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Deep Learning in Spiking Neural Networks

Amirhossein TavanaeiMasoud GhodratiSaeed Reza KheradpishehTimothee MasquelierAnthony S. Maida

article2018Neural Networks1,411 citations

Compares supervised and unsupervised training methods for deep spiking neural networks, demonstrating how biologically realistic models achieve competitive task accuracy with significantly lower computational and energy costs on hardware.

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Modern deep learning has delivered breakthroughs in pattern recognition, but conventional artificial neural networks require substantial computing power and high energy consumption. This high power demand restricts their deployment in battery-powered, edge, and embedded computing systems. In contrast, the biological brain performs complex computations with high energy efficiency by transmitting discrete, event-based electrical pulses known as spikes. Spiking neural networks aim to replicate these biological mechanisms to build power-efficient computing platforms, but training deep spiking architectures remains a fundamental technical challenge because discrete spike signals are non-differentiable and prevent the direct use of standard gradient-based optimization algorithms.

The article systematically reviews recent methods for constructing and training deep spiking neural networks across various structural configurations. It evaluates supervised and unsupervised learning techniques, conversion methods, and the resulting trade-offs in recognition accuracy, computational efficiency, and hardware viability.

To evaluate the landscape of spiking deep learning, the authors surveyed multiple architectural paradigms, including fully connected deep networks, convolutional architectures, deep belief networks, and recurrent or reservoir models. The analysis evaluated training frameworks across standard machine learning benchmarks such as handwritten digit and visual object recognition datasets, categorizing methods into direct spike-based training and the conversion of pre-trained conventional models into spiking platforms.

The findings indicate that deep spiking networks are closing the accuracy gap with traditional non-spiking neural networks while requiring substantially fewer computational operations. First, converting pre-trained conventional neural networks into spiking equivalents currently achieves the highest accuracy among spiking models, often exceeding ninety-nine percent on standard digit recognition benchmarks and over ninety percent on complex image recognition tasks. Second, direct training methods using biologically plausible mechanisms, such as spike-timing-dependent plasticity, successfully extract hierarchical features across network layers, although their classification accuracy generally trails converted networks. Third, supervised learning using surrogate approximations for non-differentiable spike thresholds allows end-to-end gradient descent, cutting computational operations by up to eighty percent compared to traditional models. Finally, recurrent spiking models, including liquid state machines and spike-based gated units, demonstrate strong processing capabilities for temporal and sequential data while matching conventional recurrent models.

These results demonstrate that spiking neural networks offer a viable path to deploying high-performance artificial intelligence within energy-constrained environments. By executing computations through sparse, asynchronous events, spiking hardware can significantly lower energy costs and operating latency for autonomous devices, wearable sensors, and portable electronics. Furthermore, the convergence of deep learning principles with biologically grounded learning rules advances our theoretical understanding of neural information processing in biological brains.

Organizations developing low-power artificial intelligence solutions should adopt a phased technical strategy. In the near term, teams aiming for immediate deployment on neuromorphic hardware should utilize conversion pipelines from pre-trained continuous networks to maximize classification accuracy with minimal performance loss. For applications requiring online adaptation and real-time temporal processing, organizations should invest in surrogate-gradient training frameworks and reservoir computing methods. Continued research is recommended to design native multi-layer learning rules that completely eliminate reliance on non-spiking training pipelines.

The evaluated architectures were predominantly tested on controlled, standardized benchmark datasets rather than large-scale, unstructured real-world data streams. Additionally, hardware-specific performance and energy efficiency metrics rely heavily on specialized neuromorphic silicon platforms. While confidence in the energy savings and core computational principles is high, stakeholders should validate real-world latency, throughput, and noise resilience in targeted pilot environments before full commercial deployment.

Kharazmi UniversityMonash UniversityUniversité Toulouse III - Paul SabatierUniversity of Louisiana at Lafayetteannclassificationcomputational efficiencyspiking neural networksdeep learningneuromorphic computing
arXiv: 1804.08150
Cover for Deep Learning in Spiking Neural Networks

Abstract

In recent years, deep learning has been a revolution in the field of machine learning, for computer vision in particular. In this approach, a deep (multilayer) artificial neural network (ANN) is trained in a supervised manner using backpropagation. Huge amounts of labeled examples are required, but the resulting classification accuracy is truly impressive, sometimes outperforming humans. Neurons in an ANN are characterized by a single, static, continuous-valued activation. Yet biological neurons use discrete spikes to compute and transmit information, and the spike times, in addition to the spike rates, matter. Spiking neural networks (SNNs) are thus more biologically realistic than ANNs, and arguably the only viable option if one wants to understand how the brain computes. SNNs are also more hardware friendly and energy-efficient than ANNs, and are thus appealing for technology, especially for portable devices. However, training deep SNNs remains a challenge. Spiking neurons' transfer function is usually non-differentiable, which prevents using backpropagation. Here we review recent supervised and unsupervised methods to train deep SNNs, and compare them in terms of accuracy, but also computational cost and hardware friendliness. The emerging picture is that SNNs still lag behind ANNs in terms of accuracy, but the gap is decreasing, and can even vanish on some tasks, while the SNNs typically require much fewer operations.

Topics

annclassificationcomputational efficiencycomputer visiondeep learningenergy-efficientportable devicespiking neural networksfeedforward neural networksartificial neural network (ann)

Keywords

spiking neural networksdeep learningneuromorphic computingspike timing-dependent plasticitydeep SNN backpropagationspike pattern association neuronspatiotemporal spike patternsreservoir computing

Table of Contents

  • I Introduction
  • II Spiking Neural Network: A Biologically Inspired Approach to Information Processing
  • II-A SNN Architecture
  • II-B Learning Rules in SNNs
  • II-B1 Unsupervised Learning via STDP
  • II-B2 Probabilistic Characterization of Unsupervised STDP
  • II-B3 Supervised Learning
  • III Deep Learning in SNNs
  • III-A Deep, Fully Connected SNNs
  • III-B Spiking CNNs
  • III-C Spiking Deep Belief Networks
  • III-D Recurrent SNNs
  • III-D1 Gated SNNs
  • III-D2 Liquid State Machines and Reservoirs
  • III-E Performance Comparisons of Contemporary Models
  • IV Summary
  • References

Knowls

  1. Knowl 1 — Performance Comparison of Deep Spiking Neural Network Paradigms across Standard Benchmarks

    data/table

    Deep learning in spiking neural networks (SNNs) bifurcates into two main operational paradigms: (1) direct online training using spike-based learning mechanisms (including surrogate-gradient backpropagation, layer-wise spike-timing-dependent plasticity (STDP), or convolutional autoencoders) and (2) offline training of continuous-valued artificial neural networks (ANNs) followed by conversion to SNNs for neuromorphic deployment. Spiking convolutional neural networks (CNNs) consistently outperform fully connected feedforward SNNs and spiking deep belief networks (DBNs) on visual pattern recognition tasks. Converted deep SNNs achieve classification accuracies on benchmarks such as MNIST, N-MNIST, CIFAR-10, and CIFAR-100 that match standard deep ANNs while operating with sparse, event-driven spike additions rather than continuous multiply-accumulate operations.

    Model Architecture Learning Method Dataset Acc (%)
    Feedforward, Fully Connected SNNs
    O'Connor (2016) Deep SNN Stochastic gradient descent MNIST 96.40
    O'Connor (2016) Deep SNN Fractional stochastic gradient descent MNIST 97.93
    Lee (2016) Deep SNN Backpropagation MNIST 98.88
    Lee (2016) Deep SNN Backpropagation N-MNIST 98.74
    Neftci (2017) Deep SNN Event-driven random backpropagation MNIST 97.98
    Liu (2017) SNN Temporal backpropagation (3-layer) MNIST 99.10
    Eliasmith (2012) SNN Spaun brain model MNIST 94.00
    Diehl (2015) SNN STDP (2-layer) MNIST 95.00
    Tavanaei (2017) SNN STDP-based backpropagation (3-layer) MNIST 97.20
    Mostafa (2017) SNN Temporal backpropagation (3-layer) MNIST 97.14
    Querlioz (2013) SNN STDP, Hardware implementation MNIST 93.50
    Brader (2007) SNN Spike-driven synaptic plasticity MNIST 96.50
    Diehl (2015) Deep SNN Offline learning, Conversion MNIST 98.60
    Neil (2016) Deep SNN Offline learning, Conversion MNIST 98.00
    Hunsberger (2015) Deep SNN Offline learning, Conversion MNIST 98.37
    Esser (2015) Deep SNN Offline learning, Conversion MNIST 99.42
    Spiking CNNs
    Lee (2016) Spiking CNN Backpropagation MNIST 99.31
    Lee (2016) Spiking CNN Backpropagation N-MNIST 98.30
    Panda (2016) Spiking CNN Convolutional autoencoder MNIST 99.05
    Panda (2016) Spiking CNN Convolutional autoencoder CIFAR-10 75.42
    Tavanaei (2017) Spiking CNN Layer-wise sparse coding and STDP MNIST 98.36
    Tavanaei (2018) Spiking CNN Layer-wise and end-to-end STDP rules MNIST 98.60
    Kheradpisheh (2016) Spiking CNN Layer-wise STDP MNIST 98.40
    Zhao (2015) Spiking CNN Tempotron MNIST 91.29
    Cao (2015) Spiking CNN Offline learning, Conversion CIFAR-10 77.43
    Neil (2016) Spiking CNN Offline learning, Conversion N-MNIST 95.72
    Diehl (2015) Spiking CNN Offline learning, Conversion MNIST 99.10
    Rueckauer (2017) Spiking CNN Offline learning, Conversion MNIST 99.44
    Rueckauer (2017) Spiking CNN Offline learning, Conversion CIFAR-10 90.85
    Hunsberger (2015) Spiking CNN Offline learning, Conversion CIFAR-10 82.95
    Garbin (2014) Spiking CNN Offline learning, Hardware MNIST 94.00
    Esser (2016) Spiking CNN Offline learning, Hardware CIFAR-10 87.50
    Esser (2016) Spiking CNN Offline learning, Hardware CIFAR-100 63.05
    Spiking RBMs and DBNs
    Neftci (2014) Spiking RBM Contrastive divergence in LIF neurons MNIST 91.90
    O'Connor (2013) Spiking DBN Offline learning, Conversion MNIST 94.09
    Stromatias (2015) Spiking DBN Offline learning, Conversion MNIST 94.94
    Stromatias (2015) Spiking DBN Offline learning, Hardware MNIST 95.00
    Merolla (2011) Spiking RBM Offline learning, Hardware MNIST 94.00
    Neil (2014) Spiking DBN Offline learning, Hardware MNIST 92.00

  2. Knowl 2 — Non-Differentiability and Weight Transport Complications in Spiking Backpropagation

    theoretical result

    Implementing gradient-based backpropagation in multi-layer spiking neural networks faces two fundamental challenges stemming from biological and physical constraints:

    1. Non-Differentiability of Spike Trains: The classical error gradient chain rule defines error sensitivity δjμ\delta_j^\mu for unit jj under pattern μ\mu as: δjμ=g′(ajμ)∑kwkjδkμ\delta_j^\mu = g'(a_j^\mu) \sum_k w_{kj} \delta_k^\mu where ajμa_j^\mu is the net input activation to unit jj, wkjw_{kj} is the feedforward weight from unit jj to unit kk, δkμ\delta_k^\mu is the error gradient at unit kk, and g(⋅)g(\cdot) is the activation function. Because a spiking neuron communicates via discrete action potentials represented by sums of Dirac delta functions, the analytical derivative g′(⋅)g'(\cdot) does not exist (it is zero almost everywhere and infinite at spike times).

    2. The Weight Transport Problem: Backpropagating errors via ∑kwkjδkμ\sum_k w_{kj} \delta_k^\mu requires utilizing the exact transpose of the forward synaptic weight matrix wkjw_{kj} in the feedback pathway. In biological neural circuits and decentralized neuromorphic hardware, reciprocal point-to-point symmetric connections do not inherently exist. While random feedback weights (feedback alignment) can suffice for simple classification tasks, complex pattern recognition problems require symmetric feedback paths.

  3. Knowl 3 — Classical Pair-Based Spike-Timing-Dependent Plasticity (STDP) Rule

    equation

    Spike-Timing-Dependent Plasticity (STDP) is a local, unsupervised synaptic learning rule in which the change in synaptic efficacy Δw\Delta w depends on the relative timing of presynaptic and postsynaptic action potentials:

    Δw={Ae−∣tpre−tpost∣τ,tpre−tpost≤0,A>0Be−∣tpre−tpost∣τ,tpre−tpost>0,B<0\Delta w = \begin{cases} A e^{-\frac{|t_{\text{pre}} - t_{\text{post}}|}{\tau}}, & t_{\text{pre}} - t_{\text{post}} \le 0, \quad A > 0 \\ B e^{-\frac{|t_{\text{pre}} - t_{\text{post}}|}{\tau}}, & t_{\text{pre}} - t_{\text{post}} > 0, \quad B < 0 \end{cases}

    where:

    • tpret_{\text{pre}} and tpostt_{\text{post}} represent the arrival time of the presynaptic spike and the firing time of the postsynaptic spike, respectively.
    • A>0A > 0 is the learning rate for Long-Term Potentiation (LTP), which strengthens the synapse when the presynaptic neuron fires before the postsynaptic neuron (tpre≤tpostt_{\text{pre}} \le t_{\text{post}}), indicating causal correlation.
    • B<0B < 0 is the learning rate for Long-Term Depression (LTD), which weakens the synapse when the presynaptic neuron fires after the postsynaptic neuron (tpre>tpostt_{\text{pre}} > t_{\text{post}}).
    • τ\tau is the decay time constant specifying the temporal learning window (typically ≈15 ms\approx 15\text{ ms}).

    Over repeated presentations of stimuli, STDP causes postsynaptic neurons to reduce their response latencies, tuning their selectivity to the earliest incoming spikes that carry the primary discriminative information.

  4. Knowl 4 — Probabilistic Expectation-Maximization Formulation of STDP in Winner-Take-All SNNs

    model/method

    Unsupervised STDP within stochastic Winner-Take-All (WTA) spiking circuits can mathematically approximate the online Expectation-Maximization (EM) algorithm for fitting mixture models and hidden Markov models (HMMs). The synaptic weight adjustment is governed by:

    Δwki={e−wki−1,0<tkf−tif<ϵ−1,otherwise\Delta w_{ki} = \begin{cases} e^{-w_{ki}} - 1, & 0 < t_k^f - t_i^f < \epsilon \\ -1, & \text{otherwise} \end{cases}

    where wkiw_{ki} is the synaptic weight connecting presynaptic neuron ii to postsynaptic neuron kk, tift_i^f and tkft_k^f are their respective spike times, and ϵ\epsilon represents the coincidence window (e.g., 10 ms10\text{ ms}).

    Under this probabilistic framework:

    • When a postsynaptic neuron in the WTA circuit fires, it generates a sample from the posterior probability distribution over latent hidden variables, realizing the E-step of the EM algorithm.
    • Applying the STDP rule to the synapses of the active postsynaptic neuron updates the generative model parameters toward maximum likelihood, realizing the M-step of the EM algorithm.
    • Excitatory negative weight updates can be shifted into strictly positive regimes by adding a constant offset parameter to the LTP rule, enabling stable feature learning for spatiotemporal sequence recognition.

  5. Knowl 5 — Supervised Spike-Based Training via ReSuMe and SPAN

    model/method

    Supervised training methods for single spiking neurons adapt synaptic weights to elicit postsynaptic spike trains matching precise target times based on the Widrow-Hoff (Delta) formulation Δw=(yd−yo)x\Delta w = (y^d - y^o)x, where xx is the presynaptic input, ydy^d is the desired target output, and yoy^o is the observed output:

    1. Remote Supervised Learning (ReSuMe): Decomposes the weight adaptation into a linear sum of standard STDP and anti-STDP operations: Δw=ΔwSTDP(Sin,Sd)+ΔwaSTDP(Sin,So)\Delta w = \Delta w^{\text{STDP}}(S^{\text{in}}, S^d) + \Delta w^{\text{aSTDP}}(S^{\text{in}}, S^o) where SinS^{\text{in}}, SdS^d, and SoS^o denote the presynaptic spike train, the desired teacher spike train, and the actual emitted spike train, respectively. The interaction between the remote teacher neuron and the input synapse requires no direct physical wiring, allowing remote supervisory signals to guide temporal training within STDP eligibility windows.

    2. Spike Pattern Association Neuron (SPAN): Converts discrete spike sequences into continuous analog waveforms by convolving each spike with an α\alpha-kernel postsynaptic potential h(t)=te−t/τh(t) = t e^{-t/\tau}. The continuous-domain Widrow-Hoff update is then integrated over the temporal pattern duration: Δw∝∫x~i(t)(y~d(t)−y~o(t))dt\Delta w \propto \int \tilde{x}_i(t) \left(\tilde{y}^d(t) - \tilde{y}^o(t)\right) dt where x~i(t)\tilde{x}_i(t), y~d(t)\tilde{y}^d(t), and y~o(t)\tilde{y}^o(t) are the continuous analog signals corresponding to the transformed presynaptic, desired target, and observed postsynaptic spike trains, respectively.

  6. Knowl 6 — Direct Multi-Layer SNN Backpropagation via Surrogate Membrane Potential Signals

    model/method

    Direct gradient-based backpropagation across multi-layer spiking neural networks resolves the non-differentiability of discrete spike events by utilizing continuous membrane potentials as differentiable proxy signals during the backward pass.

    In this scheme:

    • Forward Dynamics: Neurons operate as leaky integrate-and-fire (LIF) units accumulating postsynaptic currents over time. When the continuous membrane potential vv reaches a threshold, a discrete action potential is emitted and the membrane potential is reset.
    • Backward Pass / Surrogate Derivatives: The discontinuous derivative of the threshold function ∂s∂v\frac{\partial s}{\partial v} is replaced by a smooth surrogate gate function g(v)≥0g(v) \ge 0 satisfying ∫g(v)dv=1\int g(v) dv = 1, or by treating the continuous sub-threshold membrane potential al,ia_{l,i} of neuron ii in layer ll (incorporating excitatory inputs, lateral inhibition, and threshold offsets) as a differentiable continuous activation.
    • Optimization: Error gradients are propagated layer-by-layer using the standard chain rule. This end-to-end optimization allows deep SNNs to train directly on static images (e.g., MNIST achieving 98.88%98.88\% accuracy) and neuromorphic spike streams (e.g., N-MNIST achieving 98.74%98.74\% accuracy) while requiring up to five times fewer computational operations than conventional artificial neural networks.

  7. Knowl 7 — ANN-to-SNN Conversion and Weight Normalization for Energy-Efficient Inference

    model/method

    ANN-to-SNN conversion enables deep neural networks trained with conventional backpropagation to execute on neuromorphic hardware without training directly on discrete spike trains:

    • Activation-to-Rate Mapping: Continuous, positive-valued activations generated by rectified linear units (ReLUs) in an ANN are mapped directly to proportional firing rates of integrate-and-fire (IF) or leaky integrate-and-fire (LIF) spiking neurons.
    • Weight Normalization and Threshold Balancing: Direct parameter transfer often causes latency inflation (due to insufficient input currents) or severe saturation (due to excessive firing). Scaling synaptic weight matrices by the maximum possible activation of each layer, or dynamically adjusting neuronal thresholds, preserves the linear response ratios across deep layers and prevents degradation on complex benchmarks such as CIFAR-10 and ImageNet.
    • Computational Advantage: In converted SNNs, multiply-accumulate (MAC) computations of ANNs are replaced by asynchronous, event-driven sparse synaptic additions, drastically lowering power consumption on specialized neuromorphic chips.

  8. Knowl 8 — Hierarchical Feature Learning in Spiking Convolutional Neural Networks

    model/method

    Spiking Convolutional Neural Networks (Spiking CNNs) combine weight-shared spatial convolution and pooling operations with event-driven spike timing. Feature extraction across deep layers is achieved using three primary methodologies:

    1. Biomimetic Early Filtering: The first processing layer uses fixed Difference-of-Gaussian (DoG) or Gabor filters to emulate orientation-selective receptive fields of simple cells in the primary visual cortex (V1), converting pixel inputs into latency- or rate-coded spike maps.
    2. Unsupervised Layer-Wise Representation Learning: Intermediate convolutional kernels are trained locally without external labels using unsupervised spike rules, such as SAILnet sparse coding or STDP convolutional autoencoders. As layers deepen, units learn invariant, abstract visual features before routing outputs to competitive spiking layers or linear classifiers.
    3. Supervised Spike Backpropagation and Conversion: Full architectures (e.g., LeNet-style spiking CNNs) can be trained end-to-end via surrogate membrane potential gradient descent or converted from pre-trained continuous CNNs, achieving high classification performance on complex datasets such as CIFAR-10 (90.85%90.85\%).

  9. Knowl 9 — Spiking Restricted Boltzmann Machines and Event-Driven Contrastive Divergence

    model/method

    Spiking Restricted Boltzmann Machines (Spiking RBMs) replace the memoryless stochastic binary units of classical RBMs with stochastic leaky integrate-and-fire spiking neurons:

    • Contrastive Divergence via STDP: An event-driven variant of STDP implemented in stochastic spiking networks approximates Contrastive Divergence (CD) weight updates: Δwij∝⟨vihj⟩data−⟨vihj⟩model\Delta w_{ij} \propto \langle v_i h_j \rangle_{\text{data}} - \langle v_i h_j \rangle_{\text{model}} where viv_i and hjh_j represent visible and hidden neuron activities. The stationary spike statistics of the stochastic network converge to the exact joint probability distributions learned by non-spiking RBMs.
    • Spiking Deep Belief Networks (DBNs): Layer-wise stacking of pre-trained spiking RBMs forms Spiking Deep Belief Networks (Spiking DBNs). These networks can be mapped onto low-power neuromorphic VLSI architectures (such as TrueNorth and SpiNNaker), achieving high classification accuracy while maintaining robustness against reduced hardware bit precision and device mismatch noise.

  10. Knowl 10 — Recurrent Spiking Neural Network Architectures: Reservoirs, LSNN, and subLSTM

    model/method

    Recurrent SNNs process temporal and sequential spike signals by incorporating feedback connections and microcircuit dynamics:

    1. Liquid State Machines (LSM) / Reservoir Computing: A sparsely connected recurrent 3D reservoir containing excitatory and inhibitory spiking neurons (typically maintaining an 80:20 neocortical ratio with distance-dependent connection probabilities) creates a continuous, fading-memory spatiotemporal representation of multi-modal input spike streams. Instantaneous internal reservoir states are decoded using linear readout units.
    2. Long Short-Term Memory SNNs (LSNN): Combines a recurrent spiking reservoir with an auxiliary population of excitatory neurons featuring slow, activity-dependent adaptive thresholds. Trained using Backpropagation Through Time (BPTT) with membrane potential pseudo-derivatives, the LSNN performs complex sequence learning (such as sequential MNIST and continuous speech recognition) and retains working memory without updating weights during testing.
    3. Subtractive LSTM (subLSTM): Adapts LSTM gated architectures to cortical microcircuits by replacing non-linear multiplicative gating operations with subtractive inhibitory gating mediated by lateral recurrent inhibitory circuits, enabling bio-plausible recurrent dynamics within standard deep learning frameworks.

Coverage note — No substantial contributed material was omitted; this survey's taxonomies, mathematical formulations of STDP and surrogate gradient learning rules, comparative performance data, and architectural reviews across feedforward, convolutional, deep belief, and recurrent SNNs are fully represented.

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Citation

MLA
Tavanaei, A., et al. “Deep Learning in Spiking Neural Networks”. Neural Networks, vol. 111, 2019, pp. 47–63, https://doi.org/10.1016/j.neunet.2018.12.002.
APA
Tavanaei, A., Ghodrati, M., Kheradpisheh, S. R., Masquelier, T., & Maida, A. (2019). Deep learning in spiking neural networks. Neural Networks, 111, 47–63. https://doi.org/10.1016/j.neunet.2018.12.002
Chicago
Tavanaei, A., M. Ghodrati, S. R. Kheradpisheh, T. Masquelier, and A. Maida. 2019. “Deep Learning in Spiking Neural Networks”. Neural Networks 111: 47–63. https://doi.org/10.1016/j.neunet.2018.12.002.
Harvard
Tavanaei, A. et al. (2019) “Deep learning in spiking neural networks”, Neural Networks, 111, pp. 47–63. Available at: https://doi.org/10.1016/j.neunet.2018.12.002.
Vancouver
1. Tavanaei A, Ghodrati M, Kheradpisheh SR, Masquelier T, Maida A (2019) Deep learning in spiking neural networks. Neural Networks 111:47–63

BibTeX

@article{Tavanaei_2019, title={Deep learning in spiking neural networks}, volume={111}, ISSN={0893-6080}, url={http://dx.doi.org/10.1016/j.neunet.2018.12.002}, DOI={10.1016/j.neunet.2018.12.002}, journal={Neural Networks}, publisher={Elsevier BV}, author={Tavanaei, Amirhossein and Ghodrati, Masoud and Kheradpisheh, Saeed Reza and Masquelier, Timothée and Maida, Anthony}, year={2019}, month=Mar, pages={47–63} }
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