A Derivation of Three-Dimensional Inertial Transformations

The derivation of the transformations between inertial frames made by Mansouri and Sexl is generalised to three dimensions for an arbitrary direction of the velocity. Assuming lenght contraction and time dilation to have their relativistic values, a set of transformations kinematically equivalent to special relativity is obtained. The ``clock hypothesis'' allows the derivation to be extended to accelerated systems. A theory of inertial transformations maintaining an absolute simultaneity is shown to be the only one logically consistent with accelerated movements. Algebraic properties of these transformations are discussed. Keywords: special relativity, synchronization, one-way velocity of light, ether, clock hypothesis.Comment: 16 pages (A5), Latex, one figure, to be published in Found. Phys. Lett. (1997

The geometry of spontaneous spiking in neuronal networks

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior

Process Algebras

Process Algebras are mathematically rigorous languages with well defined semantics that permit describing and verifying properties of concurrent communicating systems. They can be seen as models of processes, regarded as agents that act and interact continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artifacts, embodied perhaps in computer hardware or software systems. Many different approaches (operational, denotational, algebraic) are taken for describing the meaning of processes. However, the operational approach is the reference one. By relying on the so called Structural Operational Semantics (SOS), labelled transition systems are built and composed by using the different operators of the many different process algebras. Behavioral equivalences are used to abstract from unwanted details and identify those systems that react similarly to external experiments

Shaping bursting by electrical coupling and noise

Gap-junctional coupling is an important way of communication between neurons and other excitable cells. Strong electrical coupling synchronizes activity across cell ensembles. Surprisingly, in the presence of noise synchronous oscillations generated by an electrically coupled network may differ qualitatively from the oscillations produced by uncoupled individual cells forming the network. A prominent example of such behavior is the synchronized bursting in islets of Langerhans formed by pancreatic \beta-cells, which in isolation are known to exhibit irregular spiking. At the heart of this intriguing phenomenon lies denoising, a remarkable ability of electrical coupling to diminish the effects of noise acting on individual cells. In this paper, we derive quantitative estimates characterizing denoising in electrically coupled networks of conductance-based models of square wave bursting cells. Our analysis reveals the interplay of the intrinsic properties of the individual cells and network topology and their respective contributions to this important effect. In particular, we show that networks on graphs with large algebraic connectivity or small total effective resistance are better equipped for implementing denoising. As a by-product of the analysis of denoising, we analytically estimate the rate with which trajectories converge to the synchronization subspace and the stability of the latter to random perturbations. These estimates reveal the role of the network topology in synchronization. The analysis is complemented by numerical simulations of electrically coupled conductance-based networks. Taken together, these results explain the mechanisms underlying synchronization and denoising in an important class of biological models

Category Theory and Model-Driven Engineering: From Formal Semantics to Design Patterns and Beyond

There is a hidden intrigue in the title. CT is one of the most abstract mathematical disciplines, sometimes nicknamed "abstract nonsense". MDE is a recent trend in software development, industrially supported by standards, tools, and the status of a new "silver bullet". Surprisingly, categorical patterns turn out to be directly applicable to mathematical modeling of structures appearing in everyday MDE practice. Model merging, transformation, synchronization, and other important model management scenarios can be seen as executions of categorical specifications. Moreover, the paper aims to elucidate a claim that relationships between CT and MDE are more complex and richer than is normally assumed for "applied mathematics". CT provides a toolbox of design patterns and structural principles of real practical value for MDE. We will present examples of how an elementary categorical arrangement of a model management scenario reveals deficiencies in the architecture of modern tools automating the scenario.Comment: In Proceedings ACCAT 2012, arXiv:1208.430

Towards a homotopy theory of process algebra

This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between them for every process name of every process algebra with any synchronization algebra using a notion of labelled flow. This interpretation of process algebra satisfies the paradigm of higher dimensional automata (HDA): one non-degenerate full $n$-dimensional cube (no more no less) in the underlying space of the time flow corresponding to the concurrent execution of $n$ actions. This result will enable us in future papers to develop a homotopical approach of process algebras. Indeed, several homological constructions related to the causal structure of time flow are possible only in the framework of flows.Comment: 33 pages ; LaTeX2e ; 1 eps figure ; package semantics included ; v2 HDA paradigm clearly stated and simplification in a homotopical argument ; v3 "bug" fixed in notion of non-twisted shell + several redactional improvements ; v4 minor correction : the set of labels must not be ordered ; published at http://intlpress.com/HHA/v10/n1/a16