Slow Switching in Globally Coupled Oscillators: Robustness and Occurrence through Delayed Coupling

The phenomenon of slow switching in populations of globally coupled oscillators is discussed. This characteristic collective dynamics, which was first discovered in a particular class of the phase oscillator model, is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population. We argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom. We also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators. Although the heteroclinic loop might seem to arise only exceptionally, we find that it appears rather easily by introducing the time-delay in the population which would otherwise exhibit perfect phase synchrony. We argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle-node bifurcations. Slow switching arises when the system with a heteroclinic loop is weakly perturbed. This will be demonstrated with a vector model by applying weak noises. Other types of weak symmetry-breaking perturbations can also cause slow switching.Comment: 10 pages, 14 figures, RevTex, twocolumn, to appear in Phys. Rev.

A Moving Bump in a Continuous Manifold: A Comprehensive Study of the Tracking Dynamics of Continuous Attractor Neural Networks

Understanding how the dynamics of a neural network is shaped by the network structure, and consequently how the network structure facilitates the functions implemented by the neural system, is at the core of using mathematical models to elucidate brain functions. This study investigates the tracking dynamics of continuous attractor neural networks (CANNs). Due to the translational invariance of neuronal recurrent interactions, CANNs can hold a continuous family of stationary states. They form a continuous manifold in which the neural system is neutrally stable. We systematically explore how this property facilitates the tracking performance of a CANN, which is believed to have clear correspondence with brain functions. By using the wave functions of the quantum harmonic oscillator as the basis, we demonstrate how the dynamics of a CANN is decomposed into different motion modes, corresponding to distortions in the amplitude, position, width or skewness of the network state. We then develop a perturbative approach that utilizes the dominating movement of the network's stationary states in the state space. This method allows us to approximate the network dynamics up to an arbitrary accuracy depending on the order of perturbation used. We quantify the distortions of a Gaussian bump during tracking, and study their effects on the tracking performance. Results are obtained on the maximum speed for a moving stimulus to be trackable and the reaction time for the network to catch up with an abrupt change in the stimulus.Comment: 43 pages, 10 figure

Dynamically-Coupled Oscillators -- Cooperative Behavior via Dynamical Interaction --

We propose a theoretical framework to study the cooperative behavior of dynamically coupled oscillators (DCOs) that possess dynamical interactions. Then, to understand synchronization phenomena in networks of interneurons which possess inhibitory interactions, we propose a DCO model with dynamics of interactions that tend to cause 180-degree phase lags. Employing an approach developed here, we demonstrate that although our model displays synchronization at high frequencies, it does not exhibit synchronization at low frequencies because this dynamical interaction does not cause a phase lag sufficiently large to cancel the effect of the inhibition. We interpret the disappearance of synchronization in our model with decreasing frequency as describing the breakdown of synchronization in the interneuron network of the CA1 area below the critical frequency of 20 Hz.Comment: 10 pages, 3 figure

Management Strategies for Double-Crop Soybean Planted After Wheat

Double-crop (DC) soybeans (Glycine max L.) are gaining popularity as an alternative system to intensify productivity without expanding the farming area and can potentially increase net return. However, the DC soybean system faces many challenges such as late planting, which decreases yield potential. A study was conducted in four site-years in Ashland Bottoms, KS, during the 2016 and 2017 growing seasons. In both years, the soybean variety planted was Asgrow 4232 (MG 4.2). The soybean was planted right after two different wheat harvest timings (Study 1, early-wheat harvest 18–20% mois­ture; and Study 2, conventional-harvest 13–14% moisture). Seven treatments were eval­uated in each of the soybean planting dates: 1) common practice; 2) no seed treatment (without seed fungicide + insecticide treatment); 3) non-stay green (without foliar fungicide + insecticide application); 4) high seeding rate (180,000 seeds/a); 5) wide rows (30-inch row-spacing); 6) nitrogen (N) fixation (without late-fertilizer N appli­cation); and 7) kitchen sink (includes all management practices). There was adequate precipitation distribution in 2016, which helped to nurture the soybean plants even when planting later in the season. In 2017, precipitation was not well distributed, and the early planting date was affected by low precipitation during early season. Overall, the high plant population and the kitchen sink treatments presented maximum yields, while the common practice scenario showed the lowest yields

Scaling and singularities in the entrainment of globally-coupled oscillators

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling. The population is described by a Fokker-Planck equation for the distribution of phases which includes the diffusive effect of noise in the oscillator frequencies. The bifurcation from the phase-incoherent state is analyzed using amplitude equations for the unstable modes with particular attention to the dependence of the nonlinearly saturated mode $|\alpha_\infty|$ on the linear growth rate $\gamma$. In general we find $|\alpha_\infty|\sim \sqrt{\gamma(\gamma+l^2D)}$ where $D$ is the diffusion coefficient and $l$ is the mode number of the unstable mode. The unusual $(\gamma+l^2D)$ factor arises from a singularity in the cubic term of the amplitude equation.Comment: 11 pages (Revtex); paper submitted to Phys. Rev. Let

Synchronization of Integrate and Fire oscillators with global coupling

In this article we study the behavior of globally coupled assemblies of a large number of Integrate and Fire oscillators with excitatory pulse-like interactions. On some simple models we show that the additive effects of pulses on the state of Integrate and Fire oscillators are sufficient for the synchronization of the relaxations of all the oscillators. This synchronization occurs in two forms depending on the system: either the oscillators evolve ``en bloc'' at the same phase and therefore relax together or the oscillators do not remain in phase but their relaxations occur always in stable avalanches. We prove that synchronization can occur independently of the convexity or concavity of the oscillators evolution function. Furthermore the presence of disorder, up to some level, is not only compatible with synchronization, but removes some possible degeneracy of identical systems and allows new mechanisms towards this state.Comment: 37 pages, 19 postscript figures, Latex 2

Effect of Management Practices on Double-Crop Soybean Yields

Double-crop soybean has great potential to increase profits and the use of agricultural land. However, there is a gap between double-crop versus full-season soybean yields. To address this yield difference, a study evaluating different management practices on double-crop soybean was conducted. A four-site-year experiment was conducted at Ottawa, KS, during the 2016 and 2017 growing season. In both years, the soybean variety planted was Asgrow 4232 (MG 4.2). The soybean was planted right after two different wheat harvest timings (Study 1, early-wheat harvest 18–20%; and Study 2, conventional-harvest 13–14%). Seven treatments were evaluated in each of the soybean planting dates: 1) common practice; 2) no seed treatment (without seed fungicide+ insecticide treatment); 3) non-stay green (without foliar fungicide + insecticide appli­cation); 4) high seeding rate (180,000 seeds/a); 5) wide rows (30-inch row-spacing); 6) nitrogen (N) fixation (without late-fertilizer N application); and 7) kitchen sink (includes all management practices). In the 2017 season, a treatment was added with the purpose of isolating the fertilizer effect, 8) no fertilization (F). Aboveground biomass and yield were recorded. For the 2016 season, there was a different response for early and late planting in relation to yield responses. For the early planting, there were no differences in yield. However, for the late planting, high plant population, wide-rows and kitchen sink showed greater yields. For the early planting, the differences in biomass were not related to differences in yield. For the late planting, greater biomass corresponded to superior yields, except for the kitchen sink treatment that presented low biomass and greater yields, potentially via increasing biomass partitioning to the seed. For the 2017 season, biomass and yield followed the same pattern, yields increased in parallel to biomass. For the early planting, greater yields were observed for the high plant population, no nitrogen applied in reproductive R3, and kitchen sink. There were no significant differences in yield among treatments for the late planting date in 2016. However, in both years’ yields were lower for late planting dates when compared with the early planting

Statistical mechanics of mutual information maximization

An unsupervised learning procedure based on maximizing the mutual information between the outputs of two networks receiving different but statistically dependent inputs is analyzed (Becker S. and Hinton G., Nature, 355 (1992) 161). By exploiting a formal analogy to supervised learning in parity machines, the theory of zero-temperature Gibbs learning for the unsupervised procedure is presented for the case that the networks are perceptrons and for the case of fully connected committees