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.

The intricate interactions, emergent behaviors, and nonlinear dynamics of complex systems are profoundly influenced by noise, which significantly contributes to the unpredictability and complexity involved in understanding and controlling such systems. This study aims to achieve accurate reconstruction of the interaction matrix of complex systems in the presence of noise. To this end, we propose a novel method, ensemble compressive sensing (ECS), which integrates multiple independent trials of the conventional L1-Regularized Least Squares (L1-RLS) approach. The proposed ECS framework demonstrates superior performance compared to L1-RLS by successfully reconstructing the interaction matrix within a well-defined noise range, while also maintaining higher accuracy beyond this range. As a case study, we validate the proposed approach using Landau-Stuart oscillators.

We propose a comparative study of chaotification techniques that enhance the complexity of one-dimensional maps for sustainable cryptographic applications. Specifically, we examine the use of modulo operation, remainder operation, and k-deep-zoom (k-DZ) transformation applied to the sine map. Each technique’s effectiveness is analyzed using bifurcation diagrams, Lyapunov exponents, and correlation results. Additionally, the statistical performance of the techniques is assessed using the NIST SP 800-22 battery of tests, and the computational efficiency is evaluated in terms of the number of operations. The study aims to identify the best chaotification method for generating robust pseudorandom bit generators (PRBGs) suitable for secure encryption and sustainable cryptography.

Oscillatory dynamics arise ubiquitously in nonlinear systems, yet identifying a physically interpretable phase and phase dynamics in nonlinear, high-dimensional oscillations remains a central unresolved problem. Here we establish the principle of a universal dynamical clock, a physical perspective in which oscillations of arbitrary dimensionality and geometry are equivalently represented as uniform rotation through an equant-induced nonlinear viewing coordinate, inspired by Ptolemy's equant and formalised through an areal-uniformity principle reminiscent of Kepler's second law. Using a machine-learning framework, we demonstrate the existence of such an equant for a broad class of oscillatory dynamics and construct the associated dynamical clock and phase dynamics under additive forces, including noise, periodic perturbations, and coupling. Its value in uncovering new physical rules and phenomena is demonstrated by four findings: (i) collective oscillations in Escherichia coli populations obey a previously unexplained superlinear scaling law, resolving a long-standing open problem posed in 2004; (ii) the response mechanisms of engineered genetic circuits to changes in gene expression and environmental conditions; (iii) a classical-mechanics counterpart of the Berry geometric phase emerges naturally from the phase of the dynamical clock; and (iv) optimal equant non-uniformity provides a geometric early-warning signal for critical transitions and enables prediction of critical parameters. By providing operational and system-agnostic phase dynamics that can be constructed directly from data, the dynamical clock enables principled classification, comparison, and control of oscillatory systems, and offers a new route to understanding how specific dynamical regimes support distinct functional behaviours in networked systems.

Remote synchronization (RS) refers to coherence among peripheral nodes that are indirectly connected and remain desynchronized from their common hub. While RS has been extensively studied in deterministic systems, the role of stochastic perturbations in enabling or enhancing RS remains poorly understood. Here, we show that noise can induce RS in star-structured networks of coupled Stuart-Landau oscillators. As the noise intensity increases, the system exhibits four distinct dynamical regimes: a disordered state, noise-induced remote synchronization (NRS), phase-only noise-induced remote synchronization (PNRS), and noise-induced death. These regimes are distinguished by their phase coherence, frequency synchronization properties, and noise-driven spectral characteristics. A phase diagram in the coupling-noise parameter space reveals that NRS typically arises near the intrinsic RS boundary, where noise selectively enhances coordination through topology-dependent resonance filtering. A linearized frequency-domain analysis supports the numerical findings, providing a mechanistic explanation of how stochastic inputs can facilitate long-range synchronization. These results reveal a constructive role of noise in generating RS and uncover a new class of noise-induced dynamical states in complex networks.

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed Neural Dynamical Transfer Learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

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.

Cell differentiation emerges as an irreversible biological transition governed by the expression of core gene circuit. While Hill functions provide phenomenological descriptions of gene network behaviors, when addressing the issue of cell differentiation, it is often inevitable to introduce a time-correlated driving term, which undermines the theoretical closure. Utilizing a prototypical gene circuit, we develop a microscopic theory for the Hill functions based on the detailed biochemical reactions and the principles of statistical physics. This approach establishes equivalence mappings between different biological scales: the complete microscopic reaction network model, reduced microscopic model, and effective mesoscopic model. Under this equivalence, without introducing external information, we identify the differentiation-driving forces, while the remaining components of the model are exactly represented by Hill functions. Our theoretical results also demonstrate that the macroscopic force described by Hill functions maintains cellular stability over small time scales, whereas intrinsic driving forces propel directed differentiation of cells across large time scales. Furthermore, numerical simulations conducted using equivalent methods agrees with our theoretical findings. The derived relationships between reaction kinetic constants and phenomenological parameters establish a physical basis for bridging genotype-phenotype mapping in developmental systems.

In dynamical systems with a time-dependent parameter, i.e., parameter drift, after crossing a saddle-node bifurcation, the so-called ghost state formed by the disappeared equilibria or periodic orbit can influence transient dynamics, causing a delayed transition. This phenomenon has been investigated previously. However, the effect of chaotic ghosts on the critical transition in drifting systems has been less studied. In this paper, we explore how chaotic ghosts and drifting rates influence critical transitions from the perspective of the ensemble. The results reveal the mechanism of the delayed transition related to chaos and how trajectories on the initial ensemble composed of a chaotic attractor transition to a qualitatively different object during the drift. In addition, we quantify the delayed transition and further find that the delay follows a power-law scaling with respect to the drifting rate. Finally, we show that the critical transition is fully avoided as long as the reversal rate of the parameter exceeds a certain critical rate, even though the bifurcation point has been crossed.

We propose a topology-aware spatiotemporal model for predicting the occurrence of extreme events both in time ("when") and in space ("where") in nonlinear physical systems. Specifically, our model adopts a unified topological perspective to bridge temporal dynamics and spatial localization. By representing spatial grids as graph nodes, the model captures spatiotemporal dependencies from two complementary views: functional connections describe global temporal evolution, while structural neighborhood connections characterize local spatial interactions. This design supports joint prediction in time and space by effectively capturing temporal precursors and spatial patterns. The model is validated on a synthetic dataset from the two dimensional complex Ginzburg-Landau equation and on ERA5 wind speeds dataset over the North Atlantic. Experiments based on graph neural network show that our model achieves great performance in both temporal prediction and spatial localization, highlighting its potential for addressing complex extreme event prediction challenges.

Critical transitions are often used to describe the abrupt transitions between stable states when the control parameter crosses a bifurcation point. Alternatively, in chaotic systems, a suddenly disappeared chaotic attractor can transform into a saddle-type chaotic invariant set, called the chaotic saddle. Understanding how chaotic saddles influence critical transitions in drifting systems is crucial for predicting and controlling long-term dynamics. In this work we consider a piecewise linear system in which the driving amplitude varies linearly with time. The analysis shows that for both homoclinic and heteroclinic crises, when the time-dependent amplitude slowly crosses the crisis, the ensemble initially generated from the chaotic attractor remains around the chaotic saddle before suddenly collapsing to another coexisting attractor, resulting in a delay in the critical transition. In addition, we derive a power-law scaling relationship between the delayed transition and the drifting rate, where the scaling exponent depends on the pre-selected parameter interval containing chaotic saddles. Based on the delayed response of the drifting system, we reveal the necessary condition under which the collapse can be avoided, and successfully reverse the critical transition. Finally, we demonstrate that the crisis leading to the reappearance of the chaotic attractor can be masked or hidden due to the separation between the drifting rate and the escape rate.

Transformer architectures have recently surged as promising solutions for nonlinear dynamical systems, proposed as foundation models capable of zero-shot dynamics reconstruction and forecasting. Despite this success, it remains unclear whether they can truly serve as reliable digital twins of dynamical systems, i.e., whether they capture the underlying physical dynamics in distinct parameter regimes, especially in parameter regimes from which no training data is taken. For parameter-space extrapolation in nonlinear dynamical systems, reservoir computing has demonstrated broad success, as proper training can turn it into an intrinsic dynamical system capable of capturing not only the dynamical climate of the target system but more importantly, how the climate changes with parameter. Transformers, in contrast, rely on permutation-invariant attention mechanisms that can limit their ability to capture how temporal structure changes with parameter. To determine if Transformers have the capability of dynamics extrapolation, we take predicting catastrophic collapse, which occurs when a bifurcation parameter crosses a critical threshold, as a benchmark task. Models are trained on trajectories in normal parameter regimes and then tested on parameters in an unseen regime with system collapse. Our results show that Transformers, across configurations, consistently fail to capture collapse, while reservoir computing reliably predicts the transitions. This surprising finding raises questions about the generalization ability of Transformers to dynamical systems, a topic warranting future research.

Sequence memory, such as memorizing digit strings and music, is a fundamental cognitive function involving multiple brain regions. It allows for both goal-based retrieval and context-based retrieval after memory formation, which are essential for memory integration and adaptability in improved multitasking performance. However, existing brain-inspired neurodynamic models face significant challenges, including the implementation of bidirectional retrieval functions and the allocation of neurons according to task-specific requirements. Inspired by physiological clues, we hypothesize that sequence memory should involve an information processing architecture corresponding to perception, memory, and semantic memory (PMS) functions. Therefore, we constructed a neurodynamic model conforming to the PMS architecture. Our findings reveal that this network not only addresses the challenges of bidirectional retrieval and task-adaptive neuron allocation, but also spontaneously utilizes inter-layer information coupling to achieve memory retrieval while enabling efficient memorization through reuse of existing spatial patterns with similar sequences during learning. Based on this network framework, we investigated the impact of connection degradation in different brain regions on both goal-based and contextual retrieval. Moreover, in the PMS model, we discovered that the three-layer network architecture and inter-layer coupling play crucial roles in sequence memory. Overall, the brain-inspired PMS model provides theoretical possibilities for simulating and explaining phenomena in sequence memory. It also demonstrates potential for achieving complex cognitive functions through implementation of deeper structural hierarchies, thereby inspiring next-generation sequence memory algorithms based on neurodynamic neural networks.

Directed cell differentiation is a stochastic process governed by gene regulatory motifs and modulated by the microenvironment. However, a rigorous physical framework to quantify how microenvironmental fluctuations interact with intrinsic transcriptional noise remains elusive. In this work, we investigate the regulatory role of the microenvironment in phenotypic transitions by bridging microscopic biological processes with macroscopic landscape modeling. We first establish a self-activating promoter switching (PS) model subject to microenvironmental influences, coupling intrinsic Poisson noise with extrinsic colored noise. By employing large deviation theory and asymptotic analysis, we characterize the limiting behavior of the quasipotential to quantify the potential landscape. Our results reveal a non-monotonic regulation of phenotypic switching by the microenvironment, which can either inhibit or enhance transitions, with a distinct optimal enhancement point identified. This analysis is further extended to an asymmetric toggle switch (TS) model to depict lineage-directed differentiation. We find that early stages of differentiation are particularly sensitive to microenvironmental factors, aligning with in vivo observations. By integrating these frameworks via model reduction, we uncover a key regulatory principle: a symmetrical microenvironment can effectively modulate differentiation trends induced by asymmetrical internal gene expression. This study provides a quantitative physical perspective on how intrinsic and extrinsic fluctuations synergistically shape cellular decision-making.

The breathing circular billiard models the free motion of a point particle within a circular enclosure whose boundary undergoes periodic motion. It is assumed that the particle experiences elastic collisions with the boundary. Previous studies have demonstrated that when the boundary motion is sufficiently regular (specifically, C-7 smooth), the energy of the particle remains bounded due to the existence of invariant curves in the phase space. In this work, under the assumption that the motion of the boundary is C-2, we prove that the set of initial conditions that give rise to escaping orbits has Lebesgue measure zero, imparting orbit stability to the breathing circular billiard.

The Hopf whole-brain model, based on structural connectivity, overcomes limitations of traditional structural or functional connectivity-focused methods by incorporating heterogeneity parameters, quantifying dynamic brain characteristics in healthy and diseased states. Traditional parameter fitting techniques lack precision, restricting broader use. To address this, we validated parameter fitting methods using simulated networks and synthetic models, introducing improvements such as individual-specific initialization and optimized gradient descent, which reduced individual data loss. We also developed an approximate loss function and gradient adjustment mechanism, enhancing parameter fitting accuracy and stability. Applying this refined method to datasets for major depressive disorder (MDD) and autism spectrum disorder (ASD), we identified differences in brain regions between patients and healthy controls, explaining related anomalies. This rigorous validation is crucial for clinical application, paving the way for precise neuropathological identification and novel treatments in neuropsychiatric research, demonstrating substantial potential in clinical neurology.

Robust decoding performance is essential for the practical deployment of brain-computer interface (BCI) systems. Existing EEG decoding models often rely on large amounts of annotated data collected through specific experimental setups, which fail to address the heterogeneity of data distributions across different domains. This limitation hinders BCI systems from effectively managing the complexity and variability of real-world data. To overcome these challenges, we propose Synchronized Self-Training Domain Adaptation (SSTDA) for cross-domain motor imagery classification. Specifically, SSTDA leverages labeled signals from a source domain and applies self-training to unlabeled signals from a target domain, enabling the simultaneous training of a more robust classifier. The raw EEG signals are mapped into a latent space by a feature extractor for discriminative representation learning. A domain-shared latent space is then learned by optimizing the feature extractor with both source and target samples, using an easy-tohard self-training process. We validate the method with extensive experiments on two public motor imagery datasets: Dataset IIa of BCI Competition IV and the High Gamma dataset. In the inter-subject task, our method achieves classification accuracies of 64.43% and 80.40%, respectively. It also outperforms existing methods in the inter-session task. Moreover, we develope a new six-class motor imagery dataset and achieve test accuracies of 77.09% and 80.18% across different datasets. All experimental results demonstrate that our SSTDA outperforms existing algorithms in inter-session, inter-subject, and inter-dataset validation protocols, highlighting its capability to learn discriminative, domain-invariant representations that enhance EEG decoding performance.

Even though compressive sensing (CS) has been successfully implemented in numerous chaotic dynamical systems, there remains a lack of in-depth understanding of how this inverse problem method performs when reconstructing periodic dynamical systems. In this work, we investigate the reconstruction of a periodic dynamical system based on CS, considering the Landau-Stuart (LS) oscillator as a paradigmatic model. Unlike chaotic systems, the periodic dynamics bring some challenges to robust reconstruction by CS. We identified that by introducing slight differences or diversity in the dynamics of coupled Landau-Stuart (LS) oscillator systems, for example, coupling asymmetry, frequency detuning, and amplitude disparity, the process of achieving robust reconstruction by CS becomes considerably smooth.

In this paper, the stability and Bautin bifurcation of a four-wheel-steering (4WS) vehicle system, by considering driver steering control, are investigated. By using the central manifold theory and projection method, the first and second Lyapunov coefficients are calculated to predict the type of Hopf bifurcation of the vehicle system. The topological structure of Bautin bifurcation, a degenerate Hopf bifurcation of the 4WS vehicle system, is presented in parameter space, and it reveals the dynamics of the vehicle system of different choices of control parameters. The influences of system parameters on critical values of the bifurcation parameter are also analyzed. It is shown that with the increase in the frontal visibility distance of the driver control strategy coefficient and the cornering stiffness coefficients of rear wheels, the critical speed increases. Nevertheless, the critical speed decreases with the increase in the distance from the center of gravity of the vehicle to the front axles, Driver's perceptual time delay, and cornering stiffness coefficients of the front wheels.