Learning a single ReLU neuron with gradient descent is a fundamental primitive in the theory of deep learning, andhasbeen extensivelystudied inrecent years.

For a range of activation functions, including ReLUs, we establish superpolynomial Statistical Query (SQ) lower bounds for this learning problem.

Closed-loop neuroscience experimentation, where recorded neural activity is used to modify the experiment on-the-fly, is critical for deducing causal connections and optimizing experimental time. Thus while new optical methods permit on-line recording (via Multi-photon calcium imaging) and stimulation (via holographic stimulation) of large neural populations, a critical barrier in creating closed-loop experiments that can target and modulate single neurons is the real-time inference of neural activity from streaming recordings. In particular, while multi-photon calcium imaging (CI) is crucial in monitoring neural populations, extracting a single neuron's activity from the fluorescence videos often requires batch processing of the video data. Without batch processing, dimmer neurons and events are harder to identify and unrecognized neurons can create false positives when computing the activity of known neurons. We solve these issues by adapting a recently proposed robust time-trace estimator---Sparse Emulation of Unused Dictionary Objects (SEUDO) algorithm---as a basis for a new on-line processing algorithm that simultaneously identifies neurons in the fluorescence video and infers their time traces in a way that is robust to as-yet unidentified neurons. To achieve real-time SEUDO (realSEUDO), we introduce a combination of new algorithmic improvements, a fast C-based implementation, and a new cell finding loop to enable realSEUDO to identify new cells on-the-fly with no warm-up period. We demonstrate comparable performance to offline algorithms (e.g., CNMF), and improved performance over the current on-line approach (OnACID) at speeds of 120 Hz on average. This speed is faster than the typical 30 Hz framerate, leaving critical computation time for the computation of feedback in a closed-loop setting.

We theoretically study the fundamental problem of learning a single neuron with a bias term ($\mathbf{x}\mapsto \sigma(\langle\mathbf{w},\mathbf{x}\rangle + b)$) in the realizable setting with the ReLU activation, using gradient descent. Perhaps surprisingly, we show that this is a significantly different and more challenging problem than the bias-less case (which was the focus of previous works on single neurons), both in terms of the optimization geometry as well as the ability of gradient methods to succeed in some scenarios. We provide a detailed study of this problem, characterizing the critical points of the objective, demonstrating failure cases, and providing positive convergence guarantees under different sets of assumptions. To prove our results, we develop some tools which may be of independent interest, and improve previous results on learning single neurons.

We consider the problem of learning the best-fitting single neuron as measured by the expected square loss $\E_{(x,y)\sim \mathcal{D}}[(\sigma(w^\top x)-y)^2]$ over some unknown joint distribution $\mathcal{D}$ by using gradient descent to minimize the empirical risk induced by a set of i.i.d.

Neural codes are inevitably shaped by various kinds of biological constraints, e.g.

Spiking neural networks (SNNs) are a promising paradigm for energy-efficient computation, yet their theoretical foundations-especially regarding stability and robustness-remain limited compared to artificial neural networks. In this work, we study discrete-time leaky integrate-and-fire (LIF) SNNs through the lens of Boolean function analysis. We focus on noise sensitivity and stability in classification tasks, quantifying how input perturbations affect outputs. Our main result shows that wide LIF-SNN classifiers are stable on average, a property explained by the concentration of their Fourier spectrum on low-frequency components. Motivated by this, we introduce the notion of spectral simplicity, which formalizes simplicity in terms of Fourier spectrum concentration and connects our analysis to the simplicity bias observed in deep networks. Within this framework, we show that random LIF-SNNs are biased toward simple functions. Experiments on trained networks confirm that these stability properties persist in practice. Together, these results provide new insights into the stability and robustness properties of SNNs.

We assume we have access to a set of i.i.d.