Integrated stationary Ornstein–Uhlenbeck process, and double integral processes

Abundo, Mario; Pirozzi, Enrica

doi:10.1016/j.physa.2017.12.043

Cited by 16 publications

(13 citation statements)

References 14 publications

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“…The first moment, the mean ISI value, was known before for the binding neuron model only, see Eq. (21). The general expressions obtained here confirm what has been found previously for the binding neuron.…”

Section: Introductionsupporting

confidence: 90%

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First Passage Time Distribution for Spiking Neuron with Delayed Excitatory Feedback

Shchur

Vidybida

2019

Fluct. Noise Lett.

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A class of spiking neuronal models with threshold 2 is considered. It is defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-andfire (LIF) or the binding neuron model and also for some artificial neurons. A neuron is stimulated with a Poisson stream of excitatory impulses. Each output impulse is conveyed through the feedback line to the neuron input after finite delay ∆. This impulse is identical to those delivered from the input stream. We have obtained a general relation allowing calculating exactly the probability density function (PDF) p(t) for distribution of the first passage time of crossing the threshold, which is the distribution of output interspike intervals (ISI) values for this neuron. The calculation is based on known PDF p 0 (t) for that same neuron without feedback, intensity of the input stream λ and properties of the feedback line. Also, we derive exact relation for calculating the moments of p(t) based on known moments of p 0 (t).The obtained general expression for p(t) is checked numerically using Monte Carlo simulation for the case of LIF model. The course of p(t) has a δ-function type peculiarity. This fact contributes to the discussion about the possibility to model neuronal activity with Poisson process, supporting the "no" answer.

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Section: Introductionsupporting

confidence: 90%

“…In reality, the exact value of the firing threshold may fluctuate for different reasons. Results for fluctuating threshold obtained in diffusion approximation can be found in [21]. All obtained results are mathematically rigorous.…”

Section: Conclusion and Discussionmentioning

confidence: 87%

First Passage Time Distribution for Spiking Neuron with Delayed Excitatory Feedback

Shchur

Vidybida

2019

Fluct. Noise Lett.

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“…Indeed, the need to describe complex phenomena, whose time evolution is affected by the history of own behavior ( [16]) and by several memory effects of different nature ( [20]), requires to design more elaborate mathematical models relying on integrals over time of stochastic processes, and some their modifications. Several aspects, particular cases and specific models can be found in [1]- [4]. In particular, in [4] we considered Gauss-Markov (GM) processes and their time integrals.…”

Section: Introductionmentioning

confidence: 99%

“…Several aspects, particular cases and specific models can be found in [1]- [4]. In particular, in [4] we considered Gauss-Markov (GM) processes and their time integrals. Due to their easy mathematical handling, such kind of processes are suitable for modeling purposes, in a wide range of applications as, for instance, in computational neuroscience (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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On the Fractional Riemann-Liouville Integral of Gauss-Markov processes and applications

Abundo,

Pirozzi

2019

Preprint

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We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order α ∈ (0, 1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central rule, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter α affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractional integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

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Information Processing in the Olfactory Bulb

Pirozzi

2022

Encyclopedia of Computational Neuroscience

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Integrated stationary Ornstein–Uhlenbeck process, and double integral processes

Cited by 16 publications

References 14 publications

First Passage Time Distribution for Spiking Neuron with Delayed Excitatory Feedback

First Passage Time Distribution for Spiking Neuron with Delayed Excitatory Feedback

On the Fractional Riemann-Liouville Integral of Gauss-Markov processes and applications

Information Processing in the Olfactory Bulb

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