# Lossless Method of Constraining Membrane Potential in Deep Spiking Neural Networks

Yijie Miao, Makoto Ikeda

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Yijie Miao, Makoto Ikeda. Lossless Method of Constraining Membrane Potential in Deep Spiking Neural Networks. 19th IFIP International Conference on Artificial Intelligence Applications and Innovations (AIAI), Jun 2023, León, Spain. pp.529-540, 10.1007/978-3-031-34107-6_42. http-05498087

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# Lossless Method of Constraining Membrane Potential in Deep Spiking Neural Networks

Yijie Miao(✉) and Makoto Ikeda(✉)

EEIS, Graduate School of Engineering, The University of Tokyo, Tokyo 113-0032, Japan

{miao,ikeda}@silicon.u-tokyo.ac.jp

**Abstract.** Biologically inspired Spiking Neural Networks (SNNs) offer a promising path toward achieving energy-efficient artificial intelligence systems. However, in the hardware field, the deployment of deep SNNs has been stagnant, where the wide range of membrane potential of spiking neuron poses a significant challenge to hardware efficiency. To address this issue, this work proposes a guideline and a novel hardware-friendly method to constrain the membrane potential, reducing the associated hardware overhead while fully maintaining the inference accuracy. Experiments demonstrate that the proposed method is effective and achieves substantial memory usage reduction for a 20-layer ResNet model. This work paves the way toward the efficient hardware implementation of even deeper SNNs.

**Keywords:** Spiking neural networks · Membrane potential · ANN-SNN conversion · Deep networks · Hardware implementations

## 1 Introduction

Spiking Neural Networks (SNNs) are novel Artificial Neural Networks (ANNs) that mimic the biological neural system, which uses discrete and sparse spike trains to convey information. Whereas SNNs have received considerable attention for many years due to their potential for energy efficiency benefits, compared to traditional ANNs, training and implementing SNNs are more challenging since they use non-differentiable neuron models with complex internal dynamics [1].

In the software field, converting pre-trained ANNs into SNNs, known as the ANN-SNN conversion method, has emerged as a successful alternative method to overcome the difficulty of directly training SNNs [2–6]. Despite the minor accuracy degradation, converting the state-of-the-art deep ANNs to SNNs has enabled competitive SNN inference performance across various image classification benchmarks from simple MNIST to challenging CIFAR-100 or ImageNet.

If only deployed on general computation hardware like CPUs or GPUs, SNNs would have no opportunity to compete with other state-of-the-art hardware ANNs in terms of energy efficiency and speed. However, the development of dedicated SNN hardware notably lags behind the software. Until recently, the
majority of SNN hardware has only been capable of handling simple tasks like MNIST dataset [7–11], with only a few designs having implemented deeper SNN models and supporting tasks of higher complexity [10, 12, 13].

Considering the overall structural similarities between converted SNNs and source ANNs, the introduction of additional membrane potential by spiking neurons constitutes a significant negative factor, if not the only one, in hardware implementation. Prior work has recognized the crucial role of the wide dynamic range of membrane potential in accurate SNN inference, especially when the network deepens [14]. In software SNNs, the high-precision floating-point format is available, and the value range is generally not a concern. However, in hardware, the wide range of membrane potential can result in excessive computational and storage overhead. To date, no method has been reported to accommodate the impact of membrane potential on inference capability and hardware efficiency in deep SNN scenarios, which limits the further development of SNN hardware.

To address this, we, for the first time, propose a lossless method to constrain the dynamic range of membrane potential in deep SNNs, which can bridge the gap between software and hardware. The contribution of this work could be summarized as follows:

- We examine the mechanism underlying the degradation of SNNs inference capability when constraining the negative membrane potential. Our analysis and observations suggest that the fluctuations in the membrane potential are the primary factor contributing to the degradation. Based on this insight, we derive a guideline for constraining the value range of membrane potential in a lossless manner.

- We propose a hardware-friendly method that utilizes a novel packed spike format, enforcing the above-mentioned lossless constraint on the membrane potential. Our method allows deeper SNNs to employ the same hardware-friendly reduced-precision quantization for membrane potential as used in weights to reduce storage overhead and thereby enable the extension of efficient hardware SNNs to higher layers in the future.

# 2 Background and Related Work

## 2.1 Membrane Potential of Spiking Neuron

We start with the Integrate-and-Fire (IF) neuron model, a popular choice for converted SNNs due to its low computational cost. Mathematical details will be presented in the section to follow later, and here we provide a brief overview of its principles. In this model, the input spike trains are multiplied by the weights to form the input potential (Fig. 1a). When positive weights dominate, the input potential integrates to the membrane potential and is released when it exceeds the threshold, firing spikes backward. Consequently, the output spike rate positively correlates with the input spike rates. On the other hand, in cases where negative weights dominate and the input potential is negative, the IF neuron generates no output spike. These principles align well with the ANN
<figure><img src="image_3.png" /><figcaption>Fig. 1. a. The dynamics of IF spiking neuron in the cases where positive and negative weights dominate. b. Rectified Linear Unit (ReLU) activation function in ANNs.</figcaption></figure>

neurons' Rectified Linear Unit (ReLU) activation function (Fig. 1b), thereby enabling the conversion between ANNs and SNNs.

Prior works have recognized the importance of the membrane potential for accurate SNN inference [4,6]. Any changes in membrane potential can cause neurons to fire more or less rapidly, which can significantly impact the network's performance. What is concerning is that the negative membrane potential cannot be released and instead accumulated during inference. Consequently, a much more extensive value range compared to positive membrane potential is necessary, especially when the inference lasts for an extended duration.

Storing such a large negative membrane potential can present a major challenge to SNN hardware. To this end, Hwang et al. [14] were the first to investigate the possibility of constraining negative membrane potential on various network models and tasks. They found that disabling negative membrane potential had little impact on a shallow 4-layer SNN on the MNIST classification task but led to significant accuracy loss for a deeper 10-layer SNN on the CIFAR-10 task. They also assessed the safe lower bounds of the membrane potential for each case, below which there would be no significant accuracy degradation. Unfortunately, they did not offer a method to really alleviate the impact of negative membrane potential. As the networks deepen, the safe lower bound of membrane potential will eventually become impractical.

## 2.2 Hardware Implementations

There is a surprising lack of information in SNN hardware regarding the handling of membrane potential, particularly for negative membrane potential. Designs in [8,9] have employed reduced-precision fixed-point formats for membrane potential while disabling negative membrane potential. However, both designs were limited to shallow SNNs and tasks with low complexity where the disabling of negative membrane potential is not significantly detrimental.

Even less information is available regarding deeper SNNs. Designs in [10] employed a 32-bit fixed-point format for membrane potential. They implemented a custom and a VGG-16 convolutional neural network (CNN) model, achieving 81.80% and 91.46% CIFAR-10 accuracies, respectively. However, the 32-bit fixed-point format for membrane potential is a step backward compared to the low-
bit format used in shallower networks, which can lead to significant hardware overhead. Overall, efficient SNN hardware implementations are largely restricted to shallow layers, limiting their applicability across various applications.

# 3 Guideline for Constraining Membrane Potential

In this section, we derive a guideline for constraining negative membrane potential in a lossless manner. Contrary to prior research, we demonstrate that disabling negative membrane potential does not necessarily result in accuracy degradation in deep SNNs. Instead, fluctuations in membrane potential are primarily responsible for this outcome, which is alleviable.

## 3.1 Error Analysis

We start with a theoretical analysis. We follow the IF neuron model described in prior work [3]. The total input potential integrated to the i-th neuron in l-th layer at time step t is defined as:

$$z_i^l(t) = \sum_{j=1}^{M^{l-1}} W_{ij}^l \Theta_j^{l-1}(t) + b_i^l, \quad (1)$$

where $M^{l-1}$ is the number of pre-synaptic neurons, $W_{ij}^l$ is the synaptic weight, $b_i^l$ is the bias and $\Theta_j^{l-1}(t)$ is the step function indicating the spike generation:

$$\Theta_i^l(t) = \Theta(V_i^l(t-1) + z_i^l(t) - V_{\text{thr}}), \text{ with } \Theta(x) = \begin{cases} 1, & \text{if } x \ge 0 \\ 0, & \text{else} \end{cases} \quad (2)$$

Meanwhile, at each time step, the input potential $z_i^l$ is integrated to the neuron's membrane potential $V_i^l$:

$$V_i^l(t) = V_i^l(t-1) + z_i^l(t) - V_{\text{thr}} \Theta_i^l(t) + \epsilon_i^l(t, V_{\text{lb}}), \quad (3)$$

where $\epsilon_i^l(t, V_{\text{lb}})$ is the error term that appears when a lower bound $V_{\text{lb}}$ is imposed on the negative membrane potential. The value of this error term is equal to the discrepancy between the membrane potential that would have been attained in the absence of $V_{\text{lb}}$ and the actual value hindered by $V_{\text{lb}}$.

The threshold potential $V_{\text{thr}}$ is normally equal to one. By summing and averaging Eq. 3 over the entire inference duration T and rearranging it, we obtain:

$$r_i^l(T) = \frac{1}{T} \sum_{t=1}^{T} \Theta_i^l(t) = \frac{1}{T} \sum_{t=1}^{T} z_i^l(t) - \frac{V_i^l(T) - V_i^l(0)}{T} + \frac{1}{T} \sum_{t=1}^{T} \epsilon_i^l(t, V_{\text{lb}}), \quad (4)$$

where $r_i^l$ is the average output spike rate during T. It may seem that the neuron tends to exhibit abnormally higher firing rates due to the presence of the error
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Two handcrafted examples demonstrate the errors in the membrane potential and their impact on the output spike.</figcaption></figure>

term in comparison to the case where no lower bound is imposed [14]. However, this is not necessarily the case. To explain that, we handcraft two examples under non-monotonic and monotonic input potentials.

*Example 1 (Non-monotonic potential).* As shown in Fig. 2a, the membrane potential descends to the lower bound $V_{lb}$ at $t = 1$, resulting in an error of $\epsilon = 0.5$. During the subsequent period, the membrane potential ascends to positive, and the error term causes the membrane potential to reach the threshold at $t = 4$, resulting in one unanticipated spike.

*Example 2 (Monotonic potential).* As shown in Fig. 2b, the membrane potential descends to the lower bound five times at $t = 2, 3, 4, 5$ and 6 during the inference, resulting in a greater cumulative error of $\epsilon = -2.5$ than in Ex. 1. Nevertheless, unlike in Ex. 1, the error does not lead to any unanticipated spike.

Both examples satisfy Eq. 4 and present errors. However, we find that the occurrence of the unanticipated spike actually depends more on the short-term behavior of the membrane potential. In the ideal scenario, for monotonically decreasing membrane potential, errors will accumulate continuously, but no spike will be generated as long as it does not return to the positive value. And for a spiking neural network, as long as the neurons do not release unanticipated spikes, errors in membrane potential will remain localized and not propagate to affect inference capability, regardless of their magnitude.

## 3.2 Underlying Cause of Accuracy Degradation

The discrepancy between the theory and the conclusions drawn by prior work [14] prompted us to conduct experiments to investigate what exactly happens within the deep SNNs when the negative membrane potential is constrained.
**Network and Data Setup.** In this work, we conduct all experiments using a custom simulator under the TensorFlow platform. We train a ResNet-20 [15] on the CIFAR-10 datasets and convert it to SNN using the method introduced in [3]. This network is deeper than other SNNs implemented on hardware to date [10, 12, 13] and should be able to reflect deep SNN scenarios. We do not use additional techniques such as membrane potential pre-charging [5], as they can also increase hardware overhead, even though they may enhance the inference capability at low time steps. As for the test inputs, we follow the method described in [2] to convert CIFAR-10 test images into corresponding poisson spike trains.

**Results.** We do confirm significant performance degradation when constraining negative membrane potential. As shown in Fig. 3, in the absence of the lower bound on the membrane potential, the converted spiking ResNet-20 achieves the highest accuracy of 87.80%. However, after applying a uniform lower bound to all neurons’ membrane potential, we observe a reduction in both short-term and peak inference accuracies as the lower bound increased. For our configuration, we recommend $V_{lb} = -16$ as the safe lower bound considering a certain margin, below which there would be no significant accuracy degradation. This value is more conservative than the one reported in [14], possibly because we employ a network that is twice as deep.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Influence of membrane potential's lower bound on the SNN inference accuracy on CIFAR-10 dataset.</figcaption></figure>

To pinpoint error occurrences, we directly observe the membrane potential variations during inference. We randomly pick 100 neurons from the first hidden layer and present the behavior of their membrane potentials in Fig. 4a. As expected, in the absence of a lower bound, the negative membrane potentials accumulated during inference.

However, upon closer inspection, we find that negative membrane potentials do not monotonically decrease but exhibit positive fluctuations (Fig. 4b) like the case demonstrated in the handcraft Ex. 1. This phenomenon is attributed to the discrete and unstable nature of poisson spike trains. If only synapses with positive weights receive spikes at specific time steps, the total input potential
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** a,b. Variations in neurons' membrane potentials during inference at the global and local scales. c,d. An example to demonstrate that neurons with fluctuating negative membrane potentials can generate unanticipated spikes when $V_{lb} = -2$ is imposed.</figcaption></figure>

can be positive in the short term. When we set a lower bound of $V_{lb}$, any positive fluctuations greater than $(V_{thr} - V_{lb})$ could cause unanticipated spike firing by exceeding the threshold. Figure 4c,d provide an example where a neuron is not supposed to release backward spikes in the absence of the lower bound. However, with a lower bound of $V_{lb} = -2$, the membrane potential surpasses the threshold and leads to unanticipated firing multiple times, which hurts accurate SNN inference.

Referring to our handcraft examples, we regard such fluctuations in membrane potential resulting from the instability in the poisson spike trains as a major factor that contributes to the degradation of inference accuracy when constraining negative membrane potential.

## 3.3 Guideline and Validation

Based on the theory and the observations, we establish the following practical guideline for constraining negative membrane potential in deep SNNs.

*Guideline: If the non-monotonic fluctuations in the membrane potential can be eliminated, it will be possible to constrain or even disable negative membrane potential without negatively affecting SNN inference accuracy.*

We design a demonstration experiment to validate the guideline. To reduce the fluctuations in the membrane potential, we aim to ensure that all synapses
<figure><img src="image_7.png" /><figcaption>Fig. 5. Results of the demonstration experiment. a,b. Negative membrane potentials with no constraint are perfectly linear during the inference. c. No change in the inference accuracy beyond the error level, even when negative membrane potential is entirely disabled.</figcaption></figure>

of neurons receive an input spike equally at each time step, thereby eliminating instability. To achieve this, we add buffers to all synapses, count the incoming spikes during inference and then pass the average spike counts to the neuron, as described in the following equation:

$$ \delta_i^l(t) = r_i^l(T) = \frac{1}{T} \sum_{t=1}^{T} \Theta_i^l(t), \qquad (5) $$

where $\delta_i^l \in [0, 1]$ represents a fractional spike whose value equals the average input spike count. It replaces the original binary spike and is sent to the neuron at each time step. Thereby, the total input potential remains the same as the original. The network needs to be computed layer by layer to accommodate this change, while the rest of the network configurations are kept the same as before.

In this case, as illustrated in Fig. 5a,b, even the slightest fluctuations in the negative membrane potentials are eliminated in the absence of a lower bound. On this basis, we set $V_{lb} = 0$ and find that even under this extreme condition, the performance is fully maintained at every time step (Fig. 5c), demonstrating that the negative membrane potentials can indeed be constrained in a lossless manner, as per our guideline. While we have only tested the case of ResNet-20 architecture, we anticipate that it will also be applicable to even deeper networks.
# 4 Hardware-Friendly Method

The above demo proves the effectiveness of the proposed guideline. However, it requires Multiply-and-Accumulate (MAC) operations to support fractional spikes, which can significantly compromise hardware efficiency. In this section, we present an alternative practical method that employs the least possible amount of additional hardware with the same effectiveness.

## 4.1 Packed Spike Format

<figure><img src="image_8.png" /><figcaption>Fig. 6. Schematic diagram illustrating the proposed hardware-friendly method.</figcaption></figure>

The proposed method is based on a novel packed spike format. As Fig. 6 illustrates, the process starts at the input layer, where input spike trains are accumulated over a period and then packed. The spike packets (blue boxes in Fig. 6) replace the spike trains that propagate backward. Synapses of the neuron in the next layer receive the spike packets and process them by extracting one spike per time step in the following period. Meanwhile, the membrane potential updates, which functions as the neuron's axon, accumulates the input potential. Only at the end of the period, depending on whether the residual membrane potential is positive (Fig. 6b) or negative (Fig. 6a), the neuron releases a spike packet of the corresponding value, or zero spike packet, to the next layer. This process is repeated for each subsequent layer.

This method is expected to be similarly effective to the demo, as it also conveys the statistics of spikes rather than discrete and unstable spike trains; thus, transient outliers in membrane potential will not affect results. The effectiveness of the proposed method is anticipated to be affected by the depth of spike packets. Meanwhile, while this method requires additional memory for spike packets, it is unnecessary to make significant modifications to the core of neurons since they still receive binary spike inputs and therefore require no multiplier.

## 4.2 Results

This section evaluates the effectiveness and efficiency of the proposed method. We use the same network configuration as before, except that additional input and inter-layer buffers are added to handle the spike packets.
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** **a.** Influence of the spike packet depth on inference accuracy when negative membrane potential is disabled. **b.** Effect of the proposed practical method. For both schemes the accuracies are well maintained when negative potential is disabled.</figcaption></figure>

**Impact on Accuracy.** Regarding the depth of spike packets, we test two schemes: **Scheme 1.** The input layer uses the same spike packet depth as all subsequent hidden layers, with $d_{input} = d_{hidden} = T/k$, where T is the total inference duration. **Scheme 2.** Hidden layers use the same spike packet depth $d_{hidden} = T/k$ as in scheme 1, while the input layer uses deeper spike packets with $d_{input} = T$ in order to reinforce the effect of smoothing. To accommodate this change, neurons in the first hidden layer extract only $1/k$ of the spikes from the input spike packets each time during inference.

Figure 7a presents the impact of packet depth on accuracy for both schemes. We set the total inference duration $T = 512$, at which the original SNN is near saturation in terms of accuracy. We also set the $V_{lb} = 0$. In both schemes, the accuracies are fully maintained at $d_{hidden} = 512$ and experience degradation as $d_{hidden}$ decreases. However, the degradation is slower in scheme 2, where the accuracy is well maintained even at $d_{hidden} = 128 = T/4$, showing that the enhancement on the input layer benefits all subsequent layers. In contrast, scheme 1 only achieves lossless performance when $d_{hidden}$ equals the full duration.

Figure 7b further demonstrates the effectiveness of the proposed method at various inference durations. We employ $d_{hidden} = T$ and $d_{hidden} = T/4$ for schemes 1 and 2, respectively, to maximize performance. With these settings, both schemes achieve optimal performance across all inference durations, even at $V_{lb} = 0$, which allows the complete disabling of negative membrane potential in deep SNN hardware.

**Impact on Storage Overhead.** In this work, we assume an 8-bit quantization for the weights. At the same time, with the proposed method, we are able to substantially constrain the membrane potential to a range of [0, 1]. Consequently, we can perform a 7-bit quantization for the membrane potential, even lower than that of the weights.

We demonstrate a substantial reduction in the required memory for membrane potential by 41.67% for both schemes (Table 1). We anticipate that the
<table><caption>**Table 1.** Impact of the proposed practical method on the storage overhead for spiking ResNet-20 on the CIFAR-10 dataset.</caption><thead><tr><th rowspan="2">Method</th><th rowspan="2">MP Range</th><th colspan="3">Memory Usage [kB]</th></tr><tr><th>MP Memory</th><th>Spike Buffer</th><th>Total</th></tr></thead><tbody><tr><td>Directly constraining</td><td>[-16, 1]</td><td>430.19</td><td>3.82</td><td>434.01</td></tr><tr><td><b>Scheme 1</b> (512/512)*</td><td>[0, 1]</td><td>250.95 (-179.24<sup>†</sup>)</td><td>34.42 (+30.60<sup>†</sup>)</td><td>285.37 (-148.64<sup>†</sup>)</td></tr><tr><td><b>Scheme 2</b> (512/128)*</td><td>[0, 1]</td><td>250.95 (-179.24<sup>†</sup>)</td><td>30.05 (+26.23<sup>†</sup>)</td><td>281.00 (-153.01<sup>†</sup>)</td></tr></tbody></table>

MP: membrane potential.

*: spike packet depths in the input layer and hidden layers ($d_{input}/d_{hidden}$).

†: difference compared to the baseline method with directly constraining.

proposed method will yield even more significant storage overhead reduction for deeper SNNs in the future as the safe $V_{lb}$ further decreases.

Buffer for spike packets require additional memory, while, unlike membrane potential, by leveraging the convolutional topology, only a few rows of spike packets need to be retained for each channel and layer simultaneously [11] (except for the input layer in scheme 2, where all spikes within the packets cannot be extracted at once). As a result, only a small amount of additional memory for spike packets is needed, especially in scheme 2, making it the more efficient option. Furthermore, in hardware implementations, smaller spike packets can be allocated to higher-level memory with lower access power, while extensive membrane potential requires lower-level memory. Therefore, the proposed method may have a more far-reaching impact on energy efficiency.

## 5 Conclusion

We propose a guideline for constraining negative membrane potential in deep SNNs without sacrificing accuracy. The key insight is that the instability of spike trains, which is eliminable, mainly contributes to the accuracy degradation. On top of that, we develop a practical method for implementing this guideline which uses a novel packed spike format and requires minimal modifications to existing SNN hardware. We demonstrate the effectiveness of our method on the ResNet-20, achieving significant memory usage reduction through reduced-precision quantization of the membrane potential. Overall, our work lays the ground for future efficient hardware implementation of even deeper SNNs, which can facilitate high-performance outcomes across a wider range of applications.

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