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A Heterogeneous General Model for Neuromorphic-Inspired Computation

Matteo Mirigliano

arXiv:2609.05482v1cond-mat.dis-nncs.NE

TL;DR

Silicon-based processing faces energy-efficiency and production-chain limitations, while biological-inspired systems can be difficult to interface with digital paradigms. This paper presents an input-dependent stochastic weighting model, and simulations show that correlations among weight updates can stabilize system response under particular conditions.

  • Problem

    Silicon-based processing has energy-efficiency and production-chain limitations, while biological-inspired systems are difficult to interface with digital information-processing paradigms.

    Source: §1 · pp. 1–2

  • Method

    The paper presents a nonlinear, input-dependent stochastic weight model and uses software simulation to explore how correlations among weight updates affect its behavior.

    Source: §1 · p. 2, §6 · p. 13

  • Results

    Correlations strongly influence performance, acting as a stabilizing mechanism that prevents output divergence when α=0.8 and s=5.

    Source: §5 · p. 13

  • Takeaways & Limitations

    The model’s weight correlations and nonlinearity can be tuned to achieve different performance, and it is intended for implementation across heterogeneous material classes.

    Source: §6 · pp. 13–14

    The software implementation assumes uncorrelated coefficients within each weight vector (Σ=I); extending Σ beyond the identity matrix is left for future work.

    Source: §3 · p. 6

Abstract

from arXiv · show

In recent years, both academia and industry have focused on the development of computational architectures inspired by the distributed, adaptive, and event-driven characteristics of biological neural systems, with the aim of reducing the computational cost associated with conventional training approaches [1]. However, a major challenge is the lack of general models and design guidelines for emerging computational systems and hardware. This work introduces a general model based on an input-dependent stochastic weight network, referred to as a substrate. The substrate weights evolve through input-triggered stochastic updates, with correlations between weight coefficients described by a matrix-valued covariance kernel. The proposed framework is implemented using quadratic polynomial weight functions, where the input amplitude controls the magnitude of the stochastic perturbation and a substrate-dependent distance determines the correlation structure. Numerical simulations show that correlations in the stochastic weight evolution significantly affect the system response, suggesting a potential mechanism for neuromorphic-inspired computation without conventional weight training. The aim of this work is to provide a general formulation of the model and identify its main properties and characteristics. 1 H. Jaeger, Towards a generalized theory comprising digital, neuromorphic and unconventional computing, Neuromorphic Comput. Eng., vol. 1, no. 1, p. 012002, Sep. 2021

1. Introduction

The introduction motivates unconventional information processing as an alternative to energy- and production-limited silicon computing, and presents a general stochastic, input-dependent model inspired by complex materials. It frames the model as a theoretical and software contribution, not a benchmark or hardware demonstration.

1. Introduction

  • Silicon-based AI systems enable high throughput and scalability, but face significant energy-efficiency and production-chain limitations.The production-chain concerns span design, fabrication, and integration.Source: §1 · p. 1
  • Biologically inspired systems motivate both silicon redesign and novel nonlinear systems that process external stimuli, including self-assembled nanostructures,,,.Nanowires, nanoparticles, and clusters are highlighted for fabrication simplicity and compatibility with large-scale industrial integration.Source: §1 · p. 1
  • Emerging systems lack a general design framework, while their random multilevel responses make them difficult to interface with digital paradigms requiring discrete 0 and 1 states.The cited limitations concern systems responding to external electrical stimuli.Source: §1 · pp. 1–2
  • The proposed model generalizes prior work,, mapping inputs into variable-dimensional spaces with a nonlinear input-dependent weight matrix and exploiting above-threshold response randomness for adaptive task solving.A software implementation illustrates the model; benchmarking against similar models and demonstrating physical hardware are explicitly left to future work.Source: §1 · p. 2

2. The Model

The model uses an input-dependent substrate whose weights evolve stochastically, with a covariance kernel governing correlated changes. A fixed reduction layer and linear combinator map the substrate output to a scalar result.

  • A fixed, a-priori reduction layer forms a lower-dimensional control vector, which a linear combinator maps to a scalar analog result or a thresholded digital value.Source: Figure 1 · p. 2, §2 · p. 4
  • The substrate is a nonlinear, input-dependent weight matrix mapping an N-dimensional input to an M-dimensional output through weighted sums.Source: §2 · pp. 2–3
  • Rather than conventional training, substrate coefficients evolve through input-triggered stochastic jumps when input components cross a threshold.Source: §2 · p. 3
  • An input-dependent covariance kernel correlates coefficient perturbations: σ(x) sets their magnitude, l(x) their spatial or topological reach, and Σ correlations among each weight’s D parameters.Source: §2 · p. 3

3. Examples of Application

A software example applies the substrate to feedback tracking and probes how input memory, noise, nonlinear weights, and correlations shape stochastic weight updates.

3. Examples of Application

  • Input conditions span α=0.2/0.8 (weak/strong memory) and s=1/5 (10%/50% noise); μ and ε are uniform on [-5,5] and [0,1].The general input memory parameter satisfies 0≤α<1.Source: §3 · pp. 4–5
  • In the tracking example, target error is added to each input; threshold-triggered evolution uses threshold = SP*tol*0.1 + Imax, or a manually set threshold.Imax is the maximum input minus the minimum input across simulation runs; this procedure makes the simulation independent of target value.Source: §3 · pp. 4–5
  • The example uses N=3 inputs, quadratic weights (Z=2, D=10), and Σ=I_D, leaving coefficient-wise covariance beyond identity for future work while modeling inter-weight correlations.Degree-two polynomials are chosen as simple nonlinear functions, but the general model is not limited to polynomial weights.Source: §3 · pp. 6–7
  • Each triggered update reuses one shared random factor across weights and samples independent per-weight noise, yielding hub-mediated correlations but only diagonal covariance from the independent term.The shared factor is drawn once per update, whereas the idiosyncratic term is drawn independently for each weight.Source: §3 · pp. 7–8
  • Distance sets inverse-distance perturbation amplitude and (1−d)-based shared coupling; nearer weights have higher variance and stronger correlation, and zero-coupling runs isolate correlation effects.Correlations arise through weights coupled to the same triggering input, not through a general pairwise structure between all weights.Source: §3 · pp. 7–8

4. Results

Weight-update correlations matter most under high-noise, highly temporally correlated inputs, improving success and reducing the sampling effort in the hardest tested condition. The reported correlation effects also hold for linear input-dependent weights, though differences between linear and nonlinear weight dependence occur only under some input conditions.

  • At s=1, uncorrelated updates yield slightly higher success, but at s=5 correlations improve success; for α=0.8, uncorrelated updates fail most trials within 2000 samplings.The α=0.8, s=5 condition is identified as the hardest case for the system.Source: Table 1 · p. 10, §4 · p. 11
  • With s=5, correlated updates require far fewer realizations than uncorrelated updates, particularly in the hardest α=0.8 input condition.The number of realizations measures the samplings needed to reach the target output.Source: §4 · p. 11, Table 2 · p. 11
  • For one α=0.5, s=5 trial, outputs without correlated updates fall outside the target tolerance more often than outputs with correlations.The figure shows output evolution over 100 temporal steps.Source: §4 · p. 12, Figure 5 · p. 12
  • For linear input-dependent weights, the observed differences between correlated and uncorrelated updates remain, while some differences from nonlinear weight dependence also appear.Tables 3 and 4 report success rates and realization counts for the linear-weight case.Source: §4 · p. 12, Table 3 · p. 12, Table 4 · p. 12
  • Appreciable differences between conditions occur only for certain input-parameter settings.Higher α corresponds to a longer input-correlation interval; the α=0.8 input has slower autocorrelation decay than α=0.2.Source: §4 · pp. 9, 12

5. Discussion

Correlation tuning can stabilize the system under some input conditions, while the quadratic-polynomial implementation provides a tractable but specific realization whose behavior differs from linear weights. The discussion identifies analytical properties of the model as future work.

5. Discussion

  • Second-degree polynomial weight functions were chosen to capture nonlinear current-voltage curves and electron-conduction mechanisms in neuromorphic materials and systems,.Source: §5 · p. 13
  • At 𝛼=0.8 and 𝑠=5, correlations stabilize output by counteracting stochastic updates, though in some other conditions uncorrelated runs perform better through unconstrained coefficient sampling.The study varied correlation degree and uniformly distributed input noise to avoid biasing results.Source: §5 · p. 13
  • Polynomial coefficients enable linear coefficient sampling and covariance modeling analogous to Gaussian processes, while quadratic terms add input curvature like nonlinear kernels.The polynomial structure is mathematically tractable, and random-coefficient polynomials are well studied. Quadratic and linear implementations differ in system evolution; curvature and the greater number of sampled coefficients plausibly contribute, and analytical properties remain for future work.Source: §5 · p. 13

6. Conclusions

The paper presents a nonlinear, input-dependent stochastic weighting model whose random updates can be shaped by correlations and nonlinearity, and proposes it as a framework spanning algorithms and hardware. Simulations illustrate how these features can be tuned to achieve different performances; analytical, implementation-efficiency, and hardware-design studies remain future work.

6. Conclusions

  • The model uses a nonlinear, input-dependent weight matrix to map N inputs to M outputs, with random updates that do not require a strongly constrained training process.Software simulations highlight how correlations among weights during sampling and nonlinearity can be tuned to achieve different performances.Source: §6 · p. 13
  • The framework aims to support algorithm strategies and hardware design for heterogeneous materials that respond to electrical, optical, or chemical stimuli.It is not tied to a particular system class and is intended for implementation across heterogeneous materials.Source: §6 · pp. 13–14
  • Future work will analyze the model’s approximation capabilities and adaptation process.The paper identifies these analytical properties as urgent next steps.Source: §6 · p. 14
  • Future work will compare implementation efficiency with state-of-the-art systems for autonomous learning and system control on selected tasks.Source: §6 · p. 14
  • Future work will assess hardware-design guidelines that relax fabrication and integration constraints while enabling more efficient neuromorphic implementations.Source: §6 · p. 14
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