Sign changes as a universal concept in first-passage-time calculations

Braun, Wilhelm; Thul, Ruediger

doi:10.1103/physreve.95.012114

Cited by 4 publications

(5 citation statements)

References 57 publications

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“…Our approach to tackle the time-dependent FPT problem is to employ the level-crossing statistics of a Gaussian process (Rice 1945;Ricciardi and Sato 1983;Verechtchaguina et al 2006;Braun and Thul 2017;Azaïs and Wschebor 2009).…”

Section: Wiener-rice Seriesmentioning

confidence: 99%

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Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach

Schwalger

2021

Biol Cybern

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Noise in spiking neurons is commonly modeled by a noisy input current or by generating output spikes stochastically with a voltage-dependent hazard rate (“escape noise”). While input noise lends itself to modeling biophysical noise processes, the phenomenological escape noise is mathematically more tractable. Using the level-crossing theory for differentiable Gaussian processes, we derive an approximate mapping between colored input noise and escape noise in leaky integrate-and-fire neurons. This mapping requires the first-passage-time (FPT) density of an overdamped Brownian particle driven by colored noise with respect to an arbitrarily moving boundary. Starting from the Wiener–Rice series for the FPT density, we apply the second-order decoupling approximation of Stratonovich to the case of moving boundaries and derive a simplified hazard-rate representation that is local in time and numerically efficient. This simplification requires the calculation of the non-stationary auto-correlation function of the level-crossing process: For exponentially correlated input noise (Ornstein–Uhlenbeck process), we obtain an exact formula for the zero-lag auto-correlation as a function of noise parameters, mean membrane potential and its speed, as well as an exponential approximation of the full auto-correlation function. The theory well predicts the FPT and interspike interval densities as well as the population activities obtained from simulations with colored input noise and time-dependent stimulus or boundary. The agreement with simulations is strongly enhanced across the sub- and suprathreshold firing regime compared to a first-order decoupling approximation that neglects correlations between level crossings. The second-order approximation also improves upon a previously proposed theory in the subthreshold regime. Depending on a simplicity-accuracy trade-off, all considered approximations represent useful mappings from colored input noise to escape noise, enabling progress in the theory of neuronal population dynamics.

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Section: Wiener-rice Seriesmentioning

confidence: 99%

“…The distribution functions f k allow for an exact series expression of the FPT density, sometimes called Wiener-Rice series (Verechtchaguina et al 2006;Braun and Thul 2017):…”

Section: Wiener-rice Seriesmentioning

confidence: 99%

Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach

Schwalger

2021

Biol Cybern

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“…Our approach to tackle the time-dependent FPT problem is to employ the level-crossing statistics of a Gaussian process [48,49,41,50,51]. To this end, let us consider the sub-set of all realizations of x(t) that cross the barrier b(t) from below in the time interval (t * ,t * + ∆t), a so-called "up-crossing" (Fig.…”

Section: Wiener-rice Seriesmentioning

confidence: 99%

“…A.3). The distribution functions f k allow for an exact series expression of the FPT density, sometimes called Wiener-Rice series [41,50]:…”

Section: Wiener-rice Seriesmentioning

confidence: 99%

Mapping Input Noise to Escape Noise in Integrate-and-fire neurons: A Level-Crossing Approach

Schwalger¹

2021

Preprint

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Noise in spiking neurons is commonly modeled by a noisy input current or by generating output spikes stochastically with a voltage-dependent hazard rate ("escape noise"). While input noise lends itself to modeling biophysical noise processes, the phenomenological escape noise is mathematically more tractable. Using the level-crossing theory for differentiable Gaussian processes, we derive an approximate mapping between colored input noise and escape noise in leaky integrate-and-fire neurons. This mapping requires the first-passage-time (FPT) density of an overdamped Brownian particle driven by colored noise with respect to an arbitrarily moving boundary. Starting from the Wiener-Rice series for the FPT density, we apply the second-order decoupling approximation of Stratonovich to the case of moving boundaries and derive a simplified hazard-rate representation that is local in time and numerically efficient. This simplification requires the calculation of the non-stationary auto-correlation function of the level-crossing process: For exponentially correlated input noise (Ornstein-Uhlenbeck process), we obtain an exact formula for the zero-lag auto-correlation as a function of noise parameters, mean membrane potential and its speed, as well as an exponential approximation of the full auto-correlation function. The theory well predicts the FPT and interspike interval densities as well as the population activities obtained from simulations with colored input noise and time-dependent stimulus or boundary. The agreement with simulations is strongly enhanced across the sub-and suprathreshold firing regime compared to a first-order decoupling approximation that neglects cor-

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“…[5]) have been successfully employed to a plethora of non-Markovian systems such as generalised Langevin and Fokker-Planck equations and coupled oscillator chains. Further approaches have been found in [42][43][44][45][46]. The vast majority of that work is concerned with MFPT, also because in many cases higher moments or the full distribution contain prohibitively complicated expressions.…”

Section: A Introductionmentioning

confidence: 99%

First passage time distribution of active thermal particles in potentials

Walter,

Pruessner,

Salbreux

2020

Preprint

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We introduce a perturbative method to calculate all moments of the first-passage time distribution in stochastic one-dimensional processes which are subject to both white and coloured noise. This class of non-Markovian processes is at the centre of the study of thermal active matter, that is self-propelled particles subject to diffusion. The perturbation theory about the Markov process considers the effect of self-propulsion to be small compared to that of thermal fluctuations. To illustrate our method, we apply it to the case of active thermal particles (i) in a harmonic trap (ii) on a ring. For both we calculate the first-order correction of the moment-generating function of first-passage times, and thus to all its moments. Our analytical results are compared to numerics.

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Sign changes as a universal concept in first-passage-time calculations

Cited by 4 publications

References 57 publications

Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach

Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach

Mapping Input Noise to Escape Noise in Integrate-and-fire neurons: A Level-Crossing Approach

First passage time distribution of active thermal particles in potentials

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