Exact Variance and Fano Factor for Arbitrary Level Crossings in Stationary Gaussian Processes

Understanding the statistics of level crossings in stochastic processes is crucial across many scientific disciplines. The traditional Kac-Rice formula gives the mean rate of level crossings and has found broad use. However, that mean rate captures only a coarse summary of the crossing process. It depends entirely on local properties of the stochastic process at a given instant and is therefore blind to the correlation structure of the process over time. To understand whether crossing events, such as neuronal spikes, tend to cluster in time, spread apart, or exhibit more complex temporal organization, one must go beyond the mean rate and study higher-order crossing statistics. Here we go beyond the mean by deriving the exact analytical formulae for the variance and Fano factor of arbitrary level crossings in smooth stationary Gaussian processes. Our exact solution reveals how the full temporal correlation structure dictates whether crossings cluster or become regular. In systems with oscillatory correlations, such as a stochastic damped harmonic oscillator, a recent crossing suppresses an immediate subsequent one, producing sub-Poissonian statistics. However, as damping increases and oscillations disappear, a large and slow excursion above the threshold can produce multiple closely spaced crossings, yielding super-Poissonian statistics. In purely relaxational, non-oscillatory systems, such as a mean-reverting process driven by Ornstein-Uhlenbeck noise, the competition between the timescales of the driving noise and system relaxation produces a richer landscape, including reentrant transitions between sub- and super-Poissonian statistics as the threshold level is varied. Taken together, the exact variance and Fano factor derived here complement the Kac-Rice mean rate, enabling more robust parameter estimation and model selection across any setting where Gaussian processes are used.

Unconditional stability of a recurrent neural circuit implementing divisive normalization

Stability in recurrent neural models poses a significant challenge, particularly in developing biologically plausible neurodynamical models that can be seamlessly trained. Traditional cortical circuit models are notoriously difficult to train due to expansive nonlinearities in the dynamical system, leading to an optimization problem with nonlinear stability constraints that are difficult to impose. Conversely, recurrent neural networks (RNNs) excel in tasks involving sequential data but lack biological plausibility and interpretability. In this work, we address these challenges by linking dynamic divisive normalization (DN) to the stability of ORGaNICs, a biologically plausible recurrent cortical circuit model that dynamically achieves DN and has been shown to simulate a wide range of neurophysiological phenomena. By using the indirect method of Lyapunov, we prove the remarkable property of unconditional local stability for an arbitrary-dimensional ORGaNICs circuit when the recurrent weight matrix is the identity. We thus connect ORGaNICs to a system of coupled damped harmonic oscillators, which enables us to derive the circuit's energy function, providing a normative principle of what the circuit, and individual neurons, aim to accomplish. Further, for a generic recurrent weight matrix, we prove the stability of the 2D model and demonstrate empirically that stability holds in higher dimensions. Finally, we show that ORGaNICs can be trained by backpropagation through time without gradient clipping/scaling, thanks to its intrinsic stability property and adaptive time constants, which address the problems of exploding, vanishing, and oscillating gradients. By evaluating the model's performance on RNN benchmarks, we find that ORGaNICs outperform alternative neurodynamical models on static image classification tasks and perform comparably to LSTMs on sequential tasks.