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Spike-Train Level Backpropagation for Training Deep Recurrent Spiking Neural Networks

Wenrui Zhang, Peng Li

arXiv:1908.06378v3cs.NEcs.LG

TL;DR

Supervised training of general recurrent spiking neural networks (RSNNs), especially deep architectures, remains limited. This paper presents ST-RSBP, a spike-train-level backpropagation algorithm, and reports that it outperforms current SNN training techniques and conventional deep-learning models across speech and image datasets.

  • Problem

    Powerful supervised training for general RSNNs is lacking, limiting exploration of deep architectures that combine feedforward and recurrent layers.

    Source: §1 · p. 2

  • Method

    ST-RSBP computes gradients of a rate-coded output loss at the spike-train level, capturing temporal effects without unfolding the network through time.

    Source: §1 · p. 2

  • Results

    Across TI46-Alpha, TI46-Digits, N-Tidigits, MNIST, and Fashion-MNIST, ST-RSBP outperforms current SNN training techniques and conventional deep-learning models.

    Source: §5 · p. 9

  • Takeaways & Limitations

    The authors expect releasing the GPU implementation to advance research on spiking neural networks and neuromorphic computing.

    Source: §5 · p. 9

Abstract

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Spiking neural networks (SNNs) well support spatiotemporal learning and energy-efficient event-driven hardware neuromorphic processors. As an important class of SNNs, recurrent spiking neural networks (RSNNs) possess great computational power. However, the practical application of RSNNs is severely limited by challenges in training. Biologically-inspired unsupervised learning has limited capability in boosting the performance of RSNNs. On the other hand, existing backpropagation (BP) methods suffer from high complexity of unrolling in time, vanishing and exploding gradients, and approximate differentiation of discontinuous spiking activities when applied to RSNNs. To enable supervised training of RSNNs under a well-defined loss function, we present a novel Spike-Train level RSNNs Backpropagation (ST-RSBP) algorithm for training deep RSNNs. The proposed ST-RSBP directly computes the gradient of a rated-coded loss function defined at the output layer of the network w.r.t tunable parameters. The scalability of ST-RSBP is achieved by the proposed spike-train level computation during which temporal effects of the SNN is captured in both the forward and backward pass of BP. Our ST-RSBP algorithm can be broadly applied to RSNNs with a single recurrent layer or deep RSNNs with multiple feed-forward and recurrent layers. Based upon challenging speech and image datasets including TI46, N-TIDIGITS, Fashion-MNIST and MNIST, ST-RSBP is able to train RSNNs with an accuracy surpassing that of the current state-of-art SNN BP algorithms and conventional non-spiking deep learning models.

1 Introduction

The introduction presents ST-RSBP as a supervised method for training general recurrent spiking neural networks (RSNNs), including deep architectures, and reports strong results on speech and image benchmarks. It motivates the method by challenges in differentiating spike events and training recurrent networks.

1 Introduction

  • SNNs compute through discrete spikes and spatiotemporal patterns, offer rate and temporal coding, and support event-driven low-power neuromorphic hardware.The introduction cites theoretical evidence that SNNs have greater computational power than traditional ANNs.Source: §1 · p. 1
  • SpikeProp [5] addressed spike discontinuity for BP but is limited to single-spike learning and has not succeeded on real-world tasks.Source: §1 · p. 2
  • Rate-coded BP approaches variously ignore temporal correlations [23], model spike changes probabilistically [33], or use BPTT and smoothed membrane gradients [39], [4], risking mismatch with the loss; [15] does not apply to LIF neurons.The introduction argues that smoothing microscopic membrane waveforms may lose all-or-none firing characteristics and make computed gradients inconsistent with the target loss [19].Source: §1 · p. 2
  • General RSNN training adds recurrent-connection and vanishing-or-exploding-gradient challenges; LSMs restrict supervision to readout layers and impose architectural constraints.The introduction notes that LSM reservoirs are fixed or trained unsupervised, while LSNNs trained with BPTT retain approximate-gradient issues [4].Source: §1 · p. 2
  • ST-RSBP computes rate-coded loss gradients at the spike-train level for arbitrary recurrent connections without smoothing spike behavior or unfolding networks through time.The authors say this avoids costly time-point-by-time-point BP and vanishing or exploding gradients, and yields more precise gradients than HM2-BP [19].Source: §1 · p. 2
  • ST-RSBP reportedly surpasses current SNN BP algorithms and conventional non-spiking models on TI46, N-TIDIGITS, and Fashion-MNIST; spiking CNNs reach 99.62% on MNIST, best among SNN BP rules.The introduction also reports that MNIST result for feedforward spiking convolutional neural networks.Source: §1 · p. 2

2 Background

Existing supervised SNN backpropagation is poorly suited to deep recurrent networks, while ST-RSBP addresses temporal interactions through spike-train-level computation. The background defines S-PSPs as a representation of those interactions and relates their weighted sum to neuronal firing counts.

2.1 SNN Architectures and Training Challenges

  • STDP and WTA are typically unsupervised, while supervised learning rules apply only to the output layer, limiting more sophisticated deep architectures.Source: §2.1 · p. 3
  • Supervised BP methods [23] [39] [19] [33] show promise but generally target feedforward SNNs, not deep recurrent networks; BPTT also faces temporal unfolding, spatiotemporal error propagation, and non-differentiable spikes.BPTT may in principle train RSNNs, but these challenges make it difficult to apply.Source: §2.1 · p. 3
  • ST-RSBP integrates the original-size RSNN in its forward pass and computes the backward pass at spike-train level, avoiding time unfolding and point-by-point backpropagation.It uses S-PSPs to capture temporal interactions between pre- and postsynaptic neurons and is described as more scalable, with benefits against exploding or vanishing gradients.Source: §2.1 · p. 4

2.2 Spike-train Level Post-synaptic Potential (S-PSP)

  • An S-PSP aggregates a presynaptic neuron's spike-train contribution to a postsynaptic neuron's potential immediately before its firing times, capturing temporal dynamics and recurrent connections across firing events.This representation avoids expensive unfolding through time and point-by-point temporal backpropagation.Source: §2.2 · p. 4
  • A neuron's T-PSP is the sum of weighted S-PSPs from its presynaptic neurons and relates to its firing count through the firing threshold; T-PSP and firing count are analogous to ANN pre-activation and activation.The output firing count is obtained by converting T-PSP through g(·).Source: §2.2 · p. 4

3 Proposed Spike-Train level Recurrent SNNs Backpropagation (ST-RSBP)

ST-RSBP derives spike-train-level backpropagation for networks combining feedforward and recurrent layers, using neuron firing counts and pre-activations to represent layer activity.

3 Proposed Spike-Train level Recurrent SNNs Backpropagation (ST-RSBP)

  • The method models feedforward and recurrent connections explicitly in each layer’s spike-train activation, alongside firing counts and pre-activations.The recurrent connection is absent for a feedforward layer.Source: §3 · p. 4
  • ST-RSBP defines a rate-coded output loss from desired and actual firing counts, differentiates it with respect to trainable weights, and updates those weights using a learning rate.The output neurons’ T-PSPs are also specified in the loss definition.Source: §3 · pp. 4–5, §2 · p. 14

3.1.1 Output Layer

  • At the output layer, the backpropagated error is obtained by differentiating the rate-coded loss with respect to output activity.The loss uses desired firing counts (labels) and actual firing counts.Source: §3.1.1 · p. 5

3.1.2 Hidden Layers

  • For a hidden recurrent layer, error propagation accounts for S-PSPs’ dependence on presynaptic and postsynaptic firing rates and times, separating contributions from the preceding and recurrent layers.The two summations cover presynaptic neurons in the previous layer and in the current recurrent layer.Source: §3.1.2 · p. 5
  • The recurrent-layer partial derivatives form a coupled linear system, which can be solved by matrix inversion or approximated with a first-order Taylor expansion to avoid matrix factorization.The resulting derivatives propagate errors from layer k + 1 to layer k.Source: §3.1.2 · p. 6

4 Experiments and Results

Across speech and image benchmarks, ST-RSBP trained recurrent and convolutional spiking networks and outperformed the comparison methods, including prior SNN backpropagation methods and, on Fashion-MNIST, non-spiking BP methods. The experiments used repeated runs and dataset-specific network and input settings.

4.1 Experimental Settings

  • Experiments used LIF networks, 1 ms simulation steps, and per-sample weight updates; comparisons included five-run mean and standard-deviation accuracies.ST-RSBP models had feedforward and recurrent layers, while other BP methods used fully connected feedforward networks; HM2-BP comparisons used similar numbers of tunable weights.Source: §4.1 · pp. 6–7

4.2 TI46-Alpha Speech Dataset

  • 93.35% vs. 90.98%: ST-RSBP’s three-hidden-layer RSNN with 363,313 weights outperformed HM2-BP’s feedforward SNN with 723,200 weights on TI46-Alpha.ST-RSBP outperformed all compared methods; the results also indicate that training only an LSM readout was inadequate for this task.Source: §4.2 · p. 7, Table 2 · p. 7

4.3 TI46-Digits Speech Datasest

  • 99.39%: ST-RSBP outperformed all other methods, including HM2-BP, on TI46-Digits; with 19,057 tunable weights it achieved 98.77% versus at 86.66% with 32,000.The dataset contains 1,594 training and 2,542 testing examples; the same preprocessing as TI46-Alpha was used.Source: §4.3 · pp. 7–8, Table 3 · p. 8

4.4 N-Tidigits Neuromorphic Speech Dataset

  • 93.90%: ST-RSBP achieved accuracy up to this level on N-Tidigits, outperforming HM2-BP and the non-spiking GRN and LSTM in with similar or fewer tunable weights.N-Tidigits comprises recorded spike responses from a 64-channel cochlea sensor, with 2,475 examples each for training and testing.Source: §4.4 · p. 8, Table 4 · p. 8

4.5 Fashion-MNIST Image Dataset

  • ST-RSBP outperformed all other SNN and non-spiking BP methods on Fashion-MNIST.Images were Poisson-encoded as 784 × 400 binary spike sequences; no other preprocessing or data augmentation was applied.Source: Table 5 · p. 8, §4.5 · p. 9

4.6 Spiking Convolution Neural Networks for the MNIST

  • 99.62%: ST-RSBP surpassed the best previously reported spiking-CNN performance [19] on MNIST at the same model complexity.The comparison used similar network settings; the network had two convolutional layers, pooling layers, a fully connected hidden layer, and elastic-distortion augmentation.Source: §4.6 · p. 9, Table 6 · p. 9

5 Discussions and Conclusion

ST-RSBP trains SNNs, including recurrent networks, with spike-train-level backpropagation without temporal unfolding, capturing temporal effects and handling spike discontinuities. The paper reports it outperforms other SNN training techniques and conventional deep-learning models on speech and image datasets.

5 Discussions and Conclusion

  • ST-RSBP applies standard backpropagation at the spike-train level, training SNNs including RSNNs without unfolding time.The S-PSP model supports spike-train-level processing and addresses temporal effects and loss-gradient computation despite inherent discontinuities.Source: §5 · p. 9
  • Rate-coded errors propagate across layers without costly temporal unfolding or time-point-by-time-point computation, aggregating the effects of all spikes.This spike-train-level processing captures and propagates rate and temporal information during training.Source: §5 · p. 9
  • ST-RSBP handles spike discontinuities without altering or smoothing microscopic spiking behavior.The method exploits both rate and temporal information during training.Source: §5 · p. 9
  • ST-RSBP outperformed current state-of-the-art SNN training techniques and conventional deep-learning models across speech and image benchmarks.The reported benchmarks include TI46-Alpha, TI46-Digits, N-Tidigits, MNIST, and Fashion-MNIST; the comparisons include LSTM, GRN, and traditional non-spiking BP.Source: §5 · p. 9

1 Detailed Description of Spike-train Level Post-synaptic Potential (S-PSP) and Total PSP (T-PSP)

S-PSP aggregates how a presynaptic spike train affects a postsynaptic neuron, enabling spike-train-level treatment of temporal dynamics and recurrent connections. T-PSP sums weighted S-PSPs and relates to the neuron’s firing count through its firing threshold.

1 Detailed Description of Spike-train Level Post-synaptic Potential (S-PSP) and Total PSP (T-PSP)

  • S-PSP captures spike-train interactions for all-or-none spiking and synaptic models, illustrated using an LIF neuron and first-order synapse.Source: §1 · p. 13
  • S-PSP aggregates a presynaptic spike train’s effect on postsynaptic membrane potential and firing, accounting for temporal dynamics and recurrent connections without BPTT’s time-point-by-time-point unfolding.This supports efficient computation across firing events at the spike-train level.Source: §1 · p. 14
  • T-PSP is the sum of weighted S-PSPs from all presynaptic neurons and relates to the postsynaptic neuron’s firing count through its firing threshold.Source: §1 · p. 14
  • T-PSP and firing count are analogous to ANN pre-activation and activation, with g(·) converting T-PSP to output firing count.Source: §1 · p. 14

2 Detailed Derivation of ST-RSBP Algorithm

ST-RSBP defines a rate-coded loss at the output layer and differentiates it with respect to trainable weights. The derivation identifies back-propagated error and activation differentiation terms, then updates the weights using a learning rate.

2 Detailed Derivation of ST-RSBP Algorithm

  • The output-layer rate-coded loss uses desired and actual neuron firing counts and output-neuron T-PSPs.The vectors y, o, and a represent desired firing counts, actual firing counts, and T-PSPs, respectively.Source: §2 · p. 14
  • Differentiating the loss with respect to trainable weights incident on a layer yields terms identified as back-propagated error and activation differentiation.Source: §2 · p. 14
  • ST-RSBP updates the trainable weights using learning rate η.Source: §2 · p. 14

2.1 Back Propagated Errors

ST-RSBP propagates errors from the output using the loss gradient and computes hidden-layer errors by chain rule. Hidden-layer backpropagation accounts for recurrent spike-train dependencies, while feedforward layers admit a simpler case.

2.1.1 Output Layer

  • Output-layer errors are obtained by differentiating the loss and involve desired and actual firing counts and the corresponding T-PSP.The passages identify the desired firing count as the label and distinguish it from the actual firing count.Source: §2.1.1 · p. 15

2.1.2 Hidden Layers

  • For a hidden recurrent layer, error propagation uses S-PSPs that depend on both firing rates and spike timing, and separates contributions from preceding-layer and recurrent neurons.The recurrent-layer derivation accounts for dependencies among pre- and postsynaptic firing information.Source: §2.1.2 · p. 15
  • Recurrent-layer partial derivatives form a coupled linear system for the errors, which the method solves to propagate errors to the preceding layer.The passages describe the system in matrix form and state that the needed S-PSP partial derivatives are determined in Section 2.3.Source: §2.1.2 · pp. 16–17
  • Because the system’s Ω matrix is diagonal, its inversion can be approximated with a first-order Taylor expansion without matrix factorization.This approximation reduces the cost of factoring the system.Source: §2.1.2 · p. 17
  • When backpropagating from a hidden feedforward layer, the absence of recurrent connections simplifies the recurrent-layer error expression.The text says the resulting errors can be directly computed.Source: §2.1.2 · p. 17

2.2 Differentiation of Activation

The section differentiates activations in feedforward and recurrent layers. For feedforward layers, ST-RSBP accounts for activation’s direct dependency and the hidden dependence of the S-PSP on post-synaptic firing count.

2.2.1 Feedforward Layers

  • ST-RSBP includes the hidden dependence of each S-PSP on post-synaptic firing count, which HM2-BP does not consider.The feedforward activation differentiation also includes a term for the activation’s direct dependency.Source: §2.2.1 · p. 17

2.2.2 Recurrent Layers

  • At recurrent layers, the activation differentiation further considers recurrent connections.The passage introduces this extension but does not provide the resulting expression.Source: §2.2.2 · p. 18

2.3 Differentiation of S-PSP w.r.t Pre/Post-Synaptic Firing Counts

The section examines how S-PSP varies with pre- and postsynaptic firing counts to derive derivatives needed by ST-RSBP. It finds that the presynaptic relation is approximately linear, but the postsynaptic relation is not, motivating an offline polynomial fit.

2.3 Differentiation of S-PSP w.r.t Pre/Post-Synaptic Firing Counts

  • With postsynaptic firing count fixed, average S-PSP increases approximately linearly with presynaptic firing count, consistent with HM2-BP.The analysis swept both firing counts from 1 to 50, generated random spike trains, and averaged S-PSP over 500 repetitions per combination.Source: §2.3 · p. 18
  • With presynaptic firing count fixed, S-PSP is not linear across a wide range of postsynaptic firing counts, so HM2-BP’s assumption can cause errors when postsynaptic rates vary substantially.The simulations used a 600 ms duration, a 1 ms Euler step, and LIF and synaptic time constants of 64 ms and 8 ms, respectively.Source: §2.3 · p. 18
  • ST-RSBP fits S-PSP as a third-order polynomial in postsynaptic firing count for each fixed presynaptic count and computes the derivative offline, enabling more precise differentiation than HM2-BP.The authors suggest this may improve results even on feedforward networks such as spiking CNNs.Source: §2.3 · p. 18

2.4 The Final Proposed ST-RSBP Algorithm

ST-RSBP propagates layer errors backward and uses them with activation-derivative matrices to update feedforward and recurrent weights. Its application follows a forward simulation, output-error computation, and backward weight-training sequence.

2.4 The Final Proposed ST-RSBP Algorithm

  • For each layer, ST-RSBP uses activation-derivative matrices to update feedforward and recurrent weights as η·diag(δk)·F.The algorithm defines separate activation-derivative matrices for feedforward and recurrent connections.Source: §2.4 · pp. 18–19
  • The output layer error is propagated backward using one transformation when the next layer is feedforward and another when it is recurrent.The output error is denoted iνk; subsequent errors use the transposed P matrix for a feedforward next layer or the specified (Ω−Θ)^−1Φ transformation for a recurrent next layer.Source: §2.4 · p. 19
  • ST-RSBP simulates the SNN layer by layer, compares output firing counts with desired labels to compute error, then backpropagates error to train recurrent and interlayer weights.The simulation can use chosen synaptic and neural models such as the LIF model.Source: §2.4 · p. 19
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