Unveiling the impact of low-frequency electrical stimulation on network synchronization and learning behavior in cultured hippocampal neural networks.

Understanding the dynamics of neural networks and their response to external stimuli is crucial for unraveling the mechanisms associated with learning processes. In this study, we hypothesized that electrical stimulation (ES) would lead to significant alterations in the activity patterns of hippocampal neuronal networks and investigated the effects of low-frequency ES on hippocampal neuronal populations using the microelectrode arrays (MEAs). Our findings revealed significant alterations in the activity of hippocampal neuronal networks following low-frequency ES trainings. Post-stimulation, the neural activity exhibited an organized burst firing pattern characterized by increased spike and burst firings, increased synchronization, and enhanced learning behaviors. Analysis of peri-stimulus time histograms (PSTHs) further revealed that low-frequency ES (1Hz) significantly enhanced neural plasticity, thereby facilitating the learning process of cultured neurons, whereas high-frequency ES (>10Hz) impeded this process. Moreover, we observed a substantial increase in correlations and connectivity within neuronal networks following ES trainings. These alterations in network properties indicated enhanced synaptic plasticity and emphasized the positive impact of low-frequency ES on hippocampal neural activities, contributing to the brain's capacity for learning and memory.

Bridge synergy and simplicial interaction in complex contagions.

Modeling complex contagion in networked systems is an important topic in network science, for which various models have been proposed, including the synergistic contagion model that incorporates coherent interference and the simplicial contagion model that involves high-order interactions. Although both models have demonstrated success in investigating complex contagions, their relationship in modeling complex contagions remains unclear. In this study, we compare the synergy and the simplest form of high-order interaction in the simplicial contagion model, known as the triangular one. We analytically show that the triangular interaction and the synergy can be bridged within complex contagions through the joint degree distribution of the network. Monte Carlo simulations are then conducted to compare simplicial and corresponding synergistic contagions on synthetic and real-world networks, the results of which highlight the consistency of these two different contagion processes and thus validate our analysis. Our study sheds light on the deep relationship between the synergy and high-order interactions and enhances our physical understanding of complex contagions in networked systems.

Investigation on the influence of heterogeneous synergy in contagion processes on complex networks.

Synergistic contagion in a networked system occurs in various forms in nature and human society. While the influence of network's structural heterogeneity on synergistic contagion has been well studied, the impact of individual-based heterogeneity on synergistic contagion remains unclear. In this work, we introduce individual-based heterogeneity with a power-law form into the synergistic susceptible-infected-susceptible model by assuming the synergistic strength as a function of individuals' degree and investigate this synergistic contagion process on complex networks. By employing the heterogeneous mean-field (HMF) approximation, we analytically show that the heterogeneous synergy significantly changes the critical threshold of synergistic strength σc that is required for the occurrence of discontinuous phase transitions of contagion processes. Comparing to the synergy without individual-based heterogeneity, the value of σc decreases with degree-enhanced synergy and increases with degree-suppressed synergy, which agrees well with Monte Carlo prediction. Next, we compare our heterogeneous synergistic contagion model with the simplicial contagion model [Iacopini et al., Nat. Commun. 10, 2485 (2019)], in which high-order interactions are introduced to describe complex contagion. Similarity of these two models are shown both analytically and numerically, confirming the ability of our model to statistically describe the simplest high-order interaction within HMF approximation.

Inferring synchronizability of networked heterogeneous oscillators with machine learning.

In the study of network synchronization, an outstanding question of both theoretical and practical significance is how to allocate a given set of heterogeneous oscillators on a complex network in order to improve the synchronization performance. Whereas methods have been proposed to address this question in the literature, the methods are all based on accurate models describing the system dynamics, which, however, are normally unavailable in realistic situations. Here, we show that this question can be addressed by the model-free technique of a feed-forward neural network (FNN) in machine learning. Specifically, we measure the synchronization performance of a number of allocation schemes and use the measured data to train a machine. It is found that the trained machine is able to not only infer the synchronization performance of any new allocation scheme, but also find from a huge amount of candidates the optimal allocation scheme for synchronization.

Wielding intermittency with cycle expansions.

As periodic orbit theory works badly on computing the observable averages of dynamical systems with intermittency, we propose a scheme to cooperate with cycle expansion and perturbation theory so that we can deal with intermittent systems and compute the averages more precisely. The periodic orbit theory assumes that the shortest unstable periodic orbits build the framework of the system and provide cycle expansion to compute dynamical quantities based on them, while the perturbation theory can locally analyze the structure of dynamical systems. The dynamical averages may be obtained more precisely by combining the two techniques together. Based on the integrability near the marginal orbits and the hyperbolicity in the part away from the singularities in intermittent systems, the chief idea of this paper is to revise intermittent maps and maintain the natural measure produced by the original maps. We get the natural measure near the singularity through the Taylor expansions, and the periodic orbit theory captures the natural measure in the other parts of the phase space. We try this method on one-dimensional intermittent maps with single singularity, and more precise results are achieved.

Phase space partition with Koopman analysis.

Symbolic dynamics is a powerful tool to describe topological features of a nonlinear system, where the required partition, however, remains a challenge for some time due to the complications involved in determining the partition boundaries. In this article, we show that it is possible to carry out interesting symbolic partitions for chaotic maps based on properly constructed eigenfunctions of the finite-dimensional approximation of the Koopman operator. The partition boundaries overlap with the extrema of these eigenfunctions, the accuracy of which is improved by including more basis functions in the numerical computation. The validity of this scheme is demonstrated in well-known 1D and 2D maps.

The role of geometric features in a germinal center.

The germinal center (GC) is a self-organizing structure produced in the lymphoid follicle during the T-dependent immune response and is an important component of the humoral immune system. However, the impact of the special structure of GC on antibody production is not clear. According to the latest biological experiments, we establish a spatiotemporal stochastic model to simulate the whole self-organization process of the GC including the appearance of two specific zones: the dark zone (DZ) and the light zone (LZ), the development of which serves to maintain an effective competition among different cells and promote affinity maturation. A phase transition is discovered in this process, which determines the critical GC volume for a successful growth in both the stochastic and the deterministic model. Further increase of the volume does not make much improvement on the performance. It is found that the critical volume is determined by the distance between the activated B cell receptor (BCR) and the target epitope of the antigen in the shape space. The observation is confirmed in both 2D and 3D simulations and explains partly the variability of the observed GC size.

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.

Optimizing synchrony with a minimal coupling strength of coupled phase oscillators on complex networks based on desynchronous clustering.

Finding the globally optimal network is an unsolved problem in synchrony optimization. In this paper, an efficient edge-adding optimization method based on global information is proposed. The edge-adding scheme is obtained through the eigenvector corresponding to the maximum eigenvalue of the system's Jacobian matrix. With different frequency distributions, we find that the optimized networks have similar features, concluded as three conditions: (i) The deviations of nodes' frequencies from the mean value are linear with the nodes' degrees, (ii) the oscillators form a bipartite network divided according to the frequencies of the oscillators, and (iii) oscillators are only connected to those with sufficiently large frequency differences. An optimal network can be constructed directly based on these three conditions for a given distribution of natural frequencies. We show that the critical coupling strengths of these constructed networks approach the theoretical lower bound. The constructed networks are or at least close to the globally optimal ones for synchrony.

Probing the phase space of coupled oscillators with Koopman analysis.

With the development of probing and computing technology, the study of complex systems has become a necessity in various science and engineering problems, which may be treated efficiently with Koopman operator theory based on observed time series. In the current paper, combined with a singular value decomposition (SVD) of the constructed Hankel matrix, Koopman analysis is applied to a system of coupled oscillators. The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures with periodic, quasiperiodic, or chaotic motion. By checking the amplitude of the principal modes along a straight line in the phase space, cusps of different sizes on the magnitude profiles are identified whenever a qualitative change of dynamics takes place. There seems to be no obstacle to extend the current analysis to high-dimensional nonlinear systems with intricate orbit structures.

Chaotic renormalization flow in the Potts model induced by long-range competition.

A proper description of spin glass remains a hard subject in theoretical physics and is considered to be closely related to the emergence of chaos in the renormalization group (RG) flow. Previous efforts concentrate on models with either complicated or nonrealistic interactions in order to achieve this chaotic behavior. Here we find that the commonly used Potts model with long-range interaction could do the job nicely in a large parameter regime as long as the competition between the ferromagnetic and antiferromagnetic interaction is maintained. With this simplicity, the appearance of chaos is observed to sensitively depend on the detailed network structure: the parity of bond number in a branch of the basic RG substituting unit; chaos only emerges for even numbers of bonds. These surprising and universal findings may shed light on the study of spin glass.

Symbolic partition in chaotic maps.

In this work, we only use data on the unstable manifold to locate the partition boundaries by checking folding points at different levels, which practically coincide with homoclinic tangencies. The method is then applied to the classic two-dimensional Hénon map and a well-known three-dimensional map. Comparison with previous results is made in the Hénon case, and Lyapunov exponents are computed through the metric entropy based on the partition to show the validity of the current scheme.

The impact of asymptomatic individuals on the strength of public health interventions to prevent the second outbreak of COVID-19.

The pandemic of coronavirus disease 2019 (COVID-19) has threatened the social and economic structure all around the world. Generally, COVID-19 has three possible transmission routes, including pre-symptomatic, symptomatic and asymptomatic transmission, among which the last one has brought a severe challenge for the containment of the disease. One core scientific question is to understand the influence of asymptomatic individuals and of the strength of control measures on the evolution of the disease, particularly on a second outbreak of the disease. To explore these issues, we proposed a novel compartmental model that takes the infection of asymptomatic individuals into account. We get the relationship between asymptomatic individuals and critical strength of control measures theoretically. Furthermore, we verify the reliability of our model and the accuracy of the theoretical analysis by using the real confirmed cases of COVID-19 contamination. Our results, showing the importance of the asymptomatic population on the control measures, would provide useful theoretical reference to the policymakers and fuel future studies of COVID-19.

Modeling COVID-19 infection in a confined space.

In this paper, we construct a stochastic model of the 2019-nCoV transmission in a confined space, which gives a detailed account of the interaction between the spreading virus and mobile individuals. Different aspects of the interaction at mesoscopic level, such as the human motion, the shedding and spreading of the virus, its contamination and invasion of the human body and the response of the human immune system, are touched upon in the model, their relative importance during the course of infection being evaluated. The model provides a bridge linking the epidemic statistics to the physiological parameters of individuals and may serve a theoretical guidance for epidemic prevention and control.

Low-dimensional projection of stochastic cell-signalling dynamics via a variational approach.

Noise and fluctuations play vital roles in signal transduction in cells. Various numerical techniques for its simulation have been proposed, most of which are not efficient in cellular networks with a wide spectrum of timescales. In this paper, based on a recently developed variational technique, low-dimensional structures embedded in complex stochastic reaction dynamics are unfolded which sheds light on new design principles of efficient simulation algorithm for treating noise in the mesoscopic world. This idea is effectively demonstrated in several popular regulation models with an empirical selection of test functions according to their reaction geometry, which not only captures complex distribution profiles of different molecular species but also considerably speeds up the computation.

Koopman analysis in oscillator synchronization.

Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems, while a global statistical approach is overly cursory. Koopman operator technique seems to balance well the two approaches. In this paper we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in the case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighboring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.

Anomalous structure and dynamics in news diffusion among heterogeneous individuals.

Previous research has suggested that well-connected nodes in a network (commonly referred to as hubs) are better at spreading information than those with fewer connections (ordinary users). Here we investigate the roles of nodes with different numbers of connections by studying how people share news online. Quantitative analysis shows that users without many connections can sometimes spread news more effectively than well-connected users when the diffusion pattern has dendrite-like paths that reach far into the network, leading to a non-Gaussian distance distribution. When the hubs dominate, however, the distribution is Gaussian. Enhanced interactions among ordinary users are the key to the emergence of non-Gaussian characteristics. Finally, we introduce a message-passing model that reproduces the observed diffusion features. This model shows that patterns dominated by either hubs or ordinary users can be clearly demarcated by measuring the average number of forwards.

Stochastic Thermodynamics of a Particle in a Box.

The piston system (particles in a box) is the simplest paradigmatic model in traditional thermodynamics. However, the recently established framework of stochastic thermodynamics (ST) fails to apply to this model system due to the embedded singularity in the potential. In this Letter, we study the ST of a particle in a box by adopting a novel coordinate transformation technique. Through comparing with the exact solution of a breathing harmonic oscillator, we obtain analytical results of work distribution for an arbitrary protocol in the linear response regime and verify various predictions of the fluctuation-dissipation relation. When applying to the Brownian Szilard engine model, we obtain the optimal protocol λ_{t}=λ_{0}2^{t/τ} for a given sufficiently long total time τ. Our study not only establishes a paradigm for studying ST of a particle in a box but also bridges the long-standing gap in the development of ST.

A Stochastic Model of the Germinal Center Integrating Local Antigen Competition, Individualistic T-B Interactions, and B Cell Receptor Signaling.

The germinal center (GC) reaction underlies productive humoral immunity by orchestrating competition-based affinity maturation to produce plasma cells and memory B cells. T cells are limiting in this process. How B cells integrate signals from T cells and BCRs to make fate decisions while subjected to a cyclic selection process is not clear. In this article, we present a spatiotemporally resolved stochastic model that describes cell behaviors as rate-limited stochastic reactions. We hypothesize a signal integrator protein integrates follicular helper T (Tfh)- and Ag-derived signals to drive different B cell fates in a probabilistic manner and a dedicated module of Tfh interaction promoting factors control the efficiency of contact-dependent Tfh help delivery to B cells. Without assuming deterministic affinity-based decisions or temporal event sequence, this model recapitulates GC characteristics, highlights the importance of efficient T cell help delivery during individual contacts with B cells and intercellular positive feedback for affinity maturation, reveals the possibility that antagonism between BCR signaling and T cell help accelerates affinity maturation, and suggests that the dichotomy between affinity and magnitude of GC reaction can be avoided by tuning the efficiency of contact-dependent help delivery during reiterative T-B interactions.

Channel based generating function approach to the stochastic Hodgkin-Huxley neuronal system.

Internal and external fluctuations, such as channel noise and synaptic noise, contribute to the generation of spontaneous action potentials in neurons. Many different Langevin approaches have been proposed to speed up the computation but with waning accuracy especially at small channel numbers. We apply a generating function approach to the master equation for the ion channel dynamics and further propose two accelerating algorithms, with an accuracy close to the Gillespie algorithm but with much higher efficiency, opening the door for expedited simulation of noisy action potential propagating along axons or other types of noisy signal transduction.