The effect of background spatial contrast on electroretinographic responses in the human retina

AbstractThe electroretinogram (ERG) was obtained to contrast modulation (CM). This stimulus is a product of temporal modulation of the contrast of a spatial sinusoid at constant mean luminance. Mean contrast (10–40%), and modulation depth (25–1.0) were modulated at 7.5Hz to record the pattern electroretinogram (PERG). The spatial pattern was a foveally fixated grating pattern with sinusoidal luminance profile with spatial frequency of 4.6c/deg. CM resulted in significant first and second harmonic ERG responses. First harmonic amplitude increases then flattens as a function of mean contrast with ΔC=constant, while the second harmonic response remains unaffected by mean contrast. Apparently the first harmonic represents summed signals of local luminance responses arising from on and off neurons. Mean spatial contrast signals modulate preganglionic local luminance responses

Semiregular sequences and other random system of equations

The security of multivariate cryptosystems and digital signature schemes relies on the hardness of solving a system of polynomial equations over a finite field. Polynomial system solving is also currently a bottleneck of index-calculus algorithms to solve the elliptic and hyperelliptic curve discrete logarithm problem. The complexity of solving a system of polynomial equations is closely related to the cost of computing Gr ̈obner bases, since computing the solutions of a polynomial system can be reduced to finding a lexicographic Gr ̈obner basis for the ideal generated by the equations. Several algorithms for computing such bases exist: We consider those based on repeated Gaussian elimination of Macaulay matrices. In this paper, we analyze the case of random systems, where random systems means either semi-regular systems, or quadratic systems in n variables which contain a regular sequence of n polynomials. We provide explicit formulae for bounds on the solving degree of semi-regular systems with m > n equations in n variables, for equations of arbitrary degrees for m = n + 1, and for any m for systems of quadratic or cubic polynomials. In the appendix, we provide a table of bounds for the solving degree of semi-regular systems of m = n + k quadratic equations in n variables for 2 ≤ k,n ≤ 100 and online we provide the values of the bounds for 2 ≤ k,n ≤ 500. For quadratic systems which contain a regular sequence of n polynomials, we argue that the Eisenbud-Green-Harris conjecture, if true, provides a sharp bound for their solving degree, which we compute explicitly