Stability and mean-field limits of age dependent Hawkes processes

Raad, Mads Bonde; Ditlevsen, Susanne; Löcherbach, Eva

doi:10.1214/19-aihp1023

Cited by 12 publications

(9 citation statements)

References 18 publications

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“…The large population analysis goes back to well-proven techniques for mean-field systems and, as we will see below, the natural limit of (1) will be described in terms of an inhomogeneous Poisson process with nonlinear intensity. Similar mean-field analysis have been successfully carried out for a large class of models (that do not restrict to interacting point processes), incorporating various relevant features in neuroscience, such as refractory periods [14,42], spatial inhomogeneities [16,3], excitability [36]. At this point, due to the nonlinear (and often non Markovian) character of the mean-field limit, one major difficulty concerns the characterization of its dynamics as t → ∞, this difficulty being all the more present when one wants to incorporate inhibition.…”

Section: Stochastic Models For Inhibitionmentioning

confidence: 95%

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Interacting Hawkes processes with multiplicative inhibition

Duval¹,

Luçon²,

Pouzat³

2021

Preprint

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In the present work, we introduce a general class of mean-field interacting nonlinear Hawkes processes modelling the reciprocal interactions between two neuronal populations, one excitatory and one inhibitory. The model incorporates two features: inhibition, which acts as a multiplicative factor onto the intensity of the excitatory population and additive retroaction from the excitatory neurons onto the inhibitory ones. We give first a detailed analysis of the well-posedness of this interacting system as well as its dynamics in large population. The second aim of the paper is to give a rigorous analysis of the longtime behavior of the mean-field limit process. We provide also numerical evidence that inhibition and retroaction may be responsible for the emergence of limit cycles in such system.

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Section: Stochastic Models For Inhibitionmentioning

confidence: 95%

“…[6,23,25,16,14] and references therein for further details. Note that this framework allows for a large versatility in the modeling, since it can accommodate for various features such as age-dependent behaviors/refractory periods [14,15,42] or spatially-structured dynamics [16].…”

Section: Hawkes Processesmentioning

confidence: 99%

Interacting Hawkes processes with multiplicative inhibition

Duval¹,

Luçon²,

Pouzat³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In the second model class (called output noise or escape noise (Gerstner 2000)), the dynamical equations for the state variables are deterministic, while spikes ("output") are generated stochastically through a hazard rate or conditional intensity (Gerstner 2000;Paninski 2004;Truccolo et al 2005;Pillow and Latham 2008;Naud and Gerstner 2012;Brea et al 2013;Galves and Löcherbach 2016;Gerhard et al 2017;Raad et al 2020). This hazard rate depends on the state variables via a link function.…”

Section: Introductionmentioning

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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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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“…In the second model class (called output noise or escape noise [3]), the dynamical equations for the state variables are deterministic while spikes ("output") are generated stochastically through a hazard rate or conditional intensity [3,[16][17][18][19][20][21][22][23][24]. This hazard rate depends on the state variables via a link function.…”

Section: Introductionmentioning

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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Stability and mean-field limits of age dependent Hawkes processes

Cited by 12 publications

References 18 publications

Interacting Hawkes processes with multiplicative inhibition

Interacting Hawkes processes with multiplicative inhibition

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

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