Storage of phase-coded patterns via STDP in fully-connected and sparse network: a study of the network capacity

We study the storage and retrieval of phase-coded patterns as stable dynamical attractors in recurrent neural networks, for both an analog and a integrate-and-fire spiking model. The synaptic strength is determined by a learning rule based on spike-time-dependent plasticity, with an asymmetric time window depending on the relative timing between pre- and post-synaptic activity. We store multiple patterns and study the network capacity. For the analog model, we find that the network capacity scales linearly with the network size, and that both capacity and the oscillation frequency of the retrieval state depend on the asymmetry of the learning time window. In addition to fully-connected networks, we study sparse networks, where each neuron is connected only to a small number z << N of other neurons. Connections can be short range, between neighboring neurons placed on a regular lattice, or long range, between randomly chosen pairs of neurons. We find that a small fraction of long range connections is able to amplify the capacity of the network. This imply that a small-world-network topology is optimal, as a compromise between the cost of long range connections and the capacity increase. Also in the spiking integrate and fire model the crucial result of storing and retrieval of multiple phase-coded patterns is observed. The capacity of the fully-connected spiking network is investigated, together with the relation between oscillation frequency of retrieval state and window asymmetry

Khinchin's theorem in Teichmüller dynamics

This thesis is concerned with two themes which are strictly linked with each other, and therefore will be developed in parallel. The first one is dynamics in Teichmueller space. The Teichmueller space of a (topological, closed and orientable) surface S is defined as the set of the complex structures one can endow S with, up to isotopies. Such a space can be given a structure of geodesic metric space. The description of this structure requires the notion of flat structures on the underlying surface, i.e. flat Riemannian metrics with conical singularities, such that a foliation in straight lines in each direction is defined. The space of all flat structures is a sort of tangent bundle to the Teichmueller space, and the geodesic flow, knows as Teichmueller flow, has a simple description in these terms. It becomes interesting from a dynamical viewpoint when projected onto the moduli space, namely the set of the complex structures up to diffeomorphisms. Invariant subspaces under the flow are called strata; we are concerned in particular with dynamics in the strata made up by translation structures, a subspecies of the flat structures. The second theme treated in this work are interval exchange maps (i.e.m.s)i.e. injective maps of an interval which are locally a translation except at finitely many singularities. They are completely determined by providing some combinatorial data as well as the length data of the sub-intervals. If one considers an adequate leftmost portion of the considered interval, the first return map of the i.e.m. on this portion is a new i.e.m.. This yields a dynamics on the parameter space for i.e.m.s, called Rauzy dynamics. The themes above are linked on two levels. First of all, if one fixes a translation surface, the first return map induced by the flow in vertical direction on a horizontal segment is an i.e.m.; and a `generic' i.e.m. can always be obtained this way. But a link at a higher level is possible, too: the Teichmueller flow admits a transverse section such that the return map can be interpreted as Rauzy dynamics. Chapter 0 of the thesis is an introduction: it includes the preliminary material from the theory of dynamical systems which will be used in this work, as well as a description of the simplest case of the theory, represented by flat tori and rotations of the circumference. In Chapter 1 Teichmueller dynamics is formally, but rapidly, introduced; whereas Chapter 2 is concerned with the formalism related to interval exchange maps and Rauzy dynamics. Moreover, it is explained how it is possible to switch from this setting to the one of translation structures, and conversely. The first half of Chapter 3 treats, still in an extremely concise manner, classical questions related to the themes above. In particular it deals with ergodicity of i.e.m.s and of the Teichmueller and Rauzy dynamics and briefly introduces the Kontsevich-Zorich cocycle. The chapter ends with a technical discussion needed for the results tackled in the following chapters: its protagonists will be the reduced triples for an i.e.m. T, namely triples (b,a;n) where b is a singular point for $T^{-1}$, a is a singular point for T, and n is a positive integer, such that no singularities for $T^{-1},...,T^-n$ appears between $T^n(b)$ and a. Chapter 4 thus deals with a first generalisation of a theorem of A.Ya. Khinchin, found by Luca Marchese (2010). The Khinchin theorem in its classical formulation states a condition for a Diophantine inequality to have finitely, or infinitely many, solutions. Its generalisation to i.e.m.s states: Let f(n) be a positive, decreasing sequence. We are concerned with the quantity of solutions (b,a;n) to the condition $|T^n(a)-b|<f(n)$ for a fixed i.e.m. T, where b is a singular point of $T^-1$, and a is a singular point for T. If the sequence f(n) has a finite sum, then solutions are finitely many for almost any T; if nf(n) is still a decreasing sequence, with infinite sum, then solutions are infinitely many for almost any T. This result will be partially proved. It is interesting not only as a property of singularities of an i.e.m., but also because it yields a weaker version of a theorem of Jon Chaika, which states a similar property for generic points. Chapter 5 is again about translation surfaces. The theorem above is restated as a property of lengths of connections, namely segments connecting two singular points on a translation surface. Hence it is possible to gain another result of Chaika, which gives a property of 'strong recurrence' of foliations. And, eventually, this restatement of the generalised Khinchin theorem yields a logarithm law for the Teichmueller flow: Let X be a translation surface, and let Sys(X) be the minimum length of a connection of X. Denote $g^t$ the Teichmueller flow. Then, for almost any X, it holds that $\limsup [-\log (Sys(g^t(X))]/(\log t)=1/2$

Scaling and universality in glass transition

Kinetic facilitated models and the Mode Coupling Theory (MCT) model B are within those systems known to exhibit a discontinuous dynamical transition with a two step relaxation. We consider a general scaling approach, within mean field theory, for such systems by considering the behavior of the density correlator and the dynamical susceptibility -^2. Focusing on the Fredrickson and Andersen (FA) facilitated spin model on the Bethe lattice, we extend a cluster approach that was previously developed for continuous glass transitions by Arenzon et al (Phys. Rev. E 90, 020301(R) (2014)) to describe the decay to the plateau, and consider a damage spreading mechanism to describe the departure from the plateau. We predict scaling laws, which relate dynamical exponents to the static exponents of mean field bootstrap percolation. The dynamical behavior and the scaling laws for both density correlator and dynamical susceptibility coincide with those predicted by MCT. These results explain the origin of scaling laws and the universal behavior associated with the glass transition in mean field, which is characterized by the divergence of the static length of the bootstrap percolation model with an upper critical dimension d_c=8.Comment: 16 pages, 9 figure

Disordered jammed packings of frictionless spheres

At low volume fraction, disordered arrangements of frictionless spheres are found in un--jammed states unable to support applied stresses, while at high volume fraction they are found in jammed states with mechanical strength. Here we show, focusing on the hard sphere zero pressure limit, that the transition between un-jammed and jammed states does not occur at a single value of the volume fraction, but in a whole volume fraction range. This result is obtained via the direct numerical construction of disordered jammed states with a volume fraction varying between two limits, $0.636$ and $0.646$. We identify these limits with the random loose packing volume fraction \rl and the random close packing volume fraction \rc of frictionless spheres, respectively

Balanced Objective-Quantifiers Method (BOQM) For Software Intensive Organizations Strategies

Spanish university facilitates a method to link the strategic management with Software and Process improvement based on measurement. The method uses the process philosophy to build measurable information in Indicators templates (Based on ISO/IEC 15939) and a Balanced Scorecard (BSC) template, the process is followed by the participation of SIO’s roles such as the CEO, TI director, CPO, and others measurement roles such as measurement analyst, measurement librarian, and the measurement user

Orthogonality criterion for banishing hydrino states from standard quantum mechanics

Orthogonality criterion is used to shown in a very simple and general way that anomalous bound-state solutions for the Coulomb potential (hydrino states) do not exist as bona fide solutions of the Schr\"{o}dinger, Klein-Gordon and Dirac equations.Comment: 6 page

Crossover properties from random percolation to frustrated percolation

We investigate the crossover properties of the frustrated percolation model on a two-dimensional square lattice, with asymmetric distribution of ferromagnetic and antiferromagnetic interactions. We determine the critical exponents nu, gamma and beta of the percolation transition of the model, for various values of the density of antiferromagnetic interactions pi in the range 0<pi<0.5. Our data are consistent with the existence of a crossover from random percolation behavior for pi=0, to frustrated percolation behavior, characterized by the critical exponents of the ferromagnetic 1/2-state Potts model, as soon as pi>0.Comment: 5 pages, 7 figs, RevTe