Bio-inspired Learning and Decision-Making with Probabilistic In-Memory Computing Hardware: Part 1

# Bio-inspired Learning and Decision-Makingwith Probabilistic In-Memory Computing Hardware: Part 1
Source: [https://arxiv.org/html/2609.11281](https://arxiv.org/html/2609.11281)
###### Abstract

Learning and decision\-making in animals are often modeled as Bayesian processes, where sensory evidence is integrated with prior beliefs to guide behavior in the face of uncertainty\. But what are the inherent neural dynamics that give rise to this ability, and how could they be replicated in computing systems? This abstract discusses a biologically grounded framework in which noisy neural and synaptic dynamics perform inference and learning via stochastic sampling from an internal energy function, capturing uncertainty over latent states and model parameters through neural and synaptic variability, respectively\. This enables approaches such as predictive coding networks to account for epistemic uncertainty via Markov chain Monte Carlo sampling\. Drawing a parallel between intrinsic noise in biological systems and electrical noise in emerging probabilistic analogue memory technologies, we highlight how analogue in\-memory computing hardware naturally emerges as the solution for massively scalable and energy\-efficient probabilistic inference\.

1CEA\-List, Grenoble, France

2formerly VERSES AI Research Lab

3now PRAESC AI4now TU Wien, Vienna, Austria

## Introduction

Animals exhibit remarkable adaptability in uncertain and dynamic environments, often behaving in ways that approximate Bayesian inference\([Knill and Richards 1996](https://arxiv.org/html/2609.11281#bib.bib26);[Knill and Pouget 2004](https://arxiv.org/html/2609.11281#bib.bib24);[Paunov et al\. 2024](https://arxiv.org/html/2609.11281#bib.bib9)\)\. This has led to the hypothesis that the brain performs probabilistic reasoning by integrating sensory evidence with prior expectations, guided by uncertainty\. Energy\-based modeling is a compelling framework for understanding this process, which views the brain as a physical system minimizing its internal free energy\([Hinton et al\. 1986](https://arxiv.org/html/2609.11281#bib.bib7);[Friston 2010](https://arxiv.org/html/2609.11281#bib.bib6)\)to perform inference and learning efficiently\. This has inspired computational formulations, such as predictive coding, in which neural populations continuously generate predictions about sensory input and update them through the minimization of prediction errors, implicitly performing Bayesian inference\([Rao and Ballard 1999](https://arxiv.org/html/2609.11281#bib.bib13);[Friston 2005](https://arxiv.org/html/2609.11281#bib.bib17);[Salvatori et al\. 2023](https://arxiv.org/html/2609.11281#bib.bib14)\)\. Biological neural systems, such as the brain, are inherently noisy\. Thermodynamic fluctuations, stochastic neurotransmitter release, and variability in synaptic transmission\([Fatt and Katz 1950](https://arxiv.org/html/2609.11281#bib.bib20);[Conti et al\. 2004](https://arxiv.org/html/2609.11281#bib.bib21);[White et al\. 2000](https://arxiv.org/html/2609.11281#bib.bib19)\)introduce randomness at both the neuronal and synaptic levels\([Buesing et al\. 2011](https://arxiv.org/html/2609.11281#bib.bib22);[Kappel et al\. 2015](https://arxiv.org/html/2609.11281#bib.bib5)\)\. Rather than being detrimental however, this noise may serve a computational role by enabling sampling\-based inference through neural and synaptic variability\. Increasing evidence suggests that the brain leverages its intrinsic stochastic dynamics to perform Bayesian inference via mechanisms akin to Markov chain Monte Carlo \(MCMC\) sampling\([Hoyer and Hyvärinen 2002](https://arxiv.org/html/2609.11281#bib.bib25);[Berkes et al\. 2011](https://arxiv.org/html/2609.11281#bib.bib23)\)\.

In this abstract, we motivate a biologically inspired framework for learning and decision\-making that uses noise as a computational resource\. Specifically, we discuss the development of a new form of predictive coding\([Oliviers et al\. 2024](https://arxiv.org/html/2609.11281#bib.bib16);[Sennesh et al\. 2024](https://arxiv.org/html/2609.11281#bib.bib15)\), capable of modeling its epistemic uncertainty\. While predictive coding has been extensively studied as a mechanism for perceptual inference, its treatment of epistemic uncertainty — uncertainty about the parameters or structure of the generative model — remains underexplored\. In particular, most implementations i\) treat synaptic weights in a deterministic fashion, neglecting the possibility that the weights themselves could represent probability distributions, or ii\) propose a variational posterior over the weights\([Tschantz et al\. 2025](https://arxiv.org/html/2609.11281#bib.bib8)\), not reflecting the persistent changes that result from biological noise according to synaptic sampling theory\([Kappel et al\. 2015](https://arxiv.org/html/2609.11281#bib.bib5)\)\.

We propose a pathway whereby applying stochastic updates to both neural and synaptic variables within an underlying energy function, inference and learning naturally emerge as biologically plausible Markov chain Monte Carlo sampling processes \(Algorithmic Formulation\)\. In addition, we draw a parallel between intrinsic biological noise in the brain and the electrical noise in semiconductor technologies \(Bayesian Neuromorphic Hardware\)\. Just as MCMC sampling offers a biologically plausible mechanism for Bayesian inference in neural systems, it also provides a framework for leveraging the inherent variability of analogue neuromorphic hardware\. Rather than being a limitation, this device\-level noise can be harnessed as a computational asset, enabling scalable energy\-efficient Bayesian learning and inference\([Dalgaty et al\. 2021](https://arxiv.org/html/2609.11281#bib.bib1);[Lin et al\. 2025](https://arxiv.org/html/2609.11281#bib.bib3)\)\.

## Algorithm Formulation

The unnormalized posterior probability distribution over the states of a population of neurons,θ\\theta, with synaptic weights,ω\\omega, given an observationXXcan be expressed as:

P~​\(θ,ω∣X\)=P⁡\(X∣θ,ω\)⋅P⁡\(θ,ω\)\.\\tilde\{P\}\(\\theta,\\omega\\mid X\)=P\(X\\mid\\theta,\\omega\)\\cdot P\(\\theta,\\omega\)\.\(1\)The Boltzmann distribution can then be used to relate this unnormalized posterior to the joint energy function ofθ\\theta,ω\\omegaandXX

P~​\(θ,ω∣X\)=exp⁡\(−E⁡\(θ,ω,X\)\)\.\\tilde\{P\}\(\\theta,\\omega\\mid X\)=\\exp\\big\(\-E\(\\theta,\\omega,X\)\\big\)\.\(2\)This, therefore, allows the log probability of the posterior to be written as the combination of two energy functions

log⁡P~​\(θ,ω∣X\)=−EL​\(X,θ,ω\)−Eprior​\(θ,ω\)\.\\log\\tilde\{P\}\(\\theta,\\omega\\mid X\)=\-E\_\{\\text\{L\}\}\(X,\\theta,\\omega\)\-E\_\{\\text\{prior\}\}\(\\theta,\\omega\)\.\(3\)Depending on the choice of energy function and prior, Markov chain Monte Carlo methods can be used to generate samples from the neuron state probability density functions,p⁡\(θ\)p\(\\theta\), and synaptic weight probability distributions,p⁡\(ω\)p\(\\omega\)\. In the case of a shallow, linear predictive coding network with weightsω\\omega, latent variables \(neurons\)θ\\theta, and prediction errorse\(t\)=X−ω\(t\)​θ\(t\)e^\{\(t\)\}=X\-\\omega^\{\(t\)\}\\theta^\{\(t\)\}, Langevin dynamics can be used to draw these samples through the updates

Δ​θ\(t\)\\displaystyle\\Delta\\theta^\{\(t\)\}=τθ​\[1σX2​ω\(t\)⊤​e\(t\)−1σθ2​\(θ\(t\)−μθ\)\]\+2​τθ​ϵθ\(t\),\\displaystyle=\\tau\_\{\\theta\}\\left\[\\tfrac\{1\}\{\\sigma\_\{X\}^\{2\}\}\\,\\omega^\{\(t\)\\top\}e^\{\(t\)\}\-\\tfrac\{1\}\{\\sigma\_\{\\theta\}^\{2\}\}\(\\theta^\{\(t\)\}\-\\mu\_\{\\theta\}\)\\right\]\+\\sqrt\{2\\tau\_\{\\theta\}\}\\,\\epsilon\_\{\\theta\}^\{\(t\)\},Δ​ω\(t\)\\displaystyle\\Delta\\omega^\{\(t\)\}=τω​\[1σX2​e\(t\)​θ\(t\)⊤−1σω2​ω\(t\)\]\+2​τω​ϵω\(t\),\\displaystyle=\\tau\_\{\\omega\}\\left\[\\tfrac\{1\}\{\\sigma\_\{X\}^\{2\}\}\\,e^\{\(t\)\}\\theta^\{\(t\)\\top\}\-\\tfrac\{1\}\{\\sigma\_\{\\omega\}^\{2\}\}\\,\\omega^\{\(t\)\}\\right\]\+\\sqrt\{2\\tau\_\{\\omega\}\}\\,\\epsilon\_\{\\omega\}^\{\(t\)\},\(4\)whereϵθ\\epsilon\_\{\\theta\}andϵω\\epsilon\_\{\\omega\}denote Gaussian unit noise,τθ\\tau\_\{\\theta\}andτω\\tau\_\{\\omega\}are the neural and synaptic time constants,μθ\\mu\_\{\\theta\}is the prior mean of the latent neurons, andσX2\\sigma\_\{X\}^\{2\},σθ2\\sigma\_\{\\theta\}^\{2\}andσω2\\sigma\_\{\\omega\}^\{2\}are the variances of the Gaussian likelihood, the latent prior and the zero\-mean weight prior, respectively\. Consistent with synaptic sampling theory, the weights evolve on a slower timescale than the neural activity,τω≪τθ\\tau\_\{\\omega\}\\ll\\tau\_\{\\theta\}\([Kappel et al\. 2015](https://arxiv.org/html/2609.11281#bib.bib5)\)\. When learning from a dataset rather than a single observation, the likelihood term of the weight update must be scaled to reflect the full dataset, as in stochastic gradient Langevin dynamics\([Welling and Teh 2011](https://arxiv.org/html/2609.11281#bib.bib12)\), so that the weights sample the posterior given all observations\.

A single well\-mixed chain of these dynamics draws samples from the weight posterior and therefore captures epistemic uncertainty, but reading this uncertainty out requires waiting for the chain to mix over time\. Running an ensemble of chains in parallel, each a particle walking over the same weight posterior, instead provides an instantaneous estimate of epistemic uncertainty through the variability across the ensemble\. Local groups of neural circuits sharing the same structural motifs\([Narayanan et al\. 2005](https://arxiv.org/html/2609.11281#bib.bib10);[Yuste et al\. 2024](https://arxiv.org/html/2609.11281#bib.bib11)\)could be a mechanism by which such ensembles are realized in biological neural networks\.

Unlike similar works that rely on the Hopfield energy, we adopt the predictive coding energy\. The popularity of the Hopfield energy is driven by two primary factors\. First, its symmetric weights avoid the need to compute the transpose of the weight matrix, an operation whose biological implementation remains debated\. Second, its dynamics have been extensively analyzed and are well understood\. The predictive coding energy nevertheless offers several advantages: its dynamics are simpler, and in the shallow linear model considered here the steady state of the network can be obtained analytically\. It also naturally operates in its generative formulation, implementing a hierarchical Gaussian model\([Friston 2005](https://arxiv.org/html/2609.11281#bib.bib17)\)\. The transpose appearing in the neural update of equation[4](https://arxiv.org/html/2609.11281#Sx2.E4)does not pose a problem for analogue hardware, where crossbar arrays can be read in both the forward and reverse directions, although its biological plausibility remains an open question shared with predictive coding more broadly\.

## Probabilistic analogue in\-memory computing

The inference and learning algorithm outlined in equation[4](https://arxiv.org/html/2609.11281#Sx2.E4)is appealing because of the resonance with neural and synaptic sampling theories\. However, it is based on running an ensemble of parallel Langevin chains over the same posterior\. The execution of MCMC is notoriously slow on conventional computer architectures, making the algorithmic latency required to perform this over a large ensemble quickly prohibitive\. For example, in deterministic predictive coding models, a solution on a small network can be reached in about1515iterations\([Salvatori et al\. 2022](https://arxiv.org/html/2609.11281#bib.bib4)\), while a similar model with Langevin dynamics needs more than200200\([Oliviers et al\. 2024](https://arxiv.org/html/2609.11281#bib.bib16)\)\.

Just as biological systems are rich with intrinsic noise, so too are the semiconducting technologies used to build computing hardware\. Thermal noise\([Johnson 1928](https://arxiv.org/html/2609.11281#bib.bib28);[Nyquist 1928](https://arxiv.org/html/2609.11281#bib.bib2)\)arises from the Brownian agitation of conducting particles, shot noise\([Schottky 1918](https://arxiv.org/html/2609.11281#bib.bib27)\)from quanta of particles overcoming energy barriers, and analogue memory devices exhibit programming variability due to the stochastic nature of mechanisms like conductive filament formation\([Dalgaty et al\. 2021](https://arxiv.org/html/2609.11281#bib.bib1);[Lin et al\. 2025](https://arxiv.org/html/2609.11281#bib.bib3)\)or magnetic spin change\([Dalgaty et al\. 2023](https://arxiv.org/html/2609.11281#bib.bib18)\)\. These physical phenomena have already been mapped onto efficient implementations of MCMC sampling, such as Metropolis\-adjusted samplers and Langevin dynamics\. They are particularly efficient because analogue memory devices can be used to draw samples from probability distributions, store these samples, and then perform vector arithmetic using these stored samples, all within the memory circuit\. This is in contrast to GPUs, where parameters must be loaded in from an external high\-bandwidth DRAM memory, subject to a limited bandwidth\. This bandwidth imposes a sequential execution on an algorithm that is intrinsically massively\-parallelizable\. In\-memory computing is, by contrast, structurally a massively parallel computing approach and offers the potential of speeding up the execution of energy\-based models by several orders of magnitude\.

## Research Outlook

Within the European network of excellence dAIEDGE, we aim to advance the probabilistic analogue in\-memory computing paradigm by showing how it can enable a massive scaling\-up of energy\-based models such as predictive coding\. Such hardware can reduce the computational cost of explicitly modeling epistemic uncertainty, opening up new possibilities for frameworks such as active inference, where actions are guided by uncertainty\-driven exploration\. Our immediate focus is twofold\. First, we will investigate the performance of our proposed Bayesian predictive coding network for learning and making decisions within the energy\-based framework\. Second, we will explore how the algorithm detailed in this abstract can be efficiently mapped onto the intrinsic stochastic physics of analogue in\-memory computing hardware, which naturally reflects the variability observed in biological systems\.

## Acknowledgments

This collaboration was supported by Horizon Europe’s dAIEDGE network of excellence \- Grant Agreement Number 101120726\.

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