Pollen coupling of forest trees: forming synchronized and periodic reproduction out of chaos

Many of the tree species in mature forests show masting; their reproductive activity has a large variance between years and is often synchronized between different individuals. In this paper, we analyse a globally coupled map model in which trees accumulate photosynthate every year, produce flowers when the energy reserve level exceeds a threshold, and set seeds and fruits at a rate limited by pollen availability. Without pollen limitation, the trees in the forest show independent chaotic fluctuation. Coupling of trees via pollen exchange results in reproduction being synchronized partially or completely over the forest. The whole forest shows diverse dynamical behaviors determined by the values of two essential parameters; the depletion coefficient k and the coupling strength beta. We find perfectly synchronized periodic reproduction, synchronized reproduction with a chaotic time series, clustering phenomena, and chaotic reproduction of trees without synchronization over individuals. There are many parameter windows in which synchronized reproduction of trees shows a stable periodic fluctuation. For perfectly synchronized forests, we can calculate all the Lyapunov exponents analytically. They show that synchronized reproduction of all the trees in the forest can only occur when trees flower at low (but positive) levels in a significant fraction of years, resulting in small fruit sets due to outcrossed pollen limitation. This is consistent with the observation that the distinction between mast years and non-mast years is often not clear cut.     

Dispersion and time delay effects in synchronized spike–burst networks

We study spike–burst neural activity and investigate its transitions to synchronized states under electrical coupling. Our reported results include the following: (1) Synchronization of spike–burst activity is a multi-time scale phenomenon and burst synchrony is easier to achieve than spike synchrony. (2) Synchrony of networks with time-delayed connections can be achieved at lower coupling strengths than within the same network with instantaneous couplings. (3) The introduction of parameter dispersion into the network destroys the existence of synchrony in the strict sense, but the network dynamics in major regimes of the parameter space can still be effectively captured by a mean field approach if the couplings are excitatory. Our results on synchronization of spiking networks are general of nature and will aid in the development of minimal models of neuronal populations. The latter are the building blocks of large scale brain networks relevant for cognitive processing.     

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.     

Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.     

Noise-Stabilized Long-Distance Synchronization in Populations of Model Neurons

Rhythmic, synchronous firing of groups of neurons is associated with behaviorally relevant states, and it is thus of interest to understand the mechanisms by which synchronization may be achieved. In hippocampal slice preparations, networks of excitatory and inhibitory neurons have been seen to synchronize when strong stimulation is applied at separated sites between which any coupling must be subject to a significant axonal delay. We extend previous work on synchronization in a model system based on the network architecture of these hippocampal slices. Our new analysis addresses the effects of heterogeneous populations and noisy inputs on the stability of synchronous solutions in the system. We find that, with experimentally motivated constraints on the coupling strength, sufficiently large heterogeneity in the input currents renders synchrony unstable. The addition of noise, however, restores stable near-synchrony. We analytically reduce the high-dimensional biophysical equations for the full population to a simple three-dimensional map, and show that the map's stability properties correctly predict both the loss of stability and the restabilizing effect of the noise.     

Potential Mechanisms for Imperfect Synchronization in Parkinsonian Basal Ganglia

Neural activity in the brain of parkinsonian patients is characterized by the intermittently synchronized oscillatory dynamics. This imperfect synchronization, observed in the beta frequency band, is believed to be related to the hypokinetic motor symptoms of the disorder. Our study explores potential mechanisms behind this intermittent synchrony. We study the response of a bursting pallidal neuron to different patterns of synaptic input from subthalamic nucleus (STN) neuron. We show how external globus pallidus (GPe) neuron is sensitive to the phase of the input from the STN cell and can exhibit intermittent phase-locking with the input in the beta band. The temporal properties of this intermittent phase-locking show similarities to the intermittent synchronization observed in experiments. We also study the synchronization of GPe cells to synaptic input from the STN cell with dependence on the dopamine-modulated parameters. Earlier studies showed how the strengthening of dopamine-modulated coupling may lead to transitions from non-synchronized to partially synchronized dynamics, typical in Parkinson''s disease. However, dopamine also affects the cellular properties of neurons. We show how the changes in firing patterns of STN neuron due to the lack of dopamine may lead to transition from a lower to a higher coherent state, roughly matching the synchrony levels observed in basal ganglia in normal and parkinsonian states. The intermittent nature of the neural beta band synchrony in Parkinson''s disease is achieved in the model due to the interplay of the timing of STN input to pallidum and pallidal neuronal dynamics, resulting in sensitivity of pallidal output to the phase of the arriving STN input. Thus the mechanism considered here (the change in firing pattern of subthalamic neurons through the dopamine-induced change of membrane properties) may be one of the potential mechanisms responsible for the generation of the intermittent synchronization observed in Parkinson''s disease.     

Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex

How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.     

Coupling and internal noise sustain synchronized oscillation in calcium system

In this work, the effects of coupling on two calcium subsystems were investigated, the cooperation between coupling and internal noise was also considered. When two non-identical subsystems are in steady state, coupling can induce oscillations, and distinctly enlarge the oscillatory region in bifurcation diagram. Besides, coupling can make the two non-identical oscillators synchronized. With the increment of the coupling strength, the cross-correlation time of the two oscillators firstly increases and then decreases to be constant, showing the synchronization without tuning coupling strength. When internal noise is considered, similar phenomena can also be obtained under the cooperation between coupling and internal noise.     

Transitory behaviors in diffusively coupled nonlinear oscillators

We study collective behaviors of diffusively coupled oscillators which exhibit out-of-phase synchrony for the case of weakly interacting two oscillators. In large populations of such oscillators interacting via one-dimensionally nearest neighbor couplings, there appear various collective behaviors depending on the coupling strength, regardless of the number of oscillators. Among others, we focus on an intermittent behavior consisting of the all-synchronized state, a weakly chaotic state and some sorts of metachronal waves. Here, a metachronal wave means a wave with orderly phase shifts of oscillations. Such phase shifts are produced by the dephasing interaction which produces the out-of-phase synchronized states in two coupled oscillators. We also show that the abovementioned intermittent behavior can be interpreted as in-out intermittency where two saddles on an invariant subspace, the all-synchronized state and one of the metachronal waves play an important role.     

Dynamics of spiking neurons connected by both inhibitory and electrical coupling

We study the dynamics of a pair of intrinsically oscillating leaky integrate-and-fire neurons (identical and noise-free) connected by combinations of electrical and inhibitory coupling. We use the theory of weakly coupled oscillators to examine how synchronization patterns are influenced by cellular properties (intrinsic frequency and the strength of spikes) and coupling parameters (speed of synapses and coupling strengths). We find that, when inhibitory synapses are fast and the electrotonic effect of the suprathreshold portion of the spike is large, increasing the strength of weak electrical coupling promotes synchrony. Conversely, when inhibitory synapses are slow and the electrotonic effect of the suprathreshold portion of the spike is small, increasing the strength of weak electrical coupling promotes antisynchrony (see Fig. 10). Furthermore, our results indicate that, given a fixed total coupling strength, either electrical coupling alone or inhibition alone is better at enhancing neural synchrony than a combination of electrical and inhibitory coupling. We also show that these results extend to moderate coupling strengths.     

Diffusion of calcium and metabolites in pancreatic islets: killing oscillations with a pitchfork

Cell coupling is important for the normal function of the beta-cells of the pancreatic islet of Langerhans, which secrete insulin in response to elevated plasma glucose. In the islets, electrical and metabolic communications are mediated by gap junctions. Although electrical coupling is believed to account for synchronization of the islets, the role and significance of diffusion of calcium and metabolites are not clear. To address these questions we analyze two different mathematical models of islet calcium and electrical dynamics. To study diffusion of calcium, we use a modified Morris-Lecar model. Based on our analysis, we conclude that intercellular diffusion of calcium is not necessary for islet synchronization, at most supplementing electrical coupling. Metabolic coupling is investigated with a recent mathematical model incorporating glycolytic oscillations. Bifurcation analysis of the coupled system reveals several modes of behavior, depending on the relative strength of electrical and metabolic coupling. We find that whereas electrical coupling always produces synchrony, metabolic coupling can abolish both oscillations and synchrony, explaining some puzzling experimental observations. We suggest that these modes are generic features of square-wave bursters and relaxation oscillators coupled through either the activation or recovery variable.     

Suppressing bursting synchronization in a modular neuronal network with synaptic plasticity

Excessive synchronization of neurons in cerebral cortex is believed to play a crucial role in the emergence of neuropsychological disorders such as Parkinson’s disease, epilepsy and essential tremor. This study, by constructing a modular neuronal network with modified Oja’s learning rule, explores how to eliminate the pathological synchronized rhythm of interacted busting neurons numerically. When all neurons in the modular neuronal network are strongly synchronous within a specific range of coupling strength, the result reveals that synaptic plasticity with large learning rate can suppress bursting synchronization effectively. For the relative small learning rate not capable of suppressing synchronization, the technique of nonlinear delayed feedback control including differential feedback control and direct feedback control is further proposed to reduce the synchronized bursting state of coupled neurons. It is demonstrated that the two kinds of nonlinear feedback control can eliminate bursting synchronization significantly when the control parameters of feedback strength and feedback delay are appropriately tuned. For the former control technique, the control domain of effective synchronization suppression is similar to a semi-elliptical domain in the simulated parameter space of feedback strength and feedback delay, while for the latter one, the effective control domain is similar to a fan-shaped domain in the simulated parameter space.     

Rapid synchronization through fast threshold modulation

Synchronization properties of locally coupled neural oscillators were investigated analytically and by computer simulation. When coupled in a manner that mimics excitatory chemical synapses, oscillators having more than one time scale (relaxation oscillators) are shown to approach synchrony using mechanisms very different from that of oscillators with a more sinusoidal waveform. The relaxation oscillators make critical use of fast modulations of their thresholds, leading to a rate of synchronization relatively independent of coupling strength within some basin of attraction; this rate is faster for oscillators that have conductance-based features than for neural caricatures such as the FitzHugh-Nagumo equations that lack such features. Computer simulations of one-dimensional arrays show that oscillators in the relaxation regime synchronize much more rapidly than oscillators with the same equations whose parameters have been modulated to yield a more sinusoidal waveform. We present a heuristic explanation of this effect based on properties of the coupling mechanisms that can affect the way the synchronization scales with array length. These results suggest that the emergent synchronization behavior of oscillating neural networks can be dramatically influenced by the intrinsic properties of the network components. Possible implications for perceptual feature binding and attention are discussed.Supported in part by NASA (NGT-50497)Supported in part by NSF (DMS-8901913), and NIMH-47150 Present address and address for correspondence: Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, E25-618, Cambridge, MA 02139, USA     

Spatio-temporal dynamics of the grey-sided vole in Hokkaido: identifying coupling using state-based Markov-chain modelling

Explaining synchronization of cyclical or fluctuating populations over geographical regions presents ecologists with novel analytical challenges. We have developed a method to measure synchrony within spatial-temporal datasets of population densities applicable to both periodic and irregularly fluctuating populations. The dynamics of each constituent population is represented by a discrete Markov model. The state of a population trajectory at each time-point is classified as one of 'increase', 'decrease', 'peak' or 'trough'. The set of populations at any time-point is characterized by the frequency distribution of these different states, and the time-evolution of this frequency distribution used to test the hypothesis that the dynamics of each population proceeds independently of the others. The analysis identifies years in which population coupling results in synchronous states and onto which states the system converges, and identifies those years in which synchrony remains high but is accounted for by coupling observed in previous years. It also enables identification of which pairs of sites show the highest levels of coupling. Applying these methods to populations of the grey-sided vole on Hokkaido reveals them to be fluctuating in greater synchrony than would be expected from independent dynamics, and that this level of synchrony is maintained through intermittent coupling acting in ca. 1 year in four or five. High synchrony occurs between sites with similar vegetation and of similar altitude indicating that coupling may be mediated through shared environmental stimuli. When coupling is indicated, convergence is equally likely to occur on a peak state as a trough, indicating that synchronization may be brought about by the response of populations to a combination of different stimuli rather than by the action of any single process.     

Connexins and olfactory synchronicity: toward the olfactory code

A striking example of neuronal synchronization occurs in the mammalian main olfactory bulb where mitral cells that project to the same glomerular unit display highly synchronized spike activity. In this issue of Neuron, Christie et al. use mice deficient for the gap junction protein connexin36 (Cx36) to demonstrate that Cx36-mediated electrical coupling underlies such synchrony.     

Robust synchronization of coupled circadian and cell cycle oscillators in single mammalian cells

Circadian cycles and cell cycles are two fundamental periodic processes with a period in the range of 1 day. Consequently, coupling between such cycles can lead to synchronization. Here, we estimated the mutual interactions between the two oscillators by time‐lapse imaging of single mammalian NIH3T3 fibroblasts during several days. The analysis of thousands of circadian cycles in dividing cells clearly indicated that both oscillators tick in a 1:1 mode‐locked state, with cell divisions occurring tightly 5 h before the peak in circadian Rev‐Erbα‐YFP reporter expression. In principle, such synchrony may be caused by either unidirectional or bidirectional coupling. While gating of cell division by the circadian cycle has been most studied, our data combined with stochastic modeling unambiguously show that the reverse coupling is predominant in NIH3T3 cells. Moreover, temperature, genetic, and pharmacological perturbations showed that the two interacting cellular oscillators adopt a synchronized state that is highly robust over a wide range of parameters. These findings have implications for circadian function in proliferative tissues, including epidermis, immune cells, and cancer.     

When to go with the crowd: Modelling synchronization of all-or-nothing activity transitions in grouped animals

For groups of animals to keep together, the group members have to perform switches between staying in one place and moving to another place in synchrony. However, synchronization imposes a cost on individual animals, because they have to switch from one to the other behaviour at a communal time rather than at their ideal times. Here we model this situation analytically for groups in which the ideal times vary quasinormally and grouping benefit increases linearly with group size. Across the parameter space consisting of variation in the grouping benefit/cost ratio and variation in how costly it is to act too early and too late, the most common optimal solutions are full synchronization with the group staying together and zero synchronization with immediate dissolution of the group, if the group is too small for the given benefit/cost ratio. Partial synchronization, with animals at the tails of the distribution switching individually and the central core of the group in synchrony, occurs only at a narrow stripe of the space. Synchronization cost never causes splitting of the group into two as either zero, partial or full synchronization is always more advantageous. Stable solutions dictate lower degree of synchrony and lower net benefits than optimal solutions for a large range of the parameter values. If groups undergo repeated synchronization challenges, they stay together or quickly dissolve, unless the animals assort themselves into a smaller group with less variation in the ideal times. We conclude with arguing that synchronization cost is different from other types of grouping costs since it does not increase much with increasing group size. As a result, larger groups may be more stable than smaller groups. This results in the paradoxical prediction that when the grouping benefit/grouping cost ratio increases, the average group sizes might decrease, since smaller groups will be able to withstand synchronization challenges.     

Dynamics of Nephron-Vascular Network

The paper presents a modeling study of the spatial dynamics of a nephro-vascular network consisting of individual nephrons connected via a tree-like vascular branching structure. We focus on the effects of nonlinear mechanisms that are?responsible for the formation of synchronous patterns in order to learn about processes not directly amenable to experimentation. We demonstrate that: (i) the nearest nephrons are synchronized in-phase due to a vascular propagated electrical coupling, (ii) the next few branching levels display a formation of phase-shifted patterns due to hemodynamic coupling and mode elimination, and (iii) distantly located areas show asynchronous behavior or, if all nephrons and branches are perfectly identical, an infinitely long transient behavior. These results contribute to the understanding of mechanisms responsible for the highly dynamic and limited synchronization observed among groups of nephrons despite of the fairly strong interaction between the individual units.     

Defining synchrony of cell cultures

A method was developed for the statistical analysis of growth data from synchronized growth experiments. The analysis provided a firm basis for the recognition of synchrony and the objective graphical presentation of the growth pattern of a synchronized culture. The latter could then supply reliably the parameters required for the calculation of a synchronization index, i.e. for the synchrony evaluation.