On a link between Dirichlet kernels and central multinomial coefficients

AbstractThe central coefficients of powers of certain polynomials with arbitrary degree in x form an important family of integer sequences. Although various recursive equations addressing these coefficients do exist, no explicit analytic representation has yet been proposed. In this article, we present an explicit form of the integer sequences of central multinomial coefficients of polynomials of even degree in terms of finite sums over Dirichlet kernels, hence linking these sequences to discrete nth-degree Fourier series expansions. The approach utilizes the diagonalization of circulant Boolean matrices, and is generalizable to all multinomial coefficients of certain polynomials with even degree, thus forming the base for a new family of combinatorial identities

Characterizing synaptic conductance fluctuations in cortical neurons and their influence on spike generation

Cortical neurons are subject to sustained and irregular synaptic activity which causes important fluctuations of the membrane potential (Vm). We review here different methods to characterize this activity and its impact on spike generation. The simplified, fluctuating point-conductance model of synaptic activity provides the starting point of a variety of methods for the analysis of intracellular Vm recordings. In this model, the synaptic excitatory and inhibitory conductances are described by Gaussian-distributed stochastic variables, or colored conductance noise. The matching of experimentally recorded Vm distributions to an invertible theoretical expression derived from the model allows the extraction of parameters characterizing the synaptic conductance distributions. This analysis can be complemented by the matching of experimental Vm power spectral densities (PSDs) to a theoretical template, even though the unexpected scaling properties of experimental PSDs limit the precision of this latter approach. Building on this stochastic characterization of synaptic activity, we also propose methods to qualitatively and quantitatively evaluate spike-triggered averages of synaptic time-courses preceding spikes. This analysis points to an essential role for synaptic conductance variance in determining spike times. The presented methods are evaluated using controlled conductance injection in cortical neurons in vitro with the dynamic-clamp technique. We review their applications to the analysis of in vivo intracellular recordings in cat association cortex, which suggest a predominant role for inhibition in determining both sub- and supra-threshold dynamics of cortical neurons embedded in active networks.Comment: 9 figures, Journal of Neuroscience Methods (in press, 2008