On Some Algorithmic and Computational Problems for Neuronal Diffusion Models

In this work we consider some one-dimensional diffusion processes arising in single neurons' activity modelling and discuss some of the related theoretical and computational first passage time problems. With reference to the Wiener and the Ornstein-Uhlenbeck processes, we outline some theoretical methods and algorithmic procedures. In particular, the relevance of the computational methods to infer about asymptotic trends of the firing pdf is pointed out.

[1]  A. G. Nobile,et al.  A new integral equation for the evaluation of first-passage-time probability densities , 1987, Advances in Applied Probability.

[2]  L. M. Ricciardi,et al.  Diffusion approximation and first-passage-time problem for a model neuron , 2004, Kybernetik.

[3]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[4]  W. Feller.  Diffusion processes in one dimension , 1954 .

[5]  A. G. Nobile,et al.  On the evaluation of first-passage-time probability densities via non-singular integral equations , 1989, Advances in Applied Probability.

[6]  J. Durbin.  Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test , 1971, Journal of Applied Probability.

[7]  F. J. Schuurmann,et al.  Evaluations of barrier-crossing probabilities of Wiener paths , 1976, Journal of Applied Probability.

[8]  Katsuhiko Shirai,et al.  On some properties of stochastic information processes in neurons and neuron populations , 1974, Kybernetik.

[9]  Y. Matsuyama.  A note on stochastic modeling of shunting inhibition , 2004, Biological Cybernetics.

[10]  P Lánský,et al.  On approximations of Stein's neuronal model. , 1984, Journal of theoretical biology.

[11]  R W RODIECK,et al.  Some quantitative methods for the study of spontaneous activity of single neurons. , 1962, Biophysical journal.

[12]  Virginia Giorno,et al.  ON THE ASYMPTOTIC BEHAVIOUR OF FIRST- PASSAGE-TIME DENSITIES FOR ONE-DIMENSIONAL DIFFUSION PROCESSES AND VARYING BOUNDARIES , 1990 .

[13]  B. Mandelbrot,et al.  RANDOM WALK MODELS FOR THE SPIKE ACTIVITY OF A SINGLE NEURON. , 1964, Biophysical journal.

[14]  R. Anderssen,et al.  On the numerical solution of Brownian motion processes , 1973, Journal of Applied Probability.

[15]  Laura Sacerdote,et al.  MEAN VARIANCE AND SKEWNESS OF THE FIRST PASSAGE TIME FOR THE ORNSTEIN-UHLENBECK PROCESS , 1981 .

[16]  A. G. Nobile,et al.  Exponential trends of Ornstein-Uhlenbeck first passage time densities , 1985 .

[17]  H. E. Daniels.  The minimum of a stationary Markov process superimposed on a U-shaped trend , 1969 .

[18]  William Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 2 , 1967 .

[19]  W. Feller.  THE PARABOLIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED SEMI-GROUPS OF TRANSFORMATIONS , 1952 .

[20]  L. Ricciardi,et al.  FIRST PASSAGE TIME PROBLEMS AND SOME RELATED COMPUTATIONAL METHODS , 1982 .

[21]  S. Amari,et al.  Competition and Cooperation in Neural Nets , 1982 .

[22]  A. G. Nobile,et al.  Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution , 1985 .

[23]  L. Ricciardi.  Diffusion Approximations and Computational Problems for Single Neurons’ Activity , 1982 .

[24]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[25]  A. Holden.  A note on convolution and stable distributions in the nervous system , 1975, Biological cybernetics.

[26]  R. Capocelli,et al.  Diffusion approximation and first passage time problem for a model neuron , 1971, Biological cybernetics.

[27]  J. Syka,et al.  Spontaneous discharge patterns of mesencephalic neurons: interval histogram and mean interval relationship , 1971, Kybernetik.

[28]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .