Unveiling the Spatial-temporal Effective Receptive Fields of Spiking Neural Networks

Q1: Concerns about the comparison between ViT-B and S-ViTs:

A1: We sincerely appreciate your attention to the clarity of Figure 2 and the ensuing concerns it raised regarding the rigor of our experimental controls. We understand your concerns and would like to clarify the original intent behind Figure 2 and its key findings.

The primary purpose of Figure 2 was to: provide a visualization and comparison of the Spatial Effective Receptive Fields (Spatial ERFs) for comparable ANN- and SNN-based Transformer models. This approach was designed to profoundly reveal the underlying learning mechanisms and behavioral patterns. Our experimental design rigorously controlled for architectural consistency:

1. ViT-like Architecture Comparison Group: The Vanilla ViT (ANN) and Spikformer/SDT-V1 (SNN) exhibit identical module structure, with the core difference residing solely in the activation function (ANN: ReLU; SNN: Spiking).
2. Metaformer Architecture Comparison Group: The CAFormer-s18 (ANN, Metaformer paradigm) and its counterpart Meta-SDT-S (SNN) are precisely matched in critical design elements such as convolutional pathways and mixed hierarchical block design.

The experimental results clearly demonstrate that, under strict control of architectural variables, the ANN and SNN models exhibit strikingly similar Spatial ERF distribution characteristics:

• ViT-like architecture's ERF manifests a checkerboard pattern, which is highly similar to the ERF patterns observed in both Spikformer (8-512) and SDT-V1.
• Similarly, the ERF of both CAFormer-s18 (ANN) and Meta-SDT-S (SNN) display a highly localized pattern, remarkably analogous to that of traditional CNN networks. This indicates the model's focus is primarily concentrated on local regions, with relatively limited interaction occurring with other image patches.

Therefore, the consistent ANN/SNN ERF patterns presented in Figure 2 precisely serve as compelling evidence for the effectiveness of our experimental design and the rigor of our variable control (particularly concerning architecture). This observed consistency directly captures the key phenomenon we aimed to introduced: Different architectures influence the learning behavior of ANN vs. SNN models, which is the foundation of our investigation on the limitation in spatial ERF distribution in current S-ViT models.

To present the aforementioned experimental design and core findings more clearly, we will undertake a comprehensive reorganization of Figure 2:

• ANN models will be presented side-by-side with their structurally identical SNN counterparts.
• Move QKFormer (10-384) visualization to Appendix, as its architecture is not quite comparable to the other models.

Q2: More Spatial ERF visualizations for other ViT models:

We have supplemented the following ViT variants to provide a more comprehensive presentation of the Spatial ERF characteristics across different ViT models. Since the rebuttal contents are not allowed to contain image links, we will give an overall comparison in the following table:

Note that all the above ANN-based models exhibit a more global receptive field distribution compared to the S-ViT models, which is consistent with the findings in our paper. The interaction intensity with other patches in ANN-based models is also higher, indicating a more global receptive field distribution. We will include these visual aids into the revised manuscript to facilitate a more comprehensive comparison of the Spatial ERF distributions across different ViT models.

Q3: Concerns about the controlled replacements of MLPixer and SR Block and unclear analysis path from spatial ERF to MLPixer/SR block design:

A3.1: We thank the reviewer for raising this point. In our detection and segmentation experiments, all blocks in Stages 1 & 2 were systematically replaced with MLPixer or SR Block, ensuring that block type was the sole variable. This controlled setup allowed us to isolate the impact of architectural design on model performance.

A3.2: We acknowledge that the connection between Spatial ERF analysis and our block design choices was not fully articulated. The key insight is that the Spatial ERF distribution of the S-ViTs, as visualized in Figure 2, reveals a more localized pattern: having a stronger focus on local features but limited global context. This suggests that current S-ViTs may not fully leverage the global receptive field capabilities of their ANN counterparts. Throughout the exploration, we observed that the MLPixer and SR Block designs, which are more suited for global feature extraction, align better with the Spatial ERF distribution of S-ViTs. This is why we chose to replace the original blocks with MLPixer and SR Block in our experiments. Our experiments demonstrated that these replacements led to improved performance in detection and segmentation tasks, further supporting the hypothesis that S-ViTs can benefit from architectural designs that enhance global context understanding.

Q4: Question on the benchmarking of S-ViTs: Provide more temporal benchmarks:

We appreciate your suggestion to provide more temporal benchmarks for S-ViTs. We have conducted additional experiments on the DVS128 Gesture datasets based on the QKFormer network, which are commonly used benchmarks for event-based vision classification tasks.

The baseline network is QKFormer-PE1(4-layer-res)-S1-PE2(2-layer-res)-S2, containing two stages of Q-K attention (S1 & S2) and the corresponding positional encoding blocks(PE1 & PE2). We apply the block design of MLPixer and SR block on the PEs. Note that because the original QKFormer has residual paths in PEs, we also added the block design of MLPixer and SR block on the residual paths to see their performance

The results are summarized in the following table:

Exp Setting:

We also have conducted additional experiments on the Event-based tracking tasks on FE108 dataset and VisEvent dataset. The results are summarized in the following tables:

Exp Setting:

These results confirm that the MLPixer and SR block designs preserve S-ViT performance. While current temporal benchmarks show promise, further improvement is possible. We appreciate your insightful suggestion, which has highlighted this important direction for exploration.

Q1: ST-ERF in commonly used LIF model :

A1: We apologize for the oversight in not including the ST-ERF visualization for the commonly used LIF model in our original submission. We have now finished this visualization. The detailed explanation of the ST-ERF for the LIF model is shown below the table since the rebuttal contents are not allowed to contain image links. We will include this visualization in the appendix to provide a more comprehensive understanding of the ST-ERF in commonly used LIF models. (SG: surrogate gradient function)

Experiment Setting:

The LIF-based ST-ERF visualization results show that the ST-ERF distribution follows the 2D Gaussian-like pattern, which is consistent with the findings in our paper. The horizontal and vertical $\sigma^2$ values are smaller than those of the I-LIF and NI-LIF models, indicating a more localized receptive field distribution.

Q2: More comparison to SOTA models on various tasks:

A2: We appreciate your suggestion. We have conducted additional experiments on event-based tracking tasks using the FE108 and VisEvent datasets to compare with SOTA models.

Event-based Tracking Experiments: To comprehensively evaluate our ERF-guided architectural improvements across different temporal processing domains, we extended our evaluation to event-based object tracking tasks. Event-based tracking presents unique challenges as it requires processing asynchronous, sparse event streams with high temporal resolution while maintaining spatial consistency across time. Unlike traditional RGB tracking, event cameras capture brightness changes asynchronously, producing sparse event streams that require specialized temporal aggregation and spatial attention mechanisms.

Our experimental setup employed the SD-Track architecture as the baseline, which integrates event-based feature extraction specifically designed for processing event streams. We systematically replaced the core processing blocks with our proposed MLPixer and SR block variants while maintaining identical experiment setups. The FE108 dataset provides high-speed tracking scenarios with rapid object motion, while VisEvent offers more diverse tracking conditions with varying lighting and occlusion patterns. The results are summarized in the following tables:

The results demonstrate that our architectural modifications not only preserve but in some cases improve tracking performance across both precision (PR) and area-under-curve (AUC) metrics. Notably, MLPixer with $\epsilon=6$ and SR blocks show consistent improvements, validating that our ERF-guided architectural choices are effective across diverse temporal processing tasks beyond static image classification.

Q3: In Figure 6, the receptive field of SR-SDT is mainly concentrated in the lower-left region. What causes this localization?

A3: The localization of the receptive field in the lower-left region of Figure 6 is primarily due to the model's learning selectivity. This phenomenon occurs because the analysis framework is based on the propagation of gradients, which can lead to a shift in the effective receptive field when the gradient stimuli backwards towards the whole network. The input data also plays a role in this localization effect, as it influences how the model learns to focus on specific regions of the input space.

To verify whether the Spatial ERF is pixel-wise invariant, we can conduct experiments by shifting the input gradient stimuli and observing the corresponding changes in the spatial ERF pattern. If we use a stimuli that simply shifts the focus by a certain amount, we would see a totally different receptive field distribution. This result demonstrates that in a properly-trained network, every single pixel has its own unique ERF pattern. To verify whether the training data is responsible for this effect, we can conduct ablation studies by sampling a subset of the original training data and observing the changes in the spatial ERF given the same gradient stimuli. We will include the visualization explanation in the updated manuscript to clarify this point further. The comparison is below the table since the rebuttal contents are not allowed to contain image links.

The exploration of the spatial ERF in S-ViTs, particularly in the context of MLPixer and SR blocks, reveals that the design of these architectures plays a crucial role in introducing a disaggregated-then-reintegrated receptive field structure. This finding highlights the importance of architectural choices in shaping the effective receptive fields and their localization properties.

Q4: How to adapt ERF-based conclusions in SNNs to practical applications?

A4: Through the comprehensive ST-ERF analytical framework, we demonstrate that SNN blocks with specialized designs (including but not limited to task-specific convolutional operators, adaptive/windmill convolutions, or attention-enhanced mechanisms such as self-attention/SE modules) exhibit distinct input feature preferences. These spatially modulated preferences facilitate the interpretability of block-level behaviors across different tasks, thereby providing principled guidance for: (1) designing more powerful and computationally efficient task-specific architectures, or (2) formulating constraints for Neural Architecture Search (NAS) to discover optimal topologies.

Temporally, our analysis reveals dynamic preference variations across timesteps in SNN implementations (under current LIF/I-LIF activation paradigms). The intrinsic membrane potential updating mechanism inherently strengthens responses to recent inputs while attenuating distant temporal information—a property we term temporal locality bias. This finding motivates further investigation via the ST-ERF framework to quantify how neuronal dynamics parameters govern spatiotemporal receptive fields in more biologically plausible SNNs. Such exploration could yield fundamental insights into the shared principles and divergences between brain-inspired networks and biological neural systems.

Q1: Corrigendum in Figure 3 where the second block design appears to lack residual connections.

A1: We sincerely appreciate your meticulous review and valuable feedback. You rightly identified the missing residual connection in Figure 3, which was indeed an oversight in our original manuscript. We sincerely apologize for this error.

We have carefully revised Figure 3 in the updated manuscript—the residual pathway is now explicitly illustrated in the MLPixer block (black solid arrows). Furthermore, our actual implementation (Code in supplementary materials: erf_compute/spatial_Erf/erf_sdt/models/sdtv3.py, Lines 357 & 395) and mathematical formulations in the original manuscript (Eq. (12) & (13): $\mathbf{X}''= \mathbf{X}'+\mathrm{Mixer}_{ch}(\mathbf{X}')$) confirm that the residual connection was consistently incorporated in the design.

Q2: Figure 5(b) legend lacks clear explanation of which experiments (a)-(d) each element represents, hindering interpretation.

A2: We apologize for the lack of clarity in the legend of Figure 5(b). In Figure 5(b), we aimed to illustrate the temporal ERF patterns of Spiking CNNs with different spike activations. We employ a 20-layer deep SCNN network to derive temporal ERFs across four distinct surrogate functions. From subfigures (a) to (d), the surrogate functions are as follows: (a): $\mathrm{arctan(\cdot)}$, (b): $\mathrm{Sigmoid}(\cdot)$, (c): $\mathrm{Rect(\cdot)}$, (d): $\mathrm{Poly(\cdot)}$.

In each subfigure, we visualize the temporal ERF patterns at different decay factors. Considering that the spike activations are achieved by LIF neurons, we can describe the forward process as follows:

$$\begin{aligned} \mathbf{v}^{\ell}[t]&=\mathbf{h}^{\ell}[t-1]+f({\mathbf{w}^{\ell}},\mathbf{x}^{\ell-1}[t-1]), &\text{(Charging function)} \\ \mathbf{s}^{\ell}[t]&=\mathbf{\Theta}(\mathbf{v}^{\ell}[t]-\vartheta), &\text{(Firing function)}\\ \mathbf{h}^{\ell}[t]&= \beta\mathbf{v}^{\ell}[t]- \vartheta \mathbf{s}^{\ell}[t], &\text{soft reset} \end{aligned}$$

Some amount of works define the LIF model as

$$\begin{aligned} \mathbf{h}^{\ell}[t]&= (1-\frac{1}{\tau})\mathbf{v}^{\ell}[t]- \vartheta \mathbf{s}^{\ell}[t], &\text{soft reset} \end{aligned}$$

where $\tau$ is the decay factor. However, in our theoretical framework, we use $\beta=1-\frac{1}{\tau}$ for convenience while ensuring consistent monotonicity between $\beta$ and $\tau$

Proven by Appendix D of the original manuscript, the temporal ERF is the temporal effective receptive field decays exponentially with time delay $\tau$, and the decay rate is primarily determined by the membrane potential decay constant $\beta$, i.e., $\underset{\mathcal{T}}{\mathrm{ERF}}(\cdot,\tau) \propto \beta^{\tau}$. The four experiments used decay factors of 2.0, 1.33, 1.2, and 1.14, respectively, which may not be the commonly-said decay factors in many essays. Throughout the experiments, we observed that the fitted curves closely followed the trends of the original data, validating our theoretical framework.

Q3: Broader task validation to demonstrate framework versatility.

A3: We appreciate your suggestion to validate our framework across a broader range of tasks. In response, we have conducted additional experiments on (a): DVS data classification tasks using DVS128Gesture, and (b): event-based tracking tasks using the FE108 and VisEvent datasets.

(a): DVS data classification tasks. The baseline network is QKFormer-PE1(4-layer-res)-S1-PE2(2-layer-res)-S2, containing two stages of Q-K attention (S1 & S2) and the corresponding positional encoding blocks(PE1 & PE2). We apply the block design of MLPixer and SR block on the PEs. Note that because the original QKFormer has residual paths in PEs, we also added the block design of MLPixer and SR block on the residual paths to see their performance. The results are summarized in the following table:

(b): Event-based tracking tasks. Our experimental setup employed the SD-Track architecture as the baseline, which integrates event-based feature extraction specifically designed for processing event streams. We systematically replaced the core processing blocks with our proposed MLPixer and SR block variants while maintaining identical experiment setups. The results are summarized in the following tables:

The results demonstrate that our architectural modifications not only preserve but in some cases improve tracking performance across both precision (PR) and area-under-curve (AUC) metrics. Notably, MLPixer with $\epsilon=6$ and SR blocks show consistent improvements, validating that our ERF-guided architectural choices are effective across diverse temporal processing tasks beyond static image classification.

Q4: Further connections between ST-ERF and other biological theories of neural dynamics?

A4.1 Connections with neural oscillation theory: While the current ST-ERF framework focuses on temporal-domain assessment of dynamic feature variations, its natural extension to frequency-domain analysis would unveil neuronal response mechanisms across spectral bands. Such a multidimensional spatiotemporal-frequency analytical approach would: (i) deepen understanding of feature extraction processes in biological neural populations, and (ii) advance bio-plausible computational models for next-generation neuromorphic chips (e.g., Loihi), particularly in cognitive tasks demanding sophisticated time-frequency processing (e.g., working memory and spatial navigation). This proposed expansion aligns with the emerging paradigm of hybrid time-frequency analysis in computational neuroscience.

A4.2 Connections with synaptic plasticity theories: Although the current ST-ERF framework is based on gradients, a simple yet potential method to incorporate STDP into the proposed ST-ERF framework to the analysis of synaptic weights can be modifying the core definition in the original manuscript Eq.(3) with an STDP window function $g(\Delta t)$, where $\Delta t = t_{\text{post}} - t_{\text{pre}}$ captures pre-/post-synaptic timing relationships. This allows explicit modeling of how synaptic efficacy modulates input-output sensitivity. The spatial (Eq.(4)) and temporal (Eq.(5)) decompositions of ST-ERF further benefit from STDP-aware formulations: Spatial ERFs should weight contributions by STDP-driven synaptic updates $w(t,\tau) \propto \eta \sum_{\Delta t} g(\Delta t) I_{\text{pre}}(t - \tau) I_{\text{post}}(t)$, while temporal ERFs can directly adopt the bi-exponential form to reflect STDP’s asymmetric time constants. This unified perspective may reveal how STDP dynamically sculpts spatiotemporal processing, which can be a critical insight bridging synaptic mechanisms with system-level information encoding.

Q5: How is the ST-ERF compatible with the scheme of encoding information in pulse timing?

A5: Our framework employs dynamic firing threshold (DFT)-based neurons, which process input spikes in two stages:

1. Membrane potential accumulation:$V_j^l(t) = \sum w_{ij}^l \cdot \mathrm{ReL}(I_i^{l-1}(t))$
2. Dynamic Spike firing (DFT mechanism):The firing time $t_j^l$ is determined by direct/standard firing or no firing based on a dynamic threshold $\theta^l[t]$ Note that the spatial ERF is similar to the original formulation. The temporal ERF captures how the timing of input spikes influences the output spike timing. As the spike time-to-first-arrival (TFA) is a primary parameter to describe the output spikes, we rewrite the temporal ERF in the manuscript Eq.(7) as: $\mathrm{ERF}_\mathcal{T}(s^{\ell}, \tau) =\sum\frac{\partial \mathcal{L}}{\partial s^{\ell}[T]} \cdot \frac{\partial s^{\ell}[T]}{\partial (T-\tau)^{\ell}}$ $\frac{\partial s^{\ell}[t]}{\partial t^{\ell}}=\frac{\partial s^l[t]}{\partial \Theta_j^l[t]} \cdot \frac{\partial \Theta_j^l[t]}{\partial V_j^{l}[t]} \cdot \frac{\partial V_j^{l}[t]}{\partial t^{l}} - \frac{\partial s^l[t]}{\partial \Theta_j^l[t]} \cdot \frac{\partial \Theta_j^l[t]}{\partial \theta^l[t]} \cdot \frac{\partial \theta^l[t]}{\partial t^{l}}$

By setting the gradient stimuli $\frac{\partial \mathcal{L}}{\partial s_{(i,j)}^{\ell}[T]}$ to 1, we can compute the temporal ERF using Pytorch's autograd functionality. This allows us to analyze how the timing of input spikes influences the output spike timing, thus providing insights into how information is encoded in the timing of spikes.

Q1: Some of the operator notations in equations (especially Eq. (12), (13)) are overloaded or ambiguously defined.

A1: We appreciate your feedback regarding the operator notations in equations, particularly Eq. (12) and (13). Here we provide a table summarizing the definitions of the operators used in these equations:

We have revised the manuscript to clarify the definitions of the operators used in these equations. The Eq.(12) and (13) is supposed to be: $\mathrm{Mixer}_{ch}(\cdot) = \mathrm{BN}(\mathrm{MLP}(\mathbb{SN}(\mathrm{BN}(\mathrm{MLP}(\mathbb{SN}(\cdot))))))$

$\mathrm{Mixer}_{ch}(\cdot) = \mathrm{BN}(\mathrm{Conv}(\mathbb{SN}(\mathrm{BN}(\mathrm{MLP}(\mathbb{SN}(\cdot))))))$

We have ensured that all operators are clearly defined and consistently used throughout the manuscript. We hope this clarification addresses your concerns. If you have any further questions or suggestions, please feel free to let us know.

Q2: ST-ERF behaviors in other spiking neuron models.

A2.1: Izhikevich's spiking neuron model:

We conducted the Temporal ERF framework on Izhikevich's spiking neuron model, which is a well-known model in computational neuroscience. The Izhikevich model is defined by the following equations:

\frac{dV}{dt} = 0.04V^2 + 5V + 140 - U + I, \frac{dU}{dt} = a(bV - U)$$ $$if \space V \geq v_{threshold}, then \space V \leftarrow c, U \leftarrow U + d$$ where $V$ is the membrane potential, $U$ is the recovery variable, and $I$ is the input current. The parameters $a$, $b$, $c$, and $d$ are constants that define the specific behavior of the neuron. We discretized the equations using Euler's method and implemented the Izhikevich model in PyTorch and Spikingjelly, which can be formulized as follows: $v(t + \Delta t) \approx v(t) + \Delta t \left[ 0.04v(t)^2 + 5v(t) + 140 - u(t) + I(t) \right]$, $u(t + \Delta t) \approx u(t) + \Delta t \left[ a(bv(t) - u(t)) \right]$ $V \leftarrow c, U \leftarrow U + d$ Description of each parameter is provided in the following table: |Parameter|Description| |---------|-----------| |$a$|Time scale of the recovery variable| |$b$|Sensitivity of the recovery variable to the membrane potential| |$c$|Reset value of the membrane potential after a spike| |$d$|Increment of the recovery variable after a spike| |$\Delta t$ or $tau_{inv}$|Time step for discretization| By properly setting the parameters, we can acquire the desired spiking behavior as we discovered in the manuscript. The parameter $\Delta t$ primarily controls the time resolution of the simulation, and influences the temporal ERF evolution (same as the parameter $\beta$ in the manuscript). The detailed parameter settings and results are summarized in the following table: | a | b | c | d | tau_inv | v_threshold | Temporal ERF | |---|---|---|---|---------|-------------|--------------| | 0.02 | 0.2 | -65.0 | 6.0 | 0.006 | 2.0 | Exponential Decay| Note that the Izhikevich model can be extremely flexible, and the parameters can be adjusted to achieve various spiking behaviors(e.g. bursting, regular spiking, etc.). And we believe that the exploration of these behaviors using the Temporal ERF framework can provide valuable insights into the dynamics of spiking neural networks. A2.2: **Parametric LIF:** We also conducted the Temporal ERF framework on the Parametric LIF model, which is a variant of the LIF model that incorporates learnable parameters to control the dynamics of the neuron. Because it is a variant of the LIF model, the Temporal ERF is computed in a similar way as described in the manuscript, which follows the exponential attenuation of the membrane potential over time. **Both the results of Izhikevich's spiking neuron model and the Parametric LIF model demonstrate that the Temporal ERF framework can effectively capture the temporal dynamics of spiking neurons, regardless of the specific neuron model used.** The exponential decay behavior observed in both models aligns with our theoretical framework, confirming the robustness and versatility of the ST-ERF approach. We have included these visual results in the updated manuscript to provide a more comprehensive evaluation of the ST-ERF framework across different spiking neuron models. > Q3: Provide a clearer explanation of what each subplot (a)-(d) in Figure 5(b) represents. What cause the jagged temporal ERF? A3.1: We apologize for the lack of clarity in the legend of Figure 5(b). In Figure 5(b), we aimed to illustrate the temporal ERF patterns of Spiking CNNs with different spike activations. We employ a 20-layer deep SCNN network to derive temporal ERFs across four distinct surrogate functions. From subfigures (a) to (d), the surrogate functions are as follows: (a): $\mathrm{arctan(\cdot)}$, (b): $\mathrm{Sigmoid}(\cdot)$, (c): $\mathrm{Rect(\cdot)}$, (d): $\mathrm{Poly(\cdot)}$. In each subfigure, we visualize the temporal ERF patterns at different decay factors. Considering that the spike activations are achieved by LIF neurons, we can describe the forward process as follows: $\mathbf{v}^{\ell}[t]=\mathbf{h}^{\ell}[t-1]+f({\mathbf{w}^{\ell}},\mathbf{x}^{\ell-1}[t-1]), \text{(Charging function)}$ $\mathbf{s}^{\ell}[t]=\mathbf{\Theta}(\mathbf{v}^{\ell}[t]-\vartheta), \text{(Firing function)}$ $\mathbf{h}^{\ell}[t]=\beta\mathbf{v}^{\ell}[t]- \vartheta \mathbf{s}^{\ell}[t], \text{soft reset}$ Accordingly, the temporal ERF is defined as:

\underset{\mathcal{T}}{\mathrm{ERF}}(s^{\ell}, \tau) = \sum_{i,j} \sum_{\hat{i},\hat{j}} \frac{\partial s_{(\hat{i},\hat{j})}^{\ell}[T]}{\partial s_{(i,j)}^{\ell-1}[T-\tau]} = \sum_{i,j} \frac{\partial \mathcal{L}}{\partial s_{(i,j)}^{\ell-1}[T-\tau]}

where $\mathcal{L}$ is the loss function, $\mathbf{v}^{\ell}[t]$ is the membrane potential, $\mathbf{s}^{\ell}[t]$ is the spike output at layer $\ell$, and $\mathbf{h}^{\ell}[t]$ is the hidden state at layer $\ell$ at time $t$. The decay factor $\beta$ controls the leakiness of the neuron, and $\vartheta$ is the threshold for spiking. Proven by Appendix D of the original manuscript, the temporal ERF is the temporal effective receptive field decays exponentially with time delay $\tau$, and the decay rate is primarily determined by the membrane potential decay constant $\beta$, i.e., $\underset{\mathcal{T}}{\mathrm{ERF}}(\cdot,\tau) \propto \beta^{\tau}$. The four experiments used decay factors of 2.0, 1.33, 1.2, and 1.14, respectively, which may not be the commonly-said decay factors in many essays. Some amount of works define the LIF model as $\mathbf{h}^{\ell}[t]=(1-\frac{1}{\tau})\mathbf{v}^{\ell}[t]- \vartheta \mathbf{s}^{\ell}[t], \text{soft reset}$ where $\tau$ is the decay factor. However, in our theoretical framework, we use $\beta=1-\frac{1}{\tau}$ for convenience while ensuring consistent monotonicity between $\beta$ and $\tau$ The shaded regions represent the $\mathrm{mean \pm std}$ deviation across 20 independent experimental trials. The solid lines represent the mean value points of the original data connected, while the dashed lines represent curves fitted using $y=a \beta^x + b$. Throughout the experiments, we observed that the fitted curves closely followed the trends of the original data, validating our theoretical framework. A3.2: In the analysis of Temporal ERF, it is essential to ensure that the network avoids vanishing or exploding gradients during backpropagation. Therefore, we meticulously controlled the surrogate gradient functions of neurons, the network’s parameter settings, and the value ranges of input tensors in our implementation to ensure stable capture of Temporal ERF characteristics during backpropagation. Detailed parameter configurations can be found in Appendix E. At the same time, we used PyTorch's automatic differentiation to compute gradients. Due to the inherent limitations of floating-point arithmetic (IEEE 754 standard), numerical errors may occur during computation, leading to jagged fluctuations in the Temporal ERF curve. This phenomenon is common in deep learning, especially when processing high-dimensional data or complex models.