Self-Organising Memristive Networks (SOMNs)

Updated 3 September 2025

Self-Organising Memristive Networks are physical computing systems composed of interconnected memristive devices that adapt through history-dependent conductance changes.
Experimental studies reveal that these networks exhibit emergent behaviors such as phase transitions, avalanche dynamics, and self-organized criticality, enabling practical applications in reservoir computing and adaptive learning.
Theoretical models leveraging nonlinear circuit theory, graph theory, and mean-field approaches offer valuable insights into controlling network dynamics for scalable, energy-efficient neuromorphic systems.

Self-Organising Memristive Networks (SOMNs) are physical computing systems composed of networks of nanoscale memristive elements—devices whose resistance (or conductance) evolves nonlinearly in response to their history of electrical stimulation. SOMNs leverage both inherent device plasticity and global network connectivity to realize adaptive, brain-like learning in hardware, embedding memory and computation within a single substrate. Their operation is governed by the interplay of local switching and global circuit topology, leading to emergent behaviors such as criticality, phase transitions, and self-organization resembling neural plasticity in biological systems (Caravelli et al., 31 Aug 2025).

1. Fundamental Principles and Definitions

A Self-Organising Memristive Network consists of interconnected memristors—two-terminal resistive memory devices—whose state evolution is described by history-dependent dynamics:

The internal state variable $x$ (e.g., conductance or filament length) obeys

$\frac{dx}{dt} = F(x, u),$

where $u$ is the applied voltage or current, and the output $y$ is typically given by $y = H(x, u)u$ .

Ideal memristors satisfy $d\Phi = M dq$ , relating flux and charge through the memristance $M$ (Caravelli et al., 31 Aug 2025).

Self-organization in these systems arises because local ionic/electronic transport (such as filament formation in a junction) is coupled through Kirchhoff’s laws and network topology, causing system-wide reconfiguration of conduction pathways in response to external stimuli.

Key characteristics:

Local nonlinear, history-dependent conductance modulation (analogous to synaptic weight change)
Emergence of global adaptive phenomena through network feedback
Operation in both analog (continuous conductance) and digital (discrete switching) modes

2. Experimental Realizations and Adaptive Network Phenomena

Recent experiments have validated the unique collective dynamics, plasticity, and robustness inherent to SOMNs:

Nanowire and nanoparticle assemblies exhibit emergent short-term plasticity, manifesting as potentiation (conductance increase) followed by relaxation (depression) akin to biological synapses (Milano et al., 2019).
Networks display scale-free avalanche switching near threshold, indicating critical behavior during conductance phase transitions—analogous to percolation and breakdown phenomena (Caravelli et al., 31 Aug 2025, Sheldon et al., 2016).
Adaptive restructuring occurs not only by tuning the resistance of individual junctions (“reweighting”) but through irreversible changes in topology (“rewiring”), supporting both short- and long-term memory.

These behaviors are evidenced through pinched hysteresis loops in I–V curves, observed conductance avalanches, and phase transitions from low- to high-conductance network states as a function of applied voltage.

3. Theoretical Foundations and Modelling Approaches

Analysis of SOMN dynamics leverages models spanning nonlinear circuit theory, graph theory, and statistical mechanics:

Lumped-element models describe dynamics of individual memristive junctions:

$\frac{dg}{dt} = F(g, \delta v(t), t),$

where $g$ is a normalized conductance and $F$ is a nonlinear, thresholded function of voltage (Caravelli et al., 31 Aug 2025).

Graph-theoretic projection operators, such as $\Omega_B = B^\top (B B^\top)^{-1} B$ (where $B$ is the graph incidence matrix), impose current conservation and enable evaluation of global network responses (Caravelli et al., 2023, Caravelli et al., 31 Aug 2025).
Mean-field theory is used to reduce the high-dimensional dynamical system to an equation for the average conductance $\langle g \rangle$ :

$\frac{d\langle g \rangle}{dt} = -\frac{d\mathcal{V}_{\Delta v}(\langle g \rangle)}{d\langle g \rangle},$

with $\mathcal{V}_{\Delta v}$ as a Landau-like effective potential controlling state transitions (Caravelli et al., 2023, Caravelli et al., 31 Aug 2025).

Near critical points, stochastic effects (noise in filament formation, device variability) are captured using statistical and stochastic frameworks, including Ornstein–Uhlenbeck processes and avalanche models.

4. Phase Transitions, Criticality, and Collective Dynamics

SOMNs display phase transitions between low and high conductance states as network parameters (e.g., voltage, device “memory” strength, or ON/OFF ratio) are varied:

Above a critical threshold, networks abruptly “switch” to a high-conductance state (potentiation), controlled by feedback between local switching events and global current redistribution (Sheldon et al., 2016).
Mean-field analysis reveals these transitions correspond to a change in minima of the effective potential $\mathcal{V}$ as a function of $\langle g \rangle$ and applied voltage:

$V(\langle g \rangle) = a\langle g \rangle + b\langle g \rangle^2 - c v \ln(1 + \chi \langle g \rangle).$

The critical regime exhibits avalanche statistics, with the branching parameter $\mu$ controlling the mean avalanche size $\langle S \rangle = 1/(1-\mu)$ (Sheldon et al., 2016, Caravelli et al., 31 Aug 2025).

Such transitions are significant for computation: near criticality, networks are maximally sensitive and can efficiently process, store, and adapt to information, akin to computational models of neural criticality.

5. Applications: Physical Learning and Neuromorphic Computing

The adaptive, nonlinear properties of SOMNs enable several classes of applications:

Physical Reservoir Computing: Intrinsic spatio-temporal network dynamics produce a high-dimensional, nonlinear response to time-varying inputs, suitable for real-time processing and classification tasks (Bürger et al., 2015).
Associative and Continual Learning: Through in situ adaptation, SOMNs can perform error-driven or self-organizing learning, with features such as memory retention, healing (restoration of damaged pathways), and one-shot adaptation (Carbajal et al., 2020, Pershin et al., 2013).
Autonomous Edge Devices: Low-power, embedded learning supports real-time decision-making in robotics, dynamic sensor networks, and on-device continual learning for personalized healthcare (Caravelli et al., 31 Aug 2025, Mao et al., 2022).
Analog and Mixed-Signal Logic: The in-memory computational capabilities of memristors enable implementation of multiply-accumulate, logic, and content-addressable memory functions in highly parallel architectures (Kavehei et al., 2011, Mao et al., 2022).

Experiments have demonstrated not only energy efficiency and scalability, but also robust performance in the presence of device-level disorder, variability, and partial synapse failure (Bouhadjar et al., 2022).

6. Design, Scaling, and Control of Network Self-Organization

Engineering SOMN properties requires consideration at both the device and network level:

Local dynamics (switching thresholds, decay rates, and nonlinearity) and their distributions determine both microstate evolution and macroscopic emergent behavior.
Global structural properties (degree distribution, modularity) can be tailored theoretically via generating function approaches, linking microscopic (re)wiring rates to desired network-level statistics (Silk et al., 2015).
Mixed analog/digital circuit strategies and hierarchical composition (modular, small-world structures) improve performance, robustness, and scalability, especially in the context of physical constraints and device non-idealities (Bürger et al., 2015).
Control strategies include external forcing (e.g., patterned voltage pulses), modular architecture, and integrating learning rules (e.g., STDP, Hebbian, mistake-driven) implemented physically in device programming.

A plausible implication is that leveraging neural network-based estimators for memristor programming can mitigate device variability and accelerate on-chip adaptation, potentially leading to real-time, self-optimizing intelligence in hardware (Yu et al., 11 Mar 2024).

7. Outlook and Future Research

Ongoing directions in the paper and application of SOMNs include:

Advanced Materials and Manufacturing: Exploration of new memristive materials and scalable, bottom-up assembly techniques (e.g., nanowire or nanoparticle networks) for improved uniformity, robustness, and functional diversity (Milano et al., 2019, Caravelli et al., 31 Aug 2025).
Sophisticated Theoretical Models: Development of mean-field and graph-theoretic approaches that capture strong disorder, fluctuation-driven phenomena, and dynamics near criticality, building toward a predictive, unifying framework for design and control (Caravelli et al., 2023, Caravelli et al., 31 Aug 2025).
Hybrid and Hierarchical Architectures: Integration of self-assembled SOMNs with CMOS technology (e.g., as “plexus” layers) and crossbar architectures, aiming for area efficiency, broadcast capability, and complex inter-neuronal coupling (Cipollini et al., 28 Nov 2024, Mao et al., 2022).
Generalization of Learning Paradigms: Extending applications beyond reservoir computing and unsupervised learning to incorporate spiking neural models, probabilistic inference, energy-based optimization, and combined function-memory hardware (Zhou et al., 2022, Caravelli et al., 31 Aug 2025).
Energy-Efficient Edge Intelligence: Deployment of SOMNs for adaptive sensing, edge AI, and personalized computation, exploiting the merging of memory, learning, and computation in a single substrate to overcome the conventional von Neumann bottleneck (Caravelli et al., 31 Aug 2025).

In summary, Self-Organising Memristive Networks represent a convergence of nanotechnology, statistical physics, complex systems theory, and neuromorphic engineering. By embedding adaptive, learning-capable computation within physical material, SOMNs provide a foundational framework for the realization of next-generation physical intelligence systems with autonomous, continual learning capabilities (Caravelli et al., 31 Aug 2025).