Neurons process information by integrating thousands of synaptic inputs along their dendrites. Understanding the computational principles underlying neuronal information processing requires a reliable measure of synaptic conductance dynamics that accurately represents the input sources before signal integration and processing. Prevailing approaches to measuring synaptic conductances typically employ a voltage clamp at the soma of a neuron and assume the neuron as an isopotential point when processing electrical signals. However, owing to the presence of the well-known space clamp effect, the measurement of synaptic conductances through these methods often leads to significant errors, impeding the elucidation of dendritic signal features and subsequent signal integration processes. To address this issue, here we first develop a two-step clamp method at the soma that separately recovers the mean and time constant information of local synaptic conductance on the dendrite with high accuracy when a neuron receives a single synaptic input. Furthermore, under in vivo conditions of multiple synaptic inputs, we propose an intercept method to extract effective net excitatory and inhibitory synaptic conductances from measurements of synaptic currents at the soma. Both methods are grounded in mathematical perturbation analyses of a conductance-based passive cable model and are validated across multiple biologically detailed multicompartment neuron models with active channels, including Purkinje neuron, pyramidal neuron, and fast-spiking interneuron. Results demonstrate that our methods effectively circumvent the space clamp effect, offering reliable means to assess the role of measured conductances and synaptic activity in neuronal information processing.

Synaptic plasticity [long-term potentiation/depression (LTP/D)], is a cellular mechanism underlying learning. Two distinct types of early LTP/D (E-LTP/D), acting on very different time scales, have been observed experimentally—spike timing dependent plasticity (STDP), on time scales of tens of ms; and behavioral time scale synaptic plasticity (BTSP), on time scales of seconds. BTSP is a candidate for a mechanism underlying rapid learning of spatial location by place cells. Here, a computational model of the induction of E-LTP/D at a spine head of a synapse of a hippocampal pyramidal neuron is developed. The single-compartment model represents two interacting biochemical pathways for the activation (phosphorylation) of the kinase (CaMKII) with a phosphatase, with ion inflow through channels (NMDAR, CaV1,Na). The biochemical reactions are represented by a deterministic system of differential equations, with a detailed description of the activation of CaMKII that includes the opening of the compact state of CaMKII. This single model captures realistic responses (temporal profiles with the differing timescales) of STDP and BTSP and their asymmetries. The simulations distinguish several mechanisms underlying STDP vs. BTSP, including i) the flow of Ca 2 + through NMDAR vs. CaV1 channels, and ii) the origin of several time scales in the activation of CaMKII. The model also realizes a priming mechanism for E-LTP that is induced by Ca 2 + flow through CaV1.3 channels. Once in the spine head, this small additional Ca 2 + opens the compact state of CaMKII, placing CaMKII ready for subsequent induction of LTP.

The causal connectivity of a network is often inferred to understand network function. It is arguably acknowledged that the inferred causal connectivity relies on the causality measure one applies, and it may differ from the network's underlying structural connectivity. However, the interpretation of causal connectivity remains to be fully clarified, in particular, how causal connectivity depends on causality measures and how causal connectivity relates to structural connectivity. Here, we focus on nonlinear networks with pulse signals as measured output, e.g., neural networks with spike output, and address the above issues based on four commonly utilized causality measures, i.e., time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy. We theoretically show how these causality measures are related to one another when applied to pulse signals. Taking a simulated Hodgkin-Huxley network and a real mouse brain network as two illustrative examples, we further verify the quantitative relations among the four causality measures and demonstrate that the causal connectivity inferred by any of the four well coincides with the underlying network structural connectivity, therefore illustrating a direct link between the causal and structural connectivity. We stress that the structural connectivity of pulse-output networks can be reconstructed pairwise without conditioning on the global information of all other nodes in a network, thus circumventing the curse of dimensionality. Our framework provides a practical and effective approach for pulse-output network reconstruction.

The causal connectivity of a network is often inferred to understand the network function. It is arguably acknowledged that the inferred causal connectivity relies on causality measure one applies, and it may differ from the network’s underlying structural connectivity. However, the interpretation of causal connectivity remains to be fully clarified, in particular, how causal connectivity depends on causality measures and how causal connectivity relates to structural connectivity. Here, we focus on nonlinear networks with pulse signals as measured output, e.g., neural networks with spike output, and address the above issues based on four intensively utilized causality measures, i.e., time-delayed correlation, time-delayedmutual information, Granger causality, and transfer entropy. We theoretically show how these causality measures are related to one another when applied to pulse signals. Taking the simulated Hodgkin-Huxley neural network and the real mouse brain network as two illustrative examples, we further verify the quantitative relations among the four causality measures and demonstrate that the causal connectivity inferred by any of the four well coincides with the underlying network structural connectivity, therefore establishing a direct link between the causal and structural connectivity. We stress that the structural connectivity of networks can be reconstructed pairwisely without conditioning on the global information of all other nodes in a network, thus circumventing the curse of dimensionality. Our framework provides a practical and effective approach for pulse-output network reconstruction.

Neurons compute by integrating spatiotemporal excitatory ( E ) and inhibitory ( I ) synaptic inputs received from the dendrites. The investigation of dendritic integration is crucial for understanding neuronal information processing. Yet quantitative rules of dendritic integration and their mathematical modeling remain to be fully elucidated. Here neuronal dendritic integration is investigated by using theoretical and computational approaches. Based on the passive cable theory, a PDE‐based cable neuron model with spatially branched dendritic structure is introduced to describe the neuronal subthreshold membrane potential dynamics, and the analytical solutions in response to conductance‐based synaptic inputs are derived. Using the analytical solutions, a bilinear dendritic integration rule is identified, and it characterizes the change of somatic membrane potential when receiving multiple spatiotemporal synaptic inputs from the dendrites. In addition, the PDE‐based cable neuron model is reduced to an ODE‐based point‐neuron model with the feature of bilinear dendritic integration inherited, thus providing an efficient computational framework of neuronal simulation incorporating certain important dendritic functions. The above results are further extended to active dendrites by numerical verification in realistic neuron simulations. Our work provides a comprehensive and systematic theoretical and computational framework for the study of spatial neuron dynamics. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Recent experiments in the developing mammalian visual cortex have revealed that gap junctions couple excitatory cells and potentially influence the formation of chemical synapses. In particular, cells that were coupled by a gap junction during development tend to share an orientation preference and are preferentially coupled by a chemical synapse in the adult cortex, a property that is diminished when gap junctions are blocked. In this work, we construct a simplified model of the developing mouse visual cortex including spike-timing-dependent plasticity of both the feedforward synaptic inputs and recurrent cortical synapses. We use this model to show that synchrony among gap-junction-coupled cells underlies their preference to form strong recurrent synapses and develop similar orientation preference; this effect decreases with an increase in coupling density. Additionally, we demonstrate that gap-junction coupling works, together with the relative timing of synaptic development of the feedforward and recurrent synapses, to determine the resulting cortical map of orientation preference.

Cortical networks are complex systems of a great many interconnected neurons that operate from collective dynamical states. To understand how cortical neural networks function, it is important to identify their common dynamical operating states from the probabilistic viewpoint. Probabilistic characteristics of these operating states often underlie network functions. Here, using multi-electrode data from three separate experiments, we identify and characterize a cortical operating state (the "probability polling" or "p-polling" state), common across mouse and monkey with different behaviors. If the interaction among neurons is weak, the p-polling state provides a quantitative understanding of how the high dimensional probability distribution of firing patterns can be obtained by the low-order maximum entropy formulation, effectively utilizing a low dimensional stimulus-coding structure. These results show evidence for generality of the p-polling state and in certain situations its advantage of providing a mathematical validation for the low-order maximum entropy principle as a coding strategy.

Recent experiments in the developing mammalian visual cortex have revealed that gap junctions couple excitatory cells and potentially influence the formation of chemical synapses. Though gap junctions between inhibitory cells are ubiquitous in the adult cortex, and their presence has been shown to promote synchronous network firing, their function among excitatory, pyramidal cells remains poorly understood. During development, pyramidal cells that were derived from the same progenitor cell, called sister cells, are preferentially connected by a gap junction during the first postnatal week, while chemical synapses are still being formed. Additionally, these sister cells tend to share an orientation preference and a chemical synapse in the adult cortex, a property that is diminished when gap junctions are blocked. In this work, we construct an idealized model of the mouse visual cortex during the first two postnatal weeks of development to analyze the response properties of gap-junction-coupled cells and their effect on synaptic plasticity. Further, as an application of this model, we investigate the interplay of gap-junction coupling and synaptic plasticity on the order, or organization, of the resulting cortical map of orientation preference. Author summary Gap junctions, or sites of direct electrical connections between neurons, have a significant presence in the cortex, both during development and in adulthood. Their primary function during either of these periods, however, is still poorly understood. In the adult cortex, gap junctions between local, inhibitory neurons have been shown to promote synchronous firing, a network characteristic thought to be important for learning, attention, and memory. During development, gap junctions between excitatory, pyramidal cells, have been conjectured to play a role in synaptic plasticity and the formation of cortical circuits. In the visual cortex, where neurons exhibit tuned responses to properties of visual input such as orientation and direction, recent experiments show that excitatory cells are coupled by gap junctions during the first postnatal week and are replaced by chemical synapses during the second week. In this work, we explore the possible contribution of gap-junction coupling during development to the formation of chemical synapses both into the visual cortex from the thalamus and within the visual cortex between cortical cells. Specifically, within a mathematical model of the visual cortex during development, we identify the response properties of gap-junction-coupled cells and their influence on the formation of the cortical map of orientation preference.

When similar visual stimuli are presented binocularly to both eyes, one perceives a fused single image. However, when the two stimuli are distinct, one does not perceive a single image; instead, one perceives binocular rivalry. That is, one perceives one of the stimulated patterns for a few seconds, then the other for few seconds, and so on – with random transitions between the two percepts. Most theoretical studies focus on rivalry, with few considering the coexistence of fusion and rivalry. Here we develop three distinct computational neuronal network models which capture binocular rivalry with realistic stochastic properties, fusion, and the hysteretic transition between. Each is a conductance-based point neuron model, which is multi-layer with two ocular dominance columns (L & R) and with an idealized “ring” architecture where the orientation preference of each neuron labels its location on a ring. In each model, the primary mechanism initiating binocular rivalry is cross-column inhibition, with firing rate adaptation governing the temporal properties of the transitions between percepts. Under stimulation by similar visual patterns, each of three models uses its own mechanism to overcome cross-column inhibition, and thus to prevent rivalry and allow the fusion of similar images: The first model uses cross-column feedforward inhibition from the opposite eye to “shut off” the cross-column feedback inhibition; the second model “turns on” a second layer of monocular neurons as a parallel pathway to the binocular neurons, rivaling out of phase with the first layer, and together these two pathways represent fusion; and the third model uses cross-column excitation to overcome the cross-column inhibition and enable fusion. Thus, each of the idealized ring models depends upon a different mechanism for fusion that might emerge as an underlying mechanism present in real visual cortex.

Over the past two decades, mathematicians and neuroscientists at New York University have developed several large-scale computational models of a layer of macaque primary visual cortex. Here we provide an overview of these models, organized by the specific questions about cortical processing that each model addressed. Each model was founded upon the available anatomical and physiological data; and not by building into the model network assumptions about theoretical mechanisms specifically designed to enable the network to produce desired response properties. Also, our aim was to use one comprehensive network, with a fixed architecture and one set of parameters, to model all experiments. The response properties of individual neurons and populations of neurons then emerge from this experimentally constrained model. This overview is dedicated to Professor David Cai, who played a leading role in several of the models described here. We are very fortunate to have had the opportunity to work with him over the past two decades.

Complex dendrites in general present formidable challenges to understanding neuronal information processing. To circumvent the difficulty, a prevalent viewpoint simplifies the neuronal morphology as a point representing the soma, and the excitatory and inhibitory synaptic currents originated from the dendrites are treated as linearly summed at the soma. Despite its extensive applications, the validity of the synaptic current description remains unclear, and the existing point neuron framework fails to characterize the spatiotemporal aspects of dendritic integration supporting specific computations. Using electrophysiological experiments, realistic neuronal simulations, and theoretical analyses, we demonstrate that the traditional assumption of linear summation of synaptic currents is oversimplified and underestimates the inhibition effect. We then derive a form of synaptic integration current within the point neuron framework to capture dendritic effects. In the derived form, the interaction between each pair of synaptic inputs on the dendrites can be reliably parameterized by a single coefficient, suggesting the inherent low-dimensional structure of dendritic integration. We further generalize the form of synaptic integration current to capture the spatiotemporal interactions among multiple synaptic inputs and show that a point neuron model with the synaptic integration current incorporated possesses the computational ability of a spatial neuron with dendrites, including direction selectivity, coincidence detection, logical operation, and a bilinear dendritic integration rule discovered in experiment. Our work amends the modeling of synaptic inputs and improves the computational power of a modeling neuron within the point neuron framework.

Recent experiments have shown that mouse primary visual cortex (V1) is very different from that of cat or monkey, including response properties-one of which is that contrast invariance in the orientation selectivity (OS) of the neurons' firing rates is replaced in mouse with contrast-dependent sharpening (broadening) of OS in excitatory (inhibitory) neurons. These differences indicate a different circuit design for mouse V1 than that of cat or monkey. Here we develop a large-scale computational model of an effective input layer of mouse V1. Constrained by experiment data, the model successfully reproduces experimentally observed response properties-for example, distributions of firing rates, orientation tuning widths, and response modulations of simple and complex neurons, including the contrast dependence of orientation tuning curves. Analysis of the model shows that strong feedback inhibition and strong orientation-preferential cortical excitation to the excitatory population are the predominant mechanisms underlying the contrast-sharpening of OS in excitatory neurons, while the contrast-broadening of OS in inhibitory neurons results from a strong but nonpreferential cortical excitation to these inhibitory neurons, with the resulting contrast-broadened inhibition producing a secondary enhancement on the contrast-sharpened OS of excitatory neurons. Finally, based on these mechanisms, we show that adjusting the detailed balances between the predominant mechanisms can lead to contrast invariance-providing insights for future studies on contrast dependence (invariance).

Commodity crop expansion in the tropics presents the challenge of preserving tropical moist forest (TMF) ecosystems and their role in carbon sequestration. We propose an algorithm, specific to the TMF biome, which identifies 125 million ha of degraded, low-carbon density land (LCDL) in the Pantropical TMF belt for agricultural expansion. About 65 million ha of LCDL are in contiguous tracts >5,000 ha and <500 m elevation, meeting the prerequisites for commercial-scale oil palm production, the fastest-expanding industrialized commodity crop in the TMF. These areas could support expansion of commercial agriculture for another 25-50 years without further conversion of TMF. Confining agricultural expansion to the LCDL can avoid the release of approximately 13 billion tons of CO2 while saving valuable tropical biodiversity. The simplicity and transparency of this easily monitored metric could prove useful to producers, governments, investors, environmental stewards, and consumers and enhance good governance in tropical regions.

One of the fundamental questions in system neuroscience is how the brain encodes external stimuli in the early sensory cortex. It has been found in experiments that even some simple sensory stimuli can activate large populations of neurons. It is believed that information can be encoded in the spatiotemporal profile of these collective neuronal responses. Here, we use a large-scale computational model of the primary visual cortex (V1) to study the population responses in V1 as observed in experiments in which monkeys performed visual detection tasks. We show that our model can capture very well spatiotemporal activities measured by voltage-sensitive-dye-based optical imaging in V1 of the awake state. In our model, the properties of horizontal long-range connections with NMDA conductance play an important role in the correlated population responses and have strong implications for spatiotemporal coding of neuronal populations. Our computational modeling approach allows us to reveal intrinsic cortical dynamics, separating them from those statistical effects arising from averaging procedures in experiment. For example, in experiments, it was shown that there was a spatially antagonistic center-surround structure in optimal weights in signal detection theory, which was believed to underlie the efficiency of population coding. However, our study shows that this feature is an artifact of data processing.

This paper reviews our recent work addressing the role of both synaptic-input and connectivity-architecture fluctuations in coarse-grained descriptions of integrate-and-fire (I&F) point- neuron network models. Beginning with the most basic coarse-grained description, the all-to-all coupled, mean-field model, which ignores all fluctuations, we add the effects of the two types of fluctuations one at a time. To study the effects of synaptic-input fluctuations, we derive a kinetic- theoretic description, first in the form of a Boltzmann equation in (2+1) dimensions, simplifying that to an advection-diffusion equation, and finally reducing the dimension to a system of two (1+1)- dimensional kinetic equations via the maximum entropy principle. In the limit of an infinitely-fast conductance relaxation time, we derive a Fokker-Planck equation which captures the bifurcation between a bistable, hysteretic operating regime of the network when the amount of synaptic-input fluctuations is small, and a stable regime when the amount of fluctuations increases. To study the ef- fects of complex neuronal-network architecture, we incorporate the network connectivity statistics in the mean-field description, and investigate the dependence of these statistics on the statistical prop- erties of the neuronal firing rates for three network examples with increasingly complex connectivity architecture.

Communications on Pure and Applied MathematicsVolume 65, Issue 7 p. 891-892 Introduction: Courant Institute at 75 David W. McLaughlin, David W. McLaughlin Courant InstituteSearch for more papers by this author David W. McLaughlin, David W. McLaughlin Courant InstituteSearch for more papers by this author First published: 25 April 2012 https://doi.org/10.1002/cpa.21403AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume65, Issue7Special Issue: First Special Issue Commemorating the 75th Anniversary of the Courant InstituteJuly 2012Pages 891-892 RelatedInformation

These Guidelines describe the University’s process for allocating additional space to schools and administrative units, developing new University space, undertaking major renovations in existing space and authorizing planning studies and capital projects. The Guidelines pertain to all units of NYU other than the School of Medicine and Abu Dhabi, and apply to projects in the immediate and the near future (in a five to ten year time frame). They ensure that:

The West Coast of Washington and the NE and SW comers of Wyoming are regions of the contiguous United States where NEXRAD coverage is incomplete. One approach to addressing these gaps is to install additional NEXRAD-class radars. Another potential approach is to install small radar networks of the type being investigated in the CASA project. This paper compares these two approaches. We provide a meteorological and user-need assessment of present radar coverage in these regions (based on a recent feasibility study led by J. Brotzge) as well as an objective assessment of the radar-coverage that would be achieved using the large radar and small radar approaches. For this evaluation we consider two classes of radar: long-range radars having similar attributes to the WSR-88D (i.e., 10 cm wavelength, >250 km maximum range, 1 degree beamwidth, -500 kW peak power); and short-range radars having attributes similar to those operating in CASA's Oklahoma prototype network (i.e., 3 cm wavelength, 40 km maximum range, 2 degree beamwidth). We first establish the number of both types of radar that would be needed to provide coverage over a given rectangular ground-domain. Next, we quantify the coverage-versus-altitude for both weather-event detection and precipitation estimation over these regions, considering the blockage caused by both the curved earth and the local terrain.