Denominators of cluster variables

Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category inducing a correspondence between clusters and cluster-tilting objects. Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T.Comment: 22 pages; one figur

Cluster structures for 2-Calabi-Yau categories and unipotent groups

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised, especially Chapter III replaces the old Chapter III and I

A geometric model of tube categories

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A-double-infinity.Comment: 15 pages, 7 figures. Discussion of maximal rigid objects and triangulations at end of Section 3. Minor correction

Spike-based coupling between single neurons and populations across rat sensory cortices, perirhinal cortex, and hippocampus

Cortical computations require coordination of neuronal activity within and across multiple areas. We characterized spiking relationships within and between areas by quantifying coupling of single neurons to population firing patterns. Single-neuron population coupling (SNPC) was investigated using ensemble recordings from hippocampal CA1 region and somatosensory, visual, and perirhinal cortices. Within-area coupling was heterogeneous across structures, with area CA1 showing higher levels than neocortical regions. In contrast to known anatomical connectivity, between-area coupling showed strong firing coherence of sensory neocortices with CA1, but less with perirhinal cortex. Cells in sensory neocortices and CA1 showed positive correlations between within- and between-area coupling; these were weaker for perirhinal cortex. All four areas harbored broadcasting cells, connecting to multiple external areas, which was uncorrelated to within-area coupling strength. When examining correlations between SNPC and spatial coding, we found that, if such correlations were significant, they were negative. This result was consistent with an overall preservation of SNPC across different brain states, suggesting a strong dependence on intrinsic network connectivity. Overall, SNPC offers an important window on cell-to-population synchronization in multi-area networks. Instead of pointing to specific information-coding functions, our results indicate a primary function of SNPC in dynamically organizing communication in systems composed of multiple, interconnected areas

Neural correlates of object identity and reward outcome in the sensory cortical-hippocampal hierarchy:coding of motivational information in perirhinal cortex

Neural circuits support behavioral adaptations by integrating sensory and motor information with reward and error-driven learning signals, but it remains poorly understood how these signals are distributed across different levels of the corticohippocampal hierarchy. We trained rats on a multisensory object-recognition task and compared visual and tactile responses of simultaneously recorded neuronal ensembles in somatosensory cortex, secondary visual cortex, perirhinal cortex, and hippocampus. The sensory regions primarily represented unisensory information, whereas hippocampus was modulated by both vision and touch. Surprisingly, the sensory cortices and the hippocampus coded object-specific information, whereas the perirhinal cortex did not. Instead, perirhinal cortical neurons signaled trial outcome upon reward-based feedback. A majority of outcome-related perirhinal cells responded to a negative outcome (reward omission), whereas a minority of other cells coded positive outcome (reward delivery). Our results highlight a distributed neural coding of multisensory variables in the cortico-hippocampal hierarchy. Notably, the perirhinal cortex emerges as a crucial region for conveying motivational outcomes, whereas distinct functions related to object identity are observed in the sensory cortices and hippocampus