Traces Propagation in SNNs

• Traces Propagation is a forward-only, memory-efficient learning rule for spiking neural networks that combines eligibility traces with a layer-wise contrastive loss.
• It replaces the backward pass and auxiliary layer matrices with local surrogate gradients and decaying traces to solve temporal and spatial credit assignment.
• TP achieves competitive performance on benchmarks with reduced memory overhead, making it ideal for edge computing and on-device learning.

Traces Propagation (TP) is a forward-only, memory-efficient, scalable, and fully local learning rule for spiking neural networks (SNNs) that combines eligibility traces with a layer-wise contrastive loss without requiring auxiliary layer-wise matrices. It was introduced as an alternative to Backpropagation Through Time (BPTT) with surrogate gradients for jointly addressing temporal and spatial credit assignment under constraints of biological locality, efficient training, and on-device learning (Pes et al., 16 Sep 2025).

1. Problem setting and rationale

TP is defined in the supervised training setting for SNNs, where membrane states evolve over time and spikes are generated through a non-differentiable threshold. In that setting, the dominant approach is BPTT with surrogate gradients, with parameter updates written as

$$\Delta \theta_l[i,j] = - \eta \cdot \frac{\partial E_L}{\partial \theta_l[i,j]}.$$

The paper summarizes the corresponding unfolded gradient as \begin{align} \left(\frac{\partial E_L}{\partial \theta_l[i,j]} \right){\text{BPTT}} &= \sum^T{t=1} \frac{\partial E_L}{\partial v_l^t[j]} \frac{\partial v_l^t[j]}{\partial W_l[i,j]} \ &= \sum^T_{t=1} \left( \underbrace{ \frac{\partial E_L}{\partial s_l^t[j]} \frac{\partial s_l^t[j]}{\partial v_l^t[j]} }{\text{spatial}} + \underbrace{ \frac{\partial E_L}{\partial v_l^{t+1}[j]} \frac{\partial v_l^{t+1}[j]}{\partial v_l^{t}[j]} }{\text{temporal}} \right) \underbrace{\frac{\partial v_l^t[j]}{\partial W_l[i,j]}}_{\text{spatial}}. \end{align}

The motivation for TP is that BPTT is effective but incurs high memory and computation costs and is in stark contrast with spatial and temporal locality in biological neural systems. The paper highlights three classes of limitation. First, hidden-layer errors depend on reverse traversal and weight transport, with \begin{align} \frac{\partial E_L}{\partial v_l^t[j]} &= \frac{\partial E_L}{\partial v_{l+1}^t[k]} \cdot \frac{\partial v_{l+1}^t[k]}{\partial s_l^t[j]} \cdot \frac{\partial s_l^t[j]}{\partial v_l^t[j]} \ &= \frac{\partial E_L}{\partial v_{l+1}^t[k]} \cdot W_{l+1}[j,k] \cdot \frac{\partial s_l^t[j]}{\partial v_l^t[j]}. \end{align} Second, update locking forces lower layers to wait until the whole forward pass is complete. Third, memory scales with sequence length, neurons, and layers as

$\mathcal{O}(THL),$

which is problematic for deep SNNs and embedded devices.

Existing local learning rules alleviate temporal nonlocality through eligibility traces, typically written as

$$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$

However, the paper argues that such rules still struggle with spatial locality unless they use auxiliary layer-wise matrices. TP is proposed precisely to combine local temporal traces with a local spatial objective while avoiding one auxiliary matrix per layer (Pes et al., 16 Sep 2025).

2. Architecture, neuron model, and trace variables

TP is formulated for layered SNNs using discrete-time Leaky Integrate-and-Fire neurons. The membrane dynamics are

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$

and spike generation is

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$

Here $v_l^t[j]$ is the membrane potential of neuron $j$ in layer $l$ at time $t$, $s_l^t[j]$ is its spike output, $\mathcal{O}(THL),$0 is the feedforward weight, $\mathcal{O}(THL),$1 is the recurrent matrix, $\mathcal{O}(THL),$2 is the membrane decay, and $\mathcal{O}(THL),$3 is the firing threshold. Because $\mathcal{O}(THL),$4 is non-differentiable, TP still uses a local surrogate gradient,

$\mathcal{O}(THL),$5

In the reported experiments, the surrogate is ArcTan with scale factor $\mathcal{O}(THL),$6.

The characteristic objects of TP are two per-neuron temporal traces. For the input-driven path,

$\mathcal{O}(THL),$7

and for the target-driven path,

$\mathcal{O}(THL),$8

These are exponentially decaying sums of spikes. The first is an input trace for batch sample $\mathcal{O}(THL),$9; the second is a target trace for the same layer.

The target path uses analogous LIF dynamics: $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$0 At the first layer, the one-hot target $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$1 is projected using a special matrix $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$2 in place of $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$3. In deeper layers, the target representation is propagated with the forward weights themselves. This is the structural reason TP does not require auxiliary layer-wise feedback matrices (Pes et al., 16 Sep 2025).

3. Layer-wise contrastive supervision and local update rule

The distinctive spatial mechanism of TP is a layer-wise contrastive objective. At each layer $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$4 and time $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$5, TP defines the pairwise logits

$$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$6

which compare the input trace of sample $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$7 with the target trace of sample $$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$8. The corresponding local targets are

$$\left(\frac{\partial E_L}{\partial \theta_l [i,j]} \right)_{\text{three-factors}} = \sum_t S_l^t[j] \epsilon_l^t[i,j], \qquad \epsilon_l^t[i,j] = \beta \epsilon_l^{t-1}[i,j] + g(s_{l-1}^t[i])f(s_{l}^t[j]).$$9

where $$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$0 is a similarity function. The local loss is then

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$1

This construction makes each layer align input traces with target traces from the same class and separate them from target traces of other classes in the batch. In the first layer, target traces are derived from one-hot labels; in deeper layers they are propagated representations. A plausible implication is that TP turns hidden-layer supervision into a representational clustering problem rather than an error-backpropagation problem.

The paper derives the update by chain rule: \begin{align} \frac{\partial E^t_l}{\partial \theta_{l}[i,j]} &= \frac{\partial E^t_l}{\partial z_l^t}\frac{\partial z_l^t}{\partial \theta_l} \ &= \frac{\partial E^t_l}{\partial z_l^t} \left[ \underbrace{ \frac{\partial z_l^t}{\partial \epsilon_l^t} \frac{\partial \epsilon_l^t}{\partial s_l^t} \frac{\partial s_l^t}{\partial v_l^t} \frac{\partial v_l^t}{\partial \theta_{l}} }{\text{input}} + \underbrace{ \frac{\partial z_l^t}{\partial \tilde{\epsilon}_l^t} \frac{\partial \tilde{\epsilon}_l^t}{\partial \tilde{s}_l^t} \frac{\partial \tilde{s}_l^t}{\partial \tilde{v}_l^t} \frac{\partial \tilde{v}_l^t}{\partial \theta{l}} }{\text{target}} \right]. \end{align} The resulting local rule is \begin{align} \frac{\partial E^t_l}{\partial \theta{l}[i,j]} &= (\text{Softmax}(z_l^t) - y_l^t)[b,b'] \Big[ \tilde{\epsilon}l^t[b',j] \, \Theta'(v_l^t[b,j] - V{\text{th}}) \, s^t_{l-1}[b,i] \ &\hspace{3.3cm}+ \epsilon_l^t[b,j] \, \Theta'(\tilde{v}l^t[b',j] - V{\text{th}}) \, \tilde{s}^t_{l-1}[b',i] \Big]. \end{align}

The appendix-style algorithm separates the update into input-path and target-path terms,

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$2

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$3

followed by

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$4

In this formulation, temporal credit assignment is handled by the decaying traces $$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$5 and $$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$6, while spatial credit assignment is handled by the local contrastive objective. TP therefore remains forward-only at the network level, but not derivative-free: it still uses local surrogate derivatives $$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$7 (Pes et al., 16 Sep 2025).

4. Computational profile, memory scaling, and algorithmic trade-offs

TP is designed to reduce both backward-pass dependence and auxiliary storage. The paper gives total TP memory as

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$8

hence

$$v^t_l[j] = \alpha^l v^{t-1}_l[j] + s^t_{l-1}[i] W_l[i,j] + s^{t-1}_{l}[j] R_l[j,j]- s_{l}^{t-1}[j] v_{th},$$9

Asymptotically, TP stores per-neuron quantities and therefore scales as

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$0

rather than $s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$1 as in BPTT.

The comparison with TESS is central. The paper gives

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$2

so the relative memory cost is

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$3

It follows directly that TP is more memory-efficient whenever

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$4

This condition becomes increasingly favorable when the number of output classes is large.

For computation, the paper isolates the cost of the modulatory signal and notes that TP depends on batchwise pairwise similarities, yielding

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$5

whereas output-projection-based local rules often scale as $s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$6. It also provides a full multiply–accumulate estimate: $s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$7

These formulas define the main trade-off. TP removes backward traversal, removes per-layer auxiliary matrices, and stores only per-neuron state variables, but it requires batchwise pairwise comparisons. Consequently, the method is not online per sample: it requires

$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$8

A plausible implication is that TP is most attractive when memory is the dominant constraint and moderate batchwise pairwise computation is acceptable (Pes et al., 16 Sep 2025).

5. Empirical behavior across datasets and fine-tuning tasks

The reported experiments cover shallow feedforward SNNs, recurrent SNNs, and deep convolutional SNNs, including VGG-9. The results show three main patterns. First, TP outperforms other fully local learning rules on N-MNIST and SHD. Second, on DVS-GESTURE it remains competitive with BPTT and TESS while scaling to VGG-9. Third, on DVS-CIFAR10 it remains functional but loses more accuracy, which the paper attributes to the contrastive mechanism being hindered by high inter-class similarity (Pes et al., 16 Sep 2025).

Dataset/task	TP result	Comparison note
N-MNIST	$s^t_l[j] = \Theta (v^t_l[j] - v_{th}).$9	Outperforms other fully local rules
SHD (feedforward)	$v_l^t[j]$0	Best among fully local methods reported
SHD (recurrent)	$v_l^t[j]$1	Best fully local result in table
DVS-GESTURE (VGG-9)	$v_l^t[j]$2	Competitive with BPTT and TESS
DVS-CIFAR10 (VGG-9)	$v_l^t[j]$3	Larger gap to BPTT/TESS

The deeper-model evidence is particularly significant. On DVS-GESTURE, TP reaches $v_l^t[j]$4 on VGG-9, compared with $v_l^t[j]$5 for BPTT and $v_l^t[j]$6 for TESS. This suggests that the absence of layer-wise auxiliary matrices does not prevent scaling to deeper convolutional SNNs.

The paper also provides a representational analysis on DVS-GESTURE using t-SNE visualizations of final-time-step input traces across the eight convolutional layers of VGG-9. The reported observation is that class traces are mixed in early layers, progressively separate in middle layers, and form tighter clusters in deeper layers; the paper also reports increasing Silhouette scores across depth. This is consistent with the intended effect of the local contrastive loss.

For practical adaptation at the edge, TP is evaluated on Google Speech Commands in a fine-tuning setting. A model is pretrained on cloud data and then adapted to a held-out user. The reported user-specific query accuracy before fine-tuning is $v_l^t[j]$7. After TP fine-tuning, the paper reports improvements of $v_l^t[j]$8 percentage points for 1-shot, $v_l^t[j]$9 percentage points for 5-shot, and $j$0 percentage points for all-shot in one part of the text, while a later discussion reports $j$1, $j$2, and $j$3 percentage points for 1-shot, 5-shot, and all-shot. The paper therefore contains an internal inconsistency in the shot-specific improvement values. It is nevertheless explicit that pretraining is essential, because training only on the user data yields much lower accuracies. The fine-tuning experiment positions TP as suitable for lightweight personalization and on-device adaptation.

6. Scope, limits, and terminological ambiguity

TP has several explicit limitations. It is not online per sample because its contrastive objective requires inter-sample comparisons over a batch. Its modulatory-signal cost scales as $j$4. It shows a larger performance gap on DVS-CIFAR10 than on N-MNIST, SHD, or DVS-GESTURE. It also stores both input-path and target-path potentials and traces, effectively doubling internal state relative to single-path local rules. At the same time, the reported results indicate that TP is a strong option when memory is constrained, fully local forward-only learning is required, and the number of classes is large relative to batch size.

The name is also potentially misleading outside the SNN literature. The acronym TP is widely overloaded. In machine learning, it also denotes Target Propagation (Meulemans et al., 2020, Lee et al., 2014, Roulet et al., 2021, Shibuya et al., 2022), Thought Propagation (Yu et al., 2023), and Tree Pruning (0710.0564). In systems and observability, closely related phrases refer to span correlation and context carriage in distributed tracing (Phan et al., 15 Aug 2025), guaranteed delivery of control-flow logs (Caulfield et al., 2024), validation of execution traces against TLA$j$5 specifications (Cirstea et al., 2024), provenance tracing in database semantics (0812.0564), and information-flow path reconstruction from RTL traces (Deutschbein et al., 11 Jun 2026). A common misconception is therefore to treat “Traces Propagation” as a generic trace-analysis term. In the specific sense defined here, however, Traces Propagation is the 2025 SNN learning rule that combines eligibility-like temporal traces with a layer-wise contrastive objective and a single target propagator to obtain forward-only, fully local learning.