Training Spiking Neural Networks with Event-driven Backpropagation
Authors: Yaoyu Zhu
Zhaofei Yu
Wei Fang
Xiaodong Xie
Tiejun Huang
Timothée Masquelier
Venue: NeurIPS 2022 Accept (2022)
Keywords: Spiking Neural Networks
Event-driven Learning
Time-based Gradient

Abstract:     Spiking Neural networks (SNNs) represent and transmit information by spatiotemporal spike patterns, which bring two major advantages: biological plausibility and suitability for ultralow-power neuromorphic implementation. Despite this, the binary firing characteristic makes training SNNs more challenging. To learn the parameters of deep SNNs in an event-driven fashion as in inference of SNNs, backpropagation with respect to spike timing is proposed. Although this event-driven learning has the advantages of lower computational cost and memory occupation, the accuracy is far below the recurrent neural network-like learning approaches. In this paper, we first analyze the commonly used temporal backpropagation training approach and prove that the sum of gradients remains unchanged between fully-connected and convolutional layers. Secondly, we show that the max pooling layer meets the above invariance rule, while the average pooling layer does not, which will suffer the gradient vanishing problem but can be revised to meet the requirement. Thirdly, we point out the reverse gradient problem for time-based gradients and propose a backward kernel that can solve this problem and keep the property of the invariable sum of gradients. The experimental results show that the proposed approach achieves state-of-the-art performance on CIFAR10 among time-based training methods. Also, this is the first time that the time-based backpropagation approach successfully trains SNN on the CIFAR100 dataset.  Our code is available at https://github.com/zhuyaoyu/SNN-event-driven-learning.

Training Spiking Neural Networks with Event-driven Backpropagation
Yaoyu Zhu', Zhaofei Yu 1.2 Wei Fang', Xiaodong Xie', Tiejun Huang', Timothée Masquelier
School of Computer Science, Peking University
Institute for Artificial Intelligence, Peking University *Centre de Recherche Cerveau et Cognition (CERCO), UMR5549 CNRS Univ. Toulouse Toulouse, France
Abstract
Spiking Neural networks (SNNs) represent and transmit information by spatiotem- poral spike patterns, which bring two major advantages: biological plausibility and suitability for sitraloo-power neuromorphic implementation. Despite this, the binary firing characteristic makes training SNNs more challenging. To learn the parameters of deep SNNs in event-driven fashion as in inference of SNNs, back- propagation with respect to spike timing is proposed. Although this event-driven learning has the advantages of lower computational cost and memory occupation, the accuracy is far below the recurrent neural network-like learning approaches. In this paper, first analyze the commonly used temporal backpropagation train- ing approach and prove that the sum of gradients remains unchanged between fully-connected and convolutional layers. Secondly, we show that the max pooling layer meets the above invariance rule, while the average pooling layer does not, which will suffer the gradient vanishing problem but can be revised to meet the requirement. Thirdly, we point out the reverse gradient problem for time-based gradients and propose a backward kernel that can solve this problem and keep the property of the invariable sum of gradients. The experimental results show that the proposed approach achieves state-of-the-art performance on CIFARIO among time-based training methods. Also, this is the first time that the time-based backpropagation approach successfully trains SNN on the CIFAR 100 dataset. Our code is available at http:///hhhuu..mmmmmuayyuuu/-oos
learning
Introduction
Motivated by the principles of brain computing. Spiking Neural Networks (SNNs) are considered as the third generation of neural networks [ 2] SNNs are developed to work in power-critical scenarios, such edge computing. When run dedicated neuromorphic chips, they can accomplish the tasks generation of neural networks Artificial Neural Networks (ANNs) 13, generally require large amount of computation resource (e.g., GPUs). This advantage of SNNS on power consumption largely relies efficient event-based computations [5). Another advantage of SNNs originates from their biological reality (compared to ANNs). The similarity between SNNs and biological brains provides excellent opportunity study how the brain computes at the neuronal circuit level (16).
5 with ultra-low power consumption 6
7 8
12]. In contrast, the last
Compared with artificial neural networks, developing supervised learning algorithms for spiking neural networks requires more effort. The main challenge for training SNNs comes from the binary nature spikes and the non-differentiabiiity the membrane potential at spike time. This difficulty
Coorespponiing author
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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in training impedes the performance of SNNs in pattern classification tasks compared to their ANN counterparts. Existing supervised learning methods of SNNs can be grouped into two categories:
The first category consists of recurrent neural network (RNN)-like learning algorithms. These algorithms treat spiking neural networks as binary-output recurrent neural networks and handle the discontinuities of membrane potential at spike times with continuous surrogate derivatives [17]. They typically train deep SNNs with surrogate gradients based on the idea of backpropagation through
time (BPTT) algorithm [8, 19 20 22 23 24 25 26
27 28 While competitive accuracies are
reported on the MNIST, CIFAR- 10, and even ImageNet datasets 29 30 B11, the gradient information is propagated each time step, whether or not a spike is emitted (as shown in Fig. IT. Therefore, these approaches do not follow the event-driven nature of spiking neural networks, which lose the asynchronous characteristic of SNNs and consume much power when trained on neuromorphic hardware.
The second category is event-driven algorithms, which propagate gradient information through spikes. Precise spiking timing acts an important role in this situation, and they are extensively used in such algorithms [32
38 [9]. Classical examples include SpikeProp [32] and its vari- These algorithms approximate the derivative of spike timing to membrane potential as the negative inverse of the time derivative of membrane potential function. This approximation is actually mathematically correct without preconditions [42 Some other works apply non non-leaky
33
34 35 36 37
ants 33
Most of these works restrict each neuron to fire at most once, which inspires 44 to take the spike time as the state of neuron. and model the relation of neurons by this spike time. As result, the SNN is trained similarly to an ANN. Among the methods trained in an event-driven fashion (not modelling the relation of spike time to train like ANNs), the state-of tate-offtthearr model is TSSL-BP [39] However, they use RNN-like surrogate gradients (a sigmoid function) to assist training. Hence, it is still challenging to train SNNs in pure event-driven fashion.
integrate-and and-fire neurons to stabilize the training process [35 38 43
In this work, we develop novel event-driven learning algorithm that can train high-performance deep SNNs. The main contributions of our work are as follows:
1. We prove that the typical SNN temporal backpropagation training approach assigns the gradient of an output spike of a neuron to the input spikes generating it. After summing this assignment rule altogether,
find that the sum of gradients is unchanged between layers.
2. We analyze the case of the pooling layer (which does not have neurons) and find that average pooling does not keep the gradient sum unchanged, but we can modify its backward formulas to meet the requirement. Meanwhile, the max-pooling layer satisfies the rule initially.
3. We point out the reverse gradient problem in event-driven learning that the direction of the temporal gradient is reversed during backpropagation when the kernel function of an input spike is decreasing. Then we propose a backward kernel function that addresses this problem while keeping the sum of gradients unchanged between layers.
4. The adjusted average pooling layer and the non-decreasing backward kernel enhances the performance of our model as well as the convergence speed. To our best knowledge, our proposed approach achieves state-of-the-an performance CIFAR 10 among event-driven training methods (with temporal gradients) for SNNs. Meanwhile, our method is the first event-driven backpropagation approach that successfully trains SNN on the larger-scale CIFARIO0 dataset.
2 Backgrounds and Related Work
The gradient- based learning of spiking neural networks contains two stages: the forward (inference) and the backward (learning) stages. In the forward stage, Leaky Integrate-and
(LIF) neurons
are most commonly used [18 21 26 39), while other types of neurons are also applicable 32 35 Typically, these neuron models can be changed to the form of the Spike Response Model (SRM) B7 45| 46). which is easily represented in event-driven fashion.
In the backward stage, the methods used by existing works exhibits more diversity. Here, we classify existing approaches from two dimensions: whether non-spike information is needed in discrete time steps (RNN-like) or not (event-driven) and whether the gradient represents spike seale indivattio-basedd spike timing (time-based).
2

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d
Ougue
Irpat
kppd
1a/12
6,63
60
4,91
Spáke
Spake
Cinclet donetion
Gudfent dixction
No Solil
Figure 1: (a) The input and output spike train, as well as membrane potential curve of neuron in event-driven learning with (b) time-based gradient: When the spiking neural network is simulated in an event-driven fashion, neuron updates its state when an input spike arrives it emits output spike. The gradients here indicate whether spikes should move leftward or rightward along the time axis. (c) The input, output, membrane potential sequence of a neuron in RNN-like learning with (d) activation based gradient: The spiking neural network is forced to be simulated in discrete time steps (since gradient should be propagated in non-spike time steps) and the gradients denote whether spikes should be 'larger' or 'smaller'.
Event-driven learning v.s. RNN-like learning: In both forward and backward computation of event-driven learning, information is only carried by spikes in SNNs. Specifically, in backward computation, gradient information is propagated through spikes B2 35] (shown in Fig. [Tbbb). On the other side, in RNN-like learning, information is not only carried by spikes in backward computation. Especially, gradient information can be propagated through a neuron that does not emit spike in backward computation (shown in Fig. [b-d). This gradient propagation is achieved by surrogate function [12] 17 18] 23 47), which is function of the membrane potential at the current time step ue, and the firing threshold
Time-based gradient v.s. activation- gradient: Time-based gradients represent the (reverse) direction that the timing of spike should move, that is, to move leftward rightward the time axis [32] In backward propagation, the derivative of the firing time of a spike to the corresponding denoting how the change of membrane potential will change the spike firing time (Fig. [b). On the other side, activation- based approaches replace the Heaviside neuron activation function (.) (spike s; = Ou, - +)) in forward propagation with derivable functions (-)) (.) in backward propagation, whether there are spikes in the current time step [18 26 31 21). Therefore, activation-bast approaches essentially regard SNNs as binary RNNs and train them with approximated gradients, where the gradients indicate whether the values in the network (including the binary spikes) should be larger or smaller (Fig. IT)..
membrane potential Zu
is often approximated as ft
32
As result, time-based gradients are event-driven by nature, since the temporal gradient could only be carried by spikes. Meanwhile, oetivation-based gradients are more suitable for the RNN-like training scheme since the diversity of surrogate gradients largely relies the fact that u, * A in discrete time steps [7], which no longer holds continuous time simulation. If we want to apply activation based gradients to event-driven learning, there should only be one value du when the membrane potential reaches the threshold.
Tab. lists whether gradient type can be used in a learning fashion. It should be noticed that although activation-based gradient is more suitable for RNN-like learning, it is still able to be used for event-driven learning.
Table 1: Whether the gradient type can be used in a learning fashion
Event-driven learning RNN-like learning
Time-based gradient
Activation- based gradient
3

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3 Methods
3.1 Forward Formulas
We use the spike response model network can be described follows:
1 for neurons in the network. The forward propagation in the
9 (t) = f
wij $j 00-11()))
Σ
-c(t-
.ctt - T)dT,
(1)
(0)
" (1) 6( (1)
Here ui ())(() denotes the membrane potential of neuron i in layer at time 4, a denotes the weight between neuron j in layer 1 and neuron in layer j,last the time of last spike of neuron in
layer and 8 (t) represents the spike emitted from neuron at time t. The function 6(.) is the Dirac Delta function and 0 is the firing threshold. The spike response kernel e(t) can be described by
Tm
Ts
where Tm and Ta the membrane time constant and the synapse time constant respectively, Notice that we do not use reset kernels as in previous works [39 41. Instead, we eliminate the influence of input spikes prior to the last output spike on membrane potentials.
3.2 Rethinking the Classical Time-based Backward Propagation Formula
In this subsection, we analyze the classical time-based backpropagation formula in SNNs. We first theoretically prove that the backpropagation rule essentially assigns gradients of output spikes of neurons to their input spikes. Then we check the pooling layer and show that the average pooling should be adjusted in backpropagation to satisfy the gradient assignment mechanism, while the max pooling naturally satisfies this mechanism.
Invariant sum of gradients among layers with weights. The most commonly used time- based gradient backpropagation method origins from 32
The two key approximations are follows:
0t (s
dc(tx - tm)
Σ
w
(4)
0u(" (tx)
(1) du (tx)/dt
Otm
(e.luve (tm((s) (1
Oug (1x)
( Oe(1e bm)
(5)
0tm (s) (s
Otm
where (s) (1) denotes the firing time of neuron i in layer 1, (t.. is the firing time of the last
(t(()!))
on the timing of a spike, and (t is the influence of changing spike timing on that variable. spike emitted by neuron before time lg. d. means the influence of changing other variables
Combining Eqs. 45 and the forward formulas, we can get an invariant equality:
(t
Σ
Σ to te ye (al) (2--- (tn(()))) Ot (-1)) = 1.
(6)
The proof is provided in Appendix. Eq. implies the fact that the reference time (t 0) is meaningless, and only relative spike times matter. If increase all the spike times layer by 1 unit along the time axis, then all the spike times in layer / are also increased by unit along the time axis. Denote the loss function L, then the gradient of with respect to tm fm(s)
(1-1), is:
aL
aL
(t
Σ
Σ
8t (1m(()00111
Amls (1 (s! (t Stm.))))mttttt (1-1)) Ot
(1) (8
Ht (s) (111))

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Lagar!
Lager
Gradiens
Loss
Output
Imput
Figure 2: Left: Spikes (vertical ones) and their corresponding gradients (horizontal arrows) in layer I and layer Right: Input and output of the neuron in layer The input integrates spikes from connected neurons in layer | - (notice the input is not multiplied by weights). In backpropagation. the gradients are transmitted from the output of the neuron to the inputs. The key property that the gradient of an output spike exactly equals the sum of gradients of input spikes between the last output spike and it.
where nent (s) 1111) denotes the firing time of the next spike emitted by neuron j after time bm- Therefore, we actually decompose the gradient aL/Otu(s) from layer into (part of) set of gradients 0L/ōt, (s) (1m )) of the last layer 1 - 1, and keep their sum unchanged. In other words, we assign the weighted sum OL/Otz () by weights Ot, 0 (s} ()) to the gradients OL/Ot.. ()) in the last layer, as shown in Fig,
If sum all the gradients together,
get another invariant in this backpropagation rule:
OL
OL
EE
(5 111)))
AL never changes between layers under this rule. which means the sum of gradients E,E,,
(9)
Gradient invariance for pooling layers. The above equations determine the gradient propaga- tion in fully-connected and convolution layers (which contain neurons). The case for pooling layers (which do not contain neurons) illustrated in Fig.
Layer!
Layer
Layer!
Laye! --1
Lager
Layer -1
Spike
No Spike
Average pooling
Max pooling
Adjusted average pooling
Figure 3: Different pooling strategies. In average pooling (left), the gradients are propagated to neurons emitting spikes, which will cause gradient vanishing problem. Meanwhile, max pooling (middle) passes all gradients to one of the neurons emitting spikes in the backward stage, keeping the total gradient unchanged. The average pooling can be adjusted (right) pass gradients all neurons emitting spikes in the last layer to avoid the gradient vanishing problem.
In average pooling with kernel size k. the gradient of one spike (at time f) in layer is averagely propagated to k x & neurons in layer / connected to it. Some of these k x k neurons may not emit spikes time f. However, the gradients propagate the gradients to the previous layers. For instance, the two white squares (neurons) in layer 1-1 in Fig. 3 gradients, but they will not further propagate the gradients layer 7 Thus, part of the gradients is lost in the backpropagation of the average pooling layer, which might cause the gradient vanishing problem. Meanwhile, among layers in this case. We can adjust the backpropagation stage in average pooling to satisfy the gradient sum invariance requirement by increasing the multiplier from 1 1/h: to 1/Uspikes where Nepike the number of spikes emitted by the & x k neurons in layer / - 1 at the current time step.
also propagated to those neurons, which cannot further
sum of gradients carried by the spikes is not kept
5

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Output
tk
Ck
Input
Lm + At
Figure 4: A comparison of temporal gradient direction in backpropagation between two different cases. Consider moving one of its input spike moving from time im to fm + At for an SRM neuron. Without loss of generality, we assume the weight multiplied on this input spike is positive. If the next output spike is fired at time fR. moving the input spike from im to tm + At will decrease the membrane potential of this neuron at time tg. which causes this spike to fire later. As result, the gradient direction from the output spike at time & will be kept. In opposite, if the next output spike is fired at time k' moving the input spike from fm to Im At will cause this spike fire earlier, so the gradient direction from the output spike at time ig will be reversed.
On the contrary, in max pooling, the gradient of a spike in layer is entirely propagated to one of the spikes emitted by its connected neurons in layer 1 (shown in the middle of Fig. 3). This maintains the property of the invariable sum of gradients.
It should be noticed that, although the backpropagation stage of max pooling is different from adjusted average pooling in discrete simulation, they are almost surely the same in the continuous simulation since (almost surely) no two spikes emit at exactly the same time in this case.
3.3 Deficiencies of The Typical Time-based Gradient Propagation and A New Approach
In this section, we first point out the reverse gradient problem in event-driven learning: the gradient direction for spike timing gets wrong when the spike response kernel is decreasing. Then propose a backward kernel that can not only solve the reverse gradient problem but also keep the property of the invariable sum of gradients.
The reverse gradient problem. Fig. Iillustrates the membrane potential response of neuron (with index i) to one of its input spikes (with positive weight) from presynaptic neuron As in equation double-exponentail function. Notice that this is not whole
(3) the spike response kernel membrane potential of the neuron i as it also receives the inputs from other presynaptic neurons. We consider two spike times (tx and t()) of neuron and one spike time fm of presynaptic ncuron j. If the next spike of neuron fires at time tg, moving the presynaptic spike from to fm. At will decrease membrane potential at time tx, which means postponing the spike at time ty (fm t u[fx] + ty 1). This result shows that if the presynaptic neuron fires earlier, the postsynaptic neuron will fire carlier in this case.
Oppositely, if the next spike of neuron i fires at time ty moving the input spike from f... to t. At will cause a increase of u(('), which further moves 14 leftward (l.. t u((k] f l4 4). As a result, if we want to move the output spike 14 leftward, we should move the input spike at 4m rightward, which reverses the direction. This might cause a problem: When we want to move it leftward, want the neuron to emit more spikes. However, in gradient backpropagation, it moves t..

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rightward (assume weight wij > 0), which may cause the neuron in the last layer to spike fewer, further causing neurons in the current layer to spike fewer.
More formally, we assume the neuron receives a input spike at time bm from presynaptic neuron with synaptic weight Waj, then the membrane potential of neuron at time l is:
(t(+&) Wsj
<(4x - (m) + C,
where c(t) denotes the spike response kernel (Eq. B). C' denotes the influence other spikes, which is not in our concern here. In backward pass, according to Eqs. 44.55) we have
aL
(s.) (tx) ÖL Ot, (s:) (tx) ms))) (s;)s;)
ÖL
de(ty -m)) tm)
Hn (s3)
dtx(t))/dt
wij
Dtm
(10)
Note that when spike is emitted by neuron i at time lg, the slope of ui(() > 0 at time la which means
dug(ts)/dt has negative sign. Considering de(tz-Im
de(r) dr where tx fmm we get:
8tm
aL.
sign (wsy) sign (dc(T) OL = sign (tt()s)) )
sign
(11)
it (s,)
dr
When sign (de(T)/dr)
1, which is the part of the spike response kernel that decreases (see the case at i in Fig. 4). the gradient direction of mmm. can be classified into two cases: When Wiy >
sign
01
sign
B1.
which means the gradient direction reversed. When wij
3l mm
[s.)
sign
aL Bt. (tm))5))
sign
which means the gradient direction is kept.
(A:)
In both cases, the sign of the gradient gets wrong in propagation between layers. Thus, the commonly used double-exponential spike response kernel is incompatible with the time-based gradient in
event-driven learning.
A smoother gradient assigning approach. Inspired by the above gradient inconsistency well as the invariance of gradient sum, we propose new gradient backpropagation approach here. Specifically, we replace the function @e(t,-[m) in Eqs. 4 and (5) with a new function h(tx tm).
8t.m
Therefore, the backpropagation formula between layers turns into:
0la(s;) @u:(ta) (8)));) Ou((k:) (s))
(12)
{
0, if km(8;) I (s.) or Im (s() > (x(s)), -1
Way litt. - km)) Wij h(lx - lm), otherwise.
(sy) will not change we multiply h(t) by an arbitrary constant, so we do not need to care about the scale of h(t). Meanwhile, the property of invariable sum of gradients is kept after this replacement.
be see from Eq. 12 that (ss)
To guarantee that the gradients are not reversed between layers, we should expect h(t) > 0 always
hold when > 0. Therefore, we choose h(t) = e "grad to simplify the calculation, where Tyred is a tunable parameter. Notice that the function h(t) is only used in backward propagation, which means the spike response kernel in the forward propagation is not necessarily the integral of h(t).
3.4 Overall Learning Rule
The loss function we use in this work is the counting loss function, which has the form L
(+ (Ntargget So (()d)))) where Nout is the number of output neurons and equals N EE
Nout
the number of classes, (t) represents the spike train emitted by neuron i. Besides, N N.farget the target of the spike number outputted by neuron and typically we set Natgget larger when i the correct answer.
7

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During the learning process, the gradient is first propagated from the loss function to the firing time of each spike from the last layer the first layer. The formula for this stage (please refer to Appendix for the detailed deduction):
0L
aL
Σ
(1) wij h(t tm)
Ōt 119-111
Ut. nem (5
coo seemlo ")
wii h h(tk,nezt (mm);
(13)
where nems is the firing time of the first spike emitted by neuron i after time 4m-
After this, the gradient to weights in each layer is calculated by summing up the multiplication of the gradients of spike firing times the same layer and the derivative of weights with respect to spike firing times. The learning rule for this stage is
ÖL Ow 13
OL
Σ
e(tu, - tm).
(14)
tmial 1y Hen (s
du, ( (fu neas))
4 Experiments
In this section, we validate the effectiveness of our method on MNIST 48 Fashion-MNIST [49].
N-MNIST [50], CIFARIO [51], and CIFARION 100 [51] datasets. This section is organized as follows: We first introduce the training details, then evaluate the performance of our algorithm and compare with the state-of-the- event-driven learning approaches. At last, we conduct ablation studies to illustrate the effectiveness of our proposed modules. More details of the configurations can be found in the Appendix.
4.1 Training Details
Initialization: When training in an event-driven fashion, gradient information is only carried by spikes. Therefore, the gradient information will be completely blocked by layer when there no spikes in that layer. To solve this problem, we start with layers of arbitrarily initialized weights and scale them by certain multiples, which can make the average firing rate be certain number for each layer. We obtain these multiple parameters by binary search and this strategy works well in practice.
Supervisory signals: Another problem we face is that output neurons corresponding to certain classes do not fire anymore after certain epochs of training. This problems makes corresponding gradients difficult to propagate in the network, further leading to these neurons no longer fire afterwards, resulting recognizing those classes correctly impossible in the following epochs. To address this problem, we utilize supervisory signals. For each neuron in the output layer corresponding to the ground-truth label, we force it to fire the end of the simulation.
Experiment settings: In our experiments, we use the real-valued spike current representing the pixel intensities of the image inputs. We list the network architecture each work uses and the accuracy they achieves on each dataset in table Notice that the output layer is, by default, a fully-connected layer containing the same number of neurons as the number of classes in the dataset, and omitted from the architecture representation. We run all experiments on a single Nvidia A100 GPU.
4.2 Comparison with the State-of-the-Art
Tab. 2 reports the accuracies of the proposed method and other comparing methods. The performance of our algorithm is lower than TSSL-BP by 0.06% on the MNIST dataset and 0.01% on the N-MNIST dataset. However, the output of their network is real-valued postsynaptic currents while the output of our network binary spikes. In addition, they use RNN-like gradients to assist learning. On the remaining datasets, we have achieved state-of -fate-of-the-art performance among these works with temporal gradients. For the Fashion-MNIST dataset, our algorithm performs 0.45% higher than the previous SOTA. On the CIFARIO dataset, we achieve 92.45% accuracy with 14-layer REW-Resnet and

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Table 2: Performance comparison
Dataset
Model
Gradient Type
Architecture
Accuracy
Mostafa
Temporal
784-800
97.5%
Cosma
34
Temporal
784-340
97.9%
S4NN 35
Temporal
784-600
97.4%
MNIST
BS4NN 52
Temporal
784-600
97.0%
STiDi-BI
36
Temporal
784-500
97.4%
TSSL-BP
39
Temporal
15C5-P2-400C5-P2-300
99.53%
STDBP
Temporal
I6CSS-P-332C5-P28800 128
99.4%
Ours
Temporal
15C5-P2-40C5-P22-2
99.47%
S4NN 33
Temporal
1000
88.0%
BS4NN 52
Temporal
784-1000
87.3%
Fashion-MNIST
STDBP 38
Temporal
90.1%
TSSL-BP
Temporal
DDC5-P2-64C5-P2
92.83%
Ours
Temporal
1155-P2-64C5-PPP
93.28%
TSSL-BP
39
Temporal
12C5-P2-64CC P2-64C5-P2
99.40%
N-MNIST
Ours
Temporal
12C5-P2-64CS-P2
99.39%
TSSL-BP BP (39
Temporal
CIFARNet
91.41%
CIFARIO
Ours
Temporal
VGGII
92.10%
Ours
Temporal
SEW-Resnet14
92.45%
CIFARI00
Ours
Temporal
VGG11
63.97%
15C5: convolution layer with 15 channels of 5 filters. P2: pooling layer with 2 filters. b CIFARNet: 12BC2266922311022222000555552112 VGG11: 1273-128C3--2 2123:1/8888P-22200---2-222222233
10% with VGGI1, which are all better than the current SOTA, 91.41%. For the CIFAR100 dataset, we are the first work successfully train SNNs with time-based gradients an event-driven fashion. We have achieved a performance of 63.97% on this dataset.
4.3 Ablation Studies
To show the effect of our proposed modules, we conduct ablation experiments on the CIFARIO
dataset. Specifically, two proposed components are taken into consideration: (1) As mentioned in
Section 3.3 we compare
proposed gradient assignment functions h(t) = e e in T2) with the
commonly used one hx(t) = dr(t) (2) We compare the results of three different types of pooling
layers (average pooling, adjusted average pooling and max pooling) mentioned Section 3.2 We have tried all combinations of gradient assignment functions and pooling layers. The test accuracy of these different settings is shown Tab.
Table 3: Test accuracy comparison between different settings on CIFAR10
Average pooling
Adjusted average pooling
Max pooling
de(t) 89.58%
91.52%
91.64%
dt
e 31
89.80%
91.88%
91.79%
The results in Tab. meets expectation. For the pooling layer, max pooling and adjusted average pooling have much better performance than the average pooling. This accords with the conclusion in Section 3.2 that pooling layers keeping the property of invariant sum of gradients better than those that do not.
The proposed gradient assignment function lc(() e -8t is also better than the commonly used one de(t) for all three types of pooling layers. In addition, as shown in Fig. h(t) -Bt
h(t)) de
converges faster than h(t)) dr(t) in early stage.

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0.9
0.8
accuracy
0.6
0.5
exponential gradient adjusted avg pooling
0.4
exponential gradient avg pooling
0.3
exponential gradient max pooling
original gradient adjusted avg pooling
original gradicut avg pooling
original gradient max pooling
0.1
100
150
200
250
300
350
400
epoch
Figure 5: Test accuracy curve comparison between different settings on CIFARIO (exponential gradient denotes h(t) = - and original gradient denotes h(t)
dr(t) db
Conclusion and Discussion
In this work, we analyze the commonly-used SNN temporal backpropagation training approach and find that follows the gradient assignment rule. We also find the average pooling layer does obey this rule while the max pooling layer does. We show that the direction of the temporal gradient will be reversed when the spike kernel is decreasing and avoid with increasing kernel in backpropagation. Our algorithm achieves state-of-the-ar performance on CIFAR 10 among time-based SNN learning approaches and successfully learns the parameters of SNN on CIFAR100 for the first time.
Compared with RNN-like methods, the proposed event- learning algorithm has a lower compu- tational cost and memory occupation when there are many time steps. Besides, algorithm also does not need bias between layers. Meanwhile, gradient propagation between spikes instead of time steps can mitigate the gradient explosion/vanishing problem along the time axis.
However, there still gap between event-driven backpropagation and biological plausible learning, since event-driven backpropagation processes the spike train in reverse time, which conflicts with the online learning in the real world and desires for future research.
6 Aeknowledgements
This work supported by the National Natural Science Foundation of China Grants 62176003 and 62088102.
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14


Reviews:
[{"review_id": "tLWbrSLRy5D", "reviewer": "Reviewer_1S4Q", "summary": "The authors propose a temporal Backprop approach for training SNNs with some interesting backward kernel fucntion.", "questions": "If the authors can highlight their technical novelty a compared to previous works, it will help me re-assess the paper's contributions. See above comments for reference.", "limitations": "See weakness section.", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "+ The authors showcase that their temporal BP methodology yields high accuracy as compared to similar related works.\n-The paper presents a direct training method using BP for SNNs. This work is very derivative and incremental. There is a lot of work from Priya Panda's group at Yale, Emre Neftci's group, and many others with regard to SNN training. The authors have failed to acknowledge most recent works and the method they are proposing is very incremental in the context of those works. Further, many recent works on SNNs have targeted larger datatsets including video segmenattion with direct training. I wonder if the author's method can even scale up, since their results are limited to CIFAR10, CIFAR100.\n\n-The authors did not comment on how many time steps their method requires to train. In the recent work [5], the authors show that they can use backward connections to train SNNs better, is there a similarity between the backward kernel and backward conenction?\n\n\nBelow is a list of publications (not exhaustive) that the author should check:\n[1] \n[2] Enabling spike-based backpropagation for training deep neural network architectures C Lee, SS Sarwar, P Panda, G Srinivasan, K Roy Frontiers in neuroscience, 119\n[3] Rate Coding Or Direct Coding: Which One Is Better For Accurate, Robust, And Energy-Efficient Spiking Neural Networks? Y Kim, H Park, A Moitra, A Bhattacharjee, Y Venkatesha, P Panda ICASSP 2022-2022\n[4] Neuromorphic Data Augmentation for Training Spiking Neural Networks Y Li, Y Kim, H Park, T Geller, P Panda arXiv preprint arXiv:2203.06145\n[5] Neural architecture search for spiking neural networks Y Kim, Y Li, H Park, Y Venkatesha, P Panda arXiv preprint arXiv:2201.10355\n[6] Optimizing deeper spiking neural networks for dynamic vision sensing Y Kim, P Panda Neural Networks 144, 686-698\n[7] Federated Learning with Spiking Neural Networks Y Venkatesha, Y Kim, L Tassiulas, P Pand IEEE Transactions on Signal Processing 2021\n[8] Beyond classification: directly training spiking neural networks for semantic segmentation Y Kim, J Chough, P Panda arXiv preprint arXiv:2110.07742\n[9] Visual explanations from spiking neural networks using interspike intervals Y Kim, P Panda Scientific Reports 11, Article number: 19037 (2021)\n[10] Revisiting batch normalization for training low-latency deep spiking neural networks from scratch Y Kim, P Panda Frontiers in neuroscience, 1638", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "dEKk844QTj", "reviewer": "Reviewer_neur", "summary": "The authors propose a modified event-driven backpropagation and investigate its performance on benchmarks. The authors also investigate if the backpropagation followed a gradient assignment rule, finding that max-pooling obeyed this rule. This is one of the most important conclusions of the paper. The algorithm achieved SOTA on CIFAR-10, and was the first to be trained on CIFAR-100.", "questions": "In line 233 you stated that more details about the configuration of training were available on appendix, I was not able to fin the appendix section, nor a appendix file on the supplement material, even though the code contains the configurations of the experiment. It is not clear for me if you want to provide more information in an appendix section or you were referring to the code provided.", "limitations": "Spiking Neural Networks has yet achieved SOTA in no benchmark.\nAlthough spiking neural networks are believed to have greater biological plausibility, it is not clear whether biological neural networks learn through backpropagation, which was the method tested in this study. Despite this, nowadays there is no alternative that works better than backpropagation.\nThe authors also stated that another gap in biological plausibility is the reverse time processing feature.", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Spiking neural networks represent a new paradigm of neural networks that, among other advantages, incorporates time into the building blocks of its own functioning. Thus, in addition to having greater biological plausibility, it is also believed to be more coherent with learning in the real-world, which contains the time dimension.\nEvent-base learning paradigm preserves de biological plausibility and the event-based features advantages of spiking neural networks when compared with surogate backpropagation.\nHowever, this new paradigm has not yet reached SOTA in any benchmark, not even in those that demand incorporation of the time dimension.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "NwpburIDs8q", "reviewer": "Reviewer_tMmb", "summary": "The focus of this paper is training of spiking neural networks. Specifically, the paper proposes back-propagation with respect to spike timing. They analyze previous methods and propose a small increment over the current state-of-the-art. The results are not clearly discussed. ", "questions": "Please refer to Strengths and Weakness for the points. ", "limitations": "There are no potential negative societal impacts. One major limitation of this work is applicability to neuromorphic hardware and how will the work shown on GPU translate to neuromorphic cores.", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The work focuses on an aspect of the learning algorithms that requires optimization and innovation. \n\nWeakness\n1.\tIt is hard to understand what the axes are for Figure 1. \n2.\tIt is unclear what the major contributions of the paper are. Analyzing previous work does not constitute as a contribution. \n3.\tIt is unclear how the proposed method enables better results. For instance, Table 1 reports similar accuracies for this work compared to the previous ones.\n4.\tThe authors talk about advantages over the previous work in terms of efficiency however the paper does not report any metric that shows it is more efficient to train with this proposed method. \n5.\tDoes the proposed method converge faster compared to previous algorithms?\n6.\tHow does the proposed methods compare against surrogate gradient techniques?\n7.\tThe paper does not discuss how the datasets are converted to spike domain. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "AFsSs-sHe-", "reviewer": "Reviewer_kF1o", "summary": "This paper focuses on the temporal, event-driven manners of training a spiking neural network from scratch. The authors first revisit the learning dynamics of event-driven learning and discover several invariance properties. Then, a problem called reverse gradient is raised and addressed. Extensive experiments are conducted to verify the effectiveness of this method. ", "questions": "TBH, I am not an expert in event-based training of SNN, therefore I cannot give useful feedback with respect to that. I have a few questions about the difference between event-based training and the \"RNN-like\" training. \n\n1. When implementing these event-based learning algorithms on the neuromorphic hardware, how to accelerate the training and why is it significantly faster than the RNN-based learning algorithms?\n\n2. Can this method be trained with residual connection?\n", "limitations": "Overall, I found this paper interesting and could be valuable for publication at the NeurIPS conference. As I am familiar with this type of training method for SNNs, I could not give conceptual limitations for this paper. I am giving borderline acceptance of this paper due to its good presentation. Meanwhile, I will set my confidence score to 2 and will look into the comments from other reviewers to finalize my rating. \n\n\n-----\n\nPOST-REBUTTAL REVIEW:\n\nI'd like to thank the authors for their detailed response. My questions are addressed, thus I increase my rating to 7.\n", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. This paper is well-presented. The writing and visualization are neat and easy to follow.\n\n2. The experimental results are sufficient for comparison with existing event-driven learning works.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept