Theta rhythmicity enhances learning in adaptive STDP

The classical STDP window captures changes of a synaptic weight in response to the relative timing of a pre and a postsynaptic spike (see e.g. Bi and Poo, 1998). Due to its static nature, however, it cannot account for nonlinear interactions between spikes. Several theoretical studies offer dynamic formulations for STDP, for example by modulating the synaptic weight change by variables like synaptic calcium concentration (Shouval et al., 2002) or somatic depolarisation (Clopath et al., 2010), or by introducing spike triplet interactions (Pfister and Gerstner, 2006). Here, we propose a new model which is formulated as a set of differential equations (Schmiedt et al., 2010). The weight change is given by a differential Hebbian learning rule, which reproduces the STDP window for spike pairs. To account for the effects of repeated neuronal firing on the synaptic weight, we introduce modulations of the spike impact, which act on exponential traces of the spiking activity. We found that this model captures a series of experiments on STDP with complex spike pattern in cortex (Froemke et al., 2006) and hippocampus (Wang et al., 2005). When applied to continuous firing rates, our approach allows us to analyze the effects of given time courses of firing rates on the synaptic weight change, i.e. the filter properties of STDP. For sinusoidal modulations of baseline firing rates we find the strongest weight changes for modulation frequencies in the theta band, which plays a key role in learning. Furthermore, weight modifications in the hippocampus are predicted to be most prominent for baseline rates of around 5Hz in striking agreement with experimental findings.
This suggests that STDP-dependent learning is mediated by theta oscillations and modulated by the background firing rate which are both testable predictions of our theory

A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology

In this article we address two issues. First, we explore to what extent the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] an isomorphism between Hamiltonian Floer homology and the singular homology is established. In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general. Depending on the minimal Maslov number, we construct for certain degrees two homomorphisms between Lagrangian Floer homology and singular homology. In degrees where both maps are defined we prove them to be isomorphisms. Examples show that this statement is sharp. Second, we construct two comparison homomorphisms between Lagrangian and Hamiltonian Floer homology. They underly no degree restrictions and are proven to be the natural analogs to the homomorphisms in singular homology induced by the inclusion map of the Lagrangian submanifold into the ambient symplectic manifold.Comment: 41 pages, 14 figures. v2: major revision, v3: included detailed transversality proofs. accepted by IMR

Comparing Survival Curves Using Rank Tests

Survival times of patients can be compared using rank tests in various experimental setups, including the two-sample case and the case of paired data. Attention is focussed on two frequently occurring complications in medical applications: censoring and tail alternatives. A review is given of the author's recent work on a new and simple class of censored rank tests. Various models for tail alternatives are discussed and the relation to censoring is demonstrated

Foundations of a Theory of Prominence in the Decimal System - Part II: Exactness Selection Rule, and Confirming Results

The information that is used to create a numerical response is typically diffuse, and cannot be described by a distribution. A criterion to describe the information is its range of reasonable alternatives, corresponding to the worst case-best case analysis of practitioners in decision situations where distributions are missing. Empirical data show, that numerical responses in such situations follow a rule that gives conditions for the exactness of the response. The rule says that the exactness is selected such that there are between 3 and 5 alternatives on this or a cruder level of exactness in the range of reasonable alternatives. This rule permits to predict the exactness of responses, but also permits to deduce on the exactness of information. Once known, it is a powerful tool to inform about information and motives of subjects from their numerical responses. - The paper introduces the rule, and gives some empirical examples that support the theory. These examples concern retail price setting of firms, subjects' estimates of numbers of inhabitants of towns, and a bearing experiment in which different degrees of diffuseness are simulated.

Translated points and Rabinowitz Floer homology

We prove that if a contact manifold admits an exact filling then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [San11b]. In addition we prove that if the Rabinowitz Floer homology of the filling is non-zero then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional then every contactmorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover if the contact manifold has dimension greater than or equal to 3, the latter option generically doesn't happen. Finally, we prove that a generic contactomorphism on $\mathbb{R}^{2n+1}$ has infinitely many geometrically distinct iterated translated points all of which lie in the interior of its support.Comment: 13 pages, v2: numerous corrections, results unchange

Efficiency and deficiency considerations in the symmetry problem

Usually, two statistical procedures A and B are compared by means of their asymptotic relative efficiency e. If e= 1, however, it is more informative to compare A and B by means of the concept of deficiency, which was introduced by Hodges and Lehmann [7]. In the present paper we use this concept for the comparison of linear rank tests and parametric tests for the symmetry problem. In this problem, the hypothesis has to be tested that a sample comes from a distribution that is symmetric about zero. The results provide new and strong edivence for the nice performance of linear rank tests for the symmetry problem. The present paper gives a survey of the results obtained by Albers, Bickel and van Zwet [1] and by Albers [2]

Negative Binomial charts for monitoring high-quality processes

Good control charts for high quality processes are often based on the number of successes between failures. Geometric charts are simplest in this respect, but slow in recognizing moderately increased failure rates p. Improvement can be achieved by waiting until r > 1 failures have occurred, i.e. by using negative binomial charts.In this paper we analyze such charts in some detail. On the basis of a fair comparison, we demonstrate how the optimal r is related to the degree of increase of p. As in practice p will usually be unknown, we also analyze the estimated version of the charts. In particular, simple corrections are derived to control the non-negligible effects of this estimation step

The Golden Mean, the Arab Spring and a 10-step analysis of American economic history

The Long-Wave theories of Nikolai Kondratiev and others claim to find mathematic waves in economic and other social data which are at present in dispute. Currently the theory is considered outside the scope of mainstream economics under several rationales. Despite the lack of mainstream acceptance, we make a strong case for the existence of long waves in the Real GNP of the United States with a 56 year cycle. Our analysis bypasses many of the issues cited by Long-Wave theory critics and in fact clarifies the mathematical structure of the theory.Real GNP; Golden Mean; Fibonacci Series; Arab Spring; Phi; Long Wave; Long Cycle; Kondratiev Wave; Economic Forecasting; Economic Model; Global Financial Crisis; Constitutional Law; American Economic History; Revolution; Consolidation; GNP Spiral; Okun's Law; “The Great Moderation”