Rui-Jie Zhu

Rui-Jie Zhu, Malu Zhang, Qihang Zhao et al.

Spiking Neural Networks (SNNs) are attracting widespread interest due to their biological plausibility, energy efficiency, and powerful spatio-temporal information representation ability. Given the critical role of attention mechanisms in enhancing neural network performance, the integration of SNNs and attention mechanisms exhibits potential to deliver energy-efficient and high-performance computing paradigms. We present a novel Temporal-Channel Joint Attention mechanism for SNNs, referred to as TCJA-SNN. The proposed TCJA-SNN framework can effectively assess the significance of spike sequence from both spatial and temporal dimensions. More specifically, our essential technical contribution lies on: 1) We employ the squeeze operation to compress the spike stream into an average matrix. Then, we leverage two local attention mechanisms based on efficient 1D convolutions to facilitate comprehensive feature extraction at the temporal and channel levels independently. 2) We introduce the Cross Convolutional Fusion (CCF) layer as a novel approach to model the inter-dependencies between the temporal and channel scopes. This layer breaks the independence of these two dimensions and enables the interaction between features. Experimental results demonstrate that the proposed TCJA-SNN outperforms SOTA by up to 15.7% accuracy on standard static and neuromorphic datasets, including Fashion-MNIST, CIFAR10-DVS, N-Caltech 101, and DVS128 Gesture. Furthermore, we apply the TCJA-SNN framework to image generation tasks by leveraging a variation autoencoder. To the best of our knowledge, this study is the first instance where the SNN-attention mechanism has been employed for image classification and generation tasks. Notably, our approach has achieved SOTA performance in both domains, establishing a significant advancement in the field. Codes are available at https://github.com/ridgerchu/TCJA.

Rui-Jie Zhu, Qihang Zhao, Guoqi Li et al.

As the size of large language models continue to scale, so does the computational resources required to run it. Spiking Neural Networks (SNNs) have emerged as an energy-efficient approach to deep learning that leverage sparse and event-driven activations to reduce the computational overhead associated with model inference. While they have become competitive with non-spiking models on many computer vision tasks, SNNs have also proven to be more challenging to train. As a result, their performance lags behind modern deep learning, and we are yet to see the effectiveness of SNNs in language generation. In this paper, inspired by the Receptance Weighted Key Value (RWKV) language model, we successfully implement `SpikeGPT', a generative language model with binary, event-driven spiking activation units. We train the proposed model on two model variants: 45M and 216M parameters. To the best of our knowledge, SpikeGPT is the largest backpropagation-trained SNN model to date, rendering it suitable for both the generation and comprehension of natural language. We achieve this by modifying the transformer block to replace multi-head self attention to reduce quadratic computational complexity O(N^2) to linear complexity O(N) with increasing sequence length. Input tokens are instead streamed in sequentially to our attention mechanism (as with typical SNNs). Our preliminary experiments show that SpikeGPT remains competitive with non-spiking models on tested benchmarks, while maintaining 20x fewer operations when processed on neuromorphic hardware that can leverage sparse, event-driven activations. Our code implementation is available at https://github.com/ridgerchu/SpikeGPT.

Wen-Jie Shu, Xuerui Qiu, Rui-Jie Zhu et al.

Recent advances in visual reasoning have leveraged vision transformers to tackle the ARC-AGI benchmark. However, we argue that the feed-forward architecture, where computational depth is strictly bound to parameter size, falls short of capturing the iterative, algorithmic nature of human induction. In this work, we propose a recursive architecture called Loop-ViT, which decouples reasoning depth from model capacity through weight-tied recurrence. Loop-ViT iterates a weight-tied Hybrid Block, combining local convolutions and global attention, to form a latent chain of thought. Crucially, we introduce a parameter-free Dynamic Exit mechanism based on predictive entropy: the model halts inference when its internal state ``crystallizes" into a low-uncertainty attractor. Empirical results on the ARC-AGI-1 benchmark validate this perspective: our 18M model achieves 65.8% accuracy, outperforming massive 73M-parameter ensembles. These findings demonstrate that adaptive iterative computation offers a far more efficient scaling axis for visual reasoning than simply increasing network width. The code is available at https://github.com/WenjieShu/LoopViT.

Bo Peng, Daniel Goldstein, Quentin Anthony et al. · harvard

We present Eagle (RWKV-5) and Finch (RWKV-6), sequence models improving upon the RWKV (RWKV-4) architecture. Our architectural design advancements include multi-headed matrix-valued states and a dynamic recurrence mechanism that improve expressivity while maintaining the inference efficiency characteristics of RNNs. We introduce a new multilingual corpus with 1.12 trillion tokens and a fast tokenizer based on greedy matching for enhanced multilinguality. We trained four Eagle models, ranging from 0.46 to 7.5 billion parameters, and two Finch models with 1.6 and 3.1 billion parameters and find that they achieve competitive performance across a wide variety of benchmarks. We release all our models on HuggingFace under the Apache 2.0 license. Models at: https://huggingface.co/RWKV Training code at: https://github.com/RWKV/RWKV-LM Inference code at: https://github.com/RWKV/ChatRWKV Time-parallel training code at: https://github.com/RWKV/RWKV-infctx-trainer

Yimeng Shan, Xuerui Qiu, Rui-jie Zhu et al.

Spiking Neural Networks (SNNs) have garnered substantial attention in brain-like computing for their biological fidelity and the capacity to execute energy-efficient spike-driven operations. As the demand for heightened performance in SNNs surges, the trend towards training deeper networks becomes imperative, while residual learning stands as a pivotal method for training deep neural networks. In our investigation, we identified that the SEW-ResNet, a prominent representative of deep residual spiking neural networks, incorporates non-event-driven operations. To rectify this, we propose a novel training paradigm that first accumulates a large amount of redundant information through OR Residual Connection (ORRC), and then filters out the redundant information using the Synergistic Attention (SynA) module, which promotes feature extraction in the backbone while suppressing the influence of noise and useless features in the shortcuts. When integrating SynA into the network, we observed the phenomenon of "natural pruning", where after training, some or all of the shortcuts in the network naturally drop out without affecting the model's classification accuracy. This significantly reduces computational overhead and makes it more suitable for deployment on edge devices. Experimental results on various public datasets confirmed that the SynA-ResNet achieved single-sample classification with as little as 0.8 spikes per neuron. Moreover, when compared to other residual SNN models, it exhibited higher accuracy and up to a 28-fold reduction in energy consumption.

Xingwei Qu, Shaowen Wang, Zihao Huang et al.

Large Language Models (LLMs) apply uniform computation to all tokens, despite language exhibiting highly non-uniform information density. This token-uniform regime wastes capacity on locally predictable spans while under-allocating computation to semantically critical transitions. We propose $\textbf{Dynamic Large Concept Models (DLCM)}$, a hierarchical language modeling framework that learns semantic boundaries from latent representations and shifts computation from tokens to a compressed concept space where reasoning is more efficient. DLCM discovers variable-length concepts end-to-end without relying on predefined linguistic units. Hierarchical compression fundamentally changes scaling behavior. We introduce the first $\textbf{compression-aware scaling law}$, which disentangles token-level capacity, concept-level reasoning capacity, and compression ratio, enabling principled compute allocation under fixed FLOPs. To stably train this heterogeneous architecture, we further develop a $\textbf{decoupled $μ$P parametrization}$ that supports zero-shot hyperparameter transfer across widths and compression regimes. At a practical setting ($R=4$, corresponding to an average of four tokens per concept), DLCM reallocates roughly one-third of inference compute into a higher-capacity reasoning backbone, achieving a $\textbf{+2.69$\%$ average improvement}$ across 12 zero-shot benchmarks under matched inference FLOPs.

Skye Gunasekaran, Téa Wright, Rui-Jie Zhu et al.

Several recent Transformer architectures expose later layers to representations computed in the earliest layers, motivated by the observation that low-level features can become harder to recover as the residual stream is repeatedly transformed through depth. The cheapest among these methods add static value residuals: learned mixing coefficients that expose the first-layer value projection V_1 uniformly across tokens and heads. More expressive dense or dynamic alternatives recover finer-grained access, but at higher memory cost and lower throughput. The usefulness of V_1 is unlikely to be constant across tokens, heads, and contexts; different positions plausibly require different amounts of access to early lexical or semantic information. We therefore treat early-representation reuse as a retrieval problem rather than a connectivity problem, and introduce Selective Access Transformer (SATFormer), which preserves the first-layer value pathway while controlling access with a context-dependent gate. Across models from 130M to 1.3B parameters, SATFormer consistently improves validation loss and zero-shot accuracy over the static value-residual and Transformer baselines. Its strongest gains appear on retrieval-intensive benchmarks, where it improves over static value residuals by approximately 1.5 average points, while maintaining throughput and memory usage close to the baseline Transformer. Gate analyses suggest sparse, depth-dependent, head-specific, and category-sensitive access patterns, supporting the interpretation that SATFormer learns selective reuse of early representations rather than uniform residual copying. Our code is available at https://github.com/SkyeGunasekaran/SATFormer.

Chunyuan Deng, Yizhe Zhang, Rui-Jie Zhu et al.

Looped Transformers (LT) have emerged as a powerful architecture by iterating their layers multiple times before decoding the final token. However, pairing them with full attention retains quadratic complexity, making them computationally expensive and slow. We introduce LT2 (Linear-Time Looped Transformers), a family of looped architectures that replace quadratic softmax attention with subquadratic, linear-time attention. We study two variants: LT2-linear with linear attention and LT2-sparse with sparse attention. We find that looping uniquely synergizes with these variants: it enables iterative memory refinement in linear attention and progressively expands the effective receptive field in sparse attention. We formalize these benefits theoretically and demonstrate consistent empirical gains across controlled recall, state-tracking, and language modeling tasks. We then explore LT2-hybrid, which combines different attention variants in a looped setting. Two variants are especially promising: LT2-hybrid (GDN+DSA), which interleaves linear and sparse attention to maximize efficiency and matches the standard looped transformer's quality at fully linear-time cost; and LT2-hybrid (Full+GDN), which interleaves GDN with a small fraction of full attention layers to maximize quality, surpassing the standard looped transformer in both performance and efficiency. We also show how to convert a pre-trained LT into an LT2-hybrid model. With about 1B tokens of training, our converted model, Ouro-hybrid-1.4B, outperforms industry-level 1B models and is competitive with industry-level 4B models while retaining the speed benefits of linear-time attention. Together, these results show a clear path toward making looped transformers more scalable and advancing efficient, capable small language models.

Rui-Jie Zhu, Tianhao Peng, Tianhao Cheng et al.

Large Language Models (LLMs) have demonstrated impressive reasoning capabilities, especially when guided by explicit chain-of-thought (CoT) reasoning that verbalizes intermediate steps. While CoT improves both interpretability and accuracy, its dependence on natural language reasoning limits the model's expressive bandwidth. Latent reasoning tackles this bottleneck by performing multi-step inference entirely in the model's continuous hidden state, eliminating token-level supervision. To advance latent reasoning research, this survey provides a comprehensive overview of the emerging field of latent reasoning. We begin by examining the foundational role of neural network layers as the computational substrate for reasoning, highlighting how hierarchical representations support complex transformations. Next, we explore diverse latent reasoning methodologies, including activation-based recurrence, hidden state propagation, and fine-tuning strategies that compress or internalize explicit reasoning traces. Finally, we discuss advanced paradigms such as infinite-depth latent reasoning via masked diffusion models, which enable globally consistent and reversible reasoning processes. By unifying these perspectives, we aim to clarify the conceptual landscape of latent reasoning and chart future directions for research at the frontier of LLM cognition. An associated GitHub repository collecting the latest papers and repos is available at: https://github.com/multimodal-art-projection/LatentCoT-Horizon/.

Xuerui Qiu, Malu Zhang, Jieyuan Zhang et al.

Spiking neural networks are emerging as a promising energy-efficient alternative to traditional artificial neural networks due to their spike-driven paradigm. However, recent research in the SNN domain has mainly focused on enhancing accuracy by designing large-scale Transformer structures, which typically rely on substantial computational resources, limiting their deployment on resource-constrained devices. To overcome this challenge, we propose a quantized spike-driven Transformer baseline (QSD-Transformer), which achieves reduced resource demands by utilizing a low bit-width parameter. Regrettably, the QSD-Transformer often suffers from severe performance degradation. In this paper, we first conduct empirical analysis and find that the bimodal distribution of quantized spike-driven self-attention (Q-SDSA) leads to spike information distortion (SID) during quantization, causing significant performance degradation. To mitigate this issue, we take inspiration from mutual information entropy and propose a bi-level optimization strategy to rectify the information distribution in Q-SDSA. Specifically, at the lower level, we introduce an information-enhanced LIF to rectify the information distribution in Q-SDSA. At the upper level, we propose a fine-grained distillation scheme for the QSD-Transformer to align the distribution in Q-SDSA with that in the counterpart ANN. By integrating the bi-level optimization strategy, the QSD-Transformer can attain enhanced energy efficiency without sacrificing its high-performance advantage. For instance, when compared to the prior SNN benchmark on ImageNet, the QSD-Transformer achieves 80.3% top-1 accuracy, accompanied by significant reductions of 6.0$\times$ and 8.1$\times$ in power consumption and model size, respectively. Code is available at https://github.com/bollossom/QSD-Transformer.

Dustin Wang, Rui-Jie Zhu, Steven Abreu et al.

Transformers face quadratic complexity and memory issues with long sequences, prompting the adoption of linear attention mechanisms using fixed-size hidden states. However, linear models often suffer from limited recall performance, leading to hybrid architectures that combine linear and full attention layers. Despite extensive hybrid architecture research, the choice of linear attention component has not been deeply explored. We systematically evaluate various linear attention models across generations - vector recurrences to advanced gating mechanisms - both standalone and hybridized. To enable this comprehensive analysis, we trained and open-sourced 72 models: 36 at 340M parameters (20B tokens) and 36 at 1.3B parameters (100B tokens), covering six linear attention variants across five hybridization ratios. Benchmarking on standard language modeling and recall tasks reveals that superior standalone linear models do not necessarily excel in hybrids. While language modeling remains stable across linear-to-full attention ratios, recall significantly improves with increased full attention layers, particularly below a 3:1 ratio. Our study highlights selective gating, hierarchical recurrence, and controlled forgetting as critical for effective hybrid models. We recommend architectures such as HGRN-2 or GatedDeltaNet with a linear-to-full ratio between 3:1 and 6:1 to achieve Transformer-level recall efficiently. Our models are open-sourced at https://huggingface.co/collections/m-a-p/hybrid-linear-attention-research-686c488a63d609d2f20e2b1e.

Weijie Zhao, Mingquan Liu, Bolun Wang et al.

Scaling Transformers typically necessitates training larger models from scratch, as standard architectures struggle to expand without discarding learned representations. We identify the primary bottleneck in the attention mechanism's linear projections, which strictly confine feature extraction to fixed-dimensional subspaces, limiting both expressivity and incremental capacity. To address this, we introduce Nexusformer, which replaces linear $Q/K/V$ projections with a Nexus-Rank layer, a three-stage nonlinear mapping driven by dual activations in progressively higher dimensional spaces. This design overcomes the linearity constraint and enables lossless structured growth: new capacity can be injected along two axes via zero-initialized blocks that preserve pretrained knowledge. Experiments on language modeling and reasoning benchmarks demonstrate that Nexusformer matches Tokenformer's perplexity using up to 41.5\% less training compute during progressive scaling (240M to 440M). Furthermore, our analysis of growth dynamics reveals that zero initialization induces a stable convergence trajectory, allowing us to derive a geometric scaling law that accurately predicts performance across expansion scales.

Rui-Jie Zhu, Zixuan Wang, Kai Hua et al. · princeton

Modern LLMs are trained to "think" primarily via explicit text generation, such as chain-of-thought (CoT), which defers reasoning to post-training and under-leverages pre-training data. We present and open-source Ouro, named after the recursive Ouroboros, a family of pre-trained Looped Language Models (LoopLM) that instead build reasoning into the pre-training phase through (i) iterative computation in latent space, (ii) an entropy-regularized objective for learned depth allocation, and (iii) scaling to 7.7T tokens. Ouro 1.4B and 2.6B models enjoy superior performance that match the results of up to 12B SOTA LLMs across a wide range of benchmarks. Through controlled experiments, we show this advantage stems not from increased knowledge capacity, but from superior knowledge manipulation capabilities. We also show that LoopLM yields reasoning traces more aligned with final outputs than explicit CoT. We hope our results show the potential of LoopLM as a novel scaling direction in the reasoning era. Our model is available here: http://ouro-llm.github.io.

Yimeng Shan, Malu Zhang, Rui-jie Zhu et al.

Recent advancements in neuroscience research have propelled the development of Spiking Neural Networks (SNNs), which not only have the potential to further advance neuroscience research but also serve as an energy-efficient alternative to Artificial Neural Networks (ANNs) due to their spike-driven characteristics. However, previous studies often neglected the multiscale information and its spatiotemporal correlation between event data, leading SNN models to approximate each frame of input events as static images. We hypothesize that this oversimplification significantly contributes to the performance gap between SNNs and traditional ANNs. To address this issue, we have designed a Spiking Multiscale Attention (SMA) module that captures multiscale spatiotemporal interaction information. Furthermore, we developed a regularization method named Attention ZoneOut (AZO), which utilizes spatiotemporal attention weights to reduce the model's generalization error through pseudo-ensemble training. Our approach has achieved state-of-the-art results on mainstream neural morphology datasets. Additionally, we have reached a performance of 77.1% on the Imagenet-1K dataset using a 104-layer ResNet architecture enhanced with SMA and AZO. This achievement confirms the state-of-the-art performance of SNNs with non-transformer architectures and underscores the effectiveness of our method in bridging the performance gap between SNN models and traditional ANN models.

Steven Abreu, Sumit Bam Shrestha, Rui-Jie Zhu et al.

Large language models (LLMs) deliver impressive performance but require large amounts of energy. In this work, we present a MatMul-free LLM architecture adapted for Intel's neuromorphic processor, Loihi 2. Our approach leverages Loihi 2's support for low-precision, event-driven computation and stateful processing. Our hardware-aware quantized model on GPU demonstrates that a 370M parameter MatMul-free model can be quantized with no accuracy loss. Based on preliminary results, we report up to 3x higher throughput with 2x less energy, compared to transformer-based LLMs on an edge GPU, with significantly better scaling. Further hardware optimizations will increase throughput and decrease energy consumption. These results show the potential of neuromorphic hardware for efficient inference and pave the way for efficient reasoning models capable of generating complex, long-form text rapidly and cost-effectively.

Mude Hui, Rui-Jie Zhu, Songlin Yang et al.

Flow models are effective at progressively generating realistic images, but they generally struggle to capture long-range dependencies during the generation process as they compress all the information from previous time steps into a single corrupted image. To address this limitation, we propose integrating autoregressive modeling -- known for its excellence in modeling complex, high-dimensional joint probability distributions -- into flow models. During training, at each step, we construct causally-ordered sequences by sampling multiple images from the same semantic category and applying different levels of noise, where images with higher noise levels serve as causal predecessors to those with lower noise levels. This design enables the model to learn broader category-level variations while maintaining proper causal relationships in the flow process. During generation, the model autoregressively conditions the previously generated images from earlier denoising steps, forming a contextual and coherent generation trajectory. Additionally, we design a customized hybrid linear attention mechanism tailored to our modeling approach to enhance computational efficiency. Our approach, termed ARFlow, achieves 6.63 FID scores on ImageNet at 256 * 256 without classifier-free guidance, reaching 1.96 FID with classifier-free guidance 1.5, outperforming the previous flow-based model SiT's 2.06 FID. Extensive ablation studies demonstrate the effectiveness of our modeling strategy and chunk-wise attention design.

Yimeng Shan, Zhenbang Ren, Haodi Wu et al.

Event cameras provide superior temporal resolution, dynamic range, power efficiency, and pixel bandwidth. Spiking Neural Networks (SNNs) naturally complement event data through discrete spike signals, making them ideal for event-based tracking. However, current approaches that combine Artificial Neural Networks (ANNs) and SNNs, along with suboptimal architectures, compromise energy efficiency and limit tracking performance. To address these limitations, we propose the first Transformer-based spike-driven tracking pipeline. Our Global Trajectory Prompt (GTP) method effectively captures global trajectory information and aggregates it with event streams into event images to enhance spatiotemporal representation. We then introduce SDTrack, a Transformer-based spike-driven tracker comprising a Spiking MetaFormer backbone and a tracking head that directly predicts normalized coordinates using spike signals. The framework is end-to-end, does not require data augmentation or post-processing. Extensive experiments demonstrate that SDTrack achieves state-of-the-art performance while maintaining the lowest parameter count and energy consumption across multiple event-based tracking benchmarks, establishing a solid baseline for future research in the field of neuromorphic vision.

Skye Gunasekaran, Assel Kembay, Hugo Ladret et al.

Accurate time-series forecasting is crucial in various scientific and industrial domains, yet deep learning models often struggle to capture long-term dependencies and adapt to data distribution shifts over time. We introduce Future-Guided Learning, an approach that enhances time-series event forecasting through a dynamic feedback mechanism inspired by predictive coding. Our method involves two models: a detection model that analyzes future data to identify critical events and a forecasting model that predicts these events based on current data. When discrepancies occur between the forecasting and detection models, a more significant update is applied to the forecasting model, effectively minimizing surprise, allowing the forecasting model to dynamically adjust its parameters. We validate our approach on a variety of tasks, demonstrating a 44.8% increase in AUC-ROC for seizure prediction using EEG data, and a 23.4% reduction in MSE for forecasting in nonlinear dynamical systems (outlier excluded).By incorporating a predictive feedback mechanism, Future-Guided Learning advances how deep learning is applied to time-series forecasting.

Rui-Jie Zhu, Yu Zhang, Steven Abreu et al.

Large Language Models (LLMs) have fundamentally altered how we approach scaling in machine learning. However, these models pose substantial computational and memory challenges, primarily due to the reliance on matrix multiplication (MatMul) within their attention and feed-forward (FFN) layers. We demonstrate that MatMul operations can be eliminated from LLMs while maintaining strong performance, even at billion-parameter scales. Our MatMul-free models, tested on models up to 2.7B parameters, are comparable to state-of-the-art pre-trained Transformers, and the performance gap narrows as model size increases. Our approach yields significant memory savings: a GPU-efficient implementation reduces memory consumption by up to 61% during training and over 10x during inference. When adapted for a multi-chip neuromorphic system, the model leverages asynchronous processing to achieve 4x higher throughput with 10x less energy than edge GPUs.

Bo Peng, Eric Alcaide, Quentin Anthony et al.

Transformers have revolutionized almost all natural language processing (NLP) tasks but suffer from memory and computational complexity that scales quadratically with sequence length. In contrast, recurrent neural networks (RNNs) exhibit linear scaling in memory and computational requirements but struggle to match the same performance as Transformers due to limitations in parallelization and scalability. We propose a novel model architecture, Receptance Weighted Key Value (RWKV), that combines the efficient parallelizable training of transformers with the efficient inference of RNNs. Our approach leverages a linear attention mechanism and allows us to formulate the model as either a Transformer or an RNN, thus parallelizing computations during training and maintains constant computational and memory complexity during inference. We scale our models as large as 14 billion parameters, by far the largest dense RNN ever trained, and find RWKV performs on par with similarly sized Transformers, suggesting future work can leverage this architecture to create more efficient models. This work presents a significant step towards reconciling trade-offs between computational efficiency and model performance in sequence processing tasks.

Cheng Jin, Rui-Jie Zhu, Xiao Wu et al.

Spiking Neural Networks (SNNs) have piqued researchers' interest because of their capacity to process temporal information and low power consumption. However, current state-of-the-art methods limited their biological plausibility and performance because their neurons are generally built on the simple Leaky-Integrate-and-Fire (LIF) model. Due to the high level of dynamic complexity, modern neuron models have seldom been implemented in SNN practice. In this study, we adopt the Phase Plane Analysis (PPA) technique, a technique often utilized in neurodynamics field, to integrate a recent neuron model, namely, the Izhikevich neuron. Based on the findings in the advancement of neuroscience, the Izhikevich neuron model can be biologically plausible while maintaining comparable computational cost with LIF neurons. By utilizing the adopted PPA, we have accomplished putting neurons built with the modified Izhikevich model into SNN practice, dubbed as the Standardized Izhikevich Tonic (SIT) neuron. For performance, we evaluate the suggested technique for image classification tasks in self-built LIF-and-SIT-consisted SNNs, named Hybrid Neural Network (HNN) on static MNIST, Fashion-MNIST, CIFAR-10 datasets and neuromorphic N-MNIST, CIFAR10-DVS, and DVS128 Gesture datasets. The experimental results indicate that the suggested method achieves comparable accuracy while exhibiting more biologically realistic behaviors on nearly all test datasets, demonstrating the efficiency of this novel strategy in bridging the gap between neurodynamics and SNN practice.