Delayed Spiking Neural Network and Exponential Time Dependent Plasticity Algorithm

1. The synaptic weight is an abstract concept that measure the connection strength between neurons, which comes from ANNs. It is indeed necessary in any SNNs. The motivation of this study is not to abandon "weights" (that's a bad expression we would correct in our revised paper). We proposed our temporal-coded SNN based on the belief that the regulation of synaptic weights can be implemented by the synapses’ effects on the input spikes. Consider different lengths and shapes of axons and dendrites, and different release and absorption rates of chemical transmitters, it may be reasonable to model them as time delays added to the input spikes. In Eq. (15), we have demonstrated that the exponential term of dendritic delays in our model is functionally analogous to the synaptic weights used in conventional IF neurons. In addition, the balance of excitatory and inhibitory synapses is indeed important in biological neurons. Currently, very few SNNs can successfully use the balance to improve their performance and biological plausibility. It is a very interesting orientation; we attempt to conduct the related research in our future works.

2. We have provided abundant biological evidence to verify the biological plausibility of our proposed neural network and learning algorithm in Section 5. First, the proposed neuron network DSNN can be considered novel and biologically plausible. Delay is an inherent property of real biological systems. Recognizing the indispensability of spiking time delays, various studies have sought to incorporate them into SNNs. Spiking time delays are typically treated as additional learnable parameters alongside synaptic weights, resulting in increased complexity and longer running times. In contrast, our proposed DSNN uniquely employs only dendritic and axon delays as learnable parameters. To the best of our knowledge, our DSNN is the first neuron network that exclusively relies on dendritic and axon delays as learnable parameters. The comparison between our proposed method and previous delay learning approaches has been discussed in Section 5. Second, the proposed learning algorithm ETDP can be considered novel and biologically plausible. The ETDP algorithm is originally inspired by the gradient-based approach, but intriguingly bears a striking resemblance to the biologically plausible learning rule outlined in [1]. We compared these update strategies of two learning algorithms, and they are very similar to each other. The realizations of each term in the update strategy have been analyzed and discussed. This observation strongly supports the assertion that ETDP can be considered a biologically plausible learning algorithm. Section 5 provides the biological plausibility analysis of the DSNN and ETDP. [1] Roelfsema, Pieter R., and Anthony Holtmaat. "Control of synaptic plasticity in deep cortical networks." Nature Reviews Neuroscience 19.3 (2018): 166-180.

3. The ETDP algorithm does not need the accurate values of gradients to update the parameters; it instead uses approximate values for simplification. According to Eq. (16), we can find that there are three terms in the update strategy of ETDP. The term Zo-ZT can be regarded as the global signal that influences all the neurons in the network. The term Zout-Zin can be considered as the local STDP signal. Both terms determine whether to strengthen or weaken the delays. And the change amplitude is related to the forward and backward propagation time of signals. Adopting these three terms do not result in credit assignment problem. That is why the ETDP can be regarded as a biologically plausible learning algorithm.

4. The datasets used in the study are indeed not complex, but they are very commonly used to verify the performance of SNNs. Currently, SNNs cannot solve complex problems, because they use fully connected architectures and biologically plausible learning algorithms. If they use the convolution or transformer architectures and the gradient-based algorithms, their performance will become better, similar to the deep learning methods. But they will lose the biological plausibility simultaneously.

About the Weaknesses:

1. Indeed, the IF neuron is a highly simplified neuron model. It is s a simple one-dimensional model of a biological neuron that needs less computational effort for modelling a biological neuron. The Hodgkin and Huxley model is a four-dimensional model that can simulate the dynamic of a biological neuron with more details. But it has a very high computational cost. Most SNNs used the IF neurons in their structures. \tau is a constant hyper-parameter, and it is common to set \tau=1 to simplify the calculation of SNNs. The biological plausibility of the proposed method has been analyzed in Section 5. We provided a great amount of biological evidence to verify its biological plausibility.
2. In theory, the proposed approach can be extended to deep architectures of DSNN with multiple hidden layers. It is because, first, the proposed learning algorithm ETDP can be extended to multiple hidden layers, and the update strategies are the same for all the layers. Second, the activation function is proven non-linear in the DSNN. But, we have to admit that, currently, the SNNs cannot solve very complex problems such as the Imagenet dataset, if they use fully connected architectures and biologically plausible learning algorithms. The benchmark problems used in this study are too simple; SNNs do not need deep architectures to solve them.
3. As mentioned above, the used benchmark problems are not complex, but they are very commonly used in the field of SNNs. Even most SNNs are not verified on the FashionMNIST dataset. They cannot solve complex problems because they use fully connected architectures and biologically plausible learning algorithms. If they use the convolution or transformer architectures and BP algorithms, their performance will be better, as the deep learning methods. But they will lose the biological plausibility in this way. Our main contributions of this study can be summarized as follows. First, a novel DSNN is proposed solely with dendritic and axon delays, without the commonly used synaptic weights, which can yield competitive performance. To the best of our knowledge, our DSNN is the first neuron network that exclusively relies on dendritic and axon delays as learnable parameters. Second, the proposed learning algorithm ETDP has a high degree of biological plausibility. It intriguingly bears a striking resemblance to the biologically plausible learning rule outlined in [2, 3]. [2] Roelfsema, Pieter R., and Anthony Holtmaat. "Control of synaptic plasticity in deep cortical networks." Nature Reviews Neuroscience 19.3 (2018): 166-180. [3] Fisher, Simon D., et al. "Reinforcement determines the timing dependence of corticostriatal synaptic plasticity in vivo." Nature communications 8.1 (2017): 334.

Answers to the questions:

1. \kappa is the kernel function of IF neurons. We have added its description to the revised manuscript.
2. i, j and k are the indexes of neurons on the input, hidden and output layers, respectively. We have added the corresponding description.
3. The non-linearity of activation function is an important property of extending the SNNs to deep architectures, similar to artificial neural networks. Thus, we discuss the non-linearity of spiking neurons in Section 4. However, due to the page limit, the part has been placed into the Appendix.
4. According to your suggestion, the best methods have been highlighted, and the results of the proposed method have been separated from other methods with horizontal lines.
5. The BP algorithm uses the gradient information directly; thus it has better optimization performance. But it is thought to be biologically unrealistic. In converse, our proposed ETDP, originally inspired by the gradient-based approach, has been verified to obtain a high degree of biological plausibility. The biological evidence has been added to Section 5. The ETDP uses the approximate values of gradients. That is why it performs worse than the BP algorithm. In fact, the phenomenon is not only observed on our DSNN, but also on other SNNs, such as [1]. [1] Payeur, Alexandre, et al. "Burst-dependent synaptic plasticity can coordinate learning in hierarchical circuits." Nature neuroscience 24.7 (2021): 1010-1019.
6. The XOR problem is the simplest nonlinear problem. Most SNNs can perform well on this problem. We used it to verify the effectiveness of our proposed method on this nonlinear problem. However, the FashionMNIST dataset is more complex than the MNIST dataset. Few SNNs trained by the STDP algorithm are tested on this dataset. That led to a lack of contrasting experimental results on this problem. But we have tried our best to compare our model with the other SNNs. The results have been presented in Table 3.

About the Weaknesses: First, the proposed neuron network DSNN is novel and biologically plausible. Delay is an inherent property of real biological systems. Recognizing the indispensability of spiking time delays, various studies have sought to incorporate them into SNNs. Spiking time delays are typically treated as additional learnable parameters alongside synaptic weights, resulting in increased complexity and longer running times. In contrast, our proposed DSNN uniquely employs only dendritic and axon delays as learnable parameters. To the best of our knowledge, our DSNN is the first neuron network that exclusively relies on dendritic and axon delays as learnable parameters. In addition, we have presented that the exponential term of dendritic delays in our model is functionally analogous to the synaptic weight used in conventional IF neurons.

Second, the proposed learning algorithm ETDP is novel and biologically plausible. The ETDP algorithm is originally inspired by the gradient-based approach, but intriguingly bears a striking resemblance to the biologically plausible learning rule outlined in [1]. We compared these update strategies of two learning algorithms; they are very similar to each other. The realizations of each term in the update strategy have been analyzed and discussed. This observation strongly supports the assertion that ETDP can be considered a biologically plausible learning algorithm. Section 5 provides the biological plausibility analysis of the DSNN and ETDP. [1] Roelfsema, Pieter R., and Anthony Holtmaat. "Control of synaptic plasticity in deep cortical networks." Nature Reviews Neuroscience 19.3 (2018): 166-180.

Third, the used benchmark problems are indeed not complex, but they are very commonly used to verify the performance of SNNs. Currently, SNNs cannot solve complex problems, because they use fully connected architectures and biologically plausible learning algorithms. If they use the convolution or transformer architectures and the gradient-based algorithms, their performance will become better, similar to the deep learning methods. But they will lose the biological plausibility simultaneously.

Answers to the questions:

1. The biological plausibility analysis of the DSNN and ETDP has been presented in Section 5
2. The ETDP algorithm was originally inspired by the gradient-based approach, but it has been simplified and modified. Each term in the update strategy has been given specific biological realizations.
3. The ablation study of dendrite and the axon delay in the SNN has been conducted. The results have been presented in Table 4.
4. Since the FashionMNIST dataset is more complex than the MNIST dataset, few SNNs trained by the STDP algorithm are tested on this dataset. That led to a lack of contrasting experimental results on this problem for us. However, we have tried our best to compare our model with the other SNNs. The results have been presented in Table 3.