A universal description of stochastic oscillators.

Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function [Formula: see text](x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function [Formula: see text] (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ1 = µ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width µ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

COVID-19 and silent hypoxemia in a minimal closed-loop model of the respiratory rhythm generator.

Silent hypoxemia, or "happy hypoxia," is a puzzling phenomenon in which patients who have contracted COVID-19 exhibit very low oxygen saturation ( SaO 2 < 80%) but do not experience discomfort in breathing. The mechanism by which this blunted response to hypoxia occurs is unknown. We have previously shown that a computational model of the respiratory neural network (Diekman et al. in J Neurophysiol 118(4):2194-2215, 2017) can be used to test hypotheses focused on changes in chemosensory inputs to the central pattern generator (CPG). We hypothesize that altered chemosensory function at the level of the carotid bodies and/or the nucleus tractus solitarii are responsible for the blunted response to hypoxia. Here, we use our model to explore this hypothesis by altering the properties of the gain function representing oxygen sensing inputs to the CPG. We then vary other parameters in the model and show that oxygen carrying capacity is the most salient factor for producing silent hypoxemia. We call for clinicians to measure hematocrit as a clinical index of altered physiology in response to COVID-19 infection.

The role of lung ultrasound for detecting atelectasis, consolidation, and/or pneumonia in the adult cardiac surgery population: A scoping review of the literature.

OBJECTIVES: Postoperative pulmonary complications (PPCs) frequently occur after cardiac surgery and may lead to adverse patient outcomes. Traditional diagnostic tools such as auscultation or chest x-ray have inferior diagnostic accuracy compared to the gold standard (chest computed tomography). Lung ultrasound (LUS) is an emerging area of research combating these issues. However, no review has employed a formal search strategy to examine the role of LUS in identifying the specific PPCs of atelectasis, consolidation, and/or pneumonia or investigated the ability of LUS to predict these complications in this cohort. The objective of this study was to collate and present evidence for the use of LUS in the adult cardiac surgery population to specifically identify atelectasis, consolidation, and/or pneumonia. REVIEW METHOD USED: A scoping review of the literature was completed using predefined search terms across six databases which identified 1432 articles. One additional article was included from reviewing reference lists. Six articles met the inclusion criteria, providing sufficient data for the final analysis. DATA SOURCES: Six databases were searched: MEDLINE, Embase, CINAHL, Scopus, CENTRAL, and PEDro. This review was not registered. REVIEW METHODS: The review followed the PRISMA Extension for Scoping Reviews. RESULTS: Several LUS methodologies were reported across studies. Overall, LUS outperformed all other included bedside diagnostic tools, with superior diagnostic accuracy in identifying atelectasis, consolidation, and/or pneumonia. Incidences of PPCs tended to increase with each subsequent timepoint after surgery and were better identified with LUS than all other assessments. A change in diagnosis occurred at a rate of 67% with the inclusion of LUS and transthoracic echocardiography in one study. Pre-established assessment scores were improved by substituting chest x-rays with LUS scans. CONCLUSION: The results of this scoping review support the use of LUS as a diagnostic tool after cardiac surgery; however, they also highlighted a lack of consistent methodologies used. Future research is required to determine the optimal methodology for LUS in diagnosing PPCs in this cohort and to determine whether LUS possesses the ability to predict these complications and guide proactive respiratory supports after extubation.

Sensitivity to Control Signals in Triphasic Rhythmic Neural Systems: A Comparative Mechanistic Analysis via Infinitesimal Local Timing Response Curves.

Similar activity patterns may arise from model neural networks with distinct coupling properties and individual unit dynamics. These similar patterns may, however, respond differently to parameter variations and specifically to tuning of inputs that represent control signals. In this work, we analyze the responses resulting from modulation of a localized input in each of three classes of model neural networks that have been recognized in the literature for their capacity to produce robust three-phase rhythms: coupled fast-slow oscillators, near-heteroclinic oscillators, and threshold-linear networks. Triphasic rhythms, in which each phase consists of a prolonged activation of a corresponding subgroup of neurons followed by a fast transition to another phase, represent a fundamental activity pattern observed across a range of central pattern generators underlying behaviors critical to survival, including respiration, locomotion, and feeding. To perform our analysis, we extend the recently developed local timing response curve (lTRC), which allows us to characterize the timing effects due to perturbations, and we complement our lTRC approach with model-specific dynamical systems analysis. Interestingly, we observe disparate effects of similar perturbations across distinct model classes. Thus, this work provides an analytical framework for studying control of oscillations in nonlinear dynamical systems and may help guide model selection in future efforts to study systems exhibiting triphasic rhythmic activity.

Cheater suppression and stochastic clearance through quorum sensing.

The evolutionary consequences of quorum sensing in regulating bacterial cooperation are not fully understood. In this study, we reveal unexpected effects of regulating public good production through quorum sensing on bacterial population dynamics, showing that quorum sensing can be a collectively harmful alternative to unregulated production. We analyze a birth-death model of bacterial population dynamics accounting for public good production and the presence of non-producing cheaters. Our model demonstrates that when demographic noise is a factor, the consequences of controlling public good production according to quorum sensing depend on the cost of public good production and the growth rate of populations in the absence of public goods. When public good production is inexpensive, quorum sensing is a destructive alternative to unconditional production, in terms of the mean population extinction time. When costs are higher, quorum sensing becomes a constructive strategy for the producing strain, both stabilizing cooperation and decreasing the risk of population extinction.

Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity.

May and Leonard (SIAM J Appl Math 29:243-253, 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic Escherichia coli), in the immune system, in neural information processing models ("winnerless competition"), and in models of neural central pattern generators. Yet as May and Leonard observed "Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two." Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard's ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.

Inferring density-dependent population dynamics mechanisms through rate disambiguation for logistic birth-death processes.

Density dependence is important in the ecology and evolution of microbial and cancer cells. Typically, we can only measure net growth rates, but the underlying density-dependent mechanisms that give rise to the observed dynamics can manifest in birth processes, death processes, or both. Therefore, we utilize the mean and variance of cell number fluctuations to separately identify birth and death rates from time series that follow stochastic birth-death processes with logistic growth. Our nonparametric method provides a novel perspective on stochastic parameter identifiability, which we validate by analyzing the accuracy in terms of the discretization bin size. We apply our method to the scenario where a homogeneous cell population goes through three stages: (1) grows naturally to its carrying capacity, (2) is treated with a drug that reduces its carrying capacity, and (3) overcomes the drug effect to restore its original carrying capacity. In each stage, we disambiguate whether the dynamics occur through the birth process, death process, or some combination of the two, which contributes to understanding drug resistance mechanisms. In the case of limited sample sizes, we provide an alternative method based on maximum likelihood and solve a constrained nonlinear optimization problem to identify the most likely density dependence parameter for a given cell number time series. Our methods can be applied to other biological systems at different scales to disambiguate density-dependent mechanisms underlying the same net growth rate.

Quantitative comparison of the mean-return-time phase and the stochastic asymptotic phase for noisy oscillators.

Seminal work by A. Winfree and J. Guckenheimer showed that a deterministic phase variable can be defined either in terms of Poincaré sections or in terms of the asymptotic (long-time) behaviour of trajectories approaching a stable limit cycle. However, this equivalence between the deterministic notions of phase is broken in the presence of noise. Different notions of phase reduction for a stochastic oscillator can be defined either in terms of mean-return-time sections or as the argument of the slowest decaying complex eigenfunction of the Kolmogorov backwards operator. Although both notions of phase enjoy a solid theoretical foundation, their relationship remains unexplored. Here, we quantitatively compare both notions of stochastic phase. We derive an expression relating both notions of phase and use it to discuss differences (and similarities) between both definitions of stochastic phase for (i) a spiral sink motivated by stochastic models for electroencephalograms, (ii) noisy limit-cycle systems-neuroscience models, and (iii) a stochastic heteroclinic oscillator inspired by a simple motor-control system.

Mean-return-time phase of a stochastic oscillator provides an approximate renewal description for the associated point process.

Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer-Schwabedal-Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin-Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.

Variational and phase response analysis for limit cycles with hard boundaries, with applications to neuromechanical control problems.

Motor systems show an overall robustness, but because they are highly nonlinear, understanding how they achieve robustness is difficult. In many rhythmic systems, robustness against perturbations involves response of both the shape and the timing of the trajectory. This makes the study of robustness even more challenging. To understand how a motor system produces robust behaviors in a variable environment, we consider a neuromechanical model of motor patterns in the feeding apparatus of the marine mollusk Aplysia californica (Shaw et al. in J Comput Neurosci 38(1):25-51, 2015; Lyttle et al. in Biol Cybern 111(1):25-47, 2017). We established in (Wang et al. in SIAM J Appl Dyn Syst 20(2):701-744, 2021. https://doi.org/10.1137/20M1344974 ) the tools for studying combined shape and timing responses of limit cycle systems under sustained perturbations and here apply them to study robustness of the neuromechanical model against increased mechanical load during swallowing. Interestingly, we discover that nonlinear biomechanical properties confer resilience by immediately increasing resistance to applied loads. In contrast, the effect of changed sensory feedback signal is significantly delayed by the firing rates' hard boundary properties. Our analysis suggests that sensory feedback contributes to robustness in swallowing primarily by shifting the timing of neural activation involved in the power stroke of the motor cycle (retraction). This effect enables the system to generate stronger retractor muscle forces to compensate for the increased load, and hence achieve strong robustness. The approaches that we are applying to understanding a neuromechanical model in Aplysia, and the results that we have obtained, are likely to provide insights into the function of other motor systems that encounter changing mechanical loads and hard boundaries, both due to mechanical and neuronal firing properties.

A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves.

Homeostasis occurs in a control system when a quantity remains approximately constant as a parameter, representing an external perturbation, varies over some range. Golubitsky and Stewart (J Math Biol 74(1-2):387-407, 2017) developed a notion of infinitesimal homeostasis for equilibrium systems using singularity theory. Rhythmic physiological systems (breathing, locomotion, feeding) maintain homeostasis through control of large-amplitude limit cycles rather than equilibrium points. Here we take an initial step to study (infinitesimal) homeostasis for limit-cycle systems in terms of the average of a quantity taken around the limit cycle. We apply the "infinitesimal shape response curve" (iSRC) introduced by Wang et al. (SIAM J Appl Dyn Syst 82(7):1-43, 2021) to study infinitesimal homeostasis for limit-cycle systems in terms of the mean value of a quantity of interest, averaged around the limit cycle. Using the iSRC, which captures the linearized shape displacement of an oscillator upon a static perturbation, we provide a formula for the derivative of the averaged quantity with respect to the control parameter. Our expression allows one to identify homeostasis points for limit cycle systems in the averaging sense. We demonstrate in the Hodgkin-Huxley model and in a metabolic regulatory network model that the iSRC-based method provides an accurate representation of the sensitivity of averaged quantities.

Subjective Information and Survival in a Simulated Biological System.

Information transmission and storage have gained traction as unifying concepts to characterize biological systems and their chances of survival and evolution at multiple scales. Despite the potential for an information-based mathematical framework to offer new insights into life processes and ways to interact with and control them, the main legacy is that of Shannon's, where a purely syntactic characterization of information scores systems on the basis of their maximum information efficiency. The latter metrics seem not entirely suitable for biological systems, where transmission and storage of different pieces of information (carrying different semantics) can result in different chances of survival. Based on an abstract mathematical model able to capture the parameters and behaviors of a population of single-celled organisms whose survival is correlated to information retrieval from the environment, this paper explores the aforementioned disconnect between classical information theory and biology. In this paper, we present a model, specified as a computational state machine, which is then utilized in a simulation framework constructed specifically to reveal emergence of a "subjective information", i.e., trade-off between a living system's capability to maximize the acquisition of information from the environment, and the maximization of its growth and survival over time. Simulations clearly show that a strategy that maximizes information efficiency results in a lower growth rate with respect to the strategy that gains less information but contains a higher meaning for survival.

Isostables for Stochastic Oscillators.

Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.

Dynamical consequences of sensory feedback in a half-center oscillator coupled to a simple motor system.

We investigate a simple model for motor pattern generation that combines central pattern generator (CPG) dynamics with a sensory feedback (FB) mechanism. Our CPG comprises a half-center oscillator with conductance-based Morris-Lecar model neurons. Output from the CPG drives a push-pull motor system with biomechanics based on experimental data. A sensory feedback conductance from the muscles allows modulation of the CPG activity. We consider parameters under which the isolated CPG system has either "escape" or "release" dynamics, and we study both inhibitory and excitatory feedback conductances. We find that increasing the FB conductance relative to the CPG conductance makes the system more robust against external perturbations, but more susceptible to internal noise. Conversely, increasing the CPG conductance relative to the FB conductance has the opposite effects. We find that the "closed-loop" system, with sensory feedback in place, exhibits a richer repertoire of behaviors than the "open-loop" system, with motion determined entirely by the CPG dynamics. Moreover, we find that purely feedback-driven motor patterns, analogous to a chain reflex, occur only in the inhibition-mediated system. Finally, for pattern generation systems with inhibition-mediated sensory feedback, we find that the distinction between escape- and release-mediated CPG mechanisms is diminished in the presence of internal noise. Our observations support an anti-reductionist view of neuromotor physiology: Understanding mechanisms of robust motor control requires studying not only the central pattern generator circuit in isolation, but the intact closed-loop system as a whole.

Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding.

Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM J Appl Math 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in Phys Rev Lett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."

Shape versus timing: linear responses of a limit cycle with hard boundaries under instantaneous and static perturbation.

When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface. Examples include heel-ground interaction in locomotion, firing rate rectification in neural networks, and stick-slip oscillators. In many rhythmic systems, robustness against external perturbations involves response of both the shape and the timing of the limit cycle trajectory. The existing methods of infinitesimal phase response curve (iPRC) and variational analysis are well established for quantifying changes in timing and shape, respectively, for smooth systems. These tools have recently been extended to nonsmooth dynamics with transversal crossing boundaries. In this work, we further extend the iPRC method to nonsmooth systems with sliding components, which enables us to make predictions about the synchronization properties of weakly coupled stick-slip oscillators. We observe a new feature of the isochrons in a planar limit cycle with hard sliding boundaries: a nonsmooth kink in the asymptotic phase function, originating from the point at which the limit cycle smoothly departs the constraint surface, and propagating away from the hard boundary into the interior of the domain. Moreover, the classical variational analysis neglects timing information and is restricted to instantaneous perturbations. By defining the "infinitesimal shape response curve" (iSRC), we incorporate timing sensitivity of an oscillator to describe the shape response of this oscillator to parametric perturbations. In order to extract timing information, we also develop a "local timing response curve" (lTRC) that measures the timing sensitivity of a limit cycle within any given region. We demonstrate in a specific example that taking into account local timing sensitivity in a nonsmooth system greatly improves the accuracy of the iSRC over global timing analysis given by the iPRC.

Holey-structured tungsten metamaterials for broadband ultrasonic sub-wavelength imaging in water.

Metamaterials exhibiting Fabry-Pérot resonances are shown to achieve ultrasonic imaging of a sub-wavelength aperture in water immersion across a broad bandwidth. Holey-structured metamaterials of different thickness were additively manufactured using a tungsten substrate and selective laser melting, tungsten being chosen so as to create a significant acoustic impedance mismatch with water. Both broadband metamaterial behavior and sub-wavelength imaging in water are demonstrated experimentally and validated with finite element simulations over the 200-300 kHz range.

Structural Design and Physical Mechanism of Axial and Radial Sandwich Resonators with Piezoelectric Ceramics: A Review.

Piezoelectric ceramics are inexpensive functional materials which are widely used in sonar detection, home appliances, meteorological detection, telemetry and environmental protection and other applications. Sensors fabricated from these materials are compact and have fast response characteristics. Their underlying functional methodology is based on the direct piezoelectric effect whereby very small mechanical vibration signals are converted into electrical signals. Piezoelectric resonators are based on the reverse piezoelectric effect and they are widely used for the control of precision instruments and precision machinery, microelectronic components, bioengineering devices and other in applications requiring components to provide precision control of the relevant functional mechanism. In this paper, the structural evolution and design mechanism of sandwich resonators based on piezoelectric materials are reviewed, and the advantages and disadvantages of different structures are compared and analyzed. The goal is to provide a comprehensive reference for the selection, application and promotion of piezoelectric resonators and for future structural innovation and mechanism research relevant to sandwich resonators.

Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics.

Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor D and suggested a Langevin formulation with noise coefficient matrix S such that SS⊤=D. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this article, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 × 28 noise coefficient matrix S. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix S that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the L1 and L∞ metric-space norms; and (5) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies S with minimal loss of accuracy under both voltage- and current-clamp conditions.

9-Acridinemethanamine and Acridine-9-Carboxaldehyde as Potential Fluorescence Lifetime pH Indicators.

A significant challenge concerning the development of fluorescence lifetime (FL) based pH sensors is the paucity of fluorophores with sufficiently large FL variation with pH. Acridine is amongst the indicators with highest fluoresce lifetime responses to pH, with a change in lifetime of about 13 ns within a pH range of 5-8. Here we examine the two acridine derivatives, 9-acridinemethanamine (9-AMA) and acridine-9-carbaldehyde (9-ACA) in terms of their FL pH sensitivity and pH sensing range. Both indicators are characterized when dissolved in buffer solutions, as well as when immobilized in support materials. 9-AMA has a change in FL of 11 ns between pH 2-5, both when dissolved in solution and when immobilized in surfactant-filled mesoporous silica. The FL of 9-ACA is not sensitive to pH when dissolved in buffer solutions; however, when covalently bound to amine-modified silica, its FL changes 15 ns between pH 3-6. 9-AMA and 9-ACA represent promising FL in the pH range of pH 2-6, and could potentially form the basis of new FL pH sensors. Graphical Abstract.