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January  2018, 23(1): 377-385. doi: 10.3934/dcdsb.2018026

Stability of stochastic semigroups and applications to Stein's neuronal model

Katarzyna PichÓr 1, and  Ryszard Rudnicki 2,,

1. 

Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland
2. 

Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

* Corresponding author: Ryszard Rudnicki

Received  October 2016 Revised  December 2016 Published  January 2018

Fund Project: This research was partially supported by the National Science Centre (Poland) Grant No. 2014/13/B/ST1/00224

A new theorem on asymptotic stability of stochastic semigroups is given. This theorem is applied to a stochastic semigroup corresponding to Stein's neuronal model. Asymptotic properties of models with and without the refractory period are compared.

Keywords: Stochastic semigroup, asymptotic stability, Stein's model, Markov process, neuron activity.
Mathematics Subject Classification: Primary: 47D06; Secondary: 60J75, 92C20.
Citation: Katarzyna PichÓr, Ryszard Rudnicki. Stability of stochastic semigroups and applications to Stein's neuronal model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 377-385. doi: 10.3934/dcdsb.2018026
References:
[1] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. An Introduction, Cambridge University Press, Cambridge, 2015. doi: 10.1017/CBO9780511614583.
[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. doi: 10.1017/CBO9781316480663. Google Scholar

[3]

A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/978-1-4612-4286-4. Google Scholar

[9]

J. R. Norris, Markov Chains Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998. Google Scholar

[10]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. Google Scholar

[11]

______, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl. , 436 (2016), 305-321. doi: 10.1016/j.jmaa.2015.12.009. Google Scholar

[12]

_____, Asymptotic decomposition of substochastic semigroups and applications Stochastics and Dynamics 18 (2018) in press. doi: 10.1142/S0219493718500016. Google Scholar

[13]

K. Rajdl and P. Lansky, Stein's neuronal model with pooled renewal input, Biol. Cybern., 109 (2015), 389-399. doi: 10.1007/s00422-015-0650-x. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/978-3-319-12145-1_15. Google Scholar

[16]

R. B. Stein, Some models of neuronal variability, Biophys. J., 7 (1967), 37-68. doi: 10.1016/S0006-3495(67)86574-3. Google Scholar

[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. doi: 10.1038/nrn1668. Google Scholar

[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. doi: 10.1007/BF00335237. Google Scholar

show all references

References:
[1] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. An Introduction, Cambridge University Press, Cambridge, 2015. doi: 10.1017/CBO9780511614583.
[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. doi: 10.1017/CBO9781316480663. Google Scholar

[3]

A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/978-1-4612-4286-4. Google Scholar

[9]

J. R. Norris, Markov Chains Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998. Google Scholar

[10]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. Google Scholar

[11]

______, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl. , 436 (2016), 305-321. doi: 10.1016/j.jmaa.2015.12.009. Google Scholar

[12]

_____, Asymptotic decomposition of substochastic semigroups and applications Stochastics and Dynamics 18 (2018) in press. doi: 10.1142/S0219493718500016. Google Scholar

[13]

K. Rajdl and P. Lansky, Stein's neuronal model with pooled renewal input, Biol. Cybern., 109 (2015), 389-399. doi: 10.1007/s00422-015-0650-x. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/978-3-319-12145-1_15. Google Scholar

[16]

R. B. Stein, Some models of neuronal variability, Biophys. J., 7 (1967), 37-68. doi: 10.1016/S0006-3495(67)86574-3. Google Scholar

[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. doi: 10.1038/nrn1668. Google Scholar

[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. doi: 10.1007/BF00335237. Google Scholar

Figure 1.  A schematic diagram of the model
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