[Paper] A Spiking Neural Network Implementation of Gaussian Belief Propagation

Source: arXiv - 2512.10638v1

Overview

This paper shows how a network of spiking (leaky‑integrate‑and‑fire) neurons can perform Gaussian belief propagation—the core message‑passing algorithm behind many Bayesian inference tasks. By translating the three basic linear operations (equality/branching, addition, multiplication) into spike‑based encodings, the authors build a fully functional spiking neural network (SNN) that matches the classic sum‑product algorithm, opening the door to neuromorphic hardware that can run probabilistic models in real time.

Key Contributions

• Spike‑based encoding/decoding scheme for Gaussian messages (means and variances) that preserves the exact arithmetic needed for belief propagation.
• Constructed SNN primitives that implement equality (branching), addition, and multiplication using only leaky‑integrate‑and‑fire neurons and synaptic weights.
• End‑to‑end validation against the standard sum‑product algorithm, demonstrating negligible error across a range of factor‑graph topologies.
• Demonstrated applications to two canonical Bayesian tasks: (1) Kalman filtering for dynamic state estimation, and (2) Bayesian linear regression for static parameter learning.
• Provided a blueprint for mapping probabilistic graphical models onto neuromorphic platforms (e.g., Loihi, SpiNNaker), highlighting energy‑efficient inference.

Methodology

1. Factor‑graph representation – The target probabilistic model is expressed as a factor graph where each factor corresponds to a linear Gaussian constraint (equality, sum, product).
2. Message representation – A Gaussian message (\mathcal{N}(\mu, \sigma^2)) is encoded as a pair of spike trains: one carrying the mean (\mu) (via firing rate) and the other the precision (\lambda = 1/\sigma^2) (via inter‑spike interval modulation).
3. Neural primitives –
   • Equality node: a branching circuit that copies incoming spike trains to multiple outputs while preserving rate/precision.
   • Addition node: a set of excitatory/inhibitory synapses that sum incoming rates and combine precisions according to Gaussian addition rules.
   • Multiplication node: a more involved microcircuit that implements the product of two Gaussians by adjusting both rate and variance through recurrent inhibition.
4. Simulation – The authors built the SNN in a custom Python/NumPy spiking simulator, using standard LIF dynamics (membrane time constant, threshold, reset). Each primitive runs in discrete time steps, and messages are decoded after a short integration window.
5. Benchmarking – The SNN’s output messages are compared to those from a textbook sum‑product implementation on the same factor graph, measuring mean‑square error of means and relative error of variances.

Results & Findings

The SNN converges within a few hundred simulation steps per message, which translates to sub‑millisecond latency on modern neuromorphic chips. Energy consumption estimates (based on Loihi’s power model) suggest 10–20× lower energy per inference compared to a CPU‑based floating‑point implementation.

Practical Implications

• Neuromorphic inference engines – Developers can now embed Bayesian reasoning directly into edge devices (e.g., IoT sensors, autonomous drones) without offloading to cloud servers.
• Robust sensor fusion – The Kalman‑filter demo shows that spiking hardware can fuse noisy measurements in real time, useful for robotics and AR/VR pipelines where power budgets are tight.
• Probabilistic programming on hardware – The primitive library (equality, addition, multiplication) can be composed to compile higher‑level probabilistic programs (e.g., Pyro, Edward) into SNN graphs, enabling a new class of “probabilistic neuromorphic compilers.”
• Explainable AI – Because the underlying computation mirrors classic Bayesian updates, the resulting models retain interpretability (posterior means/uncertainties) while benefiting from the parallelism of spiking networks.

Limitations & Future Work

• Gaussian restriction – The current implementation only handles linear Gaussian factors; extending to non‑Gaussian or discrete variables will require richer spike encodings or hybrid SNN‑digital schemes.
• Scalability – While the primitives work for modest graph sizes, the number of neurons grows linearly with the number of messages, which could become a bottleneck on limited‑size neuromorphic chips.
• Hardware validation – Experiments were performed in software simulators; real‑world deployment on Loihi, SpiNNaker, or emerging memristive SNN platforms remains to be demonstrated.
• Learning of parameters – The paper assumes known factor parameters (means, variances). Future work could integrate online learning rules (e.g., STDP‑based updates) to estimate these parameters on the fly.

Bottom line: By translating Gaussian belief propagation into spike‑based operations, this work bridges a gap between probabilistic AI and neuromorphic engineering, offering a concrete pathway for developers to run energy‑efficient Bayesian inference on next‑generation hardware.

Authors

• Sepideh Adamiat
• Wouter M. Kouw
• Bert de Vries

Paper Information

• arXiv ID: 2512.10638v1
• Categories: cs.NE
• Published: December 11, 2025
• PDF: Download PDF