A Semi-Markov Leaky Integrate-and-Fire Model
Abstract
:1. Introduction
- In Section 2 we introduce the semi-Markov Leaky Integrate-and-Fire model and discuss the semi-Markov property of the membrane potential process;
- In Section 3 we give mean and variance of the membrane potential process;
- In Section 4 we address the problem of the autocovariance function. Despite being already determined in [18], we use a different approach that leads to two independently interesting results. In particular, we have in Theorem 1 a formula for the bivariate Laplace transform of an inverse subordinator while in Theorem 2 a formula for the autocovariance function of a time-changed stationary Ornstein-Uhlenbeck as defined in [18]. This last result was obtained in the non-stationary case (for deterministic initial values) by [18]: we provide some changes in the proof given there to determine the autocovariance in the stationary case. We then use these two results to determine the autocovariance function of the membrane potential process. In the same section we show that the autocovariance function is still infinitesimal and decreasing.
- In Section 5 we focus on the effect of the time-change on the distribution of the first spiking times and the Interspike intervals of this model;
- In Section 6 we compare only qualitative (due to a lack of quantitative data) the features of the distribution of the Interspike intervals of the model and of the Unit 240-1;
- Finally, in Section 7 we give a resume of the results.
2. The Semi-Markov Leaky Integrate-and-Fire Model
3. Mean and Variance Functions of
3.1. Preliminaries on
- The process
- The function
3.2. Mean of
- the function is in ;
- for it holds:
3.3. The Variance of
4. The Autocovariance Function of
4.1. The Bivariate Laplace Transform of
4.2. The Autocovariance Function of a Time-Changed Stationary Ornstein-Uhlenbeck Process
4.3. The Autocovariance Function of
- a
- is decreasing;
- b
- .
5. First Spiking Times and Interspike Intervals
5.1. Spiking Times in Case of Excitatory Stimuli
- i
- If Φ is regularly varying at with index . Then as
- ii
- If is an absolutely continuous function, then is an absolutely continuous random variable;
- iii
- If is an absolutely continuous function and there exist and such that
- iv
- Under the hypotheses of , if Φ is regularly varying at with index , then as
5.2. The Interspike Intervals
6. Comparison with the Unit 240-1
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Theorem 1
Appendix B. Proof of Theorem 2
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Ascione, G.; Toaldo, B. A Semi-Markov Leaky Integrate-and-Fire Model. Mathematics 2019, 7, 1022. https://doi.org/10.3390/math7111022
Ascione G, Toaldo B. A Semi-Markov Leaky Integrate-and-Fire Model. Mathematics. 2019; 7(11):1022. https://doi.org/10.3390/math7111022
Chicago/Turabian StyleAscione, Giacomo, and Bruno Toaldo. 2019. "A Semi-Markov Leaky Integrate-and-Fire Model" Mathematics 7, no. 11: 1022. https://doi.org/10.3390/math7111022
APA StyleAscione, G., & Toaldo, B. (2019). A Semi-Markov Leaky Integrate-and-Fire Model. Mathematics, 7(11), 1022. https://doi.org/10.3390/math7111022