Learning dynamical systems from data: An introduction to physics-guided deep learning.

Modeling complex physical dynamics is a fundamental task in science and engineering. Traditional physics-based models are first-principled, explainable, and sample-efficient. However, they often rely on strong modeling assumptions and expensive numerical integration, requiring significant computational resources and domain expertise. While deep learning (DL) provides efficient alternatives for modeling complex dynamics, they require a large amount of labeled training data. Furthermore, its predictions may disobey the governing physical laws and are difficult to interpret. Physics-guided DL aims to integrate first-principled physical knowledge into data-driven methods. It has the best of both worlds and is well equipped to better solve scientific problems. Recently, this field has gained great progress and has drawn considerable interest across discipline Here, we introduce the framework of physics-guided DL with a special emphasis on learning dynamical systems. We describe the learning pipeline and categorize state-of-the-art methods under this framework. We also offer our perspectives on the open challenges and emerging opportunities.

Heterogeneous stochastic bifurcations explain intrinsic oscillatory patterns in entorhinal cortical stellate cells.

Stellate cells (SC) in the medial entorhinal cortex manifest intrinsic membrane potential oscillatory patterns. Although different theoretical frameworks have been proposed to explain these patterns, a robust unifying framework that jointly accounts for intrinsic heterogeneities and stochasticity is missing. Here, we first performed in vitro patch-clamp electrophysiological recordings from rat SCs and found pronounced cell-to-cell variability in their characteristic physiological properties, including peri-threshold oscillatory patterns. We demonstrate that noise introduced into two independent populations (endowed with deterministic or stochastic ion-channel gating kinetics) of heterogeneous biophysical models yielded activity patterns that were qualitatively similar to electrophysiological peri-threshold oscillatory activity in SCs. We developed spectrogram-based quantitative metrics for the identification of valid oscillations and confirmed that these metrics reliably captured the variable-amplitude and arhythmic oscillatory patterns observed in electrophysiological recordings. Using these quantitative metrics, we validated activity patterns from both heterogeneous populations of SC models, with each model assessed with multiple trials of different levels of noise at distinct membrane depolarizations. Our analyses unveiled the manifestation of stochastic resonance (detection of the highest number of valid oscillatory traces at an optimal level of noise) in both heterogeneous populations of SC models. Finally, we show that a generalized network motif comprised of a slow negative feedback loop amplified by a fast positive feedback loop manifested stochastic bifurcations and stochastic resonance in the emergence of oscillations. Together, through a unique convergence of the degeneracy and stochastic resonance frameworks, our unifying framework centered on heterogeneous stochastic bifurcations argues for state-dependent emergence of SC oscillations.

Form, function, mind: What doesn't compute (and what might).

The applicability of computational and dynamical systems models to organisms is scrutinized, using examples from developmental biology and cognition. Developmental morphogenesis is dependent on the inherent material properties of developing animal (metazoan) tissues, a non-computational modality, but cell differentiation, which utilizes chromatin-based revisable memory banks and program-like function-calling, via the developmental gene co-expression system unique to the metazoans, has a quasi-computational basis. Multi-attractor dynamical models are argued to be misapplied to global properties of development, and it is suggested that along with computationalism, classic forms of dynamicism are similarly unsuitable to accounting for cognitive phenomena. Proposals are made for treating brains and other nervous tissues as novel forms of excitable matter with inherent properties which enable the intensification of cell-based basal cognition capabilities present throughout the tree of life. Finally, some connections are drawn between the viewpoint described here and active inference models of cognition, such as the Free Energy Principle.

Cell-cell communication through FGF4 generates and maintains robust proportions of differentiated cell types in embryonic stem cells.

During embryonic development and tissue homeostasis, reproducible proportions of differentiated cell types are specified from populations of multipotent precursor cells. Molecular mechanisms that enable both robust cell-type proportioning despite variable initial conditions in the precursor cells, and the re-establishment of these proportions upon perturbations in a developing tissue remain to be characterized. Here, we report that the differentiation of robust proportions of epiblast-like and primitive endoderm-like cells in mouse embryonic stem cell cultures emerges at the population level through cell-cell communication via a short-range fibroblast growth factor 4 (FGF4) signal. We characterize the molecular and dynamical properties of the communication mechanism and show how it controls both robust cell-type proportioning from a wide range of experimentally controlled initial conditions, as well as the autonomous re-establishment of these proportions following the isolation of one cell type. The generation and maintenance of reproducible proportions of discrete cell types is a new function for FGF signaling that might operate in a range of developing tissues.

The Michaelis-Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation.

We demonstrate that the Michaelis-Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke's comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is s 0 ≪ K , where K = k 2 / k 1 is the Van Slyke-Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

The role of perceived negative partner behavior in daily snacking behavior: A dynamical systems approach.

Past work suggested that psychological stress, especially in the context of relationship stress, is associated with increased consumption of energy-dense food and when maintained for long periods of time, leads to adverse health consequences. Furthermore, this association is moderated by a variety of factors, including emotional over-eating style. That being said, few work utilized a dynamical system approach to understand the intraindividual and interindividual fluctuations within this process. The current study utilized a 14-day daily diary study, collected between January-March 2020, where participants reported their partner's negative relationship behavior and their own snacking behavior. A differential equation model was applied to the daily dairy data collected. Results showed that snacking behavior followed an undamped oscillator model while negative relationship behavior followed a damped coupled oscillator model. In other words, snacking behavior fluctuated around an equilibrium but was not coupled within dyadic partners. Negative relationship behavior fluctuated around an equilibrium and was amplified over time, coupled within dyadic partners. Furthermore, we found a two-fold association between negative relationship behavior and snacking: while the association between the displacement of negative relationship behavior and snacking was negative, change in negative relationship behavior and snacking were aligned. Thus, at any given time, one's snacking depends both on the amount of negative relationship behaviors one perceives and the dynamical state a dyad is engaging in (i.e., whether the negative relationship behavior is "exacerbating" or "resolving"). This former association was moderated by emotional over-eating style and the latter association was not. The current findings highlight the importance of examining dynamics within dyadic system and offers empirical and methodological insights for research in adult relationships.

Dynamical system model predicts when social learners impair collective performance.

A key question concerning collective decisions is whether a social system can settle on the best available option when some members learn from others instead of evaluating the options on their own. This question is challenging to study, and previous research has reached mixed conclusions, because collective decision outcomes depend on the insufficiently understood complex system of cognitive strategies, task properties, and social influence processes. This study integrates these complex interactions together in one general yet partially analytically tractable mathematical framework using a dynamical system model. In particular, it investigates how the interplay of the proportion of social learners, the relative merit of options, and the type of conformity response affect collective decision outcomes in a binary choice. The model predicts that, when the proportion of social learners exceeds a critical threshold, a bistable state appears in which the majority can end up favoring either the higher- or lower-merit option, depending on fluctuations and initial conditions. Below this threshold, the high-merit option is chosen by the majority. The critical threshold is determined by the conformity response function and the relative merits of the two options. The study helps reconcile disagreements about the effect of social learners on collective performance and proposes a mathematical framework that can be readily adapted to extensions investigating a wider variety of dynamics.

Breaking reflection symmetry: evolving long dynamical cycles in Boolean systems.

In interacting dynamical systems, specific local interaction rules for system components give rise to diverse and complex global dynamics. Long dynamical cycles are a key feature of many natural interacting systems, especially in biology. Examples of dynamical cycles range from circadian rhythms regulating sleep to cell cycles regulating reproductive behavior. Despite the crucial role of cycles in nature, the properties of network structure that give rise to cycles still need to be better understood. Here, we use a Boolean interaction network model to study the relationships between network structure and cyclic dynamics. We identify particular structural motifs that support cycles, and other motifs that suppress them. More generally, we show that the presence of dynamical reflection symmetry in the interaction network enhances cyclic behavior. In simulating an artificial evolutionary process, we find that motifs that break reflection symmetry are discarded. We further show that dynamical reflection symmetries are over-represented in Boolean models of natural biological systems. Altogether, our results demonstrate a link between symmetry and functionality for interacting dynamical systems, and they provide evidence for symmetry's causal role in evolving dynamical functionality.

A Note on Stronger Forms of Sensitivity for Non-Autonomous Dynamical Systems on Uniform Spaces.

This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding N-sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space (S,U), the system (S,fk∘⋯∘f1) exhibits N-sensitivity (or strongly multi-sensitivity) if and only if the system (S,f1,∞) displays N-sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space (S,U). If the system (S,f1,∞) is N-sensitive, then the system (S,fk,∞) is also N-sensitive. When the family f1,∞ is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space (S,U), N-sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates N-sensitivity.

A geometrical perspective on development.

Cell fate decisions emerge as a consequence of a complex set of gene regulatory networks. Models of these networks are known to have more parameters than data can determine. Recent work, inspired by Waddington's metaphor of a landscape, has instead tried to understand the geometry of gene regulatory networks. Here, we describe recent results on the appropriate mathematical framework for constructing these landscapes. This allows the construction of minimally parameterized models consistent with cell behavior. We review existing examples where geometrical models have been used to fit experimental data on cell fate and describe how spatial interactions between cells can be understood geometrically.

A mathematical study of the efficacy of possible negative feedback pathways involved in neuronal polarization.

Neuronal polarization, a process wherein nascent neurons develop a single long axon and multiple short dendrites, can occur within in vitro cell cultures without environmental cues. This is an apparently random process in which one of several short processes, called neurites, grows to become long, while the others remain short. In this study, we propose a minimum model for neurite growth, which involves bistability and random excitations reflecting actin waves. Positive feedback is needed to produce the bistability, while negative feedback is required to ensure that no more than one neurite wins the winner-takes-all contest. By applying the negative feedback to different aspects of the neurite growth process, we demonstrate that targeting the negative feedback to the excitation amplitude results in the most persistent polarization. Also, we demonstrate that there are optimal ranges of values for the neurite count, and for the excitation rate and amplitude that best maintain the polarization. Finally, we show that a previously published model for neuronal polarization based on competition for limited resources shares key features with our best-performing minimal model: bistability and negative feedback targeted to the size of random excitations.

Tumorigenesis and axons regulation for the pancreatic cancer: A mathematical approach.

The nervous system is today recognized to play an important role in the development of cancer. Indeed, neurons extend long processes (axons) that grow and infiltrate tumors in order to regulate the progression of the disease in a positive or negative way, depending on the type of neuron considered. Mathematical modeling of this biological process allows to formalize the nerve-tumor interactions and to test hypotheses in silico to better understand this phenomenon. In this work, we introduce a system of differential equations modeling the progression of pancreatic ductal adenocarcinoma (PDAC) coupled with associated changes in axonal innervation. The study of the asymptotic behavior of the model confirms the experimental observations that PDAC development is correlated with the type and densities of axons in the tissue. We study then the identifiability and the sensitivity of the model parameters. The identifiability analysis informs on the adequacy between the parameters of the model and the experimental data and the sensitivity analysis on the most contributing factors on the development of cancer. It leads to significant insights on the main neural checkpoints and mechanisms controlling the progression of pancreatic cancer. Finally, we give an example of a simulation of the effects of partial or complete denervation that sheds lights on complex correlation between the healthy, pre-cancerous and cancerous cell densities and axons with opposite functions.

Linking within- and between-host scales for understanding the evolutionary dynamics of quantitative antimicrobial resistance.

Understanding both the epidemiological and evolutionary dynamics of antimicrobial resistance is a major public health concern. In this paper, we propose a nested model, explicitly linking the within- and between-host scales, in which the level of resistance of the bacterial population is viewed as a continuous quantitative trait. The within-host dynamics is based on integro-differential equations structured by the resistance level, while the between-host scale is additionally structured by the time since infection. This model simultaneously captures the dynamics of the bacteria population, the evolutionary transient dynamics which lead to the emergence of resistance, and the epidemic dynamics of the host population. Moreover, we precisely analyze the model proposed by particularly performing the uniform persistence and global asymptotic results. Finally, we discuss the impact of the treatment rate of the host population in controlling both the epidemic outbreak and the average level of resistance, either if the within-host scale therapy is a success or failure. We also explore how transitions between infected populations (treated and untreated) can impact the average level of resistance, particularly in a scenario where the treatment is successful at the within-host scale.

Asymptotic behavior of an epidemic model with infinitely many variants.

We investigate the long-time dynamics of a SIR epidemic model with infinitely many pathogen variants infecting a homogeneous host population. We show that the basic reproduction number [Formula: see text] of the pathogen can be defined in that case and corresponds to a threshold between the persistence ([Formula: see text]) and the extinction ([Formula: see text]) of the pathogen population. When [Formula: see text] and the maximal fitness is attained by at least one variant, we show that the systems reaches an endemic equilibrium state that can be explicitly determined from the initial data. When [Formula: see text] but none of the variants attain the maximal fitness, the situation is more intricate. We show that, in general, the pathogen is uniformly persistent and any family of variants that have a fitness which is uniformly lower than the optimal fitness, eventually gets extinct. We derive a condition under which the total pathogen population converges to a limit which can be computed explicitly. We also find counterexamples that show that, when our condition is not met, the total pathogen population may converge to an unexpected value, or the system can even reach an eternally transient behavior where the total pathogen population between several values. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics.

Social Pressure from a Core Group can Cause Self-Sustained Oscillations in an Epidemic Model.

Let the individuals of a population be divided into two groups with different personal habits. The core group is associated with health risk behaviors; the non-core group avoids unhealthy activities. Assume that the infected individuals of the core group can spread a contagious disease to the whole population. Also, assume that cure does not confer immunity. Here, an epidemiological model written as a set of ordinary differential equations is proposed to investigate the infection propagation in this population. In the model, migrations between these two groups are allowed; however, the transitions from the non-core group into the core group prevail. These migrations can be either spontaneous or stimulated by social pressure. It is analytically shown that, in the scenario of spontaneous migration, the disease is either naturally eradicated or chronically persists at a constant level. In the scenario of stimulated migration, in addition to eradication and constant persistence, self-sustained oscillations in the number of sick individuals can also be found. These analytical results are illustrated by numerical simulations and discussed from a public health perspective.

Dynamic Modeling of Carbon Dioxide Transport through the Skin Using a Capnometry Wristband.

The development of a capnometry wristband is of great interest for monitoring patients at home. We consider a new architecture in which a non-dispersive infrared (NDIR) optical measurement is located close to the skin surface and is combined with an open chamber principle with a continuous circulation of air flow in the collection cell. We propose a model for the temporal dynamics of the carbon dioxide exchange between the blood and the gas channel inside the device. The transport of carbon dioxide is modeled by convection-diffusion equations. We consider four compartments: blood, skin, the measurement cell and the collection cell. We introduce the state-space equations and the associated transition matrix associated with a Markovian model. We define an augmented system by combining a first-order autoregressive model describing the supply of carbon dioxide concentration in the blood compartment and its inertial resistance to change. We propose to use a Kalman filter to estimate the carbon dioxide concentration in the blood vessels recursively over time and thus monitor arterial carbon dioxide blood pressure in real time. Four performance factors with respect to the dynamic quantification of the CO2 blood concentration are considered, and a simulation is carried out based on data from a previous clinical study. These demonstrate the feasibility of such a technological concept.

Chaotic Pulse-Shaping Filter Based on Root-Raised-Cosine Division.

Chaotic baseband wireless communication system (CBWCS) suffers bit error rate (BER) degradation due to their intrinsic intersymbol interference (ISI). To this end, an ISI-free chaotic filter based on root-raised-cosine (RRC) division is constructed to generate a chaotic signal. A wireless communication system using this chaotic signal as a baseband waveform is proposed. The chaotic property is proved by the corresponding new hybrid dynamical system with topological conjugation to symbolic sequences and a positive Lyapunov exponent. Simulation results show that under a single-path channel and multi-path channel, the proposed method outperforms CBWCS in both BER performance and computational complexity.

The Human Brain Encodes a Chronicle of Visual Events at Each Instant of Time Through the Multiplexing of Traveling Waves.

The human brain continuously processes streams of visual input. Yet, a single image typically triggers neural responses that extend beyond 1s. To understand how the brain encodes and maintains successive images, we analyzed with electroencephalography the brain activity of human subjects while they watched ∼5000 visual stimuli presented in fast sequences. First, we confirm that each stimulus can be decoded from brain activity for ∼1s, and we demonstrate that the brain simultaneously represents multiple images at each time instant. Second, we source localize the corresponding brain responses in the expected visual hierarchy and show that distinct brain regions represent, at each time instant, different snapshots of past stimulations. Third, we propose a simple framework to further characterize the dynamical system of these traveling waves. Our results show that a chain of neural circuits, which each consist of (1) a hidden maintenance mechanism and (2) an observable update mechanism, accounts for the dynamics of macroscopic brain representations elicited by visual sequences. Together, these results detail a simple architecture explaining how successive visual events and their respective timings can be simultaneously represented in the brain.SIGNIFICANCE STATEMENT Our retinas are continuously bombarded with a rich flux of visual input. Yet, how our brain continuously processes such visual streams is a major challenge to neuroscience. Here, we developed techniques to decode and track, from human brain activity, multiple images flashed in rapid succession. Our results show that the brain simultaneously represents multiple successive images at each time instant by multiplexing them along a neural cascade. Dynamical modeling shows that these results can be explained by a hierarchy of neural assemblies that continuously propagate multiple visual contents. Overall, this study sheds new light on the biological basis of our visual experience.

The mixed blessing of AMPK signaling in Cancer treatments.

BACKGROUND: Nutrient acquisition and metabolism pathways are altered in cancer cells to meet bioenergetic and biosynthetic demands. A major regulator of cellular metabolism and energy homeostasis, in normal and cancer cells, is AMP-activated protein kinase (AMPK). AMPK influences cell growth via its modulation of the mechanistic target of Rapamycin (mTOR) pathway, specifically, by inhibiting mTOR complex mTORC1, which facilitates cell proliferation, and by activating mTORC2 and cell survival. Given its conflicting roles, the effects of AMPK activation in cancer can be counter intuitive. Prior to the establishment of cancer, AMPK acts as a tumor suppressor. However, following the onset of cancer, AMPK has been shown to either suppress or promote cancer, depending on cell type or state. METHODS: To unravel the controversial roles of AMPK in cancer, we developed a computational model to simulate the effects of pharmacological maneuvers that target key metabolic signalling nodes, with a specific focus on AMPK, mTORC, and their modulators. Specifically, we constructed an ordinary differential equation-based mechanistic model of AMPK-mTORC signaling, and parametrized the model based on existing experimental data. RESULTS: Model simulations were conducted to yield the following predictions: (i) increasing AMPK activity has opposite effects on mTORC depending on the nutrient availability; (ii) indirect inhibition of AMPK activity through inhibition of sirtuin 1 (SIRT1) only has an effect on mTORC activity under conditions of low nutrient availability; (iii) the balance between cell proliferation and survival exhibits an intricate dependence on DEP domain-containing mTOR-interacting protein (DEPTOR) abundance and AMPK activity; (iv) simultaneous direct inhibition of mTORC2 and activation of AMPK is a potential strategy for suppressing both cell survival and proliferation. CONCLUSIONS: Taken together, model simulations clarify the competing effects and the roles of key metabolic signalling pathways in tumorigenesis, which may yield insights on innovative therapeutic strategies.

Hormonal regulation of ovarian follicle growth in humans: Model-based exploration of cycle variability and parameter sensitivities.

We present a modelling and simulation framework for the dynamics of ovarian follicles and key hormones along the hypothalamic-pituitary-gonadal axis throughout consecutive human menstrual cycles. All simulation results (hormone concentrations and ovarian follicle sizes) are in biological units and can easily be compared to clinical data. The model takes into account variability in follicles' response to stimulating hormones, which introduces variability between cycles. The growth of ovarian follicles in waves is an emergent property in our model simulations and further supports the hypothesis that follicular waves are also present in humans. We use Approximate Bayesian Computation and cluster analysis to construct a population of virtual subjects and to study parameter distributions and sensitivities. The model can be used to compare and optimize treatment protocols for ovarian hyperstimulation, thus potentially forming the integral part of a clinical decision support system in reproductive endocrinology.