Complemented Ternary Spiking Neuron

• CTSN is a spiking neuron model that integrates ternary signaling with a learnable, non-resetting memory to mitigate iterative information loss.
• It enhances temporal gradient flow and network heterogeneity, leading to improved performance on both static image and neuromorphic benchmarks.
• The model supports efficient hardware implementation via superconducting circuits, achieving high-speed, energy-efficient computation with state-of-the-art accuracy.

A Complemented Ternary Spiking Neuron (CTSN) is a model of spiking neuron that integrates ternary-valued signaling with a learnable memory mechanism, designed to improve information retention, temporal gradient flow, and network heterogeneity within both algorithmic and hardware SNN implementations. This model advances on prior spiking neuron architectures by embedding adaptive, non-resetting memory and supporting excitatory-inhibitory balance, while offering straightforward mapping to high-speed hardware such as superconductor-based circuits. CTSN models have been demonstrated to achieve state-of-the-art performance on a variety of neuromorphic and static image benchmarks and form the computational primitives of recent ultra-fast, cryogenic SNN accelerators (Zhang et al., 22 Jan 2026, Karamuftuoglu et al., 2024).

1. Mathematical Definition and Dynamics

The baseline ternary spiking neuron model, as formalized by Guo et al. (2024), consists of leaky membrane updates, hard resets on spike emission, and ternary output quantization:

• Membrane update:

$$u^{(l)}(t) = \tau\,u^{(l)}(t\mathbin{-}1)\,(1 - |o^{(l)}(t\mathbin{-}1)|) + x^{(l)}(t), \quad x^{(l)}(t) = W^{(l)} o^{(l-1)}(t) + b^{(l)}$$

• Ternary spike:

$$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$

The CTSN augments this framework with a learnable, non-resetting complemental term $h^{(l)}(t)$, which preserves partial state across spikes:

• Integrated update:

\begin{align*} u^{(l)}(t) &= \tau\,\tilde{u}^{(l)}(t-1)\,(1 - |o^{(l)}(t-1)|)\ \tilde{u}^{(l)}(t) &= h^{(l)}(t) + x^{(l)}(t)\ o^{(l)}(t) &= \Theta(\tilde{u}^{(l)}(t), V_{\rm th}) \end{align*}

• Complemental memory dynamics:

For static image inputs: $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$ For neuromorphic/dynamic inputs: $$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$ with $$\alpha, \beta, \gamma = \sigma(\omega_a), \sigma(\omega_b), \sigma(\omega_c) \in (0,1)$$ and typically initialized to 0.5.

The presence of $h^{(l)}(t)$ endows the neuron with heterogeneity and partial memory, counteracting iterative loss due to reset and supporting richer temporal dynamics (Zhang et al., 22 Jan 2026).

2. Biological and Algorithmic Motivation

Vanilla ternary spiking neurons experience full memory loss after each spike due to the hard reset, precluding any influence of pre-spike states on future output—this is described as iterative information loss. Additionally, in backpropagation-through-time (BPTT), two compounding effects undermine stable learning: rapid attenuation due to repeated factors of small $\tau < 1$, and abrupt gradient vanishing whenever large spikes occur ($|u| > 1.5 V_{\rm th}$). The complemental term $h(t)$ bypasses hard resets, enabling ongoing integration of past states and introducing dynamic heterogeneity at the neuron level. In gradient-based learning, $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$0 supplies alternative gradient pathways ($$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$1), preventing rapid decay and supporting reliable temporal credit assignment (Zhang et al., 22 Jan 2026).

3. Temporal Membrane Potential Regularization (TMPR)

CTSNs are typically trained jointly with a TMPR loss, designed to regularize the distribution and magnitude of time-evolving membrane potentials:

• Loss function:

$$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$2, with

$$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$3

where $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$4 is batch size, $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$5 the number of neurons in layer $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$6, $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$7 the time horizon, $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$8 the number of layers, and $$o^{(l)}(t) = \Theta(u^{(l)}(t), V_{\rm th}) = \begin{cases} +1 & \text{if } u \geq V_{\rm th} \ -1 & \text{if } u \leq -V_{\rm th} \ 0 & \text{otherwise} \end{cases}$$9 a small hyperparameter.

• The $h^{(l)}(t)$0 factor exerts strong regularization early in the sequence to prevent premature over-activation and decays temporally to favor use of later memory.
• TMPR directly creates spatiotemporal shortcut gradients ($h^{(l)}(t)$1), augmenting the gradient signal available to earlier time steps and deep layers.

This approach demonstrably stabilizes training and mitigates the temporal vanishing gradient phenomenon in recurrent SNNs (Zhang et al., 22 Jan 2026).

4. Hardware Realization and Ternary Superconducting Circuits

A complementary hardware instantiation of CTSN has been engineered using superconductor electronics, leveraging single flux quantum (SFQ) logic and Josephson Junctions (JJ) (Karamuftuoglu et al., 2024):

• Each synapse is implemented as ternary ($h^{(l)}(t)$2), configured via mutual inductive coupling to the neuron loop.
• Input spikes are encoded as SFQ voltage pulses; excitatory/inhibitory weighting is enacted by routing these pulses through positive/negative inductive loops.
• The neuron loop contains two output JJs (JJP for positive, JJN for negative). Whichever output threshold is crossed first fires the corresponding output.
• Leaky integration is realized via resistive branches in each dendrite, establishing the time constant $h^{(l)}(t)$3.

This hardware CTSN supports high fan-in architectures, asynchronous firing, and dual-polarity outputs, capturing the complemented ternary operation at the physical level (Karamuftuoglu et al., 2024).

Feature	Algorithmic CTSN (Zhang et al., 22 Jan 2026)	Superconductor CTSN (Karamuftuoglu et al., 2024)
Memory mechanism	Learnable complemental term $h^{(l)}(t)$4	Persistent loop current, no hard reset
Ternary signaling	$h^{(l)}(t)$5 output	SFQ pulses, positive/negative coupling
Reset behavior	Partial; $h^{(l)}(t)$6 never resets	Loop reset only when either JJ fires

5. Pseudocode and Training Integration

Forward and backward passes for CTSN employ the following algorithmic structure:

• For each time step:
  1. Update $h^{(l)}(t)$7,
  2. Update $h^{(l)}(t)$8 using $h^{(l)}(t)$9,
  3. Compute $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$0,
  4. Output $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$1,
  5. Post-spike hard reset: $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$2.

Surrogate gradients are applied to the non-differentiable $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$3 using a rectangular window of width $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$4. Training is performed using SGD with standard spiking data augmentation and weight decay. For hardware-oriented SNNs, quantized weights are pruned and mapped directly to SFQ-coupled circuits after training (using straight-through estimators and in-situ masking).

6. Empirical Results and Performance Characteristics

Across static image and neuromorphic datasets, CTSN consistently improves over baseline and SOTA ternary SNNs. Key results:

• Ablation (ResNet20):
  • CIFAR-100, $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$5: Baseline 73.95%, +CTSN 74.48%, +CTSN+TMPR 74.82%
  • CIFAR10-DVS, $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$6: Baseline 80.30%, +CTSN 80.57%, +CTSN+TMPR 81.23%
• Full-network SOTA Scores:
  • CIFAR-10, ResNet19: 96.28% (ternary) → 96.46% (CTSN+TMPR)
  • CIFAR-100, ResNet19: 79.68% → 81.19%
  • ImageNet-100, SEW-ResNet34: 83.27% (prior best) → 85.06% (CTSN+TMPR)
  • Neuromorphic (VGGSNN, CIFAR10-DVS): 83.20% (CTSN+TMPR open-source reimpl.)
• Temporal ablation: For short sequence lengths ($$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$7), CTSN yields negligible difference. Gains increase with $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$8, supporting the interpretation that $$h^{(l)}(t) = \alpha \operatorname{ReLU}(h^{(l)}(t-1)) + \beta(-\operatorname{ReLU}(-h^{(l)}(t-1))) + \gamma u^{(l)}(t)$$9 improves long-term credit assignment by maintaining informative state.
• Hardware platform: On MNIST, a SFQ-based CTSN SNN (784–128–96–96–10) achieves 96.1% accuracy with 64-synapse pruning, 8.92 GHz throughput, and $$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$01.5 nJ per inference (including 4 K cooling). Ternary pruning reduces static power by 82.7% with a 0.97% accuracy drop (Karamuftuoglu et al., 2024).

7. Architectural and Hyperparameter Considerations

• Surrogate gradient: rectangular window, width $$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$1
• Decay constant: $$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$2 (all experiments)
• Training: SGD, momentum 0.9, cosine-annealed LR, batch size 64, 300 epochs
• Weight decay: $$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$3 (static), $$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$4 (DVS)
• Input encoding: Direct multi-step encoding for static, 1-bit Poisson for hardware SNNs
• Data augmentation: Standard flips/crops, AutoAugment, Cutout (static), random roll+flip (DVS)

CTSNs require three additional per-neuron scalars ($$h^{(l)}(t) = \alpha h^{(l)}(t-1) + \beta \operatorname{ReLU}(u^{(l)}(t)) + \gamma(-\operatorname{ReLU}(-u^{(l)}(t)))$$5), imposing a minimal parameter overhead while furnishing substantial gains in memory retention, gradient flow, and task performance across diverse SNN architectures (Zhang et al., 22 Jan 2026, Karamuftuoglu et al., 2024).