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A two-state neuronal model with alternating exponential excitation

Facultad de Economía, Universidad del Rosario, Calle 12c, No.4-69, Bogotá, D. C. Colombia

Special Issues: Advances in Stochastic processes and Applications

We develop a stochastic neural model based on point excitatory inputs. The nerve cell depolarisation is determined by a two-state point process corresponding the two states of the cell. The model presumes state-dependent excitatory stimuli amplitudes and decay rates of membrane potential. The state switches at each stimulus time. We analyse the neural firing time distribution and the mean firing time. The limit of the firing time at a definitive scaling condition is also obtained. The results are based on an analysis of the first crossing time of the depolarisation process through the firing threshold. The Laplace transform technique is widely used.
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Keywords jump-telegraph process; first passage time; neural activity; firing probability; asymptotical behaviour

Citation: Nikita Ratanov. A two-state neuronal model with alternating exponential excitation. Mathematical Biosciences and Engineering, 2019, 16(5): 3411-3434. doi: 10.3934/mbe.2019171

References

  • 1. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its applica- tion to conduction and excitation in nerve, J. Physiol., 117 (1952), 500–544.
  • 2. M. L. Lapicque, Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation, (French) [Quantitative research on electrical excitation of nerves treated as a polarization], J. De Physiol. Pathol. Gen., 9 (1907), 620–635.
  • 3. R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173–194.
  • 4. P. Lánsk´ y and V. Lánská, Diffusion approximation of the neuronal model with synaptic reversal potentials, Biol. Cybern., 56 (1987), 19–26.
  • 5. V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes. The- ory, Models, and Applications to Finance, Biology, and Medicine, Springer-Verlag, New York, Heidelberg, Dordrecht, London, 2012.
  • 6. S. Olmi, D. Angulo-Garcia, A. Imparato, et al., Exact firing time statistics of neurons driven by discrete inhibitory noise, Sci. Rep., 7 (2017), 1577.
  • 7. A. Di Crescenzo and B. Martinucci, Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects, Math. Biosci., 209 (2007), 547–563.
  • 8. E. N. Brown, R. Barbieri, V. Ventura, et al., The time-rescaling theorem and its application to neural spike train data analysis, Neural Comput., 14 (2001), 325–346.
  • 9. H. C. Tuckwel, Introduction to Theoretical Neurobiology, Volume 2, Nonlinear and Stochastic Theories, Cambridge University Press, 1988.
  • 10. R. Rudnicki and M. Tyran-Kami´ nska, Piecewise Deterministic Processes in Biological Models, Spinger-Verlag, 2017.
  • 11. H. U. Bauer and K. Pawelzik, Alternating oscillatory and stochastic dynamics in a model for a neuronal assembly, Phys. D Nonlin. Phenom., 69 (1993), 380–393.
  • 12. M. Walczak and T. Błasiak, Midbrain dopaminergic neuron activity across alternating brain states of urethane anaesthetized rat. Eur. J. Neurosci., 45 (2017), 1068–1077.
  • 13. A. D. Kolesnik and N. Ratanov, Telegraph Processes and Option Pricing, Springer-Verlag, Heidelberg-New York-Dordrecht-London, 2013.
  • 14. S. Zacks, Sample Path Analysis and Distributions of Boundary Crossing Times, Lecture Notes in Mathematics, vol. 2203, Springer-Verlag, 2017.
  • 15. N. Ratanov, First crossing times of telegraph processes with jumps, Methodol. Comput. Appl. Probab., (2019). https://doi.org/10.1007/s11009-019-09709-5
  • 16. L. Beghin, L. Nieddu and E. Orsingher, Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations, J. Appl. Math. Stoch. Anal., 14 (2001), 11–25.
  • 17. L. Bogachev and N. Ratanov, Occupation time distributions for the telegraph process, Stoch. Process. Appl., 121 (2011), 1816–1844.
  • 18. O. López and N. Ratanov, On the asymmetric telegraph processes, J. Appl. Prob. 51 (2014), 569–589.
  • 19. N. Ratanov, Self-exciting piecewise linear processes, ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017), 445–471.
  • 20. A. A. Pogorui, R. M. Rodrguez-Dagnino and T. Kolomiets, The first passage time and estimation of the number of level-crossings for a telegraph process, Ukr. Math. J. 67 (2015), 998–1007.
  • 21. G. D'Onofrio, C. Macci and E. Pirozzi, Asymptotic results for first-passage times of some expo- nential processes, Methodol. Comput. Appl. Probab., 20 (2018), 1453–1476.
  • 22. A. Di Crescenzo and A. Meoli, On a jump-telegraph process driven by an alternating fractional Poisson process, J. Appl. Probab., 55 (2018), 94–111.
  • 23. M. Abundo, On the first hitting time of a one-dimensional diffusion and a compound Poisson process, Methodol. Comput. Appl. Probab., 12 (2010), 473–490.
  • 24. N. Ratanov, Option pricing model based on a Markov-modulated diffusion with jumps, Braz. J. Probab. Stat. 24 (2010), 413–431.
  • 25. L. Breuer, First passage times for Markov-additive processes with positive jumps of phase type, J. Appl. Prob. 45 (2008), 779–799.
  • 26. V. Srivastava, S. F. Feng, J. D. Cohen, et al., A martingale analysis of first passage times of time-dependent Wiener diffusion models, J. Math. Psychol., 77 (2017), 94–110.
  • 27. A. N. Shiryaev, On martingale methods in the boundary crossing problems for brownian motion, Sovrem. Probl. Mat., 8 (2007), 3–78.
  • 28. M. T. Giraudo and L. Sacerdote, Jump-diffusion processes as models for neuronal activity, BioSys- tems, 40 (1997), 75–82.
  • 29. N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 1 (Wiley Series in Probability and Statistics) Wiley-Interscience, 1994.
  • 30. K. S. Lomax, Business failures; another example of the analysis of failure data, J. Amer. Stat. Assoc., 49 (1954), 847–852.
  • 31. H. C. Tuckwell, Introduction to theoretical neurobiology: Volume 1, Linear Cable Theory and Dendritic Structure, Cambridge University Press, 1988.
  • 32. A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 5. Inverse Laplace Transforms, Gordon and Breach Science Publ. 1992.

 

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