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Lefebvre M. First-Passage Times and Optimal Control of Integrated Jump-Diffusion Processes. FRACTAL AND FRACTIONAL 2023; 7:152. [DOI: 10.3390/fractalfract7020152] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
Abstract
Let Y(t) be a one-dimensional jump-diffusion process and X(t) be defined by dX(t)=ρ[X(t),Y(t)]dt, where ρ(·,·) is either a strictly positive or negative function. First-passage-time problems for the degenerate two-dimensional process (X(t),Y(t)) are considered in the case when the process leaves the continuation region at the latest at the moment of the first jump, and the problem of optimally controlling the process is treated as well. A particular problem, in which ρ[X(t),Y(t)]=Y(t)−X(t) and Y(t) is a standard Brownian motion with jumps, is solved explicitly.
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Affiliation(s)
- Mario Lefebvre
- Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Montréal, QC H3T 1J4, Canada
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2
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Zhang X, Yi H, Shu H. Nonparametric estimation of periodic signal disturbed by α-stable noises. J Nonparametr Stat 2022. [DOI: 10.1080/10485252.2022.2026944] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Xuekang Zhang
- Energy Internet Engineering Research Center of Anhui Provincial Department of Education, Anhui Polytechnic University, Wuhu, People's Republic of China
- School of Mathematics-Physics and Finance, Anhui Polytechnic University, Wuhu, People's Republic of China
- College of Information Science and Technology, Donghua University, Shanghai, People's Republic of China
| | - Haoran Yi
- College of Information Science and Technology, Donghua University, Shanghai, People's Republic of China
| | - Huisheng Shu
- College of Science, Donghua University, Shanghai, People's Republic of China
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3
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Efficient inference for stochastic differential equation mixed-effects models using correlated particle pseudo-marginal algorithms. Comput Stat Data Anal 2021. [DOI: 10.1016/j.csda.2020.107151] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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4
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Holbach S. Positive Harris recurrence for degenerate diffusions with internal variables and randomly perturbed time-periodic input. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2020.07.005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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5
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Kammerer NB, Stummer W. Some Dissimilarity Measures of Branching Processes and Optimal Decision Making in the Presence of Potential Pandemics. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E874. [PMID: 33286645 PMCID: PMC7517477 DOI: 10.3390/e22080874] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/26/2020] [Revised: 07/27/2020] [Accepted: 07/28/2020] [Indexed: 11/16/2022]
Abstract
We compute exact values respectively bounds of dissimilarity/distinguishability measures-in the sense of the Kullback-Leibler information distance (relative entropy) and some transforms of more general power divergences and Renyi divergences-between two competing discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration (importation) is arbitrarily Poisson-distributed; especially, we allow for arbitrary type of extinction-concerning criticality and thus for non-stationarity. We apply this to optimal decision making in the context of the spread of potentially pandemic infectious diseases (such as e.g., the current COVID-19 pandemic), e.g., covering different levels of dangerousness and different kinds of intervention/mitigation strategies. Asymptotic distinguishability behaviour and diffusion limits are investigated, too.
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Affiliation(s)
| | - Wolfgang Stummer
- Department of Mathematics, University of Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
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6
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Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process. STATISTICAL INFERENCE FOR STOCHASTIC PROCESSES 2020. [DOI: 10.1007/s11203-020-09213-5] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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7
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D'Onofrio G, Lansky P, Pirozzi E. On two diffusion neuronal models with multiplicative noise: The mean first-passage time properties. CHAOS (WOODBURY, N.Y.) 2018; 28:043103. [PMID: 31906649 DOI: 10.1063/1.5009574] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Two diffusion processes with multiplicative noise, able to model the changes in the neuronal membrane depolarization between two consecutive spikes of a single neuron, are considered and compared. The processes have the same deterministic part but different stochastic components. The differences in the state-dependent variabilities, their asymptotic distributions, and the properties of the first-passage time across a constant threshold are investigated. Closed form expressions for the mean of the first-passage time of both processes are derived and applied to determine the role played by the parameters involved in the model. It is shown that for some values of the input parameters, the higher variability, given by the second moment, does not imply shorter mean first-passage time. The reason for that can be found in the complete shape of the stationary distribution of the two processes. Applications outside neuroscience are also mentioned.
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Affiliation(s)
- G D'Onofrio
- Institute of Physiology, Czech Academy of Sciences, Videnska 1083, 14220 Prague 4, Czech Republic
| | - P Lansky
- Institute of Physiology, Czech Academy of Sciences, Videnska 1083, 14220 Prague 4, Czech Republic
| | - E Pirozzi
- Dipartimento di Matematica e Applicazioni, University of Napoli Federico II, Via Cintia, 80126 Napoli, Italy
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Höpfner R, Löcherbach E, Thieullen M. Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems. BERNOULLI 2017. [DOI: 10.3150/16-bej820] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Höpfner R, Löcherbach E, Thieullen M. Ergodicity for a stochastic Hodgkin–Huxley model driven by Ornstein–Uhlenbeck type input. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2016. [DOI: 10.1214/14-aihp647] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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11
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BIBBONA ENRICO, DITLEVSEN SUSANNE. Estimation in Discretely Observed Diffusions Killed at a Threshold. Scand Stat Theory Appl 2012. [DOI: 10.1111/j.1467-9469.2012.00810.x] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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12
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Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process. J Comput Neurosci 2011; 31:563-79. [PMID: 21479618 PMCID: PMC3232348 DOI: 10.1007/s10827-011-0326-z] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2010] [Revised: 03/18/2011] [Accepted: 03/21/2011] [Indexed: 10/31/2022]
Abstract
Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein-Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein-Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.
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Höpfner R, Kutoyants Y. Estimating discontinuous periodic signals in a time inhomogeneous diffusion. STATISTICAL INFERENCE FOR STOCHASTIC PROCESSES 2010. [DOI: 10.1007/s11203-010-9046-7] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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14
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Flores E, León JR. Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE). STATISTICS-ABINGDON 2010. [DOI: 10.1080/02331880903235209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Bibbona E, Lansky P, Sirovich R. Estimating input parameters from intracellular recordings in the Feller neuronal model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:031916. [PMID: 20365779 DOI: 10.1103/physreve.81.031916] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2009] [Indexed: 05/29/2023]
Abstract
We study the estimation of the input parameters in a Feller neuronal model from a trajectory of the membrane potential sampled at discrete times. These input parameters are identified with the drift and the infinitesimal variance of the underlying stochastic diffusion process with multiplicative noise. The state space of the process is restricted from below by an inaccessible boundary. Further, the model is characterized by the presence of an absorbing threshold, the first hitting of which determines the length of each trajectory and which constrains the state space from above. We compare, both in the presence and in the absence of the absorbing threshold, the efficiency of different known estimators. In addition, we propose an estimator for the drift term, which is proved to be more efficient than the others, at least in the explored range of the parameters. The presence of the threshold makes the estimates of the drift term biased, and two methods to correct it are proposed.
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Affiliation(s)
- Enrico Bibbona
- Department of Mathematics G Peano, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
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Picchini U, Ditlevsen S, De Gaetano A, Lansky P. Parameters of the Diffusion Leaky Integrate-and-Fire Neuronal Model for a Slowly Fluctuating Signal. Neural Comput 2008; 20:2696-714. [DOI: 10.1162/neco.2008.11-07-653] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
Stochastic leaky integrate-and-fire (LIF) neuronal models are common theoretical tools for studying properties of real neuronal systems. Experimental data of frequently sampled membrane potential measurements between spikes show that the assumption of constant parameter values is not realistic and that some (random) fluctuations are occurring. In this article, we extend the stochastic LIF model, allowing a noise source determining slow fluctuations in the signal. This is achieved by adding a random variable to one of the parameters characterizing the neuronal input, considering each interspike interval (ISI) as an independent experimental unit with a different realization of this random variable. In this way, the variation of the neuronal input is split into fast (within-interval) and slow (between-intervals) components. A parameter estimation method is proposed, allowing the parameters to be estimated simultaneously over the entire data set. This increases the statistical power, and the average estimate over all ISIs will be improved in the sense of decreased variance of the estimator compared to previous approaches, where the estimation has been conducted separately on each individual ISI. The results obtained on real data show good agreement with classical regression methods.
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Affiliation(s)
- Umberto Picchini
- Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark; and Biomathematics Laboratory, IASI–CNR, Università Cattolica Del Sacro Cuore, Largo A. Gemelli 8, 00168 Rome, Italy
| | - Susanne Ditlevsen
- Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
| | - Andrea De Gaetano
- Biomathematics Laboratory, IASI–CNR, Università Cattolica Del Sacro Cuore, Largo A. Gemelli 8, 00168 Rome, Italy
| | - Petr Lansky
- Institute of Physiology, Academy of Sciences of the Czech Republic, Videnska 1083, 142 20 Prague 4, Czech Republic
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Bibbona E, Lansky P, Sacerdote L, Sirovich R. Errors in estimation of the input signal for integrate-and-fire neuronal models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:011918. [PMID: 18763993 DOI: 10.1103/physreve.78.011918] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2007] [Indexed: 05/26/2023]
Abstract
Estimation of the input parameters of stochastic (leaky) integrate-and-fire neuronal models is studied. It is shown that the presence of a firing threshold brings a systematic error to the estimation procedure. Analytical formulas for the bias are given for two models, the randomized random walk and the perfect integrator. For the third model considered, the leaky integrate-and-fire model, the study is performed by using Monte Carlo simulated trajectories. The bias is compared with other errors appearing during the estimation, and it is documented that the effect of the bias has to be taken into account in experimental studies.
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Affiliation(s)
- Enrico Bibbona
- Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce, 91-10135 Torino, Italy.
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Ditlevsen S, Lansky P. Parameters of stochastic diffusion processes estimated from observations of first-hitting times: application to the leaky integrate-and-fire neuronal model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041906. [PMID: 17995025 DOI: 10.1103/physreve.76.041906] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2007] [Indexed: 05/25/2023]
Abstract
A theoretical model has to stand the test against the real world to be of any practical use. The first step is to identify parameters in the model estimated from experimental data. In many applications where renewal point data are available, models of first-hitting times of underlying diffusion processes arise. Despite the seemingly simplicity of the model, the problem of how to estimate parameters of the underlying stochastic process has resisted solution. The few attempts have either been unreliable, difficult to implement, or only valid in subsets of the relevant parameter space. Here we present an estimation method that overcomes these difficulties, is computationally easy and fast to implement, and also works surprisingly well on small data sets. The method is illustrated on simulated and experimental data. Two common neuronal models--the Ornstein-Uhlenbeck and Feller models--are investigated.
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Affiliation(s)
- Susanne Ditlevsen
- Department of Biostatistics, University of Copenhagen, Øster Farimagsgade 5, 1014 Copenhagen K, Denmark.
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