Unified Complex-valued Neural Network: A Magnitude-Phase Computational Model for Event-Driven Neuromorphic Learning

Abstract: Artificial neural networks (ANN) provide accurate continuous-valued representation, whereas spiking neural networks (SNN) offer event-driven temporal processing, yet both paradigms face limitations when value encoding and timing dynamics must be learned within a single computational structure. This paper introduces a network based on Unified Complex-valued Neuron (UCN), a new neural computational model that integrates continuous activation and phase-driven event generation through an asymmetric complex-valued state. In the UCN, magnitude encodes signal strength while phase governs intrinsic temporal evolution and valued spike emission. A foundational training framework combining backpropagation (BP) and backpropagation through time (BPTT) is first developed to optimize magnitude and phase pathways in a unified way. To reduce computational complexity, an event-driven adaptive phase learning (EAPL) rule is then introduced as a more efficient alternative. The proposed model is evaluated through object tracking and Lorenz attractor learning. Results demonstrate that UCN-based Network (UCNN) provides accurate, stable, and interpretable spatiotemporal learning while preserving sparse event-driven computation for neuromorphic and edge-AI applications.

• The paper proposes the Unified Complex-valued Neuron that decouples graded magnitude encoding from phase-regulated spike timing for robust information transfer.
• It develops an event-driven adaptive phase learning (EAPL) algorithm that efficiently trains both continuous magnitude and phase channels, nearly halving computational cost.
• Empirical results demonstrate that UCNN closes over half the error gap between traditional SNNs and ANNs in object tracking and chaotic dynamics modeling.

Unified Magnitude-Phase Neural Computation for Event-Driven Neuromorphic Systems

Introduction and Motivation

The dichotomy between artificial neural networks (ANNs) and spiking neural networks (SNNs) remains pronounced in neuromorphic computing. While ANNs deliver high-fidelity continuous-valued representation, their processing paradigm is inherently dense and lacks direct modeling of temporal event-driven dynamics critical for energy-efficient edge-AI and real-time neuromorphic platforms. Pure SNNs address event-driven computation and temporal sparsity, but the binary nature of spike communication limits their capacity for graded information encoding and stable optimization in complex dynamical tasks. Previous hybrid architectures introduced modular amalgamations, but at the cost of architectural and training complexity.

This work presents a unified computational model, the Unified Complex-valued Neuron (UCN), and its network implementation (UCNN), which employs an asymmetric complex-valued state to encode both magnitude (signal strength) and phase (governing intrinsic temporal evolution and event emission). This architecture is explicitly designed for joint processing of static (frame-based) and dynamic (event-driven) visual data streams without redundant or parallel computation paths.

Model Formulation and Training

Asymmetric Complex-Valued State

The core innovation is the UCN model, formalized by the internal state $z(t) = r(t)\exp(i\theta(t))$. Here, $r(t)\geq0$ encodes the graded signal amplitude and $\theta(t)$ tracks internal phase evolution, akin to a leaky Integrate-and-Fire neuron but augmented to allow decoupling of event generation (spike timing) and valued transmission (what-information). Pre-activation input, shared for both channels, drives hyperbolic tangent-based magnitude computation and leaky phase dynamics:

• Magnitude: $r(t) = \max(0, \tanh(u(t)))$
• Phase evolution: $$\dot{\theta}(t) = -\lambda\theta(t) + \omega_0 + u(t)$$

The model supports hard gating (Heaviside function and threshold crossing for event generation) with configurable reset strategies (reset-to-zero, modulo, or soft decay) and soft gating for improved gradient-based optimization. This design allows neurons to interpolate between continuous-value and event-driven regimes, with the phase gating event emission and the magnitude channel modulating the value carried per event.

Unified End-to-End Learning

A crucial contribution is the joint learning framework combining standard backpropagation (BP) for the magnitude pathway and backpropagation through time (BPTT) with surrogate gradients for the phase-driven spike pathway. Surrogate gradients are employed near threshold events to enable timing-credit assignment in the phase dynamics. The training objective is a composite loss weighted between value (magnitude) and timing (phase/event) errors, and gradients are propagated end-to-end through both magnitude and phase pathways, integrating both continuous and sparse event-driven computation.

To address the computational cost associated with explicit temporal unrolling in BPTT, the authors introduce an event-driven adaptive phase learning (EAPL) algorithm, which treats learning as a backward dynamical process using an adjoint state that evolves efficiently without full unrolled graphs. This reduces the backward computational complexity—demonstrated to nearly halve the dominant operations in typical neuromorphic workloads—without sacrificing the model’s asymptotic scalability.

Experimental Evaluation

Case Study: Mixed-Modality Object Tracking

UCNN, ANN, and SNN were evaluated on an object tracking benchmark with mixed frame/even inputs. All models learned to predict the 2D object center given asynchronous sequences of static frames and inter-frame difference maps. The results can be summarized as follows:

• ANN converged to lowest mean squared error (MSE) due to access to dense spatial information.
• SNN, despite event sparsity and energy efficiency, exhibited the highest RMSE and strongest variance, reflecting instability in spike-only learning.
• UCNN closed over half the error gap between the SNN and ANN, delivering substantially lower tracking RMSE and error variability compared to SNN while retaining significant event-driven sparsity in activity. Most notably, the UCNN's error CDF and trajectory smoothness closely approached that of the ANN, supporting claims of improved tracking stability by jointly encoding graded and temporal information.

Case Study: Learning the Lorenz Attractor

The models were further benchmarked on learning the chaotic Lorenz attractor, providing insight into their ability to model nontrivial dynamical systems:

• SNNs showed inferior performance in all error metrics and suffered from high prediction variance, consistent with difficulties in representing continuous chaotic dynamics via binary events.
• ANN performed adequately but lagged behind the UCNN in both component-wise and aggregate RMSE.
• UCNN trained with EAPL achieved the lowest mean RMSE, tightest error distributions, and best reconstruction of the attractor's geometry and underlying state trajectories. The resultant spike/event activity was both regular and structured, a direct manifestation of the phase-determined temporal evolution and the graded-valued event outputs of the architecture.

Sparsity and Activity

UCNN maintained a mean active neuron fraction much closer to SNNs than ANNs, confirming its practical value in neuromorphic and edge-AI deployments requiring low-latency, energy-efficient operation with minimal computational redundancy.

Implications and Future Directions

The proposed UCNN fundamentally challenges the necessity of hand-crafted modularity or separate static-dynamic processing in hybrid neural systems. By leveraging a unified magnitude-phase framework, the architecture simultaneously maintains graded signal communication and phase-coded event emission, enabling accurate and stable learning across a spectrum of spatiotemporal tasks. The EAPL algorithm enhances computational tractability for deep or long-sequence use cases and is immediately relevant for deployment on neuromorphic hardware or event-driven sensor platforms.

From a theoretical perspective, the UCN model provides a precise abstraction aligning with biological evidence on phase-coded information transfer in neural populations. It also offers extensibility to more complex topologies—spiking graph networks, manifold learning—and to advanced edge-computation where joint value/timing encoding is critical.

Future research directions include:

• Scaling UCNNs to deeper/more complex architectures and tasks in robotics and adaptive control.
• Hardware implementation and benchmarking on neuromorphic processors with event-driven communication substrates.
• Exploration of plasticity, local learning, or unsupervised/variational extensions within the magnitude-phase computational framework.

Conclusion

This paper establishes the UCNN as a robust, unified computational paradigm that bridges the gap between the high-precision continuous computation of ANNs and the sparse, event-driven processing of SNNs. Through rigorous formulation, efficient training rule design, and comprehensive validation on multi-modal vision and nonlinear dynamical tasks, the UCNN demonstrates substantial improvements in error reduction, temporal stability, and computational efficiency over standard SNNs, while closely approaching the performance and interpretability of dense ANNs. The magnitude-phase modeling enables practical applications in edge-AI and neuromorphic computing, representing a significant advance in the design of scalable, interpretable, and efficient event-driven learning systems (2606.29099).