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

Abstract

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.

Intra-ripple frequency accommodation in an inhibitory network model for hippocampal ripple oscillations

“…Since already the zeroth-order approximation m 1 max ð Îi E Þ is close to the numerical solution m i max of Eq (61), and the reset mechanism remains unchanged, this implies that also m i reset is close to m 1 reset ð Îi E Þ. The duration of the upstroke t i off can be obtained from Eq (59) taking into account that I E ð0Þ ¼ Îi E À mt i off :…”

“…Methods Eq (25)). Because in the considered regime spikes are mainly driven by the mean input rather than membrane potential fluctuations, the population rate can be well approximated by the drift part of the probability current across the threshold, while diffusion-mediated spiking is ignored (Methods, Eqs ( 28) and (29); see also [56][57][58][59]):…”

“…A theory that can describe the dynamics of recurrent spiking networks both in the drift-and diffusion-dominated regimes, as well as capture the transitions in between, would be desirable. In the context of escape noise models, various approaches have been put forward to approximate the combined effect of drift-and diffusion-mediated spiking [56][57][58][59][63][64][65]. The rate expression in our Gaussian-drift approximation (Eq (8b)) is equivalent to the drift-based hazard function derived by [56,58].…”

Intra-ripple frequency accommodation in an inhibitory network model for hippocampal ripple oscillations

“…Because in the considered regime spikes are mainly driven by the mean input rather than membrane potential fluctuations, the population rate can be well approximated by the drift part of the probability current across the threshold, while diffusion-mediated spiking is ignored (Methods, Eqs. ( 30) and (31); see also Goedeke and Diesmann, 2008;Plesser and Gerstner, 2000;Chizhov and Graham, 2007;Schwalger, 2021):…”

Effect of burst spikes on linear and nonlinear signal transmission in spiking neurons

“…With respect to the latter aspect we note that several sources of noise lead to a certain degree of unreliability of the encoding process, limiting the transmission of information [Faisal et al, 2008]. Many studies have focussed on the interplay of nonlinear dynamics of neurons, intrinsic noise sources, and time-dependent stimulation [Longtin, 1993, Greenwood et al, 2000, Lindner and Schimansky-Geier, 2001, Fourcaud and Brunel, 2002, Fourcaud-Trocme and Brunel, 2005, Longtin, 2009, Gai et al, 2010, Richardson and Swarbrick, 2010, McDonnell and Ward, 2011, Tchumatchenko et al, 2011, Alijani and Richardson, 2011, Doose et al, 2016, Voronenko and Lindner, 2017, Richardson, 2018, Schwalger, 2021, Franzen et al, 2023, Gowers and Richardson, 2023.…”