arXiv:2605.25607v1 Announce Type: new Abstract: Understanding how network function constrains neural connectivity is a central challenge in neuroscience. An influential approach is to train neural networks with gradient descent on cognitive tasks and characterize the resulting connectivity. A key limitation is that the resulting structure depends on the details of the training procedure. Here we propose a complementary normative approach based on the maximum entropy principle for network connectivity, independent of any particular learning algorithm. We describe connectivity as a probability distribution over single-neuron weights, express task requirements as constraints on this distribution, and determine the unique distribution maximizing Shannon entropy subject to these constraints. A weight scale parameter controls the balance between randomness and task-induced structure. We apply this framework to context-dependent input-selection tasks in 2-layer feed-forward networks, and show that maximum entropy inference becomes analytically tractable by mapping nonlinear networks onto gain-modulated linear models. Starting from an a priori homogeneous distribution, we find that maximizing entropy under task constraints leads to the emergence of populations of neurons, each defined by its pattern of contextual gain modulation. Increasing the number of contexts drives a transition from context-specialized to unspecialized, random populations. Increasing the weight scale drives a parallel transition from structured to random stimulus selectivity. Strikingly, this maximum entropy connectivity matches both qualitatively and quantitatively the structure of networks trained with gradient descent across different learning regimes. Our results suggest that the interplay between task constraints and entropy maximization provides a fundamental principle for understanding the relationship between structure and function in neural networks.

arXiv:2605.25278v1 Announce Type: cross Abstract: 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.

arXiv:2605.26032v1 Announce Type: cross Abstract: Creating images from noise is image generation; reconstructing fine details from coarse inputs is super-resolution. Despite their practical differences, both can be understood as reversing information loss across scales. We introduce $\textbf{SKILD}$, a $\textbf{S}$cale-invariant $\textbf{K}$-Space $\textbf{I}$mage $\textbf{L}$earning $\textbf{D}$iffusion model that unifies generation and continuous super-resolution within a single unconditional framework. Both natural images and critical physical systems exhibit scale invariance, and we leverage it to design a forward process that attenuates image content from fine to coarse scales while injecting spectrum-matched Gaussian noise, making scale an explicit coordinate of the diffusion dynamics. The same trained reverse process performs generation and continuous super-resolution by varying only the starting timestep: $\textit{no task-specific architecture, no conditioning branch, no classifier-free guidance, no retraining per scale factor}$. Empirically, SKILD reaches FID $2.65$ and Inception Score $9.63$ on unconditional CIFAR-10, performs $2\times$--$8\times$ super-resolution on ImageNet from a single unconditional checkpoint while outperforming conditional models across perceptual metrics, and reconstructs critical Ising models whose connected four-point correlations closely track the ground truth.