3,554 research outputs found
What grid cells convey about rat location
We characterize the relationship between the simultaneously recorded quantities of rodent grid cell firing and the position of the rat. The formalization reveals various properties of grid cell activity when considered as a neural code for representing and updating estimates of the rat's location. We show that, although the spatially periodic response of grid cells appears wasteful, the code is fully combinatorial in capacity. The resulting range for unambiguous position representation is vastly greater than the ≈1–10 m periods of individual lattices, allowing for unique high-resolution position specification over the behavioral foraging ranges of rats, with excess capacity that could be used for error correction. Next, we show that the merits of the grid cell code for position representation extend well beyond capacity and include arithmetic properties that facilitate position updating. We conclude by considering the numerous implications, for downstream readouts and experimental tests, of the properties of the grid cell code
Shape-dependent universality in percolation
The shape-dependent universality of the excess percolation cluster number and
cross-configuration probability on a torus is discussed. Besides the aspect
ratio of the torus, the universality class depends upon the twist in the
periodic boundary conditions, which for example are generally introduced when
triangular lattices are used in simulations.Comment: 11 pages, 3 figures, to be published in Physica
Cluster Analysis of the Ising Model and Universal Finite-Size Scaling
The recent progress in the study of finite-size scaling (FSS) properties of
the Ising model is briefly reviewed. We calculate the universal FSS functions
for the Binder parameter and the magnetization distribution function
for the Ising model on two-dimensional lattices with tilted
boundary conditions. We show that the FSS functions are universal for fixed
sets of the aspect ratio and the tilt parameter. We also study the
percolating properties of the Ising model, giving attention to the effects of
the aspect ratio of finite systems. We elucidate the origin of the complex
structure of for the system with large aspect ratio by the
multiple-percolating-cluster argument.Comment: 11 pages including 6 eps figures, elsart.sty, to appear in Physica
A class of infinite convex geometries
Various characterizations of finite convex geometries are well known. This
note provides similar characterizations for possibly infinite convex geometries
whose lattice of closed sets is strongly coatomic and lower continuous. Some
classes of examples of such convex geometries are given.Comment: 10 page
Towards MKM in the Large: Modular Representation and Scalable Software Architecture
MKM has been defined as the quest for technologies to manage mathematical
knowledge. MKM "in the small" is well-studied, so the real problem is to scale
up to large, highly interconnected corpora: "MKM in the large". We contend that
advances in two areas are needed to reach this goal. We need representation
languages that support incremental processing of all primitive MKM operations,
and we need software architectures and implementations that implement these
operations scalably on large knowledge bases.
We present instances of both in this paper: the MMT framework for modular
theory-graphs that integrates meta-logical foundations, which forms the base of
the next OMDoc version; and TNTBase, a versioned storage system for XML-based
document formats. TNTBase becomes an MMT database by instantiating it with
special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
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