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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 wellspaced groups of hundreds of individuals on the water surface. By reconstructing each individual's 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, intermediaterange 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.

Cultured gonadotropin-releasing hormone (GnRH) neurons have been shown to express GnRH receptors. GnRH binding to its receptors activates three types of G-proteins at increasing doses. These G-proteins selectively activate or inhibit GnRH secretion by regulating the intracellular levels of Ca 2+ and cAMP. Based on these recent observations, we build a model in which GnRH plays the roles of a feedback regulator and a diffusible synchronizing agent. We show that this GnRH-regulated GnRH-release mechanism is sufficient for generating pulsatile GnRH release. The model reproduces the observed effects of some key drugs that disturb the GnRH pulse generator in specific ways. Simulations of 100 heterogeneous neurons revealed that the synchronization mediated by a common pool of diffusible GnRH is robust. The population can generate synchronized pulsatile signals even when all the individual GnRH neurons oscillate at different amplitudes and peak at different times. These results suggest that the positive and negative effects of the autocrine regulation by GnRH on GnRH neurons are sufficient and robust in generating GnRH pulses.

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.

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.

The timed secretion of the luteinizing hormone (LH) and follicle stimulating hormone (FSH) from pituitary gonadotrophs during the estrous cycle is crucial for normal reproductive functioning. The release of LH and FSH is stimulated by gonadotropin releasing hormone (GnRH) secreted by hypothalamic GnRH neurons. It is controlled by the frequency of the GnRH signal that varies during the estrous cycle. Curiously, the secretion of LH and FSH is differentially regulated by the frequency of GnRH pulses. LH secretion increases as the frequency increases within a physiological range, and FSH secretion shows a biphasic response, with a peak at a lower frequency. There is considerable experimental evidence that one key factor in these differential responses is the autocrine/paracrine actions of the pituitary polypeptides activin and follistatin. Based on these data, we develop a mathematical model that incorporates the dynamics of these polypeptides. We show that a model that incorporates the actions of activin and follistatin is sufficient to generate the differential responses of LH and FSH secretion to changes in the frequency of GnRH pulses. In addition, it shows that the actions of these polypeptides, along with the ovarian polypeptide inhibin and the estrogen-mediated variations in the frequency of GnRH pulses, are sufficient to account for the time courses of LH and FSH plasma levels during the rat estrous cycle. That is, a single peak of LH on the afternoon of proestrus and a double peak of FSH on proestrus and early estrus. We also use the model to identify which regulation pathways are indispensable for the differential regulation of LH and FSH and their time courses during the estrous cycle. We conclude that the actions of activin, inhibin, and follistatin are consistent with LH/FSH secretion patterns, and likely complement other factors in the production of the characteristic secretion patterns in female rats.

We have developed a mathematical model that describes several aspects of agonist-induced Ca2+signaling in single pituitary gonadotrophs. Our model is based on fast activation of the inositol 1,4,5-trisphosphate (InsP3) receptor Ca2+channels at low free cytosolic Ca2+concentration [Ca2+]i) and slow inactivation at high [Ca2+]i. Previous work has shown that these gating properties, when combined with a Ca2+-ATPase, are sufficient to generate simulated Ca2+oscillations. The Hodgkin-Huxley-like description we formulate here incorporates these different gating properties explicitly and renders their effects transparent and easy to modulate. We introduce regulatory mechanisms of channel opening which enable the model, both in the absence and in the presence of Ca2+entry, to give responses to a wide range of agonist doses that are in good agreement with experimental findings, including subthreshold responses, superthreshold oscillations with frequency determined by [InsP3], and nonoscillatory \"biphasic\" responses followed occasionally by small-amplitude oscillations. A particular added feature of our model, enhanced channel opening by reduced concentration of Ca2+in the lumen of the endoplasmic reticulum, allows oscillations to continue during pool depletion. The model predicts that ionomycin and thapsigargin can induce oscillations with basal [InsP3] and zero Ca2+entry, while Ca2+injection cannot. Responses to specific pairings of sub- or superthreshold stimuli of agonist, ionomycin, and thapsigargin are also correctly predicted. Since this model encompasses a wide range of observed dynamic behaviors within a single framework, based on well-established mechanisms, its relevance should not be restricted to gonadotrophs.

Recent experimental and theoretical studies have found that active dendritic ionic currents can compensate for the effects of electrotonic attenuation. In particular, temporal summation, the percentage increase in peak somatic voltage responses invoked by a synaptic input train, is independent of location of the synaptic input in hippocampal CA1 pyramidal neurons under normal conditions. This independence, known as normalization of temporal summation, is destroyed when the hyperpolarization-activated current, Ih, is blocked [Magee JC (1999a), Nature Neurosci. 2: 508-514]. Using a compartmental model derived from morphological recordings of hippocampal CA1 pyramidal neurons, we examined the hypothesis that Ih was primarily responsible for normalization of temporal summation. We concluded that this hypothesis was incomplete. With a model that included Ih, the persistent Na(+) current (INaP), and the transient A-type K+ current (IA), however, we observed normalization of temporal summation across a wide range of synaptic input frequencies, in keeping with experimental observations.

Excitatory coupling with a slow rise time destabilizes synchrony between coupled neurons. Thus, the fully synchronous state is usually unstable in networks of excitatory neurons. Phase-clustered states, in which neurons are divided into multiple synchronized clusters, have also been found unstable in numerical studies of excitatory networks in the presence of noise. The question arises as to whether synchrony is possible in networks of neurons coupled through slow, excitatory synapses. In this paper, we show that robust, synchronous clustered states can occur in such networks. The effects of non-uniform distributions of coupling strengths are explored. Conditions for the existence and stability of clustered states are derived analytically. The analysis shows that a multi-cluster state can be stable in excitatory networks if the overall interactions between neurons in different clusters are stabilizing and strong enough to counter-act the destabilizing interactions between neurons within each cluster. When heterogeneity in the coupling strengths strengthens the stabilizing inter-cluster interactions and/or weakens the destabilizing in-cluster interactions, robust clustered states can occur in excitatory networks of all known model neurons. Numerical simulations were carried out to support the analytical results.

The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons.

Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to do so. Observed and explored in numerous experimental and theoretical studies, STR is a complex dynamic phenomenon depending on the nature of external inputs as well as intrinsic properties of a neuron. The neuron under consideration could be either quiescent or spontaneously spiking in the absence of the external stimulus. Focusing on the situation in which the unstimulated neuron is quiescent but close to a switching point to oscillations, we numerically analyze STR treating each spike occurrence as a time localized event in a model neuron. We study both the averaged properties as well as individual features of spike-evoking epochs (SEEs). The effects of interactions between spikes is minimized by selecting signals that generate spikes with relatively long interspike intervals (ISIs). Under these conditions, the frequency content of the input signal has little impact on STR. We study two distinct cases, Type I in which the f – I relation ( f for frequency, I for applied current) is continuous and Type II where the f – I relation exhibits a jump. STR in the two types shows a number of similar features and differ in some others. SEEs that are capable of triggering spikes show great variety in amplitude and time profile. On average, reliable spike timing is associated with an accelerated increase in the “action” of the signal as a threshold for spike generation is approached. Here, “action” is defined as the average amount of current delivered during a fixed time interval. When individual SEEs are studied, however, their time profiles are found important for triggering more precisely timed spikes. The SEEs that have a more favorable time profile are capable of triggering spikes with higher precision even at lower action levels.