Representational capacity of a set of independent neurons

The capacity with which a system of independent neuron-like units represents a given set of stimuli is studied by calculating the mutual information between the stimuli and the neural responses. Both discrete noiseless and continuous noisy neurons are analyzed. In both cases, the information grows monotonically with the number of neurons considered. Under the assumption that neurons are independent, the mutual information rises linearly from zero, and approaches exponentially its maximum value. We find the dependence of the initial slope on the number of stimuli and on the sparseness of the representation.Comment: 19 pages, 6 figures, Phys. Rev. E, vol 63, 11910 - 11924 (2000

Multiwavelength Observations of the BL Lacertae Object PKS 2155-304 with XMM-Newton

The optical-UV and X-ray instruments on-board XMM-Ndewton provide an excellent opportunity to perform simultaneous observations of violently variable objects over a broad wavelength range. The UV and X-ray bright BL Lac object PKS 2155-304 has been repeatedly observed with XMM-Ndewton about twice per year. In this paper, we present a detailed analysis of the simultaneous multiwavelength variability of the source from optical to X-rays, based on the currently available XMM-Ndewton observations. These observations probed the intra-day multiwavelength variability at optical-UV and X-ray wavelengths of the source. The UV variability amplitude is substantially smaller than the X-ray one, and the hardness ratios of the UV to X-rays correlates with the X-ray fluxes: the brighter the source, the flatter the UV-X-ray spectra. On 2000 May 30-31 the UV and X-ray light curves were weakly correlated, while the UV variations followed the X-ray ones with no detectable lags on 2000 November 19-21. On 2001 November 30 the source exhibited a major X-ray flare that was not detected in the optical. The intra-day UV and X-ray variability presented here is not similar to the inter-day UV and X-ray variability obtained from the previous coordinated extensive multiwavelength campaigns on the source, indicating that different ``modes'' of variability might be operating in PKS 2155-304 on different timescales or from epoch to epoch.Comment: Accepted by Ap

Replica symmetric evaluation of the information transfer in a two-layer network in presence of continuous+discrete stimuli

In a previous report we have evaluated analytically the mutual information between the firing rates of N independent units and a set of multi-dimensional continuous+discrete stimuli, for a finite population size and in the limit of large noise. Here, we extend the analysis to the case of two interconnected populations, where input units activate output ones via gaussian weights and a threshold linear transfer function. We evaluate the information carried by a population of M output units, again about continuous+discrete correlates. The mutual information is evaluated solving saddle point equations under the assumption of replica symmetry, a method which, by taking into account only the term linear in N of the input information, is equivalent to assuming the noise to be large. Within this limitation, we analyze the dependence of the information on the ratio M/N, on the selectivity of the input units and on the level of the output noise. We show analytically, and confirm numerically, that in the limit of a linear transfer function and of a small ratio between output and input noise, the output information approaches asymptotically the information carried in input. Finally, we show that the information loss in output does not depend much on the structure of the stimulus, whether purely continuous, purely discrete or mixed, but only on the position of the threshold nonlinearity, and on the ratio between input and output noise.Comment: 19 pages, 4 figure

Solvability of subprincipal type operators

In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2$ at a nonradial involutive manifold $\Sigma_2$. We shall assume that the operator is of subprincipal type, which means that the $k$:th inhomogeneous blowup at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$, but transversal to the symplectic leaves of $\Sigma_2$ at the characteristics. When $k = \infty$ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing of the normal gradient and the Hessian of the blowup at the characteristics. Under these conditions, we show that $P$ is not solvable.Comment: Changed the formulation of Theorem 2.15, added an assuption. Corrected errors and clarified the arguments. Added reference

The birthplace and age of the isolated neutron star RX J1856.5-3754

X-ray observations unveiled various types of radio-silent Isolated Neutron Stars (INSs), phenomenologically very diverse, e.g. the Myr old X-ray Dim INS (XDINSs) and the kyr old magnetars. Although their phenomenology is much diverse, the similar periods (P=2--10 s) and magnetic fields (~10^{14} G) suggest that XDINSs are evolved magnetars, possibly born from similar populations of supermassive stars. One way to test this hypothesis is to identify their parental star clusters by extrapolating backward the neutron star velocity vector in the Galactic potential. By using the information on the age and space velocity of the XDINS RX J1856.5-3754, we computed backwards its orbit in the Galactic potential and searched for its parental stellar cluster by means of a closest approach criterion. We found a very likely association with the Upper Scorpius OB association, for a neutron star age of 0.42+/-0.08 Myr, a radial velocity V_r^NS =67+/- 13$ km s^{-1}, and a present-time parallactic distance d_\pi^NS = 123^{+11}_{-15} pc. Our result confirms that the "true" neutron star age is much lower than the spin-down age (tau_{sd}=3.8 Myrs), and is in good agreement with the cooling age, as computed within standard cooling scenarios. The mismatch between the spin-down and the dynamical/cooling age would require either an anomalously large breaking index (n~20) or a decaying magnetic field with initial value B_0 ~ 10^{14} G. Unfortunately, owing to the uncertainty on the age of the Upper Scorpius OB association and the masses of its members we cannot yet draw firm conclusions on the estimated mass of the RX J1856.5-3754 progenitor.Comment: 6 pages, accepted for publication on Monthly Notices of the Royal Astronomical Societ

Discovery of 5.16s pulsations from the isolated neutron star RBS 1223

The isolated neutron star candidate RBS 1223 was observed with the Advanced CCD Imaging Spectrometer aboard the Chandra X-ray observatory on 2000 June 24. A timing analysis of the data yielded a periodic modulation with a period P=5.1571696^(+1.57*10^(-4) -1.36*10^(-4)s. Using ROSAT HRI archived observations we detected a period P=5.1561274 \pm 4.4*10^(-4)s and determined period derivative dP/dt=(0.7 - 2.0)*10^(-11) s*s^(-1). The detection of this period and dP/dt indicates that RBS 12223 has a ``characteristic'' age of 6000-12000 years and huge magnetic field at the surface (B(dipole)~(1.7- 3.2)*10^(+14) G) typical for anomalous X-ray pulsars (AXPs).Comment: 7 pages, 9 figures, Accepted for publication in Astronomy & Astrophysic