Understanding the principles governing efficient information processing in biological neural systems requires the identification of mechanisms for the flexible integration of external sensory signals and internal spontaneous activity. Neural systems operating near criticality exhibit optimized sensitivity and reliability in processing sensory signals. We hypothesize acetylcholine (ACh), a key neuromodulator, controls the sensory speed in neural systems by modulating critical dynamics. Using a computational model based on an ACh-modulated clustered network, our simulations reveal a U-shaped relationship between ACh concentration and sensory response latency, with the fastest responses occurring near a phase transition. At this critical point, the model exhibits intermittent bursts of cluster activity characterized by irregular periods. This critical dynamics allows the network to flexibly and simultaneously combine different cluster activities, facilitating the classification of diverse sensory inputs. Basing on network theory and statistical analysis, we propose that ACh maintains neural criticality by balancing synchronization and stability. This work presents a theory connecting neuromodulation to critical dynamics, with implications for perception and the development of efficient brain-inspired computing.

Traditional associative memory models, exemplified by the Hopfield network, have established a foundational understanding of how equilibrium states encode and store static patterns. However, the neural mechanisms governing transitions between these multiple equilibria, a hallmark of flexible cognition, remain poorly understood. Here, we address this gap by developing a biologically grounded dynamical framework that integrates cholinergic neuromodulation into an extended Hopfield network. Specifically, we model the dual regulatory effects of acetylcholine (ACh): the modulation of the local-global inhibition balance via an inhibition ratio parameter, c, and the regulation of spike frequency adaptation (SFA) through activity-dependent thresholds. Our theoretical analysis and numerical simulations reveal that SFA-driven destabilization enables controlled transitions from stable attractor states to metastable mixture states, thereby allowing the network to overcome local energy barriers. Critically, we identify an optimal synergistic control strategy that combines both ACh-mediated mechanisms, achieving an 87% success rate in facilitating attractor switching. This combined approach outperforms individual mechanisms operating in isolation (85% for SFA alone and 78% for inhibition modulation alone). Mean-field analysis demonstrates that this synergy operates by transiently destabilizing current attractors, enabling trajectories to escape shallow energy basins and converge to deeper, more energetically favorable states. Furthermore, these findings provide a computational framework grounded in energy landscape optimization that explains how neural circuits balance memory stability with cognitive flexibility, offering mechanistic insights into associative recall, free association, and state-dependent information processing.

Hopfield networks are widely used models of associative memory. When the number of stored patterns exceeds the network’s storage capacity, theoretical predictions show that the overlap between final states and memorized patterns should vanish. However, numerical simulations show that a small, non-zero overlap persists, indicating that the network retains residual memory. To investigate the origin of this phenomenon, we analyze the network’s dynamics during the initial update steps. Using a signal-to-noise-ratio analysis, we demonstrate that when a node undergoes a state flip, the signal term of its neighbors is enhanced by the connecting link. This effect improves the stability of these neighboring neurons, facilitating a fraction of the network to remain aligned with the memory pattern and preventing a total loss of memory. Our findings elucidate the mechanism by which residual memory traces emerge in Hopfield networks beyond the storage limit.

Topological features significantly influence a network’s behavior and functional performance, particularly in terms of information transmission efficiency, learning, adaptation, and resource utilization. This paper focuses on the impact of network connectivity, specifically the role of reciprocal links, on the ability of neural networks in critical states to learn Boolean rules. Notably, the prevalence of reciprocal links in the cerebral cortex suggests that they play an important role in information processing. Our findings demonstrate that reciprocal links markedly enhance learning performance. As the backbone structure of the network becomes more complex with the addition of reciprocal links, there is an increase in alternative paths. This, in turn, facilitates more efficient signal transmission from input to output sites, ultimately leading to a higher learning success rate. Reciprocal links not only optimize information transmission pathways but also diminish the influence of irrelevant neurons, thus enhancing resource utilization efficiency.

In the percolation model, specific preprocessing of configuration data enables identifying phase transitions with unsupervised learning methods such as principal component analysis (PCA). However, the limitations of using original percolation configuration data and the role of preprocessing remain incompletely understood. Here, we study how PCA works in percolation. Firstly, we theoretically derive the PCA results. For the original configurations, PCA shows no insights into the percolation phase transition. The theoretical results of PCA reveal the relation between principal components and the indicators of phase transitions, i.e., order parameters and structure factors based on the preprocessing of removing unpercolating clusters. In addition, we investigate the PCA based on auxiliary Ising mapping, and the relation is also derived and validated. Secondly, we examine the intrinsic properties of the original percolation configurations and the operational principles of PCA and then reveal the challenge of PCA in percolation. The statistical independence among sites results in the absence of necessary lattice structure information, and further, the negligible threshold number of occupied (unoccupied) sites required for forming (destructing) percolating cluster results in the invisibility of the phase transition to the cluster analysis or statistical approaches. Both removing unpercolating clusters and auxiliary Ising mapping can remove this invisibility. Our work suggests that yielding discernible differences between percolating state and unpercolating state, or further introducing percolation statistical correlation among sites, gives effective data preprocessing methods.

Avalanche sizes and durations following power-law distributions are observed in many systems and are considered hallmarks of criticality. Time-series thresholding is a commonly used method to define avalanches, but this method is controversial. In this study, we use the time-series thresholding method to define avalanches and investigate the statistical properties of avalanches in the Kinouchi-Copelli (KC) model. We consider two definitions of avalanche size, (1) total area above threshold value reference and (2) total area above zero, and then analyze the size and duration distributions. Our results show that while avalanche size and duration obey a power-law distribution, the exponents of size and duration differ from those of the critical branching process. The scaling relation holds for avalanches defined by the first method but fails for those avalanches tdefined by the second. This study provides new insights into avalanche definition methods and avalanche distribution exponents in continuous time-series.

Segregation and integration coexist in the dynamics of complex networked systems, such as neural networks in the brain, where their balance corresponds to the highest dynamical complexity. Coupled oscillators can achieve the balance between segregation and integration on modular networks. In this study, we investigate how the degradation of balance network affects the segregation and integration of coupled oscillators. Degradation is modeled by removing nodes or links, and by weakening links. We show that, in balance networks, degradation decreases segregation while increasing integration. This behavior arises from a structural feature of the network: inter-modular connections between a pair of modules are sparse. When degradation occurs, all nodes lose a part of intra-modular links, whereas some nodes retain all of their inter-modular links.

Cortical neuronal activity varies over time and across repeated trials, yet consistently represents stimulus features. The dynamical mechanism underlying this reliable representation and computation remains elusive. This study uncovers a mechanism for reliable neural information processing, leveraging a biologically plausible network model incorporating neural heterogeneity. First, we investigate neuronal timescale diversity, revealing that it disrupts intrinsic coherent spatiotemporal patterns, induces firing rate heterogeneity, enhances local responsive sensitivity, and aligns network activity closely with input. The system exhibits globally input-slaved transient dynamics, essential for reliable neural information processing. Other neural heterogeneities, such as nonuniform input connections, spike threshold heterogeneity, and network in-degree heterogeneity, play similar roles, highlighting the importance of neural heterogeneity in shaping consistent stimulus representation. This mechanism offers a potentially general framework for understanding neural heterogeneity in reliable computation and informs the design of reservoir computing models endowed with liquid wave reservoirs for neuromorphic computing.

Critical state plays an important role in emerged behavior of neural networks. Excitable networks are widely used in simulation studies. However, the simulation of critical state is sophisticated because the state sensitively depends on dynamical and topological features. We study the effect of refractory period of dynamical units and connection symmetry on the excitable network with critical branching ratio. In bidirectional random networks with refractory period, the critical branching ratio is well known. Here, we show that the collective behavior changes from critical state into ceaseless activity, when refractory period vanishes. Instead of mean degree of all nodes, we show that the control parameter is determined by the mean degree of active nodes which is larger than the average degree and induces the transition. When refractory period is present, the number of effective links connected to resting nodes is equal to average degree. The mechanism ensures the critical branching ratio is unity. In directed networks, collective behavior keeps the critical state when refractory period vanished. However, a few reciprocal links make the network without refractory period becoming ceaseless.

Despite the success of data-driven machine learning in forecasting complex nonlinear dynamics, predicting future evolution based on incomplete historical data remains challenging. Reservoir Computing (RC), a widely adopted approach, suffers from incomplete past observations since it typically requires complete data for accurate predictions. In this paper, a novel data processing scheme is introduced to improve the predictive performance of the RC when the input time series or dynamic trajectories are incomplete, for example, a portion of elements or states are randomly omitted or removed. It is a purification strategy, in which the input data are purified by selecting data or data sequences that are one step ahead of the segments of missing data. The selected data are positioned in turn in a new input, which is no longer indexed by the temporal order in the original time series. This approach matches the one-step-head nature of the convention RC and is thus very simple and efficient, without changing anything in the architecture of RC and avoiding sophisticated pretreatment on the incomplete input. It has been successfully employed to predict the chaotic dynamics in the Logistic map, Lorenz and Rössler systems, when the machine is trained by the purified input. The effect of the missing data on the predictive performance of the RC is also discussed. The results suggest that the purification of input can significantly improve its efficiency of predictive performance.

Reservoir computing (RC) has been widely applied to predict the chaotic dynamics in many systems. Yet much broader areas related to nonsmooth dynamics have seldom been touched by the RC community which have great theoretical and practical importance. The generalization of RC to this kind of system is reported in this paper. The numerical work shows that the conventional RC with a hyperbolic tangent activation function is not able to predict the dynamics of nonsmooth systems very well, especially when reconstructing attractors (long-term prediction). A nonsmooth activation function with a piecewise nature is proposed. A kind of physics-informed RC scheme is established based on this activation function. The feasibility of this scheme has been proven by its successful application to the predictions of the short- and long-term (reconstructing chaotic attractor) dynamics of four nonsmooth systems with different complexity, including the tent map, piecewise linear map with a gap, both noninvertible and discontinuous compound circle maps, and Lozi map. The results show that RC with the new activation function is efficient and easy to run. It can make perfectly both short- and long-term predictions. The precision of reconstructing attractors depends on their complexity. This work reveals that, to make efficient predictions, the activation function of an RC approach should match the smooth or nonsmooth nature of the dynamical systems.

In the Ising model, principal component analysis (PCA) serves as an unsupervised learning method for identifying phase transitions. The PCA results of the two-dimensional square -lattice Ising model with periodic boundary conditions are theoretically deduced. By utilizing the Hamiltonian and lattice structure of the Ising model, the structure of the sample covariance matrix can be determined, enabling the realization of PCA by solving the eigenvalue problem of the sample covariance matrix. All symmetries of the lattice are implicit in the sample covariance matrix, with translational symmetry determining the eigenvector corresponding to the Fourier mode. Moreover, based on translational symmetry, other symmetries determine the degeneracy of eigenvalues. When combined with knowledge regarding the Ising model, the meaning of the PCA results becomes clarified. Our theoretical derivation elucidates the PCA results from previous studies on the Ising model. Based on theoretical derivation, we also investigated the ability of PCA to identify phase transitions in the Ising model. From the machine learning (ML) perspective, the theoretical derivation of PCA results sheds light on the mechanism through which PCA identifies phase transitions. This clarity contributes to removing opacity in ML and increasing trust in the application of ML in physics.

Predicting future evolution based on incomplete information of the past is still a challenge even though data-driven machine learning approaches have been successfully applied to forecast complex nonlinear dynamics. The widely adopted reservoir computing (RC) can hardly deal with this since it usually requires complete observations of the past. In this paper, a scheme of RC with (D+1)-dimension input and output (I/O) vectors is proposed to solve this problem, i.e., the incomplete input time series or dynamical trajectories of a system, in which certain portion of states are randomly removed. In this scheme, the I/O vectors coupled to the reservoir are changed to (D+1)-dimension, where the first D dimensions store the state vector as in the conventional RC, and the additional dimension is the corresponding time interval. We have successfully applied this approach to predict the future evolution of the logistic map and Lorenz, Rössler, and Kuramoto-Sivashinsky systems, where the inputs are the dynamical trajectories with missing data. The dropoff rate dependence of the valid prediction time (VPT) is analyzed. The results show that it can make forecasting with much longer VPT when the dropoff rate θ is lower. The reason for the failure at high θ is analyzed. The predictability of our RC is determined by the complexity of the dynamical systems involved. The more complex they are, the more difficult they are to predict. Perfect reconstructions of chaotic attractors are observed. This scheme is a pretty good generalization to RC and can treat input time series with regular and irregular time intervals. It is easy to use since it does not change the basic architecture of conventional RC. Furthermore, it can make multistep-ahead prediction just by changing the time interval in the output vector into a desired value, which is superior to conventional RC that can only do one-step-ahead forecasting based on complete regular input data.

We investigate the relationship between the synchronous transition and the power law behavior in spiking networks which are composed of inhibitory neurons and balanced by dc current. In the region of the synchronous transition, the avalanche size and duration distribution obey a power law distribution. We demonstrate the robustness of the power law for event sizes at different parameters and multiple time scales. Importantly, the exponent of the event size and duration distribution can satisfy the critical scaling relation. By changing the network structure parameters in the parameter region of transition, quasicriticality is observed, that is, critical exponents depart away from the criticality while still hold approximately to a dynamical scaling relation. The results suggest that power law statistics can emerge in networks composed of inhibitory neurons when the networks are balanced by external driving signal.

在使用分子动力学方法对微正则系综进行模拟的情况下,以单原子粒子系统为例,给出了标度因子在国际单位制下的数学表达式.然后说明了如何把物理量重新标度得到无量纲公式,使得在模拟中计算更加方便.给出了无量纲化操作中时间单位的物理意义.最后,推导了趋衡过程的速度标度因子在教材中常用的数学表达式.

The disappearance and reappearance of chaos by adjusting the internal parameters of dynamics in Lorenz system are studied. We observe monotonous and periodic time-dependent changes of Rayleigh number. There exists relaxation time for the disappearance of chaos, when we use the snapshot attractors to observe the change of the system attractors. We show that the rate of disappearance and reappearance of chaos is positively correlated with the control parameters. To reflect the relaxation phenomenon of chaotic disappearance and the sensitivity of trajectory, the concept of finite-time Lyapunov exponent is used. Then the statistical characteristics of the system can be presented by standard deviation. The chaotic disappearance and reappearance are manifested in the decrease and increase of the standard deviation. The standard deviation decreases continuously during chaotic disappearance, but increases discontinuously during chaotic reappearance. A distinctive scenario is that no matter which parameter changes, when we use the same rate of change in the process of chaotic disappearance and reappearance, their paths are different.

Hopfield neural networks on scale-free networks display the power law relation between the stability of patterns and the number of patterns. The stability is measured by the overlap between the output state and the stored pattern which is presented to a neural network. In simulations the overlap declines to a constant by a power law decay. Here we provide the explanation for the power law behavior through the signal-to-noise ratio analysis. We show that on sparse networks storing a plenty of patterns the stability of stored patterns can be approached by a power law function with the exponent -0.5. There is a difference between analytic and simulation results that the analytic results of overlap decay to 0. The difference exists because the signal and noise term of nodes diverge from the mean-field approach in the sparse finite size networks.

A model of particle-cluster aggregation on the rewired small-world network is studied. In this model, particles can adjoin to the end of the rewired long-range edges. The rewired long-range edge is controlled by two parameters, the reconnecting probability and distance controlling parameter. The simulation results indicate that, with the increasing of the length and the number of rewired long-range edges, the patterns become diffuse and the fractal dimensions of the aggregates decrease. The range of the value of dimension and the feature of patterns obtained in the model are consistent with the amyloid deposits.

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.

Spatiotemporal patterns in the transition of phase synchronization in modular networks of coupled logistic maps are studied. The phase diagram of spatiotemporal patterns is presented by analyzing both the collective behavior of direction-phase and the changes of links connecting clusters of different phases. The spatiotemporal chaos is obtained when the coupling strength is weak. We show that the spatiotemporal chaos can be composed by clusters in periodic states. The region of periodic behaviors is independent of modularity. Then with decreasing coupling or increasing modularity, the system presents the same transition path from complete synchronization to cluster synchronization, except the network is close to fully connected networks. There are two distinctive scenarios from disordered behavior to an ordered state when the modularity ratio varies from one to zero. First, for networks with small modularity, the number of phase synchronized clusters decreases with the increasing of the coupling strength. Second, for networks with large modularity, the number of phase-synchronized clusters non-monotonically changes with the coupling strength.