Analysis Of Spatial Point Processes Research Articles

Abstract 3648: The spatial organization of cells expressing MYC and BCL2 affects immune microenvironment composition and prognosis in DLBCL

Abstract Point process analyses are widely used in ecology and geography to understand the impact and relevance of spatial distribution of points in a complex system. To date, only few studies have explored point process analyses in the context of tumor heterogeneity. Malignancies often show variable patterns of cellular distribution, and the relationship of these topographic variables with underlying biological processes and clinical outcomes is not well understood. Here, we performed spatial point process analyses in diffuse large B-cell lymphoma (DLBCL), leveraging advances in multiplexed immunohistochemistry and cellular phenotyping, with downstream multi-omic and clinicopathological analyses. We have demonstrated that survival in DLBCL is strongly associated with the fraction of malignant cells showing co-expression of the oncogenes MYC and BCL2 in the absence of BCL6 (M+2+6-). These cells display non-random spatial organization within tumor infiltrates, deviating from a poisson distribution. We therefore modeled the spatial organization of M+2+6- cells within DLBCL, using x-y coordinate information from multiplexed fluorescent immunohistochemistry (mfIHC) in four independent cohorts (N=449 patients). We derived spatial point patterns, upon which Geyers point process models were applied. We observed that patients could be divided into two groups based on these: one group showed “clustered” spatial organization, while the other displayed a “dispersed” M+2+6- cell distribution. Cases with predominantly “dispersed” M+2+6- cells showed shorter overall survival in all analyzed cohorts (P < 0.05 in 4/4 cohorts). We performed multi-omic analyses to identify potential biological explanations for this association between topography and clinical outcome. Using digital spatial profiling of the transcriptome from CD20, CD3 and CD68 compartments in 47 DLBCL samples, we observed that lymphomas with an M+2+6- ‘dispersed’ phenotype had an immunologically cold microenvironment enriched in Tregs and exhausted CD4+ and CD8+ T cells. Through an integration with single-cell transcriptomic analyses (N=22 samples) we observed that malignant B-cells from cases with M+2+6- ‘dispersed’ phenotype expressed genes implicated in cell migration and adhesion, but also in potentially targetable immune checkpoints such as LAG3. In summary, by integrating multiplexed immunophenotyping and point process analysis with molecular profiling in the setting of DLBCL, we show that a dispersed pattern of distribution of the M+2+6- subclonal fraction uniquely confers poor prognosis, and associates with a potentially reversible immune cold microenvironment composition. This represents the first demonstration that the spatial distribution of malignant cell subpopulations in lymphoma can embody biological and clinical significance. Citation Format: Shruti Shridar, Michal Marek Hoppe, Patrick Jaynes, Gayatri Kumar, Siddham Jasoria, Ziwei Meng, Vaibhav Rajan, Kasthuri Kannan, Claudio Tripodo, Anand Devaprasath Jeyasekharan. The spatial organization of cells expressing MYC and BCL2 affects immune microenvironment composition and prognosis in DLBCL [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 3648.

A Python package for the local multiscale analysis of spatial point processes (LomPy)

The consideration of multiscale characteristics has emerged as a popular component of geospatial data analysis and modeling. However, the practical implementation of such analysis tasks involves time-consuming and computationally intensive processes that require the integration of knowledge and methods from different disciplines (e.g., quantitative geography, signal processing, and natural sciences) and in which large amounts of data have to be processed. Yet, to date, there is few open-source software that enables an efficient and transparent computational workflow. This paper introduces a Python package for the local multiscale analysis of spatial point processes (LomPy). LomPy is specifically designed for processing and analyzing data that is either irregularly spaced or has large data holes over the spatial territory: a common and methodologically challenging property of geoprocessing operations. The first main function of the package computes the multiresolution quantities, while the second applies them to the extraction of fractal and multifractal features at arbitrarily “local” spatial scales. The third function extends the univariate analysis option to multivariate settings. Powerful tools for regression modeling, density estimation, and visualization are also provided. It should be emphasized that the package is efficient on both vector and raster data structures, ensuring a wide range of applicability in urban data science and beyond.

A new approach for investigating spatial relationships of ichnofossils: a case study of Ediacaran–Cambrian animal traces

AbstractTrace fossils record foraging behaviors, the search for resources in patchy environments, of animals in the rock record. Quantification of the strength, density, and nature of foraging behaviors enables the investigation of how these may have changed through time. Here, we present a novel approach to explore such patterns using spatial point process analyses to quantify the scale and strength of ichnofossil spatial distributions on horizontal bedding planes. To demonstrate the utility of this approach, we use two samples from the terminal Ediacaran Shibantan Member in South China (between 551 and 543 Ma) and the early Cambrian Nagaur Sandstone in northwestern India (between 539 and 509 Ma). We find that ichnotaxa on both surfaces exhibited significant nonhomogeneous lateral patterns, with distinct levels of heterogeneity exhibited by different types of trace fossils. In the Shibantan, two ichnotaxa show evidence for mutual positive aggregation over a shared resource, suggesting the ability to focus on optimal resource areas. Trace fossils from the Nagaur Sandstone exhibit more sophisticated foraging behavior, with greater niche differentiation. Critically, mark correlation functions highlight significant spatial autocorrelation of trace fossil orientations, demonstrating the greater ability of these Cambrian tracemakers to focus on optimal patches. Despite potential limitations, these analyses hint at changes in the development and optimization of foraging at the Ediacaran/Cambrian transition and highlight the potential of spatial point process analysis to tease apart subtle differences in behavior in the trace fossil record.

Facilitating corals in an early Silurian deep‐water assemblage

AbstractCorals are powerful ecosystem engineers and can form reef communities with extraordinary biodiversity through time. Understanding the processes underlying the spatial distribution of corals allows us to identify the key biological and physical processes that structure coral communities and how these processes and interactions have evolved. However, few spatial ecology studies have been conducted on coral assemblages in the fossil record. Here we use spatial point process analysis (SPPA) to investigate the ecological interactions of an in situ tabulate and rugose coral community (n = 199), preserved under volcanic ash in the Silurian of Ireland. SPPA is able to identify many different sorts of interactions including dispersal limitation and competition within and between taxa. Our SPPA found that the spatial distribution of rugose corals were best modelled by Thomas clusters (pd = 0.834), indicating a single dispersal episode and that the tabulate corals were best modelled by double Thomas clusters (pd = 0.820), indicating two dispersal episodes. Further, the bivariate distribution was best modelled by linked double clusters (pd = 0.970), giving significant evidence of facilitation between the tabulate and rugose populations, and identifying the facilitators in this community to be the tabulate corals. This interaction could be an important ecological driver for enabling the aggregation of sessile organisms over long temporal periods and facilitation may help to explain trends in reef diversity and abundance during the Ordovician biodiversification and in the early Silurian.

Mortality, Population and Community Dynamics of the Glass Sponge Dominated Community “The Forest of the Weird” From the Ridge Seamount, Johnston Atoll, Pacific Ocean

ecosystem dynamics of benthic communities depend on the relative importance of organism reproductive traits, environmental factors, inter-specific interactions and mortality processes. fine-scale community ecology of sessile organisms can be investigated using spatial analyses because the position of the specimens on the substrate (their spatial positions) reflects the biological and ecological processes that they were subject to in-life. Consequently, spatial point process analyses (SPPA) and Bayesian network inference (BNI) can be used to reveal key insights into the ecological dynamics of these deep-sea communities. Here we use these analyses to investigate the ecology of deep-sea glass sponge dominated community The Forest of the Weird (2442m depth, Ridge Seamount, Johnston Atoll, Pacific Ocean). A 3D reconstruction was made of this community using photogrammetry of video stills taken from high-resolution ROV video. community was dominated by two genera of Hexactinellids: Farreidae Aspidoscopulia sp. and Euplectellidae Bolosominae sp. with octocorals Narella bowersi, Narella macrocalyx and Rhodaniridogorgia also present in large proportions. SPPA of the dead versus alive organisms revealed a random distribution of dead amongst the living, showing a non-density dependent cause of death for the majority of taxa. However, in the high-density ridge crest region there was non-random aggregation of dead specimens, revealing density-dependent mortality for Aspidoscopulia. SPPA showed that the glass sponges and octocorals were each most strongly influenced by different underlying processes, and reacted to the environmental conditions differently. octocorals responded to higher density areas with increased intra-specific competition, whilst the glass-sponges seemed impervious to a doubling of specimen density. BNI found that mutual habitat associations between different taxa resulted in inter-specific competition at larger (2-4m) spatial scales, with instances of competition at small-spatial scales (<0.75m) in the higher-density ridge crest section. To our knowledge, this study is the first to analyze the mortality, population and community dynamics of a deep-sea sponge community using spatial point process analyses. Our results provide the first insight into the variety of ecological behaviors of these different glass sponges and octocorals, and show how these different organisms have developed diverse responses to the biological and environmental gradients within their habitat.

Reconstructing the ecology of a Jurassic pseudoplanktonic raft colony

Pseudoplanktonic crinoid raft colonies are an enigma of the Jurassic. These raft colonies are thought to have developed as floating filter-feeding communities due to an exceptionally rich oceanic niche, high in the water column enabling them to reach large densities on these log rafts. However, this pseudoplanktonic hypothesis has not been quantitatively tested, and there remains some doubt that this mode of life was possible. The ecological structure of the crinoid colony is resolved using spatial point process analyses and the duration estimates of the floating system until sinking using moisture diffusion models. Using spatial analysis, we found that the crinoids would have trailed preferentially positioned at the back of the floating log in the regions of least resistance, consistent with a floating, not benthic ecology. Additionally, we found using a series of moisture diffusion models at different log densities and sizes that ecosystem collapse did not take place solely due to colonies becoming overladen as previously assumed. Our analyses have found that these crinoid colonies studied could have existed for more than 10 years, even up to 20 years, exceeding the life expectancy of modern documented raft systems with possible implications for the role of modern raft communities in the biotic colonization of oceanic islands and intercontinental dispersal of marine and terrestrial species.

The influence of environmental setting on the community ecology of Ediacaran organisms.

The broad-scale environment plays a substantial role in shaping modern marine ecosystems, but the degree to which palaeocommunities were influenced by their environment is unclear. To investigate how broad-scale environment influenced the community ecology of early animal ecosystems, we employed spatial point process analyses (SPPA) to examine the community structure of seven late Ediacaran (558-550 Ma) bedding-plane assemblages drawn from a range of environmental settings and global localities. The studied palaeocommunities exhibit marked differences in the response of their component taxa to sub-metre-scale habitat heterogeneities on the seafloor. Shallow-marine (nearshore) palaeocommunities were heavily influenced by local habitat heterogeneities, in contrast to their deeper-water counterparts. The local patchiness within shallow-water communities may have been further accentuated by the presence of grazers and detritivores, whose behaviours potentially initiated a propagation of increasing habitat heterogeneity of benthic communities from shallow to deep-marine depositional environments. Higher species richness in shallow-water Ediacaran assemblages compared to deep-water counterparts across the studied time-interval could have been driven by this environmental patchiness, because habitat heterogeneities increase species richness in modern marine environments. Our results provide quantitative support for the 'Savannah' hypothesis for early animal diversification-whereby Ediacaran diversification was driven by patchiness in the local benthic environment.

Attraction or Repulsion?: Testing Coagglomeration of Innovation between Firm and University

Agglomeration theory supports and existing findings confirm the geographical proximity of similar firms and spatial attraction of firms to universities. In addition to that, we are able to identify whether universities as one type of innovative units are attracted by firm-type innovators and the size of such attraction. Testing the bidirectional spatial innovation linkage contributes to the debate on firm- or university-led innovation. Using a large patent dataset from Shenzhen, the first innovation-led city in the People’s Republic of China, and employing a spatial point process analysis technique, underutilized in the literature that allows the bidirectional testing of coagglomeration, we find varying attraction distances between the same type of innovative units and across university–firm innovation pairs. Attractions are not only limited to identical technology fields but also generate coagglomerations across different technology fields of firms and universities. We find the attraction from firms to universities is more than that from universities to firms. Support is offered to the integration of firms into the university-led innovation clusters in science parks; firm innovation in patent fields like human necessities, physics, and electrical deserve more policy focus to benefit university research and innovation.

Spatial Point Pattern Analysis of Congenital Heart Defects in Lanzhou, China

Background/Aim: Analysis of spatial point processes has been used to investigate spatial variation of disease risks, and their relationships with environmental factors; however, few studies have been conducted in China, where emissions from industries and heavy traffic pose serious public health concerns. The objective of this work is to investigate the impacts of major point sources and road networks on congenital heart defects (CHD) risks in Lanzhou, China, where maternal exposure to ambient air pollution was linked to increased risks of CHD. Methods: From a Lanzhou birth cohort during 2010-2012, 8,227 singleton live births with home addresses in the city urban area were included in this study. K, L, and Kcross functions were used to detect clustering tendency of the CHD (n=65) and healthy infants (n=8162). Kernel density ratio was used to identify potential clusters of CHD cases adjusting for the distribution of healthy infants. Poisson point process model was used to model intensity of CHD cases as a function of emission-weighted distance to major point sources (power plants and cement factories), road length density within 100m buffers, intensity of healthy infants, maternal income and education. Results: Similar significant clustering patterns were identified for CHD cases and healthy infants. Adjusting for the distribution of healthy infants, maternal income and education, CHD risks were significantly associated with increased road length density within 100 m buffer (RR: 1.07) and decreased emission-weighted distance from major point sources (RR: 0.07). Chi-square test with quadrat counts and Kolmogorov-Smirnov test validated the model and showed no spatial clustering in residuals. Conclusions: Results indicate proximity to major point sources and road network have adverse impacts on newborn’s health in Lanzhou, China. The identified clusters of CHD cases might allow policy makers or future mothers to make informed decisions regarding exposure control and risk management.

Spatial analyses of Ediacaran communities at Mistaken Point

AbstractBedding-plane assemblages of Ediacaran fossils from Mistaken Point, Newfoundland, are among the oldest known records of complex multicellular life on Earth (dated to ~565 Ma). The in situ preservation of these sessile but otherwise deeply enigmatic organisms means that statistical analyses of specimen positions can be used to illuminate their underlying ecological dynamics, including the interactions between taxa.Fossil assemblages on Mistaken Point D and E surfaces were mapped to millimeter accuracy using differentiated GPS. Spatial correlations between 10 well-defined taxa (Bradgatia, Charniid,Charniodiscus,Fractofusus, Ivesheadiomorphs, Lobate Discs,Pectinifrons,Plumeropriscum,Hiemalora, andThectardis) were identified using Bayesian network inference (BNI), and then described and analyzed using spatial point-process analysis. BNI found that the E-surface community had a complex web of interactions and associations between taxa, with all but one taxon (Thectardis) interacting with at least one other. The unique spatial distribution ofThectardissupports previous, morphology-based arguments for its fundamentally distinct nature. BNI revealed that the D-surface community showed no interspecific interactions or associations, a pattern consistent with a homogeneous environment.On the E surface, all six of the abundant taxonomic groups (Fractofusus,Bradgatia, Charniid,Charniodiscus,Thectardis, andPlumeropriscum) were found to have a unique set of interactions with other taxa, reflecting a broad range of underlying ecological responses. Four instances of habitat associations were detected between taxa, of which two (Charniodiscus–PlumeropriscumandPlumeropriscum–Fractofusus) led to weak competition for resources. One case of preemptive competition between Charniid and Lobate Discs was detected. There were no instances of interspecific facilitation. Ivesheadiomorph interactions mirror those ofFractofususandCharniodiscus, identifying them as a form-taxonomic grouping of degradationally homogenized taphomorphs. The absence of increased fossil abundance in proximity to these taphomorphs argues against scavenging or saprophytic behaviors dominating the E-surface community.

Application of optimal data-based binning method to spatial analysis of ecological datasets

Investigation of highly structured datasets to unveil statistical regularities is of major importance in complex system research. The first step is to choose the scale at which to observe the process, the most informative scale being the one that includes the important features while disregarding noisy details in the data. In the investigation of spatial patterns, the optimal scale defines the optimal bin size of the histogram in which to visualize the empirical density of the pattern. In this paper we investigate a method proposed recently by K.H. Knuth to find the optimal bin size of an histogram as a tool for statistical analysis of spatial point processes. We test it through numerical simulations on various spatial processes which are of interest in ecology. We show that Knuth optimal bin size rule reducing noisy fluctuations performs better than standard kernel methods to infer the intensity of the underlying process. Moreover it can be used to highlight relevant spatial characteristics of the underlying distribution such as space anisotropy and clusterization. We apply these findings to analyse cluster-like structures in plants’ arrangement of Barro Colorado Island rainforest.

Fast food and liquor store density, co-tenancy, and turnover: Vice store operations in Chicago, 1995–2008

Fast food restaurants and liquor stores—vice stores—have been shown to be more prevalent in predominantly Black and low income U.S. neighborhoods, and are associated with a number of health risks and social ills. The purpose of this study was to investigate in the City of Chicago vice store density and spatial distribution as a function of racial, socioeconomic, and other population characteristics; to examine spatial clustering among these outlets; and to study how store turnover follows population change over a 13-year period. We used spatial point process analysis to fit linear and non-linear models for the intensity function of stores. Spatial clustering was estimated using the K function. We found heterogeneous associations between stores and population characteristics, with the most consistent finding being a positive association between percent Black and liquor store exposure. A high degree of spatial clustering was evident, and liquor stores were more likely to stay in business over time than fast food restaurants. However, when liquor stores closed, they were more likely to be replaced by non-vice businesses. Results suggest that vice stores are associated with lower positions in racial and socioeconomic hierarchies, and this patterning is often durable over time.

What’s a Stream Without Water? Disproportionality in Headwater Regions Impacting Water Quality

Headwater streams are critical components of the stream network, yet landowner perceptions, attitudes, and property management behaviors surrounding these intermittent and ephemeral streams are not well understood. Our research uses the concept of watershed disproportionality, where coupled social-biophysical conditions bear a disproportionate responsibility for harmful water quality outcomes, to analyze the potential influence of riparian landowner perceptions and attitudes on water quality in headwater regions. We combine social science survey data, aerial imagery, and an analysis of spatial point processes to assess the relationship between riparian landowner perceptions and attitudes in relation to stream flow regularity. Stream flow regularity directly and positively shapes landowners' water quality concerns, and also positively influences landowners' attitudes of stream importance-a key determinant of water quality concern as identified in a path analysis. Similarly, riparian landowners who do not notice or perceive a stream on their property are likely located in headwater regions. Our findings indicate that landowners of headwater streams, which are critical areas for watershed-scale water quality, are less likely to manage for water quality than landowners with perennial streams in an obvious, natural channel. We discuss the relationships between streamflow and how landowners develop understandings of their stream, and relate this to the broader water quality implications of headwater stream mismanagement.

Determinants of tree mortality in mixed old-growth Nothofagus forest

Rates and spatial patterns of tree mortality were examined using long-term data from old-growth, mixed-species forests of the Maruia Valley, South Island, New Zealand. The aim of the study was to investigate patterns of tree mortality in two common, co-occurring species, Nothofagus fusca (Hook. f.) Oerst. and Nothofagusmenziesii (Hook. f.) Oerst. The dynamics of three old-growth stands were followed over a 23-year period, using plots sized 0.8–1.0ha. In total the fates of 1138 individual N. fusca and 1611 N. menziesii were recorded, which had annual mortality rates of 0.016and 0.0089per year, respectively. Differing spatial and size-related patterns of mortality were found between species. For both species, individual-based logistic models showed that slower growing trees were more likely to die than faster growing trees. N. fusca trees growing in previously disturbed stands were also more likely to die than those in undisturbed stands. Spatial point process analysis showed that dead N. fusca trees were spatially aggregated, and were segregated from living trees, a pattern that was consistent across both small and large trees. Dead N. menziesii were spatially aggregated, but were not segregated from living trees. Aggregated mortality of N. fusca trees should favor regeneration of this light-demanding species in large canopy gaps, perpetuating its aggregated distribution, whereas the lower mortality of shade-tolerant N. menziesii allows this species to persist. Our results demonstrate that performance differences in coexisting tree species can be manifested spatially. Between species, different mortality patterns may have implications for sustainable forest management.

Hierarchical spatial point process analysis for a plant community with high biodiversity

A complex multivariate spatial point pattern of a plant community with high biodiversity is modelled using a hierarchical multivariate point process model. In the model, interactions between plants with different post-fire regeneration strategies are of key interest. We consider initially a maximum likelihood approach to inference where problems arise due to unknown interaction radii for the plants. We next demonstrate that a Bayesian approach provides a flexible framework for incorporating prior information concerning the interaction radii. From an ecological perspective, we are able both to confirm existing knowledge on species’ interactions and to generate new biological questions and hypotheses on species’ interactions.

Bayesian analysis of spatial point processes in the neighbourhood of Voronoi networks

A model for an inhomogeneous Poisson process with high intensity near the edges of a Voronoi tessellation in 2D or 3D is proposed. The model is analysed in a Bayesian setting with priors on nuclei of the Voronoi tessellation and other model parameters. An MCMC algorithm is constructed to sample from the posterior, which contains information about the unobserved Voronoi tessellation and the model parameters. A major element of the MCMC algorithm is the reconstruction of the Voronoi tessellation after a proposed local change of the tessellation. A simulation study and examples of applications from biology (animal territories) and material science (alumina grain structure) are presented.

Beyond the semivariogram: Patterns, scale, and hydrology in a semi-arid landscape

As an alternative to geostatistical modeling, we characterized the hydrology of a semi-arid landscape in southeastern Washington state, USA, by coupling spatial patterns identified in the distributions of relative relief and vegetation with the influence each has on soil moisture storage and evapotranspiration at the appropriate scale. Gauging precipitation, soil moisture, and evapotranspiration over a two-year period while concurrently mapping relative relief and vegetation distributions at three scales ranging from centimeters to 90 m, we determined that soil moisture and soil moisture storage are significantly greater in topographic concavities than in convexities at the microrelief (20–50 cm) scale but are not significantly different in relief features at larger scales. A generalized microrelief surface produced using a two-dimensional Fourier transformation provided a good representation of the distribution of soil moisture within microrelief when scaled to soil moisture values. Applying a spatial point process analysis we determined that big sage are randomly distributed across the landscape at all scales, suggesting that lysimeter-derived sage evapotranspiration rates also be distributed randomly across the landscape. Where sage were not present, we applied an autoregressive moving-average model conditioned on grass lysimeter measurements to derive evapotranspiration rates. Combining these hydrologic spatial patterns derived from distributions in relief and vegetation with measured precipitation inputs and evapotranspiration outputs, we created a spatially distributed model of soil moisture which we tested against measured values over an eight-week period. The model provides accurate characterization of soil moisture, allows estimates of soil moisture between measurement points, permits extrapolation of soil moisture distributions outside the gauged area, and maintains small-scale variability when aggregating soil moisture to successively larger scales.

Spatial point-process statistics: concepts and application to the analysis of lead contamination in urban soil

This article explores the use of spatial point-process analysis as an aid to describe topsoil lead distribution in urban environments. The data used were collected in Glebe, an inner suburb of Sydney. The approach focuses on the locations of punctual events defining a point pattern, which can be statistically described through local intensity estimates and between-point distance functions. F-, G- and K-surfaces of a marked spatial point pattern were described and used to estimate nearest distance functions over a sliding band of quantiles belonging to the marking variable. This provided a continuous view of the point pattern properties as a function of the marking variable. Several random fields were simulated by selecting points from random, clustered or regular point processes and diffusing them. Recognition of the underlying point process using variograms derived from dense sampling was difficult because, structurally, the variograms were very similar. Point-event distance functions were useful complimentary tools that, in most cases, enabled clear recognition of the clustered processes. Spatial sampling quantile point pattern analysis was defined and applied to the Glebe data set. The analysis showed that the highest lead concentrations were strongly clustered. The comparison of this data set with the simulation confidence limits of a Poisson process, a short-radius clustered point process and a geostatistical simulation showed a random process for the third quartile of lead concentrations but strong clustering for the data in the upper quartile. Thus the distribution of topsoil lead concentrations over Glebe may have resulted from several contamination processes, mainly from regular or random processes with large diffusion ranges and short-range clustered processes for the hot spots. Point patterns with the same characteristics as the Glebe experimental pattern could be generated by separate additive geostatistical simulation. Spatial sampling quantile point patterns statistics can, in an easy and accurate way, be used complementarily with geostatistical methods. Copyright © 2005 John Wiley & Sons, Ltd.

Application of Total Least Squares for Spatial Point Process Analysis

The total-least-squares approach is a relatively new adjustment method of estimating parameters in linear models that include error in all variables. Specifically, given an overdetermined set of linear equations y≈Aξ where y is the observation vector, A is a positive defined data matrix, and ξ is the vector of unknown parameters, the total-least-squares problem is concerned with estimating ξ providing that the number of observations n is larger than the number of parameters to be estimated and given that both the observation vector y and the data matrix A are subjected to errors and need to be adjusted. This model is different from the classical least-squares model where only the observation vector y is subjected to errors. This paper starts with a brief summary of the least-squares approach and then explains how one can modify the approach to include error in all variables using the generalized least-squares technique. Then the total-least-squares problem is presented along with its formulas and the procedures used to solve it. Finally, the total-least-squares approach is used to determine the trend in a spatial point process.

Development of intrinsic connections in cat somatosensory cortex

We explored the development of clustered connections in cat somatosensory cortex by using intracortical injections of biocytin and carbocyanine dye (DiI). Biocytin injections in adults revealed clusters of retrogradely labeled cells, affirming earlier reports of the patchy nature of corticocortical connectivity in the adult cat somatosensory cortex. On postnatal days (PNDs) 1 and 3, a diffuse distribution of axons and labeled cells were found after DiI injections. A dramatic rearrangement of connections had taken place by PND 6. Well-defined clusters of labeled cells surrounded the injection site. Intercluster zones were relatively free of labeled cells and contained few labeled axons. At later ages (PNDs 12 and 15), a clear patchy distribution of intrinsic connections was seen. We analyzed neuronal clustering by using spatial point process analysis, which corroborated our qualitative observations. The density of labeled neurons was significantly higher in clusters than in the intercluster zones on PND 6. Autocorrelations run on profile plots of optical density values, along paths parallel to the edge of the injection sites (reflecting axonal distributions), revealed significant periodicity (P < .05; center-to-center approximately 750 pm) by PND 6. These data demonstrate that corticocortical connections in cat somatosensory cortex develop from a diffuse network at birth, which transforms into a patchy network by the end of the 1st postnatal week. This is substantially earlier than the development of local circuits in the cat visual cortex and may reflect fundamental differences in the organization of the two cortices.