Canonic Signed Spike Coding for Efficient Spiking Neural Networks

Thank you for your feedback. We have conducted comprehensive supplementary experiments and will do our best to address your concerns.

• The concept of firing negative spikes has already been proposed by [1].

Indeed, we are not the first to introduce negative spikes in SNNs. However, the role that negtive spikes play is very different between the two papers. The primary contribution of [1] lies in converting quantized ANNs, which enables rate coding schemes to function effectively. In their work, negative spikes are used to allow neurons to respond to negative inputs, as described in their paper:

A negative spike will be generated when the membrane potential is smaller than zero and the total spike count generated by this neuron is greater than zero.

In contrast, our contribution focuses on proposing a novel encoding scheme. Due to spike weighting, we observe that spikes tend to concentrate in later time steps. To address this, we reduce the threshold and introduce negative spikes to counteract encoding errors caused by early firing. This approach primarily aims to adjust the spike distribution and resolve the issue of temporal coupling.
To better illustrate this, we conducted comparative experiments, the results of which are included in the public comment section. Our findings indicate that under rate coding, negative spikes primarily respond to negative inputs and are not critical for performance improvement. However, in our encoding scheme, negative spikes play a pivotal role in correcting excessive firing and are essential for final accuracy. To the best of our knowledge, we are the first to incorporate negative spikes into a nonlinear encoding scheme, which breaks temporal coupling and significantly reduces output latency.

• This paper involves firing negative spikes, however, most of the comparative works selected by the authors in Table 2 are based on firing binary spikes. Furthermore, training high-performance converted SNN models under low time latency is currently one of the research focuses in the domain of ANN-SNN Conversion, but the authors do not reveal the performance of CSS under low time latency (time-steps) as demonstrated in [1, 2, 3].

Thank you for your suggestions. As mentioned in the Introduction, recent works (e.g., [1, 2, 3]) have successfully reduced the number of time steps required for encoding by converting quantized ANNs. This reduction is largely attributed to the significantly smaller number of activation values in quantized networks. While effective, this approach addresses a different aspect of the conversion process and operates independently from our method. The primary contribution of this paper lies in introducing a novel encoding method that does not rely on quantization.

We have included comparisons with state-of-the-art methods (Fast-SNN and [3]) in our public comment. All methods were tested by converting the same low-precision ANN. From the experimental results, our method achieves optimal performance with fewer time steps compared to Fast-SNN, underscoring the inherent advantage of spike weighting. Offset performs relatively well at 1 or 2 time steps but at the cost of significantly increased latency. Furthermore, our method surpasses its accuracy when the number of time steps increases to around 3. We encourage you to refer to our public comment for more detailed information.

From a theoretical perspective, our encoding scheme is expected to outperform rate coding because we enhance the encoding capacity of spikes by weighting them. We have visually illustrated the differences between the two encoding schemes in Fig 1. of the supplementary material, and we hope you find it helpful for reference.

Thanks for the authors' response. I have comprehensively considered the contributions of this work in terms of innovation and experimental performance, but I ultimately tend to believe that this work has not yet reached the acceptance criteria of ICLR. Therefore, I will tend to maintain my rating.

Dear Reviewer DPaT,

We hope this message finds you well.

We would like to kindly follow up on the feedback regarding our rebuttal submission for the manuscript titled “Canonic Signed Spike Coding for Efficient Spiking Neural Networks.” We truly appreciate the thorough review and valuable comments you have provided thus far.

Understanding that the review process can be time-consuming, we would be grateful if you could share an update on the current status of our rebuttal. If there is anything further we can do or clarify, please don’t hesitate to let us know.

We sincerely appreciate your time and effort in reviewing our work and look forward to your feedback.

Best regards,

Authors of Submission 10929

Thank you for your valuable questions. We apologize for the delayed response. Below, we will address your concerns to the best of our ability.

• The so-called "silent period" and "fire ternary spikes", as shown in Algorithm 1, can it be effectively implemented on neuromorphic hardware without causing additional time latency and energy consumption?

Thank you for your thoughtful question. We would like to clarify that our approach is indeed quite hardware-friendly. Compared to traditional rate coding with IF neurons, our method requires additional logic, which includes: multiplying the membrane potential by $\beta$, controlling the silent period, and handling input and output of negative spikes.

The control signal for the silent period can be efficiently generated by a state register. Specifically, when a neuron starts its computation, the register outputs a value of 0, and one clock cycle later, it stays at 1 until the computation is reset. This control signal is shared across many neurons, resulting in minimal hardware overhead.

For the membrane potential amplification, setting $\beta=2$ allows us to achieve the desired effect through a simple shift operation. We use shift registers to perform the operation, which also double as the registers storing the membrane potential. A MUX is added to select between the shift operation and accumulation.

handling input negative spikes is straightforward and only requires a two's complement addition, which can be efficiently performed by the adder. For negative spike emission, we use the most significant bit (MSB) of the membrane potential to determine its polarity. If the MSB is 1, we take the two’s complement and compare it with the positive threshold in the comparator. This approach effectively compares the absolute value of the membrane potential with the threshold, requiring only a single comparator, which incurs minimal hardware overhead. Based on the comparison and the MSB value, we can determine both the presence and the polarity of the spike.

Thus, the added hardware mainly consists of a set of multiplexers and a two's complement unit. The silent period control module and the spike emission determination module contribute minimal hardware overhead, which is negligible. The shift operation contributes most to the increased energy consumption, and we have accounted for it in the Energy Consumption Analysis section. Specifically, we estimate the energy consumption of the shift operation to be roughly equivalent to a SyOP (AC), which is an overestimate.

Based on our experimental results and the analysis above, overall, our method is hardware-friendly and does not have a large impact on latency or energy consumption.

• Is it fair to directly compare CSS with other vanilla conversion methods, which are based on firing binary spikes and do not introduce special mechanisms (e.g. silent period) ?

Thank you for your insightful question. Our core contribution lies in a novel encoding scheme, where negative spikes are an integral part of the method. Therefore, we believe that our practice is reasonable and effectively highlights the novelty of the proposed method, particularly when compared to other non-linear coding schemes.

In fact, mechanisms analogous to the silent period exist in other coding schemes as well. For example, TTFS coding and FS coding both employ a layer-by-layer processing approach, which can be interpreted as setting $L$ (the length of the silent period) equal to $T$ (the coding time steps). Additionally, negative spikes are also utilized in Fast-SNN (rate coding) and TSC (temporal switch coding), which are included as comparison methods in our study.

Thank you for your constructive feedback. Based on your suggestions, we have added two experiments, which are detailed in our public comment. Furthermore, we have enhanced the paper's structure by adding a preliminary section that outlines the fundamental principles of ANN-to-SNN conversion, emphasizing the critical roles of encoding methods and neuron models. Additionally, we have provided detailed proofs for the theorems and revised the summation indices in the equations to ensure clarity and avoid potential confusion. Below, we address your concerns in detail.

• Can the author provide a comparison between this work and other SOTA SNN conversion works such as (Fast-SNN[1], Offset[2])?

Yes, we have included comparisons with these two works. The two compared works are based on rate coding and rely on low-precision ANNs (for Offset, it specifically depends on the QCFS activation function, where output quantization is a key feature of this activation function). All three methods were tested by converting the same low-precision ANN to better evaluate the differences introduced by their respective encoding schemes.

From the experimental results, our method achieves optimal performance with fewer time steps compared to Fast-SNN, which highlights the inherent advantage of spike weighting. Offset performs relatively well at 1 or 2 time steps but at the cost of significantly increased latency. Additionally, the calibrating effect of Offset appears to be limited, as its accuracy is surpassed by ours when the time steps increase to around 3. For more detailed information, please refer to our public comment.

• Can the author provide the number of synaptic operations of the SNN using CSS? The calculation code can be found [here]

Yes, we conducted experiments using the code you provided, and the results have been included in our public comment. Although our encoding scheme is not inherently sparse, it benefits from a significantly compressed number of time steps, which contributes to its energy efficiency.

We first compared our approach with the pre-conversion ANN, where we observed a reduction in energy consumption by approximately fivefold. Unfortunately, we could not find the code for the TTFS-based approach for a direct comparison. As a result, we conservatively estimated the energy consumption of TTFS encoding based on relevant works, as detailed in our official comment. The result shows that by minimizing the number of time steps, CSS can achieve even lower energy consumption than TTFS (a 10% reduction). Finally, we selected Fast-SNN as a strong baseline for rate coding. Our results demonstrated that our encoding scheme reduces energy consumption by over 30%, while maintaining competitive accuracy.

• Can the authors provide a more simple explanation of the equivalence of ANNs and SNNs?

We've now incorporated the basic concept of ANN-SNN conversion in the Preliminary section. Below, we briefly explain the principle. The connection between ANN and SNN lies in the mapping of activation values to spike sequences. Through the encoding method, we can represent a spike sequence as a value equivalent to the activation value in the ANN.

To be more specific, please refer to Eq. (5) in the newly uploaded PDF. By interpreting the spike sequence through rate coding and defining the "spike frequency" as $r_i^l[T]\coloneqq\frac{\sum_{t=1}^T S_i^i[t]}{T}$, Eq. (5) can be rewritten as:
$r_i^l[T]=\sum_j w_{ij}^{l}r_j^{l-1}[T]+\sum_{t=1}^T\frac{b_i^l}{T}-\frac{u_i^l[T]}{T}$ When $T$ is sufficiently large, we can ignore the last term on the right-hand side. Given that $r_i^l[T] \geq 0$, this simplifies to:
$r_i^l[T]=\max\left(\sum_{j}w_{ij}^{l}r_j^{l-1}[T]+\sum_{t=1}^{T}\frac{b_{i}^{l}}{T},0\right)$ Comparing this to the forward propagation process in an ANN:
$a_i^l=\max\left(\sum_{j}\hat w_{ij}^{l}a_j^{l-1}+\hat b_{i}^{l},0\right)$ we observe that equivalence between the two types of networks can be achieved by directly using the ANN weights and scaling the bias linearly! The importance of the encoding method and the neuron model stems from the fact that: 1) we need to interpret the spike sequence using some encoding scheme (here, rate coding), and 2) Eq. (5) is derived using some neuron model (here, IF model). In our work, it is precisely through 1) designing an appropriate encoding method and 2) developing a suitable neuron model for neural computation that we address the issues present in existing encoding schemes.

Thank you for your valuable suggestions. We are delighted that you recognized the innovation in our approach. Indeed, the initial manuscript lacked smooth logic and clear descriptions, which posed challenges to truly understanding our method. Based on your advice, we have reviewed and revised the mathematical expressions and added the pseudocode for the TSA forward propagation process in the main text (while keeping the mathematical formulation in the appendix). We have uploaded a fully revised PDF for your reference. Additionally, we conducted experiments to study the impact of $\beta$, and the results have been included in the public comment. These results validate our analysis, and you can find more detailed information there. Below, we will address your concerns to the best of our ability.

• Equation 3 makes no sense since the dirac function is a generalized function and is only well-defined within an integral over a specified range. The authors should consider using heaviside step function to describe the neuron firing.

Indeed, we have replaced all parentheses with square brackets to emphasize the discrete nature of the domain. Specifically, $\delta[\cdot]$ denotes a unit impulse (we've now provided the definition of $\delta[\cdot]$ directly in a footnote in the main text).

• Equation 3 appears to represent the discrete form of a LIF neuron function. However, based on the experiments, the scaling factor is consistently set to values greater than one, which is atypical for a traditional LIF neuron model. To improve clarity, it would be beneficial if the authors provide the explicit mathematical function for the TSA neuron model.

Thank you for your suggestion. We have currently included the pseudocode for TSA forward propagation in the main text, as we believe it provides a more intuitive explanation, while the mathematical description remains in the appendix.

We have now explicitly clarified that $\beta>1$ when introducing $\beta$. While settings like $\beta=p$ (where $p$ is a positive value) and $\beta = \frac{1}{p}$ can both theoretically ensure a weight difference of $p$ times between adjacent spikes, only $\beta>1$ is in practice feasible. To illustrate this, let $T$ represent the number of time steps and $\theta^l$ denote the spike amplitude. After weighting, the maximum value that the sequence can encode is $\sum_{t=0}^{T-1}\beta^t\theta^l$. Therefore, for encoding the same range, a smaller $\theta^l$ is achievable when $\beta>1$. Since $\theta^l$ determines the quantization error during encoding (as it represents the minimum value that a single spike can express), minimizing $\theta^l$ is essential. As a result, the proposed weighted scheme cannot be implemented using LIF neurons.

• It confuses me whether CSS coding serves as an input encoding method for the SNN or if it refers to the encoding capabilities inherent in the TSA neuron itself.

It refers to the encoding capabilities inherent in the TSA neuron itself. Input encoding is performed by the TSA neuron, as defined and proven in Theorem 2.

• In lines 305-306, it is mentioned that a new term is added to the right side of Equation 8 after a silent period. Could the authors clarify the purpose of doing so?

The reason we made this adjustment is to make it easier to satisfy Equation (8) (now Equation (10) in the revised version). In Corollary 1., we proved that meeting this condition ensures the accuracy of the converted SNN. In the revised version, we have added textual explanations following the theorems, clarifying how they guide the design of our method. We believe the updated version facilitates a smoother and more intuitive understanding.

• It would be better if the authors can provide the pseudo code for the conversion and the inference process.

That’s correct. We have now included the pseudocode for the conversion process in the appendix. The inference process essentially corresponds to the layer-by-layer propagation of spikes through TSA after conversion, as outlined in the TSA pseudocode (or alternatively, you may refer to the mathematical description of TSA).

Thank you for your precise and valuable feedback. We are glad that you recognize the significance of our work, and we will address your questions below.

• What is the energy consumption of this SNN compared to ANN? It is suggested to add comparative experiments on energy consumption.

Thank you for your suggestion. We have included this section of experiments in the public comment. The results show that, although our encoding scheme lacks sparsity, the significant reduction in time steps provides a clear energy advantage. For example, the energy consumption of the converted full-precision VGG-structured SNNs are more than five times lower than that of the corresponding ANNs.

Additionally, we compared our method with other neural coding approaches. Compared to the state-of-the-art rate coding method (Fast-SNN), our approach achieves over a 30% reduction in energy consumption. We also evaluated our approach against TTFS coding. However, by further reducing the time steps in CSS (e.g., by converting a quantized ANN), we were able to achieve approximately 10% lower energy consumption than TTFS.

Overall, our coding scheme provides substantial energy savings while avoiding the large output latency typically associated with TTFS-based approaches.

• What is the level of support for this type of neuron in existing neuromorphic hardware? Is deployment and implementation feasible in the future?

Thank you for your questions. We are currently focused on the design and implementation of dedicated hardware. We will release the hardware design once the validation phase is successful. Our design builds upon existing neuromorphic computing platforms, ensuring high generality and flexibility.

Specifically, the amplification coefficient $\beta$ can be efficiently implemented using shift operations. Given that the silent period for each layer is predetermined, a simple control signal is used to determine when to fire a spike. Additionally, we have developed specialized encoding methods to differentiate between positive and negative spikes.