A Wiener-type neuronal model in the presence of exponential refractoriness

Albano, Giuseppina; Giorno, Virginia; Nobile, Amelia Giuseppina; Ricciardi, Luigi M.

doi:10.1016/j.biosystems.2006.07.010

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…Expressions for such probabilities have been obtained in [3] for the Wiener process for exponentially distributed refractory periods. In the general time homogeneous case…”

Section: Refractoriness and Return Process Modelsmentioning

confidence: 99%

Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications

Sacerdote

Giraudo

2012

Lecture Notes in Mathematics

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Mathematical models are an important tool for neuroscientists. During the last thirty years many papers have appeared on single neuron description and specifically on stochastic Integrate and Fire models. Analytical results have been proved and numerical and simulation methods have been developed for their study. Reviews appeared recently collect the main features of these models but do not focus on the methodologies employed to obtain them. Aim of this paper is to fill this gap by upgrading old reviews on this topic. The idea is to collect the existing methods and the available analytical results for the most common one dimensional stochastic Integrate and Fire models to make them available for studies on networks. An effort to unify the mathematical notations is also made. This review is divided in two parts:1. Derivation of the models with the list of the available closed forms expressions for their characterization; 2. Presentation of the existing mathematical and statistical methods for the study of these models.

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“…Expressions for such probabilities have been obtained in [3] for the Wiener process for exponentially distributed refractory periods. In the general time homogeneous case…”

Section: Refractoriness and Return Process Modelsmentioning

confidence: 99%

Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications

Sacerdote

Giraudo

2012

Lecture Notes in Mathematics

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“…For instance, let us consider the diffusion Z with holding and jumping boundary c ≥ Z(0), associated to Z(t) = Z(0) + µt + B t with µ > 0, Z(0) > 0, and U ∈ (0, c). Z is similar to the Wiener-type neuronal model in the presence of refractoriness, considered in [8], [18], for which the neuron fires when its voltage exceeds the threshold c, and after the refractoriness period, the voltage is reset to U ∈ (0, c). We observe that the process Z can be reduced to our case; indeed, by using that −B t is distributed as B t , it follows that the distribution of the first hitting time of Z to c, when starting from Z(0) ≤ c, is nothing but the distribution of the first hitting time of X −µ to zero, when starting from c − Z(0) ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

A randomized first-passage problem for drifted Brownian motion subject to hold and jump from a boundary

Abundo

2015

Stochastic Analysis and Applications

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“…In general, the correct identification of refractory period substantially influences a correct inference of firing rate (see Barbieri et al, 2001 andBrown et al, 2002). The problem has also been studied frequently in neuronal models, as discussed in Albano et al (2007), Buonocore et al (2002Buonocore et al ( , 2003, Ricciardi and Esposito (1966) and Teich et al (1978).…”

Section: Introductionmentioning

confidence: 99%