Spiking neural networks (SNNs) are promising brain-inspired energy-efficient models. Recent progress in training methods has enabled successful deep SNNs on large-scale tasks with low latency. Particularly, backpropagation through time (BPTT) with surrogate gradients (SG) is popularly used to enable models to achieve high performance in a very small number of time steps. However, it is at the cost of large memory consumption for training, lack of theoretical clarity for optimization, and inconsistency with the online property of biological learning rules and rules on neuromorphic hardware. Other works connect the spike representations of SNNs with equivalent artificial neural network formulation and train SNNs by gradients from equivalent mappings to ensure descent directions. But they fail to achieve low latency and are also not online.

A.3 Proof of Theorem 1 In this subsection, we prove Theorem 1 with Assumption 1. Assumption 1. l = 1,, N, t = 1,, T, diag null As described in Sections 4.1 and 4.2, for gradients of OTTT, we have Remark 2. The above conclusion mainly focuses on the gradients for connection weights Remark 3. Note that the gradients based on spike representation may also include small errors since A.4 Proof of Theorem 2 In this subsection, we prove Theorem 2. Theorem 2. If Assumption 1 holds, As described in Sections 4.1 and 4.2 and similar to the proof of Theorem 1, let Remark 4. The above conclusion considers the single-layer condition. It can be generalized to the multi-layer condition. Therefore, the conclusion can be directly generalized to these conditions as well. L} based on the gradient-based optimizer. For VGG network structures, we directly impose sWS on all weights. For more illustrations and other details, please directly refer to [4].

A.3 Proof of Theorem 1 In this subsection, we prove Theorem 1 with Assumption 1. Assumption 1. l = 1,, N, t = 1,, T, diag null As described in Sections 4.1 and 4.2, for gradients of OTTT, we have Remark 2. The above conclusion mainly focuses on the gradients for connection weights Remark 3. Note that the gradients based on spike representation may also include small errors since A.4 Proof of Theorem 2 In this subsection, we prove Theorem 2. Theorem 2. If Assumption 1 holds, As described in Sections 4.1 and 4.2 and similar to the proof of Theorem 1, let Remark 4. The above conclusion considers the single-layer condition. It can be generalized to the multi-layer condition. Therefore, the conclusion can be directly generalized to these conditions as well. L} based on the gradient-based optimizer. For VGG network structures, we directly impose sWS on all weights. For more illustrations and other details, please directly refer to [4].

Spiking neural networks (SNNs) are promising brain-inspired energy-efficient models. Recent progress in training methods has enabled successful deep SNNs on large-scale tasks with low latency. Particularly, backpropagation through time (BPTT) with surrogate gradients (SG) is popularly used to enable models to achieve high performance in a very small number of time steps. However, it is at the cost of large memory consumption for training, lack of theoretical clarity for optimization, and inconsistency with the online property of biological learning rules and rules on neuromorphic hardware. Other works connect the spike representations of SNNs with equivalent artificial neural network formulation and train SNNs by gradients from equivalent mappings to ensure descent directions. But they fail to achieve low latency and are also not online.

The field of neuromorphic computing has gained significant attention in recent years, aiming to bridge the gap between the efficiency of biological neural networks and the performance of artificial intelligence systems. This paper introduces Mamba-Spike, a novel neuromorphic architecture that integrates a spiking front-end with the Mamba backbone to achieve efficient and robust temporal data processing. The proposed approach leverages the event-driven nature of spiking neural networks (SNNs) to capture and process asynchronous, time-varying inputs, while harnessing the power of the Mamba backbone's selective state spaces and linear-time sequence modeling capabilities to model complex temporal dependencies effectively. The spiking front-end of Mamba-Spike employs biologically inspired neuron models, along with adaptive threshold and synaptic dynamics. These components enable efficient spatiotemporal feature extraction and encoding of the input data. The Mamba backbone, on the other hand, utilizes a hierarchical structure with gated recurrent units and attention mechanisms to capture long-term dependencies and selectively process relevant information. To evaluate the efficacy of the proposed architecture, a comprehensive empirical study is conducted on both neuromorphic datasets, including DVS Gesture and TIDIGITS, and standard datasets, such as Sequential MNIST and CIFAR10-DVS. The results demonstrate that Mamba-Spike consistently outperforms state-of-the-art baselines, achieving higher accuracy, lower latency, and improved energy efficiency. Moreover, the model exhibits robustness to various input perturbations and noise levels, highlighting its potential for real-world applications.

Continuous graph neural networks (CGNNs) have garnered significant attention due to their ability to generalize existing discrete graph neural networks (GNNs) by introducing continuous dynamics. They typically draw inspiration from diffusion-based methods to introduce a novel propagation scheme, which is analyzed using ordinary differential equations (ODE). However, the implementation of CGNNs requires significant computational power, making them challenging to deploy on battery-powered devices. Inspired by recent spiking neural networks (SNNs), which emulate a biological inference process and provide an energy-efficient neural architecture, we incorporate the SNNs with CGNNs in a unified framework, named Continuous Spiking Graph Neural Networks (COS-GNN). We employ SNNs for graph node representation at each time step, which are further integrated into the ODE process along with time. To enhance information preservation and mitigate information loss in SNNs, we introduce the high-order structure of COS-GNN, which utilizes the second-order ODE for spiking representation and continuous propagation. Moreover, we provide the theoretical proof that COS-GNN effectively mitigates the issues of exploding and vanishing gradients, enabling us to capture long-range dependencies between nodes. Experimental results on graph-based learning tasks demonstrate the effectiveness of the proposed COS-GNN over competitive baselines.

In this article, we propose a new paradigm for training spiking neural networks (SNNs), spike accumulation forwarding (SAF). It is known that SNNs are energy-efficient but difficult to train. Consequently, many researchers have proposed various methods to solve this problem, among which online training through time (OTTT) is a method that allows inferring at each time step while suppressing the memory cost. However, to compute efficiently on GPUs, OTTT requires operations with spike trains and weighted summation of spike trains during forwarding. In addition, OTTT has shown a relationship with the Spike Representation, an alternative training method, though theoretical agreement with Spike Representation has yet to be proven. Our proposed method can solve these problems; namely, SAF can halve the number of operations during the forward process, and it can be theoretically proven that SAF is consistent with the Spike Representation and OTTT, respectively. Furthermore, we confirmed the above contents through experiments and showed that it is possible to reduce memory and training time while maintaining accuracy.

Spiking Neural Network (SNN) is a promising energy-efficient AI model when implemented on neuromorphic hardware. However, it is a challenge to efficiently train SNNs due to their non-differentiability. Most existing methods either suffer from high latency (i.e., long simulation time steps), or cannot achieve as high performance as Artificial Neural Networks (ANNs). In this paper, we propose the Differentiation on Spike Representation (DSR) method, which could achieve high performance that is competitive to ANNs yet with low latency. First, we encode the spike trains into spike representation using (weighted) firing rate coding. Based on the spike representation, we systematically derive that the spiking dynamics with common neural models can be represented as some sub-differentiable mapping. With this viewpoint, our proposed DSR method trains SNNs through gradients of the mapping and avoids the common non-differentiability problem in SNN training. Then we analyze the error when representing the specific mapping with the forward computation of the SNN. To reduce such error, we propose to train the spike threshold in each layer, and to introduce a new hyperparameter for the neural models. With these components, the DSR method can achieve state-of-the-art SNN performance with low latency on both static and neuromorphic datasets, including CIFAR-10, CIFAR-100, ImageNet, and DVS-CIFAR10.