complex-valued-neuromorphic-magnitude-phase

id	complex-valued-neuromorphic-magnitude-phase
name	complex-valued-neuromorphic-magnitude-phase
description	Complex-valued neural network with magnitude-phase decomposition for event-driven neuromorphic learning, enabling efficient spiking computation with rich representational capacity
tags	["neuromorphic-computing","complex-valued-networks","spiking-neural-networks","event-driven-learning","magnitude-phase"]
created	2026-07-01T00:00:00.000Z
source	arxiv:2606.29099
authors	["Reza Ahmadvand","Sarah Safura Sharif","Yaser Mike Banad"]

Complex-Valued Neuromorphic Learning via Magnitude-Phase Decomposition

Core Framework

Problem Setting

Design efficient event-driven neural networks for neuromorphic hardware using:

• Complex-valued neurons: Encode information in both magnitude and phase
• Magnitude-phase decomposition: Separate amplitude and timing information
• Event-driven processing: Spike-based computation for energy efficiency

Mathematical Formulation

Complex Neuron Model

Each neuron state z ∈ ℂ is represented as:

z = r · exp(iφ)

Where:

• r: magnitude (firing rate / amplitude)
• φ: phase (timing information)
• i: imaginary unit

Network Dynamics

dz_j/dt = -z_j + Σ_k W_jk · σ(z_k) + I_j

Where W ∈ ℂ^{N×N} are complex synaptic weights.

Magnitude-Phase Split

Decouple dynamics into two real equations:

dr/dt = -r + |Σ W·σ(z)| · cos(θ) + Re(I)
dφ/dt = arg(Σ W·σ(z)) + Im(I)

Where θ is the phase difference between pre and post synaptic terms.

Key Insights

1. Dual Information Channels

• Magnitude channel: Encodes rate-based information (traditional)
• Phase channel: Encodes temporal/relative timing information
• Multiplexing: Both channels operate simultaneously on same hardware

2. Event-Driven Efficiency

• Sparse updates: Phase changes trigger selective updates
• Phase-only spikes: Some events carry only phase information
• Energy reduction: Magnitude updates are more expensive → reduce frequency

3. Representational Advantages

• Rotation invariance: Phase naturally encodes rotation/translation
• Interference patterns: Complex multiplication enables binding operations
• Frequency multiplexing: Different oscillation frequencies for different features

Learning Rules

Complex Hebbian Learning

ΔW_jk = η · (z_j · z_k* - λW_jk)

Where z_k* is complex conjugate, λ is weight decay.

Phase-Gradient Learning

∂L/∂φ_j = ∂L/∂r_j · ∂r_j/∂φ_j + ∂L/∂φ_j|direct

Separate gradients for magnitude and phase enable independent optimization.

Implementation

Neuromorphic Mapping

class ComplexSpikingNeuron:
    def __init__(self, N, tau=20.0):
        self.r = np.zeros(N)      # magnitude (membrane potential)
        self.phi = np.zeros(N)    # phase (oscillator phase)
        self.W_re = np.random.randn(N, N) / np.sqrt(N)
        self.W_im = np.random.randn(N, N) / np.sqrt(N)
    
    def forward(self, spikes_re, spikes_im):
        # Complex multiplication via real arithmetic
        h_re = self.W_re @ spikes_re - self.W_im @ spikes_im
        h_im = self.W_re @ spikes_im + self.W_im @ spikes_re
        
        # Magnitude-phase update
        self.r = 0.95 * self.r + np.sqrt(h_re**2 + h_im**2)
        self.phi = np.angle(h_re + 1j * h_im)
        
        # Spike generation (event-driven)
        spikes = (self.r > threshold).astype(float)
        self.r[spikes > 0] = 0  # reset
        return spikes, self.phi

Hardware Considerations

• Memristor crossbars: Real/imaginary parts on separate crossbar arrays
• Phase detection: Time-to-digital converters for phase measurement
• Event routing: Address-event representation (AER) with phase tags

Applications

1. Signal Processing

• Beamforming: Phase differences encode spatial information
• Frequency analysis: Natural FFT-like computation
• Coherent detection: Phase-sensitive pattern matching

2. Associative Memory

• Complex Hopfield networks: Higher capacity via phase coding
• Sequence memory: Phase progression encodes temporal order
• Pattern binding: Complex multiplication binds features

3. Sensory Processing

• Auditory: Phase encodes frequency/ITD cues
• Visual: Phase encodes edge orientation/position
• Olfactory: Phase synchrony encodes odor identity

Comparison with Prior Work

Method	Capacity	Energy	Biological Plausibility
Real-valued SNN	Baseline	Low	High
Rate-coded ANN	High	High	Low
Complex SNN	2× real	Medium	Medium
Phase-only coding	Limited	Very Low	Medium

Open Questions

1. Optimal phase coding: What phase schemes maximize information?
2. Learning stability: How to prevent phase drift during training?
3. Hardware primitives: What neuromorphic devices best support complex operations?
4. Biological evidence: Do real neurons use complex-valued computation?

References

• Original paper: arXiv:2606.29099
• Complex-valued networks: Hirose (2012)
• Event-driven learning: Tavanaei et al. (2019)
• Phase coding in neuroscience: Buzsáki (2006)