Yue-Xian Li

Inferring individual rules from collective behavior

Social organisms form striking aggregation patterns, displaying cohesion, polarization, and collective intelligence. Determining how they do so in nature is challenging; a plethora of simulation studies displaying life-like swarm behavior lack rigorous comparison with actual data because collecting field data of sufficient quality has been a bottleneck. Here, we bridge this gap by gathering and analyzing a high-quality dataset of flocking surf scoters, forming well-spaced groups of hundreds of individuals on the water surface. By reconstructing each individuals position, velocity, and trajectory, we generate spatial and angular neighbor-distribution plots, revealing distinct concentric structure in positioning, a preference for neighbors directly in front, and strong alignment with neighbors on each side. We fit data to zonal interaction models and characterize which individual interaction forces suffice to explain observed spatial patterns. Results point to strong short-range repulsion, intermediate-range alignment, and longer-range attraction (with circular zones), as well as a weak but significant frontal-sector interaction with one neighbor. A best-fit model with such interactions accounts well for observed group structure, whereas absence or alteration in any one of these rules fails to do so. We find that important features of observed flocking surf scoters can be accounted for by zonal models with specific, well-defined rules of interaction.

Frequency specificity in intercellular communication. Influence of patterns of periodic signaling on target cell responsiveness.

Cells often communicate by means of periodic signals, as exemplified by a large number of hormones and by the aggregation of Dictyostelium discoideum amebas in response to periodic pulses of cyclic AMP. Periodic signaling allows bypassing the phenomenon of desensitization brought about by constant stimuli. To gain further insight into the efficiency of pulsatile signaling, we analyze the effect of periodic stimulation on the dynamic behavior of a receptor system capable of desensitization toward its ligand. We first show that the receptor system adapts to square-wave stimuli, i.e., the response eventually reaches a steady, periodic pattern after a transient phase. By analyzing the dependence of the response on the characteristics of the square-wave stimulation, we show that there exist a waveform and a period of that signal that result in maximum responsiveness of the target system. Similar results are obtained when the signal takes the more realistic form of a periodically repeated stimulation followed by exponential decay of the ligand. The results are discussed with respect to the role of pulsatile secretion of gonadotropin-releasing hormone (GnRH) by the hypothalamus and of periodic signaling by cyclic AMP pulses in Dictyostelium. The analysis accounts for the existence, in both cases, of an optimal frequency and waveform of the periodic stimulus that correspond to maximum target cell responsiveness.

Pulsatile signaling in intercellular communication periodic stimuli are more efficient than random or chaotic signals in a model based on receptor desensitization

The efficiency of various patterns of pulsatile stimulation is determined in a model in which a receptor becomes desensitized in the presence of its stimulatory ligand. The effect of stochastic or chaotic changes in the duration and/or interval between successive pulses in a series of square-wave stimuli is investigated. Before addressing the effect of random variations in the pulsatile signal, we first extend the results of a previous analysis (Li, Y.X., and A. Goldbeter. 1989. Biophys. J. 55:125-145) by demonstrating the existence of an optimal periodic signal that maximizes target cell responsiveness whatever the magnitude of stimulation. As to the effect of stochastic or chaotic variations in the pulsatile stimulus, three kinds of random distributions are used, namely, a Gaussian and a white-noise distribution, and a chaotic time series generated by the logistic map. All these random distributions are symmetrically centered around the reference value of the duration or interval that characterizes the optimal periodic stimulus yielding maximal responsiveness in target cells. Stochastically or chaotically varying pulses are less effective than the periodic signal that corresponds to the optimal pattern of pulsatile stimulation. The response of the receptor system is most sensitive to changes in the time interval that separates successive stimuli. Similar conclusions hold when stochastic or chaotic signals are compared to a reference periodic stimulus differing from the optimal one, although the effect of random variations is then reduced. The decreased efficiency of stochastic pulses accounts for the observed superiority of periodic versus stochastic pulses of cyclic AMP (cAMP) in Dictyostelium amoebae. The results are also discussed with respect to the efficiency of periodic versus stochastic or chaotic patterns of hormone secretion.

A Conceptual Model for Milling Formations in Biological Aggregates

Collective behavior of swarms and flocks has been studied from several perspectives, including continuous (Eulerian) and individual-based (Lagrangian) models. Here, we use the latter approach to examine a minimal model for the formation and maintenance of group structure, with specific emphasis on a simple milling pattern in which particles follow one another around a closed circular path.We explore how rules and interactions at the level of the individuals lead to this pattern at the level of the group. In contrast to many studies based on simulation results, our model is sufficiently simple that we can obtain analytical predictions. We consider a Newtonian framework with distance-dependent pairwise interaction-force. Unlike some other studies, our mill formations do not depend on domain boundaries, nor on centrally attracting force-fields or rotor chemotaxis.By focusing on a simple geometry and simple distant-dependent interactions, we characterize mill formations and derive existence conditions in terms of model parameters. An eigenvalue equation specifies stability regions based on properties of the interaction function. We explore this equation numerically, and validate the stability conclusions via simulation, showing distinct behavior inside, outside, and on the boundary of stability regions. Moving mill formations are then investigated, showing the effect of individual autonomous self-propulsion on group-level motion. The simplified framework allows us to clearly relate individual properties (via model parameters) to group-level structure. These relationships provide insight into the more complicated milling formations observed in nature, and suggest design properties of artificial schools where such rotational motion is desired.

Tango waves in a bidomain model of fertilization calcium waves

Abstract Fertilization of an egg cell is marked by one or several Ca 2+ waves that travel across the intra-cellular space, called fertilization Ca 2+ waves. Patterns of Ca 2+ waves observed in mature or immature oocytes include traveling fronts and pulses as well as concentric and spiral waves. These patterns have been studied in other excitable media in physical, chemical, and biological systems. Here, we report the discovery of a new wave phenomenon in the numerical study of a bidomain model of fertilization Ca 2+ waves. This wave is a front that propagates in a back-and-forth manner that resembles the movement of tango dancers, thus is called a tango wave. When the medium is excitable, a forward-moving tango wave can generate traveling pulses that propagate down the space without reversal. The study shows that the occurrence of tango waves is related to spatial inhomogeneity in the local dynamics. This is tested and confirmed by simulating similar waves in a medium with stationary spatial inhomogeneity. Similar waves are also obtained in a FitzHugh–Nagumo system with a linear spatial ramp. In both the bidomain model of Ca 2+ waves and the FitzHugh–Nagumo system, the front is stable when the slope of a linear ramp is large. As the slope decreases beyond a critical value, front oscillations occur. The study shows that tango waves facilitate the dispersion of localized Ca 2+ . Key features of the bidomain model underlying the occurrence of tango waves are revealed. These features are commonly found in egg cells of a variety of species. Thus, we predict that tango waves can occur in real egg cells provided that a slowly varying inhomogeneity does occur following the sperm entry. The observation of tango wave-like waves in nemertean worm and ascidian eggs seems to support such a prediction.

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.

An Integrated Model of Electrical Spiking, Bursting, and Calcium Oscillations in GnRH Neurons

The plasma membrane electrical activities of neurons that secrete gonadotropin-releasing hormone (GnRH) have been studied extensively. A couple of mathematical models have been developed previously to explain different aspects of these activities. The goal of this article is to develop a single model that accounts for the previously modeled experimental results and some more recent results that have not been accounted for. The latter includes two types of membrane potential bursting mechanisms and their associated cytosolic calcium oscillations. One bursting mechanism has not been reported in experiments and is thus regarded as a model prediction. Although the model is mainly based on data collected in immortalized GnRH cell lines, it is capable of explaining some properties of GnRH neurons observed in several other preparations including mature GnRH neurons in hypothalamic slices. We present a spatial model that incorporates a detailed description of calcium dynamics in a three-dimensional cell body with the ion channels evenly distributed on the cell surface. A phenomenological reduction of the spatial model into a simplified form is also presented. The simplified model will facilitate the study of the roles of plasma membrane electrical activities in the pulsatile release of GnRH.

Stability of front solutions in inhomogeneous media

Abstract We use the FitzHugh–Nagumo (F–N) model to study front solutions in inhomogeneous media. Inhomogeneity is modeled by adding a space-dependent term, z(x), in the equations. Changes in z shift the cubic nullcline up and down in the phase-plane of the F–N model. This is used to mimic shifts in the cubic-shaped nullcline of a bidomain model of calcium waves in egg cells [Physica D, in press] caused by changes in the total calcium concentration. Conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear. We show that spatial inhomogeneities cause pinning and oscillations of front solutions. This is best demonstrated when z(x) is a linear ramp. When the slope is large, a linear ramp breaks the translational invariance of the stationary front and stabilizes the front. The front becomes less stable as the slope decreases. At a critical slope, the front becomes unstable through a Hopf bifurcation beyond which oscillations in the front occur. This also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness. This bifurcation has been found previously [Physica D 106 (1997) 270; Phys. Rev. E 55 (1997) 366] in a local analysis of the F–N model near a co-dimension two point (sometimes called a non-equilibrium Ising–Bloch (NIB) bifurcation) and with inhomogeneities of small magnitude. The analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude. We show that this bifurcation need not occur in conjunction with the NIB point. It is caused exclusively by spatial inhomogeneity. A rich variety of wave phenomena including front oscillation, front pinning, and front reflection are shown to occur in inhomogeneous media.

Robust Synchrony and Rhythmogenesis in Endocrine Neurons via Autocrine Regulations In Vitro and In Vivo

Episodic pulses of gonadotropin-releasing hormone (GnRH) are essential for maintaining reproductive functions in mammals. An explanation for the origin of this rhythm remains an ultimate goal for researchers in this field. Some plausible mechanisms have been proposed among which the autocrine-regulation mechanism has been implicated by numerous experiments. GnRH binding to its receptors in cultured GnRH neurons activates three types of G-proteins that selectively promote or inhibit GnRH secretion (Krsmanovic et al. in Proc. Natl. Acad. Sci. 100:2969–974, 2003). This mechanism appears to be consistent with most data collected so far from both in vitro and in vivo experiments. Based on this mechanism, a mathematical model has been developed (Khadra and Li in Biophys. J. 91:74–3, 2006) in which GnRH in the extracellular space plays the roles of a feedback regulator and a synchronizing agent. In the present study, we show that synchrony between different neurons through sharing a common pool of GnRH is extremely robust. In a diversely heterogeneous population of neurons, the pulsatile rhythm is often maintained when only a small fraction of the neurons are active oscillators (AOs). These AOs are capable of recruiting nonoscillatory neurons into a group of recruited oscillators while forcing the nonrecruitable neurons to oscillate along. By pointing out the existence of the key elements of this model in vivo, we predict that the same mechanism revealed by experiments in vitro may also operate in vivo. This model provides one plausible explanation for the apparently controversial conclusions based on experiments on the effects of the ultra-short feedback loop of GnRH on its own release in vivo.

Finding complex oscillatory phenomena in biochemical systems An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.