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.
Highlights
Neurons in the brain must operate under highly nonstationary conditions
We developed a level-crossing theory for the hazard rate of a leaky integrate-and-fire neuron driven by colored input noise
Because higher-order correlations between upcrossings are approximated through their pair-wise correlations, we referred to this theory as the second-order decoupling approximation (DA)
Summary
Neurons in the brain must operate under highly nonstationary conditions. most behaviorally relevant sensory stimuli as well as internal signals are rarely constant in time but may change rapidly. An important example of non-homogeneous renewal models in neuroscience are one-dimensional integrate-and-fire neurons driven by white input noise (Gerstner et al 2014) For this model class one can formally construct the hazard rate via the formula λ(t|t) = P(t|t)/ 1 −. We allow an explicit dependence on the speed of the membrane potential u(t) (in accordance with previous studies (Plesser and Gerstner 2000; Herrmann and Gerstner 2001; Chizhov and Graham 2007; Goedeke and Diesmann 2008)), the time since the last spike t − t, and possibly further auxiliary variables {zi } whose dynamics between spikes is given by ordinary differential equations. Given these variables at time t, a spike is fired independently in the time step with probability
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