Temporal Misinformation and Conversion through Probabilistic Spiking Neurons

According to the author`s response, it seems that the author has understood why "ANN-SNN" is not a scientific term that is usually used. This is great, and I am happy that the author finally understands this. Please make sure to revise it correctly, as it is where a scientific term is introduced and explained in the paper. I read 30 papers to help the author solve this issue, I do not want my efforts to mean anything to the author, and this problem is still not revised.

Reply to "Anyways, we are grateful for the insights about terminology, but we remain sceptic about your expertise in training SNNs with ANN-SNN (conversion)." Good, thanks for your replay and it is OK that you keep your "sceptic" (though I guess you mean skeptic )

I am disappointed that the authors did not address all the questions I raised in my initial review. Additionally, I find it concerning that despite being informed about issues such as mis-references, improper capitalization, and the use of unprofessional terms, the authors questioned the expertise of the reviewer. This response is unwarranted and does not contribute to a constructive dialogue.

Regarding the term "ANN-SNN," the authors appear to misunderstand why it is not widely accepted as a scientific term in the context of SNN literature. As per the author's suggestion, I conducted a search on Google Scholar and reviewed approximately 30 papers. None of these papers used "ANN-SNN" as a standard term for ANN-to-SNN conversion technology. Instead, terms like "ANN-to-SNN conversion" or "ANN-SNN conversion" are commonly used in previous research.

I hope this clarification provides sufficient context for the authors to understand the concerns raised and reflect on the professionalism of their submission.

I recommend that the authors review at least one SNN paper before responding to the question I raised regarding the use of the term "ANN-SNN. I am very surprised that the author just asked me to review other papers on Google Scholar to help the author correct the identified problem of the author`s own paper. It is ridiculous and very unprofessional.

Thank you for pointing out some typos in our paper. We will correct these in the updated version, hopefully making the presentation more clear. The Figure 1 you mention is erroneously referred as Table 1 in the paper. It is first referred to in the Introduction, in the very first paragraph we start to describe our method (starting with line 094). It is further commented on in Section 3.1. Unfortunately, even after thorough reading of the paper after it is finished, these kind of mistakes still manage to go unnoticed.

We do not doubt good intentions in your comments, but we do question their professionalism. Especially in repeated questions on the same topics. But, what perplexed us the most is your question about what ANN-SNN means. This is a standard terminology in the area of Spiking Neural Networks (meaning Conversion of Artificial Neural Networks to Spiking Neural Networks, sometimes also abbreviated as ANN-to-SNN or similarly). A search on Google Scholar would give dozens of papers with ANN-SNN in their title. This comment/question from your side, and the fact that you gave confidence 4 to your assessment of our paper, makes us doubtful.

Of course, in case you have any constructive remarks and questions that pertain to the method we proposed, we will gladly try to assess them.

Weakness 1 and Q1:

We respectfully disagree with the claim that the ideas in this paper are similar to those in the two referenced works ([A] and [B]). While our work lies within the ANN-SNN conversion domain, the approaches and focuses are fundamentally different:

1. [A]: This work specifically addresses ANN-SNN conversion using quantized activations in ANN, relying on a 2-bit quantizer ( b = 2 ) and tailored quantization techniques. In contrast, our work has nothing to do with this method and it applies to general ReLU activated ANNs without quantization. We further emphasize that quantization of activations yields ANN models which are underperforming compared to those trained using ReLU activation, which further underscore the general applicability of our method. Our method is already ready to use for a pretrained ANN model (whether pretrained with ReLU or quantized ReLU activation), unlike the method in [A] which is only applicable for quantized models. Furthermore, even within the realm of ANN-SNN conversion, our proposed method has nothing to do with the baseline [A]. We are pointing to a common drawback of baseline ANN-SNN methods, which has to do with instable firing rates in early time steps, and propose a remedy for it in form of probabilistic spiking neuron.

2. [B]: This paper focuses on direct training of spiking neural networks (SNNs) with integer-valued spike-driven training, followed by ANN-SNN conversion framework, which lies outside the scope of our work (except for the conversion part). Our research remains fully within the ANN-SNN conversion paradigm, with a focus on bridging the performance gap between ANN (pretrained with ReLU or quantized ReLU activations) and SNN by optimizing spike dynamics during the conversion process.

These distinctions demonstrate that our work provides unique contributions within the ANN-SNN domain that are not addressed by the referenced papers. The novelty of our work lies in studying the effect of permutations on spike trains in converted SNN models, and temporal processing strategies to improve the accuracy and energy efficiency of SNNs in the conversion framework, without relying on quantization or direct training methods.

Q2:

There are significant differences in methodology and focus:

1. Quantization vs. ReLU: Paper [A] explicitly uses a 2-bit quantizer to facilitate conversion, whereas our work does not rely on any prior quantization in ANN models. We do combine our proposed approach to some of the baselines that use quantization, but this is just to show its applicability and not because of the necessity. This fundamental methodological difference highlights our focus on optimizing temporal dynamics rather than precision quantization. We once again emphasize that the quantized models tend to have weaker performance compared to those trained with ReLU activatoins, which consequently leads to lower performing SNN models.

2. Signed neurons in [A]: Another fundamental difference is that [A] use signed neurons to address the accumulation error in the conversion. In other words, they use spiking neurons that are designed to produce negative spikes as well. Our neurons produce only positive spikes, and in a probabilistic manner which yields them biologically plausible as we discussed in the paper.

We do point that the paper [A] relies on direct combination of two ideas: Quantization and Signed Neurons, both of which appeared elsewhere prior to their publication.

Q3:

Our method has not been tested in direct training, as its primary focus lies in the ANN-SNN conversion framework. Direct training (which in [B] is a prerequisite to perform ANN-SNN conversion) and ANN-SNN conversion represent fundamentally different paradigms with distinct challenges and goals.

1. Differences: Direct training methods, such as the one described in [B], optimize spiking activity during training, often relying on integer-valued updates or other specific mechanisms. In contrast, our method focuses on post-training optimization of spike order and temporal dynamics during conversion from ANN to SNN.

2. Advantages of Our Approach: Unlike direct training, our method does not require retraining the network or modifying the training pipeline, making it highly scalable and applicable to pre-trained ANN models. Furthermore, our approach achieves significant improvements in accuracy and energy efficiency with minimal additional computational cost, as demonstrated in Appendix E.4.

In summary, while our work is not directly comparable to direct training methods like [B], it offers unique advantages and contributions within the ANN-SNN conversion domain specifically, and domain of training of SNN generally, emphasizing scalability and efficiency.

As the deadline for the rebuttal is approaching, we wanted to kindly check if we have adequately addressed your concerns regarding our paper. If there are any outstanding issues or additional points that you feel need clarification, please do not hesitate to let us know.

Authors

Q1. You are absolutely correct. We rearrange the firing times according to the permutation of time steps.

Q2. In addition to what we said about the instable firing rate of the baseline models in lower latency, and how permutations overcome this, we try to address how they potentially cope with the errors identified in [1] and [2]. For the SIN (spikes of innactive neurons) errors in reference [1], we recall that the SIN refers to occurrence when a spiking neuron emits a spike, even though the corresponding ANN pre-activation was negative. SIN can happen if some neurons in the previous spiking layer $(l-1)$ emitted spike trains modulated with positive weights $w_{+}^{l}$, before some other neurons in $(l-1)$ emitted spike trains modulated with negative weights $w^{l}_{-}$, which would eventually annulate the positive spike trains. In other words, we have two types of spike trains, one where the spikes are clustered early in time, and the other, where the spikes are clustered later on, so that the two spike streams do not integrate together in order to produce the correct input. Of course, this is a rather vast oversimplification of a potential situation, but it serves a point to show how permutations may overcome this situation.

In fact, the unevenness errors in [2], describe a similar situation, where we have two stream of spikes, one where the spikes are clustered in the earlier and the other where the spikes are clustered in the later time steps. When combined together, the spike streams should produce the correct input, but due to their temporal misalignment, the subsequent neurons end up firing extra spikes or underfiring in general.

Permutations, in general, tend to break clustering of spikes. In a simplified situation, if we have $m$ spikes in the course of $T$ time steps, the probability after a permutation of $m$ spikes being clustered together in $m$ time steps is $\frac{(T-m+1)\cdot m!}{T!}$. (Permutations have the similar effect on "voids", long subintervals in the spike trains without spikes). Overall, breaking clustering contributes to the uniformity of the input, and overall, elimination of said errors.

We also note that our answer here pertains to permutations in baseline models. For the TPP neurons, we kindly point your attention to the conversion error analysis in new Appendix I.

Weakness 1. We hope that our common answer addresses the details of what we mean by a temporal misinformation and the reasons why we named it like that. We will update the lines in the introduction so that they are more informative and clear in this aspect.

Weakness 2. We argue that the source of temporal misinformation is the instable firing rate in the early time steps of the baseline SNN models.

SNN models converted from a pretrained ANN aim to approximate the ANN activation values with firing rates. The approximation is given by $\mathbf{x}^{(\ell)} \approx \bar{\mathbf{s}}^{(\ell)} = \frac{1}{T} \sum_{t=0}^{T} \mathbf{s}^{(\ell)}(t)$. However, in lower time steps, the approximation is too coarse as we can only use few spikes in order to approximate the ANN (continuous) values. For example, in $T=1$, the baselines are attempting to approximate ANN activations with binary values $0$ and $\theta$.

What is crucial is that, at each spiking layer, the spiking neurons at early time steps, use only the outputs of the previous spiking layer from the same, early, time steps. As this information is already too coarse, the approximation error accumulates throughout the network, finally yielding in models that are underperforming in low latencies.

Finally, with longer latencies, the model is using more spikes and is able to approximate the ANN values more accurately, and to correct the results from the first time steps. (One can argue that at long enough latencies, all reasonable conversion methods work equally well.)

How do permutations fix this common occurrence?

When performing permutations on spike trains after a spiking layer in the baseline models, the input to the next spiking layer in lower time steps no longer depends only on the outputs of the previous layer in the same, lower time steps, but it depends on the outputs in all time steps $T$. In particular, when a spiking layer is producing spike at time step $t=1$, it does so "taking into account" (via permutation) outputs at all the time steps from the previous spiking layer.

As a way of example, consider two connected spiking neurons $N_1$ and $N_2$, where $N_1$ is sending the weighted output to $N_2$. If a spiking neuron $N_1$ in one layer has produced spike train $s = [1,0,0,0]$, in approximating ANN value of $.25$, then a spiking neuron $N_2$ at the first time step will use 1 as the approximation and will receive the input $W\cdot 1$ from neuron $N_1$. However, after a generic permutation of $s$, the probability of having zero at the first time step of output of neuron $N_1$ is $\frac{3}{4}$ (as oppose to having 1 with probability $\frac{1}{4}$), and at the first time step neuron $N_2$ will most likely receive the input $W\cdot 0=0$ from neuron $N_1$, which is a rather better approximation for $W\cdot .25$ than $W \cdot 1$.

This property of receiving input at lower $t$ but taking into account the spike outputs of the previous layer at all the time steps is not only exclusive to lower $t$. Indeed, at every time step $t\leq T$, the input at a spiking layer is formed by taking into account spiking train outputs from the previous layer at all the time steps, but having already accounted for for the observed input at the first $t-1$ steps.

In general, the permutations overall increase the performance of the baselines because the spike trains are "uniformized" in accordance to their rate, and the accumulation error is reduced. If a layer $l$ has produced spike outputs that well approximate the $l$ layer in ANN, then, after a generic permutation, at each time step starting with the first, the next layer is receiving the most likely binary approximation of those rates.

This is nothing but Theorem 2 of the paper in visible action.

We provide several evidence for our claims (new appendices) which are as follows: Appendix F observes what is the accuracy of the model at $t=1$, when we applied permutations on spike trains of higher length $T$; Appendix G shows the membrane potential distribution before firing in the baselines and permuted and TPP models. Permuted and TPP models show higher variance, hence their enhancement of the model's ability to produce spikes in low latency. Appendix G, which shows the experiments when particular classes of permutations are applied on the baseline models. In particular, the permutation that anchor first time step show the least increment in performance.

Weakness 3.

We kindly point the Reviewer to the newly added Appendix I in our paper and Theorem 3, therein. We point that Theorem 3 ammeliorates the findings of Theorem 1 (which becomes a consequence of statement (c) in Theorem 3). We also point to the Comments following Theorem 3, where we contrast our results with some similar ones in the literature.

Q1:

In general, we do not think so. For example, in our early experiments we permuted each output spike train after every layer, separately. Later on, we apply the same permutation on every spike train that belongs to the same layer, and we did not see major differences in the performance of the two. However, if we stick to the first case, where each spike train is permuted independently, then it is not difficult to argue that carefully choosing permutations would degrade the performance of the model. For example, We can permute the spike trains which are modulated with positive weights in such a way that all the spikes are clustered in the beginning of the time frame, while we can permute the spike trains which are modulated with negative weights in such a way that all the spikes are clustered at the very end of the time frame. This way, the spike trains are completely misaligned and would not integrate together at the correct timings (in the result, every layer would produce superfluous spikes, in general).

Having said this, we have conducted several additional experiments, detailed in Appendix H, to address the reviewer’s inquiry about the impact of spike order on performance.

Our implementation is based on QCFS (T=4). The baseline accuracy of the ANN VGG-16 model is 76.21%. For the SNN model, we evaluated all 24 possible permutations of the four timesteps to systematically analyze their effect on accuracy (on every spike train, in every layer, the same permutation has been applied in one run) . To our surprise, all the permutations increased the performance of the baseline model. Why did this happen is beyond our current understanding. However, we notice that the least performance increment comes from the permutations that are anchoring the first time step: configurations starting with the unshuffled 0-th timestep, such as (0, 1, 2, 3), (0, 1, 3, 2), and (0, 2, 1, 3), yielded the lowest accuracies among all permutations. This pattern suggests a significant sensitivity to the initial timestep, likely due to the role of early spikes in establishing the network’s dynamic state.

Furthermore, we considered the same baseline for $T=8$ and we performed systematically permutations which were anchoring pairs of time steps $(i,j)$. This is in the Appendix Figure 11. Once again, the plot shows the the permutations that are fixing the first time step $(0,j)$ show the least performance increment.

Finally, we also investigated the affect of permutations on a single layer of the baselines. Figure 12 and 13 show the effect of a permutation when it is applied only on a single layer in the baseline model. We notice that the permutation show the least effect when they are applied in the early layers (as the SNN model is still approximating the ANN well), and in the late layers (as it is already too late to fix the accumulated approximation error from the previous layers).

Q2:

Our TPP neuron model works in two phases. In the first phase it accumulates the incoming spike trains of length $T$. This operation of accumulation potentially can be done in less than $T$ steps (as the neuron is receiving them all at once) on specific hardware, and we used $c$ to denote this time.

Q3

We provide an additional experimental table in Appendix E.3, comparing our proposed methods with the ANN-SNN conversion QCFS method using VGG-16 and ResNet-20 on CIFAR-10, CIFAR-100, and ImageNet. The table includes a detailed comparison of accuracy and standard deviation, with results averaged over five experiments for the TPP method.

Both the permutation and TPP methods demonstrate consistent and significant improvements in accuracy over the baseline QCFS approach and they are comparable between themselves. One visible case where TPP outperforms the permutations is on ImageNet dataset. These results highlight the effectiveness of our proposed methods in addressing the inherent limitations of traditional ANN-SNN conversion methods. Our methods achieve higher performance across diverse datasets and network architectures, underscoring their robustness and generalization.

Q4

The energy cost for the random sampling which is crucial for the implementation of our TPP neurons is unfortunately not readily reported for neuromorphic chips we enlisted in the paper, but is usually integrated in the overall energy efficiency of the chip. However, any additional cost is compensated with the superior performance of our models, as they are achieving close to ANN performance two times or faster than the baselines, hence producing approximately twice as little spikes and SOPs.

Weakness 1. While we acknowledge that probabilistic spiking neurons have been explored in prior work within the general domain of Spiking Neural Networks, to the best of our knowledge, this is the first work to propose and rigorously analyze their usage within the ANN-SNN conversion paradigm. Beside this, we point to the common phenomenon in ANN-SNN methods, and exploit it for the tailored construction of our probabilistic neurons.

Weakness 2. We hope that the common answer above clarifies the notion of temporal misinformation. As for the theoretical basis, we point to Theorem 1 of why the permutation increase the overall performance of the baseline models (uniformization of the input), but we also try to explain this in more details here.

We claim that the reason behind temporal misinformation phenomenon is the instability of firing rates of baseline SNN models in early time steps.

SNN models converted from a pretrained ANN aim to approximate the ANN activation values with firing rates. The approximation is given by $\mathbf{x}^{(\ell)} \approx \bar{\mathbf{s}}^{(\ell)} = \frac{1}{T} \sum_{t=0}^{T} \mathbf{s}^{(\ell)}(t)$. However, in lower time steps, the approximation is too coarse as we can only use few spikes in order to approximate the ANN (continuous) values. For example, in $T=1$, the baselines are attempting to approximate ANN activations with binary values $0$ and $\theta$.

What is crucial is that, at each spiking layer, the spiking neurons at early time steps, use only the outputs of the previous spiking layer from the same, early, time steps. As this information is already too coarse, the approximation error accumulates throughout the network, finally yielding in models that are underperforming in low latencies.

Finally, with longer latencies, the model is using more spikes and is able to approximate the ANN values more accurately, and to correct the results from the first time steps. (One can argue that at long enough latencies, all reasonable conversion methods work equally well.)

How do permutations fix this common occurrence?

When performing permutations on spike trains after a spiking layer in the baseline models, the input to the next spiking layer in lower time steps no longer depends only on the outputs of the previous layer in the same, lower time steps, but it depends on the outputs in all time steps $T$. In particular, when a spiking layer is producing spike at time step $t=1$, it does so "taking into account" (via permutation) outputs at all the time steps from the previous spiking layer.

As a way of example, consider two connected spiking neurons $N_1$ and $N_2$, where $N_1$ is sending the weighted output to $N_2$. If a spiking neuron $N_1$ in one layer has produced spike train $s = [1,0,0,0]$, in approximating ANN value of $.25$, then a spiking neuron $N_2$ at the first time step will use 1 as the approximation and will receive the input $W\cdot 1$ from neuron $N_1$. However, after a generic permutation of $s$, the probability of having zero at the first time step of output of neuron $N_1$ is $\frac{3}{4}$ (as oppose to having 1 with probability $\frac{1}{4}$), and at the first time step neuron $N_2$ will most likely receive the input $W\cdot 0=0$ from neuron $N_1$, which is a rather better approximation for $W\cdot .25$ than $W \cdot 1$.

This property of receiving input at lower $t$ but taking into account the spike outputs of the previous layer at all the time steps is not only exclusive to lower $t$. Indeed, at every time step $t\leq T$, the input at a spiking layer is formed by taking into account spiking train outputs from the previous layer at all the time steps, but having already accounted for for the observed input at the first $t-1$ steps.

In general, the permutations overall increase the performance of the baselines because the spike trains are "uniformized" in accordance to their rate, and the accumulation error is reduced. If a layer $l$ has produced spike outputs that well approximate the $l$ layer in ANN, then, after a generic permutation, at each time step starting with the first, the next layer is receiving the most likely binary approximation of those rates.

This is nothing but Theorem 2 of the paper in visible action.

We provide several evidence for our claims (new appendices) which are as follows: Appendix F observes what is the accuracy of the model at $t=1$, when we applied permutations on spike trains of higher length $T$; Appendix G shows the membrane potential distribution before firing in the baselines and permuted and TPP models. Permuted and TPP models show higher variance, hence their enhancement of the model's ability to produce spikes in low latency. Appendix H, which shows the experiments when particular classes of permutations are applied on the baseline models. In particular, the permutation that anchor first time step show the least increment in performance.

As the deadline for the rebuttal is approaching, we wanted to kindly check if we have adequately addressed your concerns regarding our paper. If there are any outstanding issues or additional points that you feel need clarification, please do not hesitate to let us know.

Authors

1. Thank you for the effort put into this work. The results in Figures 10, 11, and 12 are quite interesting. The figures illustrate the accuracy under $T!$ different permutation scenarios. I am curious about that did the authors perform the $T!$ tests on only a single layer? This is because I believe the search space would be $O(N \cdot T!)$ , where $N$ is the number of layers in the network. The results in Table 12 seem important, but I am still unclear about the meanings of $T$ and $t$ in the table. Could the authors provide further clarification?

2. I found the authors' response to weak3 very intriguing. Perhaps there exists some optimal permutation that could be identified through a certain search method. However, that might be a separate line of work.

3. If the authors can address my questions, I will raise my score by 1.

Thank you for your comments and for the opportunity to clarify further some points.

1. We clarify the details for Figures 10, 11, 12 and 13 in order.

Figure 10: We consider $T=4$ and there are 4!=24 possible permutations. Once a permutation is fixed (as can be observed on the $x$-axis), then it is consistently applied throughout all the layers in the baseline model. Finally, the accuracy of the model is recorded.

Figure 11: We consider $T=8$ and we group permutations according to the pair of time steps that they are leaving fixed. For example, if a pair $(i,j)$ is present on the $x$-axis, it means that we consider the permutations that are leaving time steps $i$ and $j$ fixed, and this is the only constraint on the permutation. Furthermore, at each layer, the choice of permutation (that is fixing $(i,j)$) is arbitrary, that is, permutations can vary from layer to layer as long as they are fixing $(i,j)$. Then, for each such fixed pair, we recorded the accuracy of the baseline model.

Figures 12 and 13: We consider $T=8$ and two baseline methods (QCFS and SNNC, respectively). Then, for the respective baseline, we chose one layer (the position of the layer is on the $x$-axis) and we only permute the spike trains after that chosen layers (the permutation is arbitrary). The permutations are not applied on any other layers. Then, we observe the final accuracy of the model.

We did not perform experiments where we apply all the possible $T!$ permutations on one single layer in the baseline models.

Table 12: The goal of the experiments performed therein was to support our initial hypothesis for why permutations improve the performance of the baselines. As we wrote in the Comments following the table, when performing permutations on spike trains after spiking layers in the baseline models, the input to the next spiking layer in lower time steps, no longer depends only on the outputs of the previous layer in the same lower time steps, but it depends on the outputs in all time steps $T$.

In particular, when spiking layer is producing spikes at time step $t=1$, it does so "taking into account" (via permutation) outputs at all the time steps from the previous spiking layer. To test that this is indeed the case, we performed experiments presented in Table 12. For a fixed latency $T$, and after applying permutations on spike trains in the baseline model, we recorded the accuracy of the ``permuted'' model at $t<T$. In other words, in the output layer, we do not take the whole spike trains of length $T$ but we cut them after $t$. Then we reported those accuracies. Because of permutations, the accuracies at all time steps $t$ are more stable and close to each other, as compared to the baseline models.

2. This indeed could be the case. We also find very interesting the results of Figures 12 and 13, where an effect of a single permutation at a particularly well chosen hidden layer could improve the performance of the model drastically. But, this, as you already hinted, could be a separate line of work.

Thank you for your review and for increasing the contribution score following our revisions. We appreciate the time you’ve taken to provide feedback on our work.

We believe we have addressed the weaknesses and questions you initially raised, and we would like to kindly ask if there are any remaining concerns or areas in the paper that would need further improvement? Additionally, we were wondering if our past and potential new revisions might lead you to reconsider the overall evaluation score, as it currently places our work below the acceptance threshold.