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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
| 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 |
- Abstract
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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 and 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. |
| [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. |
| [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.
doi: 10.1007/978-1-4612-4286-4. |
| [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.
doi: 10.1016/j.jmaa.2015.12.009. |
| [12] |
_____, Asymptotic decomposition of substochastic semigroups and applications Stochastics and Dynamics 18 (2018) in press.
doi: 10.1142/S0219493718500016. |
| [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. |
| [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.
doi: 10.1007/978-3-319-12145-1_15. |
| [16] |
R. B. Stein,
Some models of neuronal variability, Biophys. J., 7 (1967), 37-68.
doi: 10.1016/S0006-3495(67)86574-3. |
| [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. |
| [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. |
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. |
| [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. |
| [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.
doi: 10.1007/978-1-4612-4286-4. |
| [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.
doi: 10.1016/j.jmaa.2015.12.009. |
| [12] |
_____, Asymptotic decomposition of substochastic semigroups and applications Stochastics and Dynamics 18 (2018) in press.
doi: 10.1142/S0219493718500016. |
| [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. |
| [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.
doi: 10.1007/978-3-319-12145-1_15. |
| [16] |
R. B. Stein,
Some models of neuronal variability, Biophys. J., 7 (1967), 37-68.
doi: 10.1016/S0006-3495(67)86574-3. |
| [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. |
| [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. |
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