Horizontal cells reveal cone type-specific adaptation in primate retina

The human cone visual system maintains contrast sensitivity over a wide range of ambient illumination, a property known as light adaptation. The first stage in light adaptation is believed to take place at the first neural step in vision, within the long, middle, and short wavelength sensitive cone photoreceptors. To determine the properties of adaptation in primate outer retina, we measured cone signals in second-order interneurons, the horizontal cells, of the macaque monkey. Horizontal cells provide a unique site for studying early adaptational mechanisms; they are but one synapse away from the photoreceptors, and each horizontal cell receives excitatory inputs from many cones. Light adaptation occurred over the entire range of light levels evaluated, a luminance range of 15–1,850 trolands. Adaptation was demonstrated to be independent in each cone type and to be spatially restricted. Thus, in primates, a major source of sensitivity regulation occurs before summation of cone signals in the horizontal cell.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Detection of synchrony in the activity of auditory nerve fibers by octopus cells of the mammalian cochlear nucleus

The anatomical and biophysical specializations of octopus cells allow them to detect the coincident firing of groups of auditory nerve fibers and to convey the precise timing of that coincidence to their targets. Octopus cells occupy a sharply defined region of the most caudal and dorsal part of the mammalian ventral cochlear nucleus. The dendrites of octopus cells cross the bundle of auditory nerve fibers just proximal to where the fibers leave the ventral and enter the dorsal cochlear nucleus, each octopus cell spanning about one-third of the tonotopic array. Octopus cells are excited by auditory nerve fibers through the activation of rapid, calcium-permeable, α-amino-3-hydroxy-5-methyl-4-isoxazole-propionate receptors. Synaptic responses are shaped by the unusual biophysical characteristics of octopus cells. Octopus cells have very low input resistances (about 7 MΩ), and short time constants (about 200 μsec) as a consequence of the activation at rest of a hyperpolarization-activated mixed-cation conductance and a low-threshold, depolarization-activated potassium conductance. The low input resistance causes rapid synaptic currents to generate rapid and small synaptic potentials. Summation of small synaptic potentials from many fibers is required to bring an octopus cell to threshold. Not only does the low input resistance make individual excitatory postsynaptic potentials brief so that they must be generated within 1 msec to sum but also the voltage-sensitive conductances of octopus cells prevent firing if the activation of auditory nerve inputs is not sufficiently synchronous and depolarization is not sufficiently rapid. In vivo in cats, octopus cells can fire rapidly and respond with exceptionally well-timed action potentials to periodic, broadband sounds such as clicks. Thus both the anatomical specializations and the biophysical specializations make octopus cells detectors of the coincident firing of their auditory nerve fiber inputs.