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.

Natural carboxyterminal truncation of human CXCL10 attenuates glycosaminoglycan binding, CXCR3A signaling and lymphocyte chemotaxis, while retaining angiostatic activity.

BACKGROUND: Interferon-γ-inducible protein of 10 kDa (IP-10/CXCL10) is a dual-function CXC chemokine that coordinates chemotaxis of activated T cells and natural killer (NK) cells via interaction with its G protein-coupled receptor (GPCR), CXC chemokine receptor 3 (CXCR3). As a consequence of natural posttranslational modifications, human CXCL10 exhibits a high degree of structural and functional heterogeneity. However, the biological effect of natural posttranslational processing of CXCL10 at the carboxy (C)-terminus has remained partially elusive. We studied CXCL10(1-73), lacking the four endmost C-terminal amino acids, which was previously identified in supernatant of cultured human fibroblasts and keratinocytes. METHODS: Relative levels of CXCL10(1-73) and intact CXCL10(1-77) were determined in synovial fluids of patients with rheumatoid arthritis (RA) through tandem mass spectrometry. The production of CXCL10(1-73) was optimized through Fmoc-based solid phase peptide synthesis (SPPS) and a strategy to efficiently generate human CXCL10 proteoforms was introduced. CXCL10(1-73) was compared to intact CXCL10(1-77) using surface plasmon resonance for glycosaminoglycan (GAG) binding affinity, assays for cell migration, second messenger signaling downstream of CXCR3, and flow cytometry of CHO cells and primary human T lymphocytes and endothelial cells. Leukocyte recruitment in vivo upon intraperitoneal injection of CXCL10(1-73) was also evaluated. RESULTS: Natural CXCL10(1-73) was more abundantly present compared to intact CXCL10(1-77) in synovial fluids of patients with RA. CXCL10(1-73) had diminished affinity for GAG including heparin, heparan sulfate and chondroitin sulfate A. Moreover, CXCL10(1-73) exhibited an attenuated capacity to induce CXCR3A-mediated signaling, as evidenced in calcium mobilization assays and through quantification of phosphorylated extracellular signal-regulated kinase-1/2 (ERK1/2) and protein kinase B/Akt. Furthermore, CXCL10(1-73) incited significantly less primary human T lymphocyte chemotaxis in vitro and peritoneal ingress of CXCR3+ T lymphocytes in mice. In contrast, loss of the four endmost C-terminal residues did not affect the inhibitory properties of CXCL10 on migration, proliferation, wound closure, phosphorylation of ERK1/2, and sprouting of human microvascular endothelial cells. CONCLUSION: Our study shows that the C-terminal residues Lys74-Pro77 of CXCL10 are important for GAG binding, signaling through CXCR3A, T lymphocyte chemotaxis, but dispensable for angiostasis.

Proteolytic inactivation of CXCL12 in the lungs and circulation of COVID-19 patients.

The human chemokine stromal cell-derived factor-1 (SDF-1) or CXCL12 is involved in several homeostatic processes and pathologies through interaction with its cognate G protein-coupled receptor CXCR4. Recent research has shown that CXCL12 is present in the lungs and circulation of patients with coronavirus disease 2019 (COVID-19). However, the question whether the detected CXCL12 is bioactive was not addressed. Indeed, the activity of CXCL12 is regulated by NH2- and COOH-terminal post-translational proteolysis, which significantly impairs its biological activity. The aim of the present study was to characterize proteolytic processing of CXCL12 in broncho-alveolar lavage (BAL) fluid and blood plasma samples from critically ill COVID-19 patients. Therefore, we optimized immunosorbent tandem mass spectrometry proteoform analysis (ISTAMPA) for detection of CXCL12 proteoforms. In patient samples, this approach uncovered that CXCL12 is rapidly processed by site-specific NH2- and COOH-terminal proteolysis and ultimately degraded. This proteolytic inactivation occurred more rapidly in COVID-19 plasma than in COVID-19 BAL fluids, whereas BAL fluid samples from stable lung transplantation patients and the non-affected lung of lung cancer patients (control groups) hardly induced any processing of CXCL12. In COVID-19 BAL fluids with high proteolytic activity, processing occurred exclusively NH2-terminally and was predominantly mediated by neutrophil elastase. In low proteolytic activity BAL fluid and plasma samples, NH2- and COOH-terminal proteolysis by CD26 and carboxypeptidases were observed. Finally, protease inhibitors already approved for clinical use such as sitagliptin and sivelestat prevented CXCL12 processing and may therefore be of pharmacological interest to prolong CXCL12 half-life and biological activity in vivo.

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.

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.

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.

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.

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.

Graphop mean-field limits and synchronization for the stochastic Kuramoto model.

Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator's phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov-Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence-coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.

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.

EDTA/gelatin zymography method to identify C1s versus activated MMP-9 in plasma and immune complexes of patients with systemic lupus erythematosus.

Gelatin zymography analysis is a sensitive method and commonly used to characterize and quantify the presence of the gelatinases (MMP-2 and MMP-9) in biological samples. In human plasma samples from healthy controls and systemic lupus erythematosus (SLE) patients, we observed a gelatinolytic molecule at 80 kDa, suggestive for activated human MMP-9. However, by developing and using the EDTA/gelatin zymography method and after purification of the 80 kDa entity, we proved that this molecule was the C1s subunit of the complement system. The zymolytic capacity of C1s was validated and found to be enhanced, in the absence of calcium and in the presence of EDTA. Our findings indicate that for correct identification of gelatinolytic proteins in complex biological samples the use of EDTA/gelatin zymography for enzyme development is advised. In addition, by quantification of EDTA/gelatin zymography analysis and ELISA, we observed that the levels of C1s were higher in plasma and immune complexes of SLE patients than of healthy individuals. Therefore, our data imply that C1s may become a marker for the diagnosis of SLE.

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.

Endotoxemia shifts neutrophils with TIMP-free gelatinase B/MMP-9 from bone marrow to the periphery and induces systematic upregulation of TIMP-1.

Lipopolysaccharides or endotoxins elicit an excessive host inflammatory response and lead to life-threatening conditions such as endotoxemia and septic shock. Lipopolysaccharides trigger mobilization and stimulation of leukocytes and exaggerated production of pro-inflammatory molecules including cytokines and proteolytic enzymes. Matrix metalloproteinase-9 (MMP-9) or gelatinase B, a protease stored in the tertiary granules of polymorphonuclear leukocytes, has been implicated in such inflammatory reactions. Moreover, several studies even pinpointed MMP-9 as a potential target molecule to counter excessive inflammation in endotoxemia. Whereas the early effect of lipopolysaccharide-induced inflammation in vivo on the expression of MMP-9 in various peripheral organs has been described, the effects on the bone marrow and during late stage endotoxemia remain elusive. We demonstrate that TIMP-free MMP-9 is a major factor in bone marrow physiology and pathology. By using a mouse model for late-stage endotoxemia, we show that lipopolysaccharides elicited a depletion of neutrophil MMP-9 in the bone marrow and a shift of MMP-9 and MMP-9-containing cells towards peripheral organs, a pattern which was primarily associated with a relocation of CD11bhighGr-1high cells. In contrast, analysis of the tissue inhibitors of metalloproteinases was in line with a natural, systematic upregulation of TIMP-1, the main tissue inhibitor of TIMP-free MMP-9, and a general shift toward control of matrix metalloproteinase activity by tissue inhibitors of metalloproteinases.

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.

Zymography methods for visualizing hydrolytic enzymes.

Zymography is a technique for studying hydrolytic enzymes on the basis of substrate degradation. It is a powerful, but often misinterpreted, tool yielding information on potential hydrolytic activities, enzyme forms and the locations of active enzymes. In this Review, zymography techniques are compared in terms of advantages, limitations and interpretations. With in gel zymography, enzyme forms are visualized according to their molecular weights. Proteolytic activities are localized in tissue sections with in situ zymography. In vivo zymography can pinpoint proteolytic activity to sites in an intact organism. Future development of novel substrate probes and improvement in detection and imaging methods will increase the applicability of zymography for (reverse) degradomics studies.

Circular trimers of gelatinase B/matrix metalloproteinase-9 constitute a distinct population of functional enzyme molecules differentially regulated by tissue inhibitor of metalloproteinases-1.

Gelatinase B/matrix metalloproteinase-9 (MMP-9) (EC 3.4.24.35) cleaves many substrates and is produced by most cell types as a zymogen, proMMP-9, in complex with the tissue inhibitor of metalloproteinases-1 (TIMP-1). Natural proMMP-9 occurs as monomers, homomultimers and heterocomplexes, but our knowledge about the overall structure of proMMP-9 monomers and multimers is limited. We investigated biochemical, biophysical and functional characteristics of zymogen and activated forms of MMP-9 monomers and multimers. In contrast with a conventional notion of a dimeric nature of MMP-9 homomultimers, we demonstrate that these are reduction-sensitive trimers. Based on the information from electrophoresis, AFM and TEM, we generated a 3D structure model of the proMMP-9 trimer. Remarkably, the proMMP-9 trimers possessed a 50-fold higher affinity for TIMP-1 than the monomers. In vivo, this finding was reflected in a higher extent of TIMP-1 inhibition of angiogenesis induced by trimers compared with monomers. Our results show that proMMP-9 trimers constitute a novel structural and functional entity that is differentially regulated by TIMP-1.

Chimera states in mechanical oscillator networks.

The synchronization of coupled oscillators is a fascinating manifestation of self-organization that nature uses to orchestrate essential processes of life, such as the beating of the heart. Although it was long thought that synchrony and disorder were mutually exclusive steady states for a network of identical oscillators, numerous theoretical studies in recent years have revealed the intriguing possibility of "chimera states," in which the symmetry of the oscillator population is broken into a synchronous part and an asynchronous part. However, a striking lack of empirical evidence raises the question of whether chimeras are indeed characteristic of natural systems. This calls for a palpable realization of chimera states without any fine-tuning, from which physical mechanisms underlying their emergence can be uncovered. Here, we devise a simple experiment with mechanical oscillators coupled in a hierarchical network to show that chimeras emerge naturally from a competition between two antagonistic synchronization patterns. We identify a wide spectrum of complex states, encompassing and extending the set of previously described chimeras. Our mathematical model shows that the self-organization observed in our experiments is controlled by elementary dynamical equations from mechanics that are ubiquitous in many natural and technological systems. The symmetry-breaking mechanism revealed by our experiments may thus be prevalent in systems exhibiting collective behavior, such as power grids, optomechanical crystals, or cells communicating via quorum sensing in microbial populations.

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.