations (LRTCs) index the capacity of the brain to optimally process information. Previous research has shown that patients with chronic schizophrenia present altered LRTCs at alpha and beta oscillations. However, it is currently unclear at which stage of schizophrenia aberrant LRTCs emerge. To address this question, we investigated LRTCs in resting-state magnetoencephalographic (MEG) recordings obtained from patients with affective disorders and substance abuse (clinically at low-risk of psychosis, CHR-N), patients at clinical high-risk of psychosis (CHR-P) (n = 115), as well as patients with a first episode (FEP) (n = 25). Matched healthy controls (n = 47) served as comparison group. LRTCs were obtained for frequencies from 4 to 40 Hz and correlated with clinical and neuropsychological data. In addition, we examined the relationship between LRTCs and transition to psychosis in CHR-P participants, and the relationship between LRTC and antipsychotic medication in FEP participants. Our results show that participants from the clinical groups have similar LRTCs to controls. In addition, LRTCs did not correlate with clinical and neurocognitive variables across participants nor did LRTCs predict transition to psychosis. Therefore, impaired LRTCs do not reflect a feature in the clinical trajectory of psychosis. Nevertheless, reduced LRTCs in the beta-band over posterior sensors of medicated FEP participants indicate that altered LRTCs may appear at the onset of the illness. Future studies are needed to elucidate the role of anti-psychotic medication in altered LRTCs.

Gamma Oscillations Facilitate Effective Learning in Excitatory-Inhibitory Balanced Neural Circuits

Gamma oscillation in neural circuits is believed to associate with effective learning in the brain, while the underlying mechanism is unclear. This paper aims to study how spike-timing-dependent plasticity (STDP), a typical mechanism of learning, with its interaction with gamma oscillation in neural circuits, shapes the network dynamics properties and the network structure formation. We study an excitatory-inhibitory (E-I) integrate-and-fire neuronal network with triplet STDP, heterosynaptic plasticity, and a transmitter-induced plasticity. Our results show that the performance of plasticity is diverse in different synchronization levels. We find that gamma oscillation is beneficial to synaptic potentiation among stimulated neurons by forming a special network structure where the sum of excitatory input synaptic strength is correlated with the sum of inhibitory input synaptic strength. The circuit can maintain E-I balanced input on average, whereas the balance is temporal broken during the learning-induced oscillations. Our study reveals a potential mechanism about the benefits of gamma oscillation on learning in biological neural circuits.

Signature of consciousness in brain-wide synchronization patterns of monkey and human fMRI signals

During the sleep-wake cycle, the brain undergoes profound dynamical changes, which manifest subjectively as transitions between conscious experience and unconsciousness. Yet, neurophysiological signatures that can objectively distinguish different consciousness states based are scarce. Here, we show that differences in the level of brain-wide signals can reliably distinguish different stages of sleep and anesthesia from the awake state in human and monkey fMRI resting state data. Moreover, a whole-brain computational model can faithfully reproduce changes in global synchronization and other metrics such as functional connectivity, structure-function relationship, integration and segregation across vigilance states. We demonstrate that the awake brain is close to a Hopf bifurcation, which naturally coincides with the emergence of globally correlated fMRI signals. Furthermore, simulating lesions of individual brain areas highlights the importance of connectivity hubs in the posterior brain and subcortical nuclei for maintaining the model in the awake state, as predicted by graph-theoretical analyses of structural data.