Revealing directed effective connectivity of cortical neuronal networks from measurements.

In the study of biological networks, one of the major challenges is to understand the relationships between network structure and dynamics. In this paper, we model in vitro cortical neuronal cultures as stochastic dynamical systems and apply a method that reconstructs directed networks from dynamics [Ching and Tam, Phys. Rev. E 95, 010301(R) (2017)2470-004510.1103/PhysRevE.95.010301] to reveal directed effective connectivity, namely, the directed links and synaptic weights, of the neuronal cultures from voltage measurements recorded by a multielectrode array. The effective connectivity so obtained reproduces several features of cortical regions in rats and monkeys and has similar network properties as the synaptic network of the nematode Caenorhabditis elegans, whose entire nervous system has been mapped out. The distribution of the incoming degree is bimodal and the distributions of the average incoming and outgoing synaptic strength are non-Gaussian with long tails. The effective connectivity captures different information from the commonly studied functional connectivity, estimated using statistical correlation between spiking activities. The average synaptic strengths of excitatory incoming and outgoing links are found to increase with the spiking activity in the estimated effective connectivity but not in the functional connectivity estimated using the same sets of voltage measurements. These results thus demonstrate that the reconstructed effective connectivity can capture the general properties of synaptic connections and better reveal relationships between network structure and dynamics.

Heterogeneous Responses to Changes in Inhibitory Synaptic Strength in Networks of Spiking Neurons.

How does the dynamics of neurons in a network respond to changes in synaptic weights? Answer to this question would be important for a full understanding of synaptic plasticity. In this article, we report our numerical study of the effects of changes in inhibitory synaptic weights on the spontaneous activity of networks of spiking neurons with conductance-based synapses. Networks with biologically realistic features, which were reconstructed from multi-electrode array recordings taken in a cortical neuronal culture, and their modifications were used in the simulations. The magnitudes of the synaptic weights of all the inhibitory connections are decreased by a uniform amount subjecting to the condition that inhibitory connections would not be turned into excitatory ones. Our simulation results reveal that the responses of the neurons are heterogeneous: while the firing rate of some neurons increases as expected, the firing rate of other neurons decreases or remains unchanged. The same results show that heterogeneous responses also occur for an enhancement of inhibition. This heterogeneity in the responses of neurons to changes in inhibitory synaptic strength suggests that activity-induced modification of synaptic strength does not necessarily generate a positive feedback loop on the dynamics of neurons connected in a network. Our results could be used to understand the effects of bicuculline on spiking and bursting activities of neuronal cultures. Using reconstructed networks with biologically realistic features enables us to identify a long-tailed distribution of average synaptic weights for outgoing links as a crucial feature in giving rise to bursting in neuronal networks and in determining the overall response of the whole network to changes in synaptic strength. For networks whose average synaptic weights for outgoing links have a long-tailed distribution, bursting is observed and the average firing rate of the whole network increases upon inhibition suppression or decreases upon inhibition enhancement. For networks whose average synaptic weights for outgoing links are approximately normally distributed, bursting is not found and the average firing rate of the whole network remains approximately constant upon changes in inhibitory synaptic strength.

Reconstructing links in directed networks from noisy dynamics.

We address the long-standing challenge of how to reconstruct links in directed networks from measurements, and present a general method that makes use of a noise-induced relation between network structure and both the time-lagged covariance of measurements taken at two different times and the covariance of measurements taken at the same time. When the coupling functions have certain additional properties, we can further reconstruct the weights of the links.

Turbulent Rayleigh-Bénard convection with polymers: Understanding how heat flux is modified.

We study how polymers affect the heat flux in turbulent Rayleigh-Bénard convection at moderate Rayleigh numbers using direct numerical simulations with polymers of different relaxation times. We find that heat flux is enhanced by polymers and the amount of heat enhancement first increases and then decreases with the Weissenberg number, which is the ratio of the polymer relaxation time to the typical time scale of the flow. We show that this nonmonotonic behavior of the heat flux enhancement is the combined effect of the decrease in the viscous energy dissipation rate due to the viscosity of the Newtonian fluid and the increase in the energy dissipation rate due to polymers when Weissenberg number is increased. We explain why the viscous energy dissipation rate decreases with the Weissenberg number. Then by carrying out a generalized boundary layer analysis supplemented by a space-dependent effective viscosity from the numerical simulations, we provide a theoretical understanding of the change of the heat flux when the viscous energy dissipation rate is held constant. Our analysis thus provides a physical way to understand the numerical results.

Reconstructing weighted networks from dynamics.

We present a method that reconstructs both the links and their relative coupling strength of bidirectional weighted networks. Our method requires only measurements of node dynamics as input. Using several examples, we demonstrate that our method can give accurate results for weighted random and weighted scale-free networks with both linear and nonlinear dynamics.

Thermal boundary layer equation for turbulent Rayleigh-Bénard convection.

We report a new thermal boundary layer equation for turbulent Rayleigh-Bénard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr≳1 and (b) Pr≫1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol), respectively.

Polymer-induced change in scaling behavior in two-dimensional homogeneous turbulent thermal convection.

We study the effects of polymers in two-dimensional turbulent thermal convection using a shell model. In the absence of polymers, the inverse energy cascade in two dimensions leads to the observed Bolgiano-Obukhov scaling. When polymers are added, energy is extracted from the flow by the polymers, and as a result, the thermal balance between buoyancy and inertia in Bolgiano-Obukhov scaling is destroyed around the scales at which polymers interact strongly with the flow. This results in an increase in the Bolgiano scale and leads to a change in the scaling behavior of the velocity and temperature fluctuations for scales below the modified Bolgiano scale. We make theoretical estimates of the dependence of the mean rate of energy extracted by the polymers and the mean energy dissipation rate on the polymer relaxation time. Our theoretical analysis further leads to the prediction that the heat transport is not altered much by the polymers in two dimensions. We show that our theoretical estimates and prediction are in good agreement with the numerical results.

Extracting connectivity from dynamics of networks with uniform bidirectional coupling.

In the study of networked systems, a method that can extract information about how the individual nodes are connected with one another would be valuable. In this paper, we present a method that can yield such information of network connectivity using measurements of the dynamics of the nodes as the only input data. Our method is built upon a noise-induced relation between the Laplacian matrix of the network and the dynamical covariance matrix of the nodes, and applies to networked dynamical systems in which the coupling between nodes is uniform and bidirectional. Using examples of different networks and dynamics, we demonstrate that the method can give accurate connectivity information for a wide range of noise amplitude and coupling strength. Moreover, we can calculate a parameter Δ using again only the input of time-series data, and assess the accuracy of the extracted connectivity information based on the value of Δ.

Scaling behavior in turbulent Rayleigh-Bénard convection revealed by conditional structure functions.

We show that the nature of the scaling behavior can be revealed by studying the conditional structure functions evaluated at given values of the locally averaged thermal dissipation rate. These conditional structure functions have power-law dependence on the value of the locally averaged thermal dissipation rate, and such dependence for the Bolgiano-Obukhov scaling is different from the other scaling behaviors. Our analysis of experimental measurements verifies the power-law dependence and reveals the Bolgiano-Obukhov scaling behavior at the center of the bottom plate of the convection cell.

Effect of polymer additives on heat transport in turbulent thermal convection.

In this Letter, we explore the possible effects of polymer additives on heat transport in turbulent thermal convective flows. Using both direct numerical simulations and shell-model calculations, we show that polymer additives can significantly enhance the heat transport in homogeneous turbulent thermal convection, which mimics the bulk of turbulent Rayleigh-Bénard convection. We also discuss the implication of our results for turbulent Rayleigh-Bénard convection, in which there are boundary layers in addition to the central bulk.

Refined similarity hypotheses in shell models of homogeneous turbulence and turbulent convection.

A major challenge in turbulence research is to understand from first principles the origin of the anomalous scaling of velocity fluctuations in high-Reynolds-number turbulent flows. One important idea was proposed by Kolmogorov [J. Fluid Mech. 13, 82 (1962)], which attributes the anomaly to variations of the locally averaged energy dissipation rate. Kraichnan later pointed out [J. Fluid Mech. 62, 305 (1973)] that the locally averaged energy dissipation rate is not an inertial-range quantity and a proper inertial-range quantity would be the local energy transfer rate. As a result, Kraichnan's idea attributes the anomaly to variations of the local energy transfer rate. These ideas, generally known as refined similarity hypotheses, can also be extended to study the anomalous scaling of fluctuations of an active scalar, such as the temperature in turbulent convection. We examine the validity of these refined similarity hypotheses and their extensions to an active scalar in shell models of homogeneous turbulence and turbulent convection. We find that Kraichnan's refined similarity hypothesis and its extension are valid.

Ultimate-state scaling in a shell model for homogeneous turbulent convection.

An interesting question in turbulent convection is how the heat transport depends on the strength of thermal forcing in the limit of very large thermal forcing. Kraichnan predicted [Phys. Fluids 5, 1374 (1962)] that for fluids with low Prandtl number (Pr), the heat transport measured by the Nusselt number (Nu) would depend on the strength of thermal forcing measured by the Rayleigh number (Ra) as Nu approximately Ra(1/2) with logarithmic corrections at very high Ra. According to Kraichnan, the shear boundary layers play a crucial role in giving rise to this so-called ultimate-state scaling. A similar scaling result is predicted by the Grossmann-Lohse theory [J. Fluid Mech. 407, 27 (2000)], but with the assumption that the ultimate state is a bulk-dominated state in which both the average kinetic and thermal dissipation rates are dominated by contributions from the bulk of the flow with the boundary layers either broken down or playing no role in the heat transport. In this paper, we study the dependence of Nu and the Reynolds number (Re) measuring the root-mean-squared velocity fluctuations on Ra and Pr, for low Pr, using a shell model for homogeneous turbulent convection where buoyancy is acting directly on most of the scales. We find that Nu approximately Ra(1/2)Pr(1/2) and Re approximately Ra(1/2)Pr(-1/2) , which resemble the ultimate-state scaling behavior for fluids with low Pr, and show that the presence of a drag acting on the large scales is crucial in giving rise to such scaling. As a large-scale drag cannot exist by itself in the bulk of turbulent thermal convection, our results indicate that if buoyancy acts on most of the scales in the bulk of turbulent convection at very high Ra, then the ultimate state cannot be bulk dominated.

Comparison of theory and direct numerical simulations of drag reduction by rodlike polymers in turbulent channel flows.

Numerical simulations of turbulent channel flows, with or without additives, are limited in the extent of the Reynolds number (Re) and Deborah number (De). The comparison of such simulations to theories of drag reduction, which are usually derived for asymptotically high Re and De, calls for some care. In this paper we present a study of drag reduction by rodlike polymers in a turbulent channel flow using direct numerical simulation and illustrate how these numerical results should be related to the recently developed theory.

Anomalous scaling and refined similarity of an active scalar in a shell model of homogeneous turbulent convection.

Anomalous scaling in the statistics of an active scalar is studied in a shell model of homogeneous turbulent convection. We extend refined similarity ideas for homogeneous and isotropic turbulence to homogeneous turbulent convection and attribute the origin of the anomalous scaling to variations of the entropy transfer rate. We verify the consequences and thus the validity of our hypothesis by showing that the conditional statistics of the active scalar and the velocity at fixed values of entropy transfer rate are not anomalous but have simple scaling with exponents given by dimensional considerations, and that the intermittency corrections are given by the scaling exponents of the moments of the entropy transfer rate.

Multifractality and scale invariance in human heartbeat dynamics.

Human heart rate is known to display complex fluctuations. Evidence of multifractality in heart rate fluctuations in healthy state has been reported [Ivanov, Nature (London) 399, 461 (1999)]. This multifractal character could be manifested as the dependence of the probability density functions (PDFs) of the interbeat interval increments, which are the differences in two interbeat intervals that are separated by n beats, on n . On the other hand, "scale invariance in the PDFs of detrended healthy human heart rate increments" was recently reported [Kiyono, Phys. Rev. Lett. 93, 178103 (2004)]. In this paper, we clarify that the scale invariance reported is actually exhibited by the PDFs of the increments of the "detrended" integrated healthy interbeat interval and should, therefore, be more accurately referred as the scale invariance or n independence of the PDFs of the sum of n detrended interbeat intervals. Indeed, we demonstrate explicitly that the PDFs of detrended healthy interbeat interval increments are scale or n dependent in accord with its multifractal character. Our work also establishes that this n independence of the PDFs of the sum of n detrended interbeat intervals is a general feature of human heartbeat dynamics, shared by heart rate fluctuations in both healthy and pathological states.

Scaling laws in the central region of confined turbulent thermal convection.

In confined turbulent thermal convection, the velocity is separated into two parts: one that is correlated with some function of the temperature fluctuations, and thus associated with the plume velocity, and the other part, the background velocity, which is uncorrelated with any function of the temperature fluctuations. As a result, one should focus on the plume velocity, and not the whole velocity, and the temperature when studying the scaling behavior. In this paper, a phenomenological theory for the scaling behavior in the central region of confined turbulent thermal convection is presented. The spatial (temporal) plume velocity structure functions are found to have the same scaling behavior as the spatial (temporal) temperature structure functions. For tau> or = taub, where the buoyant scale taub is determined in terms of measurable quantities, the scaling exponents of the temporal temperature structure functions and hence those of the temporal plume velocity structure functions are obtained. These results are checked against experimental measurements, and good agreement is found.

Turbulent drag reduction by flexible and rodlike polymers: Crossover effects at small concentrations.

Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

Additive equivalence in turbulent drag reduction by flexible and rodlike polymers.

We address the additive equivalence discovered by Virk and co-workers: drag reduction affected by flexible and rigid rodlike polymers added to turbulent wall-bounded flows is limited from above by a very similar maximum drag reduction (MDR) asymptote. Considering the equations of motion of rodlike polymers in wall-bounded turbulent ensembles, we show that although the microscopic mechanism of attaining the MDR is very different, the macroscopic theory is isomorphic, rationalizing the interesting experimental observations.

Extraction of plumes in turbulent thermal convection.

We present a scheme to extract the velocity of buoyant structures in turbulent thermal convection from simultaneous local velocity and temperature measurements. Applying this scheme to measurements taken at positions within the convection cell where the buoyant structures are dominated by plumes, we obtain the temperature dependence of the plume velocity and understand our results using the equations of motion. We further obtain the scaling behavior of the average local heat flux in the vertical direction at the cell center with the Rayleigh number and find that the scaling exponent is different from that measured for the Nusselt number. This difference leads to the conclusion that heat cannot be mainly transported through the central region of the convection cell.

Drag reduction in homogeneous turbulence by scale-dependent effective viscosity.

We demonstrate, by using suitable shell models, that drag reduction in homogeneous turbulence is usefully discussed in terms of a scale-dependent effective viscosity. The essence of the phenomenon of drag reduction found in models that couple the velocity field to the polymers can be recaptured by an "equivalent" equation of motion for the velocity field alone, with a judiciously chosen scale-dependent effective viscosity that succinctly summarizes the important aspects of the interaction between the velocity and the polymer fields. Finally, we clarify the differences between drag reduction in homogeneous and in wall bounded flows.