Functional Data Approximation on Bounded Domains using Polygonal Finite Elements.

We construct and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare approximation qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates, Poisson coordinates, mean value coordinates, and quadratic serendipity bases over polygonal meshes on the domain. For a convex n-sided polygon, the quadratic serendipity elements have 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, rather than the usual n(n + 1)/2 basis functions to achieve quadratic convergence. Two greedy algorithms are proposed to generate Voronoi meshes for adaptive functional/scattered data approximations. Experimental results show space/accuracy advantages for these quadratic serendipity finite elements on polygonal domains versus traditional finite elements over simplicial meshes. Polygonal meshes and parameter coefficients of the quadratic serendipity finite elements obtained by our greedy algorithms can be further refined using an L2-optimization to improve the piecewise functional approximation. We conduct several experiments to demonstrate the efficacy of our algorithm for modeling features/discontinuities in functional data/image approximation.

Causal and structural connectivity of pulse-coupled nonlinear networks.

We study the reconstruction of structural connectivity for a general class of pulse-coupled nonlinear networks and show that the reconstruction can be successfully achieved through linear Granger causality (GC) analysis. Using spike-triggered correlation of whitened signals, we obtain a quadratic relationship between GC and the network couplings, thus establishing a direct link between the causal connectivity and the structural connectivity within these networks. Our work may provide insight into the applicability of GC in the study of the function of general nonlinear networks.

Biphasic Cholinergic Modulation of Reverberatory Activity in Neuronal Networks.

Acetylcholine (ACh) is an important neuromodulator in various cognitive functions. However, it is unclear how ACh influences neural circuit dynamics by altering cellular properties. Here, we investigated how ACh influences reverberatory activity in cultured neuronal networks. We found that ACh suppressed the occurrence of evoked reverberation at low to moderate doses, but to a much lesser extent at high doses. Moreover, high doses of ACh caused a longer duration of evoked reverberation, and a higher occurrence of spontaneous activity. With whole-cell recording from single neurons, we found that ACh inhibited excitatory postsynaptic currents (EPSCs) while elevating neuronal firing in a dose-dependent manner. Furthermore, all ACh-induced cellular and network changes were blocked by muscarinic, but not nicotinic receptor antagonists. With computational modeling, we found that simulated changes in EPSCs and the excitability of single cells mimicking the effects of ACh indeed modulated the evoked network reverberation similar to experimental observations. Thus, ACh modulates network dynamics in a biphasic fashion, probably by inhibiting excitatory synaptic transmission and facilitating neuronal excitability through muscarinic signaling pathways.

High-throughput mapping of a whole rhesus monkey brain at micrometer resolution.

Whole-brain mesoscale mapping in primates has been hindered by large brain sizes and the relatively low throughput of available microscopy methods. Here, we present an approach that combines primate-optimized tissue sectioning and clearing with ultrahigh-speed fluorescence microscopy implementing improved volumetric imaging with synchronized on-the-fly-scan and readout technique, and is capable of completing whole-brain imaging of a rhesus monkey at 1 × 1 × 2.5 µm3 voxel resolution within 100 h. We also developed a highly efficient method for long-range tracing of sparse axonal fibers in datasets numbering hundreds of terabytes. This pipeline, which we call serial sectioning and clearing, three-dimensional microscopy with semiautomated reconstruction and tracing (SMART), enables effective connectome-scale mapping of large primate brains. With SMART, we were able to construct a cortical projection map of the mediodorsal nucleus of the thalamus and identify distinct turning and routing patterns of individual axons in the cortical folds while approaching their arborization destinations.

Emergence of spatially periodic diffusive waves in small-world neuronal networks.

It has been observed in experiment that the anatomical structure of neuronal networks in the brain possesses the feature of small-world networks. Yet how the small-world structure affects network dynamics remains to be fully clarified. Here we study the dynamics of a class of small-world networks consisting of pulse-coupled integrate-and-fire (I&F) neurons. Under stochastic Poisson drive, we find that the activity of the entire network resembles diffusive waves. To understand its underlying mechanism, we analyze the simplified regular-lattice network consisting of firing-rate-based neurons as an approximation to the original I&F small-world network. We demonstrate both analytically and numerically that, with strongly coupled connections, in the absence of noise, the activity of the firing-rate-based regular-lattice network spatially forms a static grating pattern that corresponds to the spatial distribution of the firing rate observed in the I&F small-world neuronal network. We further show that the spatial grating pattern with different phases comprise the continuous attractor of both the I&F small-world and firing-rate-based regular-lattice network dynamics. In the presence of input noise, the activity of both networks is perturbed along the continuous attractor, which gives rise to the diffusive waves. Our numerical simulations and theoretical analysis may potentially provide insights into the understanding of the generation of wave patterns observed in cortical networks.

Causal inference in nonlinear systems: Granger causality versus time-delayed mutual information.

The Granger causality (GC) analysis has been extensively applied to infer causal interactions in dynamical systems arising from economy and finance, physics, bioinformatics, neuroscience, social science, and many other fields. In the presence of potential nonlinearity in these systems, the validity of the GC analysis in general is questionable. To illustrate this, here we first construct minimal nonlinear systems and show that the GC analysis fails to infer causal relations in these systems-it gives rise to all types of incorrect causal directions. In contrast, we show that the time-delayed mutual information (TDMI) analysis is able to successfully identify the direction of interactions underlying these nonlinear systems. We then apply both methods to neuroscience data collected from experiments and demonstrate that the TDMI analysis but not the GC analysis can identify the direction of interactions among neuronal signals. Our work exemplifies inference hazards in the GC analysis in nonlinear systems and suggests that the TDMI analysis can be an appropriate tool in such a case.

Spike-Triggered Regression for Synaptic Connectivity Reconstruction in Neuronal Networks.

How neurons are connected in the brain to perform computation is a key issue in neuroscience. Recently, the development of calcium imaging and multi-electrode array techniques have greatly enhanced our ability to measure the firing activities of neuronal populations at single cell level. Meanwhile, the intracellular recording technique is able to measure subthreshold voltage dynamics of a neuron. Our work addresses the issue of how to combine these measurements to reveal the underlying network structure. We propose the spike-triggered regression (STR) method, which employs both the voltage trace and firing activity of the neuronal population to reconstruct the underlying synaptic connectivity. Our numerical study of the conductance-based integrate-and-fire neuronal network shows that only short data of 20 ~ 100 s is required for an accurate recovery of network topology as well as the corresponding coupling strength. Our method can yield an accurate reconstruction of a large neuronal network even in the case of dense connectivity and nearly synchronous dynamics, which many other network reconstruction methods cannot successfully handle. In addition, we point out that, for sparse networks, the STR method can infer coupling strength between each pair of neurons with high accuracy in the absence of the global information of all other neurons.

Granger causality analysis with nonuniform sampling and its application to pulse-coupled nonlinear dynamics.

The Granger causality (GC) analysis is an effective approach to infer causal relations for time series. However, for data obtained by uniform sampling (i.e., with an equal sampling time interval), it is known that GC can yield unreliable causal inference due to aliasing if the sampling rate is not sufficiently high. To solve this unreliability issue, we consider the nonuniform sampling scheme as it can mitigate against aliasing. By developing an unbiased estimation of power spectral density of nonuniformly sampled time series, we establish a framework of spectrum-based nonparametric GC analysis. Applying this framework to a general class of pulse-coupled nonlinear networks and utilizing some particular spectral structure possessed by these nonlinear network data, we demonstrate that, for such nonlinear networks with nonuniformly sampled data, reliable GC inference can be achieved at a low nonuniform mean sampling rate at which the traditional uniform sampling GC may lead to spurious causal inference.

Analysis of sampling artifacts on the Granger causality analysis for topology extraction of neuronal dynamics.

Granger causality (GC) is a powerful method for causal inference for time series. In general, the GC value is computed using discrete time series sampled from continuous-time processes with a certain sampling interval length τ, i.e., the GC value is a function of τ. Using the GC analysis for the topology extraction of the simplest integrate-and-fire neuronal network of two neurons, we discuss behaviors of the GC value as a function of τ, which exhibits (i) oscillations, often vanishing at certain finite sampling interval lengths, (ii) the GC vanishes linearly as one uses finer and finer sampling. We show that these sampling effects can occur in both linear and non-linear dynamics: the GC value may vanish in the presence of true causal influence or become non-zero in the absence of causal influence. Without properly taking this issue into account, GC analysis may produce unreliable conclusions about causal influence when applied to empirical data. These sampling artifacts on the GC value greatly complicate the reliability of causal inference using the GC analysis, in general, and the validity of topology reconstruction for networks, in particular. We use idealized linear models to illustrate possible mechanisms underlying these phenomena and to gain insight into the general spectral structures that give rise to these sampling effects. Finally, we present an approach to circumvent these sampling artifacts to obtain reliable GC values.

Granger causality network reconstruction of conductance-based integrate-and-fire neuronal systems.

Reconstruction of anatomical connectivity from measured dynamical activities of coupled neurons is one of the fundamental issues in the understanding of structure-function relationship of neuronal circuitry. Many approaches have been developed to address this issue based on either electrical or metabolic data observed in experiment. The Granger causality (GC) analysis remains one of the major approaches to explore the dynamical causal connectivity among individual neurons or neuronal populations. However, it is yet to be clarified how such causal connectivity, i.e., the GC connectivity, can be mapped to the underlying anatomical connectivity in neuronal networks. We perform the GC analysis on the conductance-based integrate-and-fire (I&F) neuronal networks to obtain their causal connectivity. Through numerical experiments, we find that the underlying synaptic connectivity amongst individual neurons or subnetworks, can be successfully reconstructed by the GC connectivity constructed from voltage time series. Furthermore, this reconstruction is insensitive to dynamical regimes and can be achieved without perturbing systems and prior knowledge of neuronal model parameters. Surprisingly, the synaptic connectivity can even be reconstructed by merely knowing the raster of systems, i.e., spike timing of neurons. Using spike-triggered correlation techniques, we establish a direct mapping between the causal connectivity and the synaptic connectivity for the conductance-based I&F neuronal networks, and show the GC is quadratically related to the coupling strength. The theoretical approach we develop here may provide a framework for examining the validity of the GC analysis in other settings.