Integrability of a Globally Coupled Complex Riccati Array: Quadratic Integrate-and-Fire Neurons, Phase Oscillators, and All in Between.

We present an exact dimensionality reduction for dynamics of an arbitrary array of globally coupled complex-valued Riccati equations. It generalizes the Watanabe-Strogatz theory [Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70, 2391 (1993).PRLTAO0031-900710.1103/PhysRevLett.70.2391] for sinusoidally coupled phase oscillators and seamlessly includes quadratic integrate-and-fire neurons as the real-valued special case. This simple formulation reshapes our understanding of a broad class of coupled systems-including a particular class of phase-amplitude oscillators-which newly fall under the category of integrable systems. Precise and rigorous analysis of complex Riccati arrays is now within reach, paving a way to a deeper understanding of emergent behavior of collective dynamics in coupled systems.

Bifurcations in adaptive vascular networks: Toward model calibration.

Transport networks are crucial for the functioning of natural and technological systems. We study a mathematical model of vascular network adaptation, where the network structure dynamically adjusts to changes in blood flow and pressure. The model is based on local feedback mechanisms that occur on different time scales in the mammalian vasculature. The cost exponent γ tunes the vessel growth in the adaptation rule, and we test the hypothesis that the cost exponent is γ=1/2 for vascular systems [D. Hu and D. Cai, Phys. Rev. Lett. 111, 138701 (2013)]. We first perform bifurcation analysis for a simple triangular network motif with a fluctuating demand and then conduct numerical simulations on network topologies extracted from perivascular networks of rodent brains. We compare the model predictions with experimental data and find that γ is closer to 1 than to 1/2 for the model to be consistent with the data. Our study, thus, aims at addressing two questions: (i) Is a specific measured flow network consistent in terms of physical reality? (ii) Is the adaptive dynamic model consistent with measured network data? We conclude that the model can capture some aspects of vascular network formation and adaptation, but also suggest some limitations and directions for future research. Our findings contribute to a general understanding of the dynamics in adaptive transport networks, which is essential for studying mammalian vasculature and developing self-organizing piping systems.

Complex dynamics in adaptive phase oscillator networks.

Networks of coupled dynamical units give rise to collective dynamics such as the synchronization of oscillators or neurons in the brain. The ability of the network to adapt coupling strengths between units in accordance with their activity arises naturally in a variety of contexts, including neural plasticity in the brain, and adds an additional layer of complexity: the dynamics on the nodes influence the dynamics of the network and vice versa. We study a minimal model of Kuramoto phase oscillators including a general adaptive learning rule with three parameters (strength of adaptivity, adaptivity offset, adaptivity shift), mimicking learning paradigms based on spike-time-dependent plasticity. Importantly, the strength of adaptivity allows to tune the system away from the limit of the classical Kuramoto model, corresponding to stationary coupling strengths and no adaptation and, thus, to systematically study the impact of adaptivity on the collective dynamics. We carry out a detailed bifurcation analysis for the minimal model consisting of N=2 oscillators. The non-adaptive Kuramoto model exhibits very simple dynamic behavior, drift, or frequency-locking; but once the strength of adaptivity exceeds a critical threshold non-trivial bifurcation structures unravel: A symmetric adaptation rule results in multi-stability and bifurcation scenarios, and an asymmetric adaptation rule generates even more intriguing and rich dynamics, including a period-doubling cascade to chaos as well as oscillations displaying features of both librations and rotations simultaneously. Generally, adaptation improves the synchronizability of the oscillators. Finally, we also numerically investigate a larger system consisting of N=50 oscillators and compare the resulting dynamics with the case of N=2 oscillators.

Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators.

Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked-a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period-doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.

The glymphatic system: Current understanding and modeling.

We review theoretical and numerical models of the glymphatic system, which circulates cerebrospinal fluid and interstitial fluid around the brain, facilitating solute transport. Models enable hypothesis development and predictions of transport, with clinical applications including drug delivery, stroke, cardiac arrest, and neurodegenerative disorders like Alzheimer's disease. We sort existing models into broad categories by anatomical function: Perivascular flow, transport in brain parenchyma, interfaces to perivascular spaces, efflux routes, and links to neuronal activity. Needs and opportunities for future work are highlighted wherever possible; new models, expanded models, and novel experiments to inform models could all have tremendous value for advancing the field.

First-order like phase transition induced by quenched coupling disorder.

We investigate the collective dynamics of a population of X Y model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value and subject to thermal noise controlled by temperature T. We find that the system at T = 0 exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state; when thermal noise is present ( T > 0 ), the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case ( T = 0 ). We derive an exact formula for the critical transition from incoherent to coherent oscillations for the deterministic and stochastic case based on both stability analysis for finite oscillators as well as for the thermodynamic limit ( N → ∞) based on a rigorous mean-field theory using graphons, valid for heterogeneous graph structures. Our theoretical results are supported by extensive numerical simulations. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies, which consider uniform input or output strengths for each oscillator node [H. Hong and S. H. Strogatz, Phys. Rev. E 84(4), 046202 (2011); Phys. Rev. Lett. 106(5), 054102 (2011)], which suggests that these systems display a "universal" character for the onset of synchronization.

Cerebrospinal fluid is a significant fluid source for anoxic cerebral oedema.

Cerebral oedema develops after anoxic brain injury. In two models of asphyxial and asystolic cardiac arrest without resuscitation, we found that oedema develops shortly after anoxia secondary to terminal depolarizations and the abnormal entry of CSF. Oedema severity correlated with the availability of CSF with the age-dependent increase in CSF volume worsening the severity of oedema. Oedema was identified primarily in brain regions bordering CSF compartments in mice and humans. The degree of ex vivo tissue swelling was predicted by an osmotic model suggesting that anoxic brain tissue possesses a high intrinsic osmotic potential. This osmotic process was temperature-dependent, proposing an additional mechanism for the beneficial effect of therapeutic hypothermia. These observations show that CSF is a primary source of oedema fluid in anoxic brain. This novel insight offers a mechanistic basis for the future development of alternative strategies to prevent cerebral oedema formation after cardiac arrest.

A two-frequency-two-coupling model of coupled oscillators.

We considered the phase coherence dynamics in a Two-Frequency and Two-Coupling (TFTC) model of coupled oscillators, where coupling strength and natural oscillator frequencies for individual oscillators may assume one of two values (positive/negative). The bimodal distributions for the coupling strengths and frequencies are either correlated or uncorrelated. To study how correlation affects phase coherence, we analyzed the TFTC model by means of numerical simulations and exact dimensional reduction methods allowing to study the collective dynamics in terms of local order parameters [S. Watanabe and S. H. Strogatz, Physica D 74(3-4), 197-253 (1994); E. Ott and T. M. Antonsen, Chaos 18(3), 037113 (2008)]. The competition resulting from distributed coupling strengths and natural frequencies produces nontrivial dynamic states. For correlated disorder in frequencies and coupling strengths, we found that the entire oscillator population splits into two subpopulations, both phase-locked (Lock-Lock) or one phase-locked, and the other drifting (Lock-Drift), where the mean-fields of the subpopulations maintain a constant non-zero phase difference. For uncorrelated disorder, we found that the oscillator population may split into four phase-locked subpopulations, forming phase-locked pairs which are either mutually frequency-locked (Stable Lock-Lock-Lock-Lock) or drifting (Breathing Lock-Lock-Lock-Lock), thus resulting in a periodic motion of the global synchronization level. Finally, we found for both types of disorder that a state of Incoherence exists; however, for correlated coupling strengths and frequencies, incoherence is always unstable, whereas it is only neutrally stable for the uncorrelated case. Numerical simulations performed on the model show good agreement with the analytic predictions. The simplicity of the model promises that real-world systems can be found which display the dynamics induced by correlated/uncorrelated disorder.

Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons.

We study the macroscopic dynamics of large networks of excitable type 1 neurons composed of two populations interacting with disparate but symmetric intra- and inter-population coupling strengths. This nonuniform coupling scheme facilitates symmetric equilibria, where both populations display identical firing activity, characterized by either quiescent or spiking behavior, or asymmetric equilibria, where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior. Oscillations in the firing rate are possible if neurons emit pulses with non-zero width but are otherwise quenched. Here, we explore how collective oscillations emerge for two statistically identical neuron populations in the limit of an infinite number of neurons. A detailed analysis reveals how collective oscillations are born and destroyed in various bifurcation scenarios and how they are organized around higher codimension bifurcation points. Since both symmetric and asymmetric equilibria display bistable behavior, a large configuration space with steady and oscillatory behavior is available. Switching between configurations of neural activity is relevant in functional processes such as working memory and the onset of collective oscillations in motor control.

First-order synchronization transition in a large population of strongly coupled relaxation oscillators.

Onset and loss of synchronization in coupled oscillators are of fundamental importance in understanding emergent behavior in natural and man-made systems, which range from neural networks to power grids. We report on experiments with hundreds of strongly coupled photochemical relaxation oscillators that exhibit a discontinuous synchronization transition with hysteresis, as opposed to the paradigmatic continuous transition expected from the widely used weak coupling theory. The resulting first-order transition is robust with respect to changes in network connectivity and natural frequency distribution. This allows us to identify the relaxation character of the oscillators as the essential parameter that determines the nature of the synchronization transition. We further support this hypothesis by revealing the mechanism of the transition, which cannot be accounted for by standard phase reduction techniques.

Optimal Self-Induced Stochastic Resonance in Multiplex Neural Networks: Electrical vs. Chemical Synapses.

Electrical and chemical synapses shape the dynamics of neural networks, and their functional roles in information processing have been a longstanding question in neurobiology. In this paper, we investigate the role of synapses on the optimization of the phenomenon of self-induced stochastic resonance in a delayed multiplex neural network by using analytical and numerical methods. We consider a two-layer multiplex network in which, at the intra-layer level, neurons are coupled either by electrical synapses or by inhibitory chemical synapses. For each isolated layer, computations indicate that weaker electrical and chemical synaptic couplings are better optimizers of self-induced stochastic resonance. In addition, regardless of the synaptic strengths, shorter electrical synaptic delays are found to be better optimizers of the phenomenon than shorter chemical synaptic delays, while longer chemical synaptic delays are better optimizers than longer electrical synaptic delays; in both cases, the poorer optimizers are, in fact, worst. It is found that electrical, inhibitory, or excitatory chemical multiplexing of the two layers having only electrical synapses at the intra-layer levels can each optimize the phenomenon. Additionally, only excitatory chemical multiplexing of the two layers having only inhibitory chemical synapses at the intra-layer levels can optimize the phenomenon. These results may guide experiments aimed at establishing or confirming to the mechanism of self-induced stochastic resonance in networks of artificial neural circuits as well as in real biological neural networks.

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review.

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

Cerebrospinal fluid influx drives acute ischemic tissue swelling.

Stroke affects millions each year. Poststroke brain edema predicts the severity of eventual stroke damage, yet our concept of how edema develops is incomplete and treatment options remain limited. In early stages, fluid accumulation occurs owing to a net gain of ions, widely thought to enter from the vascular compartment. Here, we used magnetic resonance imaging, radiolabeled tracers, and multiphoton imaging in rodents to show instead that cerebrospinal fluid surrounding the brain enters the tissue within minutes of an ischemic insult along perivascular flow channels. This process was initiated by ischemic spreading depolarizations along with subsequent vasoconstriction, which in turn enlarged the perivascular spaces and doubled glymphatic inflow speeds. Thus, our understanding of poststroke edema needs to be revised, and these findings could provide a conceptual basis for development of alternative treatment strategies.

Chaos in Kuramoto oscillator networks.

Kuramoto oscillators are widely used to explain collective phenomena in networks of coupled oscillatory units. We show that simple networks of two populations with a generic coupling scheme, where both coupling strengths and phase lags between and within populations are distinct, can exhibit chaotic dynamics as conjectured by Ott and Antonsen [Chaos 18, 037113 (2008)]. These chaotic mean-field dynamics arise universally across network size, from the continuum limit of infinitely many oscillators down to very small networks with just two oscillators per population. Hence, complicated dynamics are expected even in the simplest description of oscillator networks.

Chimera states in two populations with heterogeneous phase-lag.

The simplest network of coupled phase-oscillators exhibiting chimera states is given by two populations with disparate intra- and inter-population coupling strengths. We explore the effects of heterogeneous coupling phase-lags between the two populations. Such heterogeneity arises naturally in various settings, for example, as an approximation to transmission delays, excitatory-inhibitory interactions, or as amplitude and phase responses of oscillators with electrical or mechanical coupling. We find that breaking the phase-lag symmetry results in a variety of states with uniform and non-uniform synchronization, including in-phase and anti-phase synchrony, full incoherence (splay state), chimera states with phase separation of 0 or π between populations, and states where both populations remain desynchronized. These desynchronized states exhibit stable, oscillatory, and even chaotic dynamics. Moreover, we identify the bifurcations through which chimeras emerge. Stable chimera states and desynchronized solutions, which do not arise for homogeneous phase-lag parameters, emerge as a result of competition between synchronized in-phase, anti-phase equilibria, and fully incoherent states when the phase-lags are near ±π2 (cosine coupling). These findings elucidate previous experimental results involving a network of mechanical oscillators and provide further insight into the breakdown of synchrony in biological systems.

Modeling of Kidney Hemodynamics: Probability-Based Topology of an Arterial Network.

Through regulation of the extracellular fluid volume, the kidneys provide important long-term regulation of blood pressure. At the level of the individual functional unit (the nephron), pressure and flow control involves two different mechanisms that both produce oscillations. The nephrons are arranged in a complex branching structure that delivers blood to each nephron and, at the same time, provides a basis for an interaction between adjacent nephrons. The functional consequences of this interaction are not understood, and at present it is not possible to address this question experimentally. We provide experimental data and a new modeling approach to clarify this problem. To resolve details of microvascular structure, we collected 3D data from more than 150 afferent arterioles in an optically cleared rat kidney. Using these results together with published micro-computed tomography (µCT) data we develop an algorithm for generating the renal arterial network. We then introduce a mathematical model describing blood flow dynamics and nephron to nephron interaction in the network. The model includes an implementation of electrical signal propagation along a vascular wall. Simulation results show that the renal arterial architecture plays an important role in maintaining adequate pressure levels and the self-sustained dynamics of nephrons.

Intermittent chaotic chimeras for coupled rotators.

Two symmetrically coupled populations of N oscillators with inertia m display chaotic solutions with broken symmetry similar to experimental observations with mechanical pendulums. In particular, we report evidence of intermittent chaotic chimeras, where one population is synchronized and the other jumps erratically between laminar and turbulent phases. These states have finite lifetimes diverging as a power law with N and m. Lyapunov analyses reveal chaotic properties in quantitative agreement with theoretical predictions for globally coupled dissipative systems.

Size structures sensory hierarchy in ocean life.

Survival in aquatic environments requires organisms to have effective means of collecting information from their surroundings through various sensing strategies. In this study, we explore how sensing mode and range depend on body size. We find a hierarchy of sensing modes determined by body size. With increasing body size, a larger battery of modes becomes available (chemosensing, mechanosensing, vision, hearing and echolocation, in that order) while the sensing range also increases. This size-dependent hierarchy and the transitions between primary sensory modes are explained on the grounds of limiting factors set by physiology and the physical laws governing signal generation, transmission and reception. We theoretically predict the body size limits for various sensory modes, which align well with size ranges found in literature. The treatise of all ocean life, from unicellular organisms to whales, demonstrates how body size determines available sensing modes, and thereby acts as a major structuring factor of aquatic life.

Model for polygonal hydraulic jumps.

We propose a phenomenological model for the polygonal hydraulic jumps discovered by Ellegaard and co-workers [Nature (London) 392, 767 (1998); Nonlinearity 12, 1 (1999); Physica B 228, 1 (1996)], based on the known flow structure for the type-II hydraulic jumps with a "roller" (separation eddy) near the free surface in the jump region. The model consists of mass conservation and radial force balance between hydrostatic pressure and viscous stresses on the roller surface. In addition, we consider the azimuthal force balance, primarily between pressure and viscosity, but also including nonhydrostatic pressure contributions from surface tension in light of recent observations by Bush and co-workers [J. Fluid Mech. 558, 33 (2006); Phys. Fluids 16, S4 (2004)]. The model can be analyzed by linearization around the circular state, resulting in a parameter relationship for nearly circular polygonal states. A truncated but fully nonlinear version of the model can be solved analytically. This simpler model gives rise to polygonal shapes that are very similar to those observed in experiments, even though surface tension is neglected, and the condition for the existence of a polygon with N corners depends only on a single dimensionless number φ. Finally, we include time-dependent terms in the model and study linear stability of the circular state. Instability occurs for sufficiently small Bond number and the most unstable wavelength is expected to be roughly proportional to the width of the roller as in the Rayleigh-Plateau instability.

Interfering waves of adaptation promote spatial mixing.

A fundamental problem of asexual adaptation is that beneficial substitutions are not efficiently accumulated in large populations: Beneficial mutations often go extinct because they compete with one another in going to fixation. It has been argued that such clonal interference may have led to the evolution of sex and recombination in well-mixed populations. Here, we study clonal interference, and mechanisms of its mitigation, in an evolutionary model of spatially structured populations with uniform selection pressure. Clonal interference is much more prevalent with spatial structure than without, due to the slow wave-like spread of beneficial mutations through space. We find that the adaptation speed of asexuals saturates when the linear habitat size exceeds a characteristic interference length, which becomes shorter with smaller migration and larger mutation rate. The limiting speed is proportional to µ(1/2) and µ(1/3) in linear and planar habitats, respectively, where the mutational supply µ is the product of mutation rate and local population density. This scaling and the existence of a speed limit should be amenable to experimental tests as they fall far below predicted adaptation speeds for well-mixed populations (that scale as the logarithm of population size). Finally, we show that not only recombination, but also long-range migration is a highly efficient mechanism of relaxing clonal competition in structured populations. Our conservative estimates of the interference length predict prevalent clonal interference in microbial colonies and biofilms, so clonal competition should be a strong driver of both genetic and spatial mixing in those contexts.