Self-Organising Memristive Networks as Physical Learning Systems

The gist: Self-Organising Memristive Networks (SOMNs) offer a promising route toward embedded edge intelligence by harnessing physical networks comprised of resistive memory nanoscale components with dynamically reconfigurable, self-organising electrical circuitry to realize brain-like continual learning.

Introduction to SOMNs

Physical learning systems can be broadly described as systems comprised of physical substrates whose intrinsic nonlinear dynamics can be exploited for learning. A particularly promising approach aims to harness native brain–like, or neuromorphic, properties exhibited by certain nanoscale electronic devices, such as memristors (see Box 1). Among the most promising examples of such systems are Self-Organising Memristive Networks (SOMNs). These physical networks are described as self–organising because under external electrical stimulation, they can spontaneously acquire specific novel structural and functional properties, effectively making them dynamically reconfigurable electrical circuits. SOMNs are unique physical learning substrates because their ability to encode, store, and process information arises directly from their material and structural properties.

Brain-like Dynamics in SOMNs

SOMNs exhibit internal conductance states that adaptively respond to external electrical signals (Fig. 1 (a)). Emergent brain-like properties include plasticity, e.g. the ability to adapt to changing input signals; and avalanche criticality, e.g. signal propagation exhibiting long-range spatio-temporal correlations.

1. Physics of ionic/electronic transport: The dynamics of NW networks are governed by ionic transport in the junctions between NWs. Under externally applied electrical stimulation, the strong electric field in nanoscale junctions can force Ag+cations to electromigrate towards the opposite wire, where they nucleate back to metallic Ag, and form conductive nanofilaments across the junction.

2. NP networks: The dominant microscopic conduction mechanisms in NP networks can differ from those in NW networks. Under an external electrical bias, the strong electric field in the gaps between NPs causes surface atoms to move. At high electric fields (or if the field is applied for a long time), the hillocks extend all the way across the gaps and form filaments.

Plasticity in SOMNs

Plasticity refers to the capacity of a system to modulate its internal states or connectivity in response to external stimuli. The taxonomy of plasticity includes:

** Homosynaptic plasticity: this is a stimulus-specific modulation of synaptic weight (input-specific “synaptic” strengthening/weakening). 1. Heterosynaptic plasticity: a form of synaptic plasticity where the strength of a synapse changes even though that specific synapse was not directly activated. 2. Short-term plasticity (STP): a temporary change in synaptic strength caused by recent presynaptic activity, producing a fading-memory trace of recent inputs. 3. Metaplasticity: plasticity that depends on the prior history of modifications (past activity changes the future plastic response). Structural plasticity refers to persistent topological changes (creation/removal of conductive connections) that encode longer-term memory. Experimental results show that emergent behaviour of NW networks can emulate input-specific changes in synaptic conductance pathways, resembling homosynaptic plasticity. Multiterminal memristive devices demonstrate that plasticity can also extend to synaptic pathways not directly stimulated, effectively mimicking heterosynaptic plasticity effects. Short-term memory effects, experimentally demonstrated to be controllable across multiple timescales, enable the emulation of working memory processes observed in biological brains. Metaplasticity and structural plasticity can be exploited to emulate long-term plasticity effects, a crucial aspect for storing knowledge in the system during learning. Notably, stimulation of a pair of electrodes results in potentiation of the corresponding synaptic pathway, followed by spontaneous relaxation due to volatile effects. Temporally correlated inputs produce a gradual increase in the effective synaptic conductance, emulating paired-pulse facilitation (PPF). Stimulation of a pair of electrodes also gives rise to changes in non-directly stimulated synaptic pathways (heterosynaptic plasticity) due to the connectivity of the system. The low field hillock and high field filament modes are useful for different kinds of computation. Moreover, short-term and long-term memory functionalities have also been reported in NP networks. 3. Criticality: Critical phenomena play a fundamental role in various physical systems, characterised by divergent spatio-temporal correlations and the breakdown of mean-field theory (MFT) approximations. NW-based SOMNs exhibit criticality when networks are electrically stimulated just above a critical applied voltage. Avalanche criticality observed in Ag-based NW networks is due to electrical percolation when the network dynamics is dominated by electron tunnelling in the junctions. NP systems, fabricated at the percolation threshold, exhibit scale-free responses to electrical stimulation. Conductance activity patterns (avalanches of switching events) are self-similar across multiple time scales and can be analysed through the probability density function, the inter-event intervals statistics and autocorrelation function. Memory-driven conductance transitions arise from the interplay between memristive internal states, external driving, and Kirchhoff-constrained network feedback. Mean-field descriptions of the percolation transition in memristive networks have been developed by approximating each junction as a threshold-based conductance switch. A dynamical MFT has been formulated that attempts to characterise the voltage-induced dynamics of SOMNs by reducing equations of the type (4) to a single mean-field dynamical equation. The resulting equation of motion is of the form: d⟨g⟩/dt = -d/d⟨g⟩V∆v(⟨g⟩), where ∆v is the voltage drop on the device. Near transition points, e.g. when the effective potential develops multiple minima, the system becomes susceptible to noise, which can be amplified and can induce transitions between these two states and a failure of the MFT. 4. Physical Learning with SOMNs**

With an understanding of the physical mechanisms that underlie adaptive dynamics in SOMNs, we now show how these properties can be exploited in physical learning systems. Neuromorphic approaches aim to realise such nonlinearities by mimicking biological synapses and/or neurons, either through emulation or simulation. Two key learning paradigms are described below and illustrated in Fig. 6: physical reservoir computing and associative learning.

4.1 Physical reservoir computing

Reservoir computing is a machine learning paradigm where a reservoir (a recurrent neural network) maps an input signal into a higher dimensional manifold that is then analysed by a single output layer, which is the only part that needs to be trained. In the physical or in materia implementation, a nonlinear physical system is used as the reservoir. SOMNs have been demonstrated to satisfy the key requirements of a physical reservoir33, 52, 54, 86, 90, 91. These properties ensure that physical reservoirs can generate rich dynamical features from input signals with sufficient expressivity to fit a model to be learned in the output layer. SOMNs produce collective memory dynamics by virtue of their memristive elements.

4.2 Associative learning

Associative learning, i.e. the formation of associations between stimuli and responses, can be implemented in SOMNs by taking advantage of their adaptive conductance changes (i.e. plasticity, cf. sec. 2.2). External electrical stimuli applied to SOMNs drive local conductance changes that can lead to global reconfigurations in network states as unique conductance pathways are established for different patterns in the stimuli. As repeated stimulation leads to material-level changes in conductance pathways, SOMNs self-organise and recall associations between input patterns and global conductance states. This is more analogous to biological synaptic plasticity than statistical optimisation.

Outlook

The brain demonstrates that intelligence can emerge from the physical properties of matter. Central to their role in physical learning systems is the emergence of global, collective dynamics from relatively simple, local interactions. A key opportunity for future work is to move SOMNs from intriguing functional materials toward engineerable learning substrates. Hierarchical and multi-scale approaches offer a promising route toward this goal.

1. textbf Physical Learning Substrate for Embedded Intelligence (SOMNs): The improved AI system can learn continuously in resource-constrained environments by embedding continuous learning in resource-constrained environments within SOMNs, enabling real-time decision-making for autonomous systems, dynamic sensing, and personalised healthcare.

2. textbf Associative Learning via Adaptive Conductance Changes: The system can form associations between stimuli and responses directly inside the SOMN via adjustment of junction conductance states, allowing it to perform self-organise and recall associations between input patterns and global conductance states.

3. textbf Physical Reservoir Computing for Feature Extraction: The AI system can map input signals into a higher dimensional manifold using SOMNs, enabling it to extract expressive features for tasks like waveform regression, image classification, audio classification, time series prediction.

Learning with physical systems is an emerging paradigm that seeks to harness the intrinsic nonlinear dynamics of physical substrates for learning. The impetus for a paradigm shift in how hardware is used for computational intelligence stems largely from the unsustainability of artificial neural network software implemented on conventional transistor-based hardware. This Perspective highlights one promising approach using physical networks comprised of resistive memory nanoscale components with dynamically reconfigurable, self-organising electrical circuitry. Experimental advances have revealed the non-trivial interactions within these Self-Organising Memristive Networks (SOMNs), offering insights into their collective nonlinear and adaptive dynamics, and how these properties can be harnessed for learning using different hardware implementations. Theoretical approaches, including mean-field theory, graph theory, and concepts from disordered systems, reveal deeper insights into the dynamics of SOMNs, especially during transitions between different conductance states where criticality and other dynamical phase transitions emerge in both experiments and models. Furthermore, parallels between adaptive dynamics in SOMNs and plasticity in biological neuronal networks suggest the potential for realising energy-efficient, brain-like continual learning. SOMNs thus offer a promising route toward embedded edge intelligence, unlocking real-time decision-making for autonomous systems, dynamic sensing, and personalised healthcare, by embedding continuous learning in resource-constrained environments. The overarching aim of this Perspective is to show how the convergence of nanotechnology, statistical physics, complex systems, and self-organising principles offers a unique opportunity to advance a new generation of physical intelligence technologies.