Other power series for the Gauss hypergeometric function 30 2.3.3 Removable singularities 33 2.4 Asymptotic expansions 34 2.4.1 Watson's lemma 36 2.4.2 Estimating the remainders of asymptotic expansions ... 2.4.3 Exponentially improved asymptotic expansions 2.4.4 Alternatives of asymptotic expansions 3 Chebyshev Expansions 3.1 Introduction 3.2 Basic results on interpolation 3.2.1 The Runge phenomenon and the Chebyshev nodes .... 3.3 Chebyshev polynomials: Basic properties 3.3.1 Properties of the Chebyshev polynomials T n (x) 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds vii viii Contents 3.4 Chebyshev interpolation 3.4.1 Computing the Chebyshev interpolation polynomial ... 3.5 Expansions in terms of Chebyshev polynomials 3.5.1 Convergence properties of Chebyshev expansions .... 3.6 Computing the coefficients of a Chebyshev expansion 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients 3.7 Evaluation of a Chebyshev sum 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum 3.8 Economization of power series 3.9 Example: Computation of Airy functions of real variable 3.10 Chebyshev expansions with coefficients in terms of special functions. 4 Linear Recurrence Relations and Associated Continued Fractions 4.1 Contents 7.5.2 The general case : 7.6 Asymptotic methods: Further examples 7.6.1 Airy functions 7.6.2 Scorer functions 7.6.3 The error functions 7.6.4 The parabolic cylinder function 7.6.5 Bessel functions 7.6.6 Orthogonal polynomials Uniform Asymptotic Expansions 8.1 Asymptotic expansions for the incomplete gamma functions 8.2 Uniform asymptotic expansions 8.3 Uniform asymptotic expansions for the incomplete gamma functions. 8.3.1 The uniform expansion 8.3.2 Expansions for the coefficients 8.3.3 Numerical algorithm for small values of JJ 8.3.4 A simpler uniform expansion 8.4 Airy-type expansions for Bessel functions 8.4.1 The Airy-type asymptotic expansions 8.4.2 Representations of a s {S),b s (S),c s (S),d s (O 8.4.3 Properties of the functions A v , B v , C v , D v 8.4.4 Expansions for a v (f)A(f).c s-(0>^(f) 8.4.5 Evaluation of the functions A v {t;), B P (£) by iteration. . .258 8.5 Airy-type asymptotic expansions obtained from integrals 8.5.1 Airy-type asymptotic expansions 8.5.2 How to compute the coefficients a n , fi n 8.5.3 Application to parabolic cylinder functions Other Methods 9.1

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Buffered Ca(2+) diffusion in the cytosol of neuroendocrine cells is a plausible explanation for the slowness and latency in the secretion of hormones. We have developed a Monte Carlo simulation to treat the problem of 3-D diffusion and kinetic reactions of ions and buffers. The 3-D diffusion is modeled as a random walk process that follows the path of each ion and buffer molecule, combined locally with a stochastic treatment of the first-order kinetic reactions involved. Such modeling is able to predict [Ca(2+)] and buffer concentration time courses regardless of how low the calcium influx is, and it is therefore a convenient method for dealing with physiological calcium currents and concentrations. We study the effects of the diffusional and kinetic parameters of the model on the concentration time courses as well as on the local equilibrium of buffers with calcium. An in-mobile and fast endogenous buffer as described by, Biophys. J. 72:674-690) was able to reach local equilibrium with calcium; however, the exogenous buffers considered are displaced drastically from equilibrium at the start of the calcium pulse, particularly below the pores. The versatility of the method also allows the effect of different arrangements of calcium channels on submembrane gradients to be studied, including random distribution of calcium channels and channel clusters. The simulation shows how the particular distribution of channels or clusters can be of relevance for secretion in the case where the distribution of release granules is correlated with the channels or clusters.

Exocytosis in neuroendocrine cells is a process triggered by Ca(2+). A Monte Carlo simulation of secretion has been developed which, together with the diffusion of calcium, buffered by endogenous and/or exogenously added chelators, also accounts for the dynamics of exocytosis for a pool of readily releasable vesicles. Different distributions of channels and vesicles (random or correlated) are studied. A local study of exocytosis is carried out by obtaining capacitance time courses for the different types of release-ready vesicle pools (correlated or not with Ca(2+) channels). Also, depending upon the kinetic constants for the exocytotic process, we study the levels of local Ca(2+) needed to trigger secretion. Our simulations show that a strong heterogeneity in the calcium concentrations at the different sites of exocytosis is a requirement for reproducing the experimentally observed biphasic response in chromaffin cells in situ (Voets, T., E. Neher, and T. Moser. 1999. Neuron. 23:607-615). Correlated nonuniform distributions of channels and vesicles and the existence of diffusion barriers are shown to quantitatively explain the experimental data on chromaffin cells in situ. The first description requires a deeply heterogeneous distribution, with vesicles attached to the channels or far from them, but never at middle distances. The second description is able to reproduce biphasic release even for uniformly (readily releasable) distributed vesicles. We quantify the degree of inhomogeneity in the distribution of vesicles and how porous the diffusion barriers should be to account for the observed biphasic response.

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