Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Stability of stochastic semigroups and applications to Stein's neuronal model
Pages: 377 - 385, Issue 1, January 2018

doi:10.3934/dcdsb.2018026      Abstract        References        Full text (346.0K)                  Related Articles

Katarzyna Pichór - Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland (email)
Ryszard Rudnicki - Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland (email)

1 A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. An Introduction, Cambridge University Press, Cambridge, 2005.       
2 A. Bobrowski, Convergence of One-Parameter Operator Semigroups: In Models of Mathematical Biology and Elsewhere, New Mathematical Monographs, 30, Cambridge University Press, Cambridge, 2016.       
3 A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19.       
4 V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes. Theory, Models and Applications to Finance, Biology and Medicine, Birkhäuser, Boston, 2005.       
5 M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.       
6 G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press, Oxford, 2001.       
7 P. Hrubý, Analysis of bursting in Stein's model with realistic synapses, Gen. Physiol. Biophys., 14 (1995), 305-311.
8 A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, II edition, Springer Applied Mathematical Sciences, 97, New York, 1994.       
9 J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998.       
10 K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685.       
11 _______, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl., 436 (2016), 305-321.       
12 _______, Asymptotic decomposition of substochastic semigroups and applications, Stochastics and Dynamics, 18 (2018) in press.
13 K. Rajdl and P. Lansky, Stein's neuronal model with pooled renewal input, Biol. Cybern., 109 (2015), 389-399.       
14 R. Rudnicki, Stochastic operators and semigroups and their applications in physics and biology, in J. Banasiak, M. Mokhtar-Kharroubi (eds.), Evolutionary Equations with Applications in Natural Sciences, Lecture Notes in Mathematics Springer, Heidelberg, 2126 (2015), 255-318.       
15 R. Rudnicki and M. Tyran-Kamińska, Piecewise deterministic Markov processes in biological models, in: Semigroups of Operators - Theory and Applications, J. Banasiak et al. (eds.), Springer Proceedings in Mathematics & Statistics 113, Springer, Heidelberg, 2015, 235-255.       
16 R. B. Stein, Some models of neuronal variability, Biophys. J., 7 (1967), 37-68.
17 R. B. Stein, E. R. Gossen and K. E. Jones, Neuronal variability: Noise or part of the signal?, Nat. Rev. Neurosci., 6 (2005), 389-397.
18 H. Tuckwell, Introduction to Theoretical Neurobiology, Cambridge University Press, Cambridge 1988.       
19 W. J. Wilbur and J. Rinzel, An analysis of Stein's model for stochastic neuronal excitation, Biol. Cybern., 45 (1982), 107-114.       

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