On the Treves theorem for the AKNS equation

According to a theorem of Treves, the conserved functionals of the AKNS equation vanish on all pairs of formal Laurent series of a specified form, both of them with a pole of the first order. We propose a new and very simple proof for this statement, based on the theory of B\"acklund transformations; using the same method, we prove that the AKNS conserved functionals vanish on other pairs of Laurent series. The spirit is the same of our previous paper on the Treves theorem for the KdV, with some non trivial technical differences.Comment: LaTeX, 16 page

Stability of the replica symmetric solution for the information conveyed by by a neural network

The information that a pattern of firing in the output layer of a feedforward network of threshold-linear neurons conveys about the network's inputs is considered. A replica-symmetric solution is found to be stable for all but small amounts of noise. The region of instability depends on the contribution of the threshold and the sparseness: for distributed pattern distributions, the unstable region extends to higher noise variances than for very sparse distributions, for which it is almost nonexistant.Comment: 19 pages, LaTeX, 5 figures. Also available at http://www.mrc-bbc.ox.ac.uk/~schultz/papers.html . Submitted to Phys. Rev. E Minor change

Pulsar Wind Nebulae as a source of the observed electron and positron excess at high energy: the case of Vela-X

We investigate, in terms of production from pulsars and their nebulae, the cosmic ray positron and electron fluxes above $\sim10$ GeV, observed by the AMS-02 experiment up to 1 TeV. We concentrate on the Vela-X case. Starting from the gamma-ray photon spectrum of the source, generated via synchrotron and inverse Compton processes, we estimated the electron and positron injection spectra. Several features are fixed from observations of Vela-X and unknown parameters are borrowed from the Crab nebula. The particle spectra produced in the pulsar wind nebula are then propagated up to the Solar System, using a diffusion model. Differently from previous works, the omnidirectional intensity excess for electrons and positrons is obtained as a difference between the AMS-02 data and the corresponding local interstellar spectrum. An equal amount of electron and positron excess is observed and we interpreted this excess (above $\sim$100 GeV in the AMS-02 data) as a supply coming from Vela-X. The particle contribution is consistent with models predicting the gamma-ray emission at the source. The input of a few more young pulsars is also allowed, while below $\sim$100 GeV more aged pulsars could be the main contributors.Comment: Accepted for publication in Journal of High Energy Astrophysics (2015

An associative network with spatially organized connectivity

We investigate the properties of an autoassociative network of threshold-linear units whose synaptic connectivity is spatially structured and asymmetric. Since the methods of equilibrium statistical mechanics cannot be applied to such a network due to the lack of a Hamiltonian, we approach the problem through a signal-to-noise analysis, that we adapt to spatially organized networks. The conditions are analyzed for the appearance of stable, spatially non-uniform profiles of activity with large overlaps with one of the stored patterns. It is also shown, with simulations and analytic results, that the storage capacity does not decrease much when the connectivity of the network becomes short range. In addition, the method used here enables us to calculate exactly the storage capacity of a randomly connected network with arbitrary degree of dilution.Comment: 27 pages, 6 figures; Accepted for publication in JSTA

On Decoding the Responses of a Population of Neurons from Short Time Windows

The effectiveness of various stimulus identification (decoding) procedures for extracting the information carried by the responses of a population of neurons to a set of repeatedly presented stimuli is studied analytically, in the limit of short time windows. It is shown that in this limit, the entire information content of the responses can sometimes be decoded, and when this is not the case, the lost information is quantified. In particular, the mutual information extracted by taking into account only the most likely stimulus in each trial turns out to be, if not equal, much closer to the true value than that calculated from all the probabilities that each of the possible stimuli in the set was the actual one. The relation between the mutual information extracted by decoding and the percentage of correct stimulus decodings is also derived analytically in the same limit, showing that the metric content index can be estimated reliably from a few cells recorded from brief periods. Computer simulations as well as the activity of real neurons recorded in the primate hippocampus serve to confirm these results and illustrate the utility and limitations of the approach