Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity
The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passa...
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| Language: | English |
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AIMS Press
2015-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2016011 |
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| author | Massimiliano Tamborrino |
| author_facet | Massimiliano Tamborrino |
| author_sort | Massimiliano Tamborrino |
| collection | DOAJ |
| description | The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data. |
| format | Article |
| id | doaj-art-067ba1d7358740329c7246bbb7ed7f55 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2015-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-067ba1d7358740329c7246bbb7ed7f552025-01-24T02:35:23ZengAIMS PressMathematical Biosciences and Engineering1551-00182015-12-0113361362910.3934/mbe.2016011Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activityMassimiliano Tamborrino0Johannes Kepler University, Altenbergerstraße 69, 4040 LinzThe first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data.https://www.aimspress.com/article/doi/10.3934/mbe.2016011spike timehitting timetime-varying thresholdbrownian motionmaximum likelihood estimator.firing statisticadaptive-threshold modelpiecewise-linear thresholdboundary crossing probability |
| spellingShingle | Massimiliano Tamborrino Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity Mathematical Biosciences and Engineering spike time hitting time time-varying threshold brownian motion maximum likelihood estimator. firing statistic adaptive-threshold model piecewise-linear threshold boundary crossing probability |
| title | Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity |
| title_full | Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity |
| title_fullStr | Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity |
| title_full_unstemmed | Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity |
| title_short | Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity |
| title_sort | approximation of the first passage time density of a wiener process to an exponentially decaying boundary by two piecewise linear threshold application to neuronal spiking activity |
| topic | spike time hitting time time-varying threshold brownian motion maximum likelihood estimator. firing statistic adaptive-threshold model piecewise-linear threshold boundary crossing probability |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2016011 |
| work_keys_str_mv | AT massimilianotamborrino approximationofthefirstpassagetimedensityofawienerprocesstoanexponentiallydecayingboundarybytwopiecewiselinearthresholdapplicationtoneuronalspikingactivity |