The function of the neocortex is fundamentally determined by its repeating microcircuit motif, but also by its rich, hierarchical, interregional structure with a highly specific laminar architecture. The last decade has seen the emergence of extensive new data sets on anatomy and connectivity at the whole brain scale, providing promising new directions for studies of cortical function that take into account the inseparability of whole-brain and microcircuit architectures. Here, we present a data-driven computational model of the anatomy of non-barrel primary somatosensory cortex of juvenile rat, which integrates whole-brain scale data while providing cellular and subcellular specificity. This multiscale integration was achieved by building the morphologically detailed model of cortical circuitry embedded within a volumetric, digital brain atlas. The model consists of 4.2 million morphologically detailed neurons belonging to 60 different morphological types, placed in the nonbarrel subregions of the Paxinos and Watson atlas. They are connected by 13.2 billion synapses determined by axo-dendritic overlap, comprising local connectivity and long-range connectivity defined by topographic mappings between subregions and laminar axonal projection profiles, both parameterized by whole brain data sets. Additionally, we incorporated core- and matrix-type thalamocortical projection systems, associated with sensory and higher-order extrinsic inputs, respectively. An analysis of the modeled synaptic connectivity revealed a highly nonrandom topology with substantial structural differences but also synergy between local and long-range connectivity. Long-range connections featured a more divergent structure with a comparatively small group of neurons serving as hubs to distribute excitation to far away locations. Taken together with analyses at different spatial granularities, these results support the notion that local and interregional connectivity exist on a spectrum of scales, rather than as separate and distinct networks, as is commonly assumed. Finally, we predicted how the emergence of primary sensory cortical maps is constrained by the anatomy of thalamo-cortical projections. A subvolume of the model comprising 211,712 neurons in the front limb, jaw, and dysgranular zone has been made freely and openly available to the community.

A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau.We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1 correspondence with the minimal recurrent configurations of the Abelian sandpile model on the corresponding Ferrers graph.Another bijection between EW-tableaux and tree-like tableaux, via spanning trees on the corresponding Ferrers graphs, connects the tree-like tableaux to the minimal recurrent configurations of the Abelian sandpile model on these graphs. We introduce a variation on the EW-tableaux, which we call NEW-tableaux, and present bijections from these to Le-tableaux and tree-like tableaux. We also present results on various properties of and statistics on EW-tableaux and NEW-tableaux, as well as some open problems on these.1. The top row of T has a 1 in every cell. 2. Every other row has at least one cell containing a 0. 3. No four cells of T that form the corners of a rectangle have 0s in two diagonally opposite corners and 1s in the other two.The size of a EW-tableau is one less than the sum of its number of rows and number of columns.An example of a EW-tableau is shown in Figure 2.1.Remark 1.2. Ehrenborg and van Willigenburg considered colorings corresponding to tableaux that in our definition would have a unique fixed, but arbitrary, row of all 1s, and no column with all 0s. As all our results naturally restrict to tableaux of a fixed shape, it is irrelevant where this fixed all-1s row is chosen, and as it facilitates the presentation of our results, we choose to always make this the top row. That implies, of course, that no column has all 0s. The choice of this row being arbitrary is directly linked to the fact, proved by Greene and Zaslavsky [16, Thm. 7.3], that the number of acyclic orientations, with a unique sink, of a graph is independent of which vertex is chosen as the unique sink. In fact, it is easy to show that given an acyclic orientation with a unique

We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of which corresponds to a canonical toppling of the corresponding configurations.

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals [1, π] in this poset, for any permutation π with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in π that are consecutive in value. As a result of this we show that the Möbius function is unbounded on the poset of all permutations. We show that the Möbius function is zero on any interval [1, π] where π has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the Möbius function on some other intervals of permutations with at most one descent.1. If i < d and π 1 = 1 then µ(π) = ±i,

In motor-related brain regions, movement intention has been successfully decoded from in-vivo spike train by isolating a lower-dimension manifold that the high-dimensional spiking activity is constrained to. The mechanism enforcing this constraint remains unclear, although it has been hypothesized to be implemented by the connectivity of the sampled neurons. We test this idea and explore the interactions between local synaptic connectivity and its ability to encode information in a lower dimensional manifold through simulations of a detailed microcircuit model with realistic sources of noise. We confirm that even in isolation such a model can encode the identity of different stimuli in a lower-dimensional space. We then demonstrate that the reliability of the encoding depends on the connectivity between the sampled neurons by specifically sampling populations whose connectivity maximizes certain topological metrics. Finally, we developed an alternative method for determining stimulus identity from the activity of neurons by combining their spike trains with their recurrent connectivity. We found that this method performs better for sampled groups of neurons that perform worse under the classical approach, predicting the possibility of two separate encoding strategies in a single microcircuit.