Weight-Activation Dynamics in Neural Networks

• Weight-activation dynamics is the simultaneous evolution of both synaptic weights and neuronal activations, capturing key biological and computational characteristics.
• In memristor-based architectures, weights and activations adapt concurrently under continuous input, supporting real-time signal processing and content-addressable memory.
• Rigorous convergence analysis using Lyapunov methods confirms that, under appropriate conditions, the coupled dynamics reliably settle into stable equilibrium points even in multi-stable systems.

Weight-activation dynamics describes the coupled temporal evolution of both synaptic weights and neuron activations in neural networks—a modeling approach that is more representative of biological systems than classical artificial networks, which typically update one set while freezing the other. In the context of feedback dynamic neural networks implemented with memristor devices, weight-activation dynamics refers to simultaneous adaptation of interconnection strengths (weights) and activations under continuous input, a process crucial for the emergence of content-addressable memories and robust real-time signal processing. The theory rigorously examines under what conditions the joint dynamics will converge to stable equilibrium points (EPs), even when the network possesses multiple such equilibria, as is the case for associative memory systems.

1. Fundamental Models and Dynamical Equations

In convergent feedback neural networks such as the Cohen-Grossberg, Hopfield, cellular neural networks (CNNs), and their memristor analogues, the system is modeled by a set of ordinary differential equations:

• Activation dynamics: The state vector $x$ (activations) evolves according to

$\frac{dx}{dt} = -f(x) + Wg(x) + I$

where $f$ is a monotonically increasing dissipation term, $W$ is the weight (interconnection) matrix, $g$ is the activation function (often monotonic), and $I$ is external input.

• Weight dynamics: In weight-activation dynamics, $W$ evolves in time as well, for example, via an ODE of the form

$\frac{dW}{dt} = h(x, W)$

where $h$ encodes a plasticity rule dependent on both current activations and potentially $W$ itself (such as a Hebbian or memristive law).

For memristor neural networks (MNNs), the memristor device's conductance serves as the dynamic physical implementation of synaptic weight, allowing for true co-evolution of $x$ and $W$.

2. Convergence Analysis and Equilibrium Points

The central analytical result is a rigorous proof that, under suitable assumptions on network structure and device physics, the composite weight-activation system

$$\left\{ \begin{align} \frac{dx}{dt} &= F(x, W), \ \frac{dW}{dt} &= G(x, W) \end{align} \right.$$

converges to an equilibrium point $(x^*, W^*)$ for almost all initial conditions—that is, except for a set of measure zero.

Specifically, the paper shows that if the memristor interconnection network is built with appropriate monotonicity and feedback properties, the global attractor consists of the set of EPs, each corresponding to a possible stored memory or pattern. The proof leverages Lyapunov function construction and invariance principles tailored to the coupled dynamics, extending classical results for activation-only or weight-only updates to the simultaneous case.

The result directly generalizes to systems with multiple stable EPs, making it relevant for content-addressable (associative) memory implementations.

3. Contrasting Weight, Activation, and Weight-Activation Dynamics

Classically, convergent NNs have been used in two ways:

• Fixing weights and inputs, letting neuron activations evolve until convergence (activation dynamics, as in standard Hopfield NNs).
• Fixing neuron activations and inputs, updating weights according to a learning rule until they converge (weight dynamics, as in certain biology-inspired learning scenarios).

However, as noted by Hirsch, and emphasized in this work, the co-evolution—weight-activation dynamics—is both theoretically rich and more biologically plausible. This dynamic is especially central to memristor-based architectures, where, by device nature, both the "weight" (conductance) and the "activation" (node voltage/current) are physical and evolving variables.

The following table summarizes these dynamics in classical and memristor-based systems:

Mode	Weights $W$	Activations $x$	Real-world Analogy
Activation Dynamics	Fixed	Evolving	Standard Hopfield / Cellular NN
Weight Dynamics	Evolving	Fixed	Synaptic learning with static activity
Weight-Activation Dyn.	Evolving	Evolving	Bio-mimetic / Memristor networks

4. Structural Requirements: Memristor Interconnections

Memristor NNs are characterized by feedback interconnections whose conductance changes as a function of voltage/current (i.e., activity-dependent plasticity). The network interconnection topology, together with the memristive device characteristics, determines the form of $G(x, W)$. For convergence, certain conditions are required:

• Boundedness and monotonicity of device response,
• Feedback structure that prohibits pathological limit cycles or unbounded growth,
• Existence of a Lyapunov (energy-like) function that decreases along the joint trajectories except at EPs.

The analysis in the paper generalizes previous proofs for electronic or analog realizations to the memristor case, emphasizing that physical device constraints must be respected.

5. Role of Multi-Stable Equilibria

Multiple stable EPs are a defining feature of associative memory networks and are crucial for implementing content-addressable memories and robust signal processing in real time. The weight-activation dynamics guarantee that, regardless of whether the system starts from an arbitrary input and weight configuration, the joint dynamics will settle into one of the stored patterns (an EP), barring a set of initializations of zero measure.

This property is essential for practical applications such as pattern completion and restoration, and is one of the main motivations for studying combined weight-activation adaptation as opposed to isolated dynamics.

6. Context within Analog and Cellular Neural Network Research

The paper situates its contributions within the broader lineage of analog and cellular neural network (CNN) theory (Marco et al., 28 Jul 2025). The "Cellular wave computing library (version 2.1)" (with Tamás Roska as a key contributor) is highlighted as an earlier framework that implemented templates and algorithms for analog, locally connected NNs with convergent activation dynamics. This software supports the implementation of real-time signal processing and image analysis, and acts as a reference for evaluating the architectural and convergence properties of alternative models, such as those based on memristors.

The present work extends these ideas by systematically handling the theoretical and application-level implications of fully coupled weight-activation dynamics in networks with memristive interconnections, broadening the toolkit for real-time, hardware-efficient neural computation.

7. Implications and Applications

The convergence result for weight-activation dynamics in memristor NNs has substantial impact for both theoretical neuroscience and neuromorphic engineering:

• Associative Memory: Ensures robust retrieval and restoration of stored patterns.
• Real-Time Signal Processing: Facilitates fast, adaptive processing where both synaptic parameters and node states can change in tandem.
• Hardware Implementations: Justifies the use of memristor arrays and similar devices for in-memory computation, where analog resistive states directly encode both weight and activation evolution.
• Comparisons with Classical CNNs: Establishes a rigorous foundation for benchmarking and integrating newly developed memristor-based networks with legacy analog and digital neural network architectures.

In conclusion, systematic analysis of weight-activation dynamics, with rigorous convergence guarantees under suitable network and device conditions, represents a crucial advance in the development and understanding of both theoretical and real-world neural processing systems.