Training Feedback Spiking Neural Networks by Implicit Differentiation on the Equilibrium State
Authors: Mingqing Xiao
Qingyan Meng
Zongpeng Zhang
Yisen Wang
Zhouchen Lin
Venue: NeurIPS 2021 Spotlight (2021)
Keywords: spiking neural networks
feedback connections
equilibrium state
training method

Abstract: Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks, and use surrogate derivatives or compute gradients with respect to the spiking time to deal with the problem. These approaches either accumulate approximation errors or only propagate information limitedly through existing spikes, and usually require information propagation along time steps with large memory costs and biological implausibility. In this work, we consider feedback spiking neural networks, which are more brain-like, and propose a novel training method that does not rely on the exact reverse of the forward computation. First, we show that the average firing rates of SNNs with feedback connections would gradually evolve to an equilibrium state along time, which follows a fixed-point equation. Then by viewing the forward computation of feedback SNNs as a black-box solver for this equation, and leveraging the implicit differentiation on the equation, we can compute the gradient for parameters without considering the exact forward procedure. In this way, the forward and backward procedures are decoupled and therefore the problem of non-differentiable spiking functions is avoided. We also briefly discuss the biological plausibility of implicit differentiation, which only requires computing another equilibrium. Extensive experiments on MNIST, Fashion-MNIST, N-MNIST, CIFAR-10, and CIFAR-100 demonstrate the superior performance of our method for feedback models with fewer neurons and parameters in a small number of time steps. Our code is available at https://github.com/pkuxmq/IDE-FSNN.

Training Feedback Spiking Neural Networks by Implicit Differentiation on the Equilibrium State
Mingqing Xiao', Qingyan Meng Zongpeng Zhang', Yisen Wang Zhouchen Lin L5n
Key Laboratory of Machine Perception (MoE), School of AI, Peking University
Thh Chinese University of Hong Kong. Shenzhen
$Shenzhen Research Institute of Big Data
*Center for Data Science, Academy for Advanced Interdisciplinary Studies, Peking University Institute for Artificial Intelligence, Peking University
Pazhou Lab, Guangzhou 510330, China
xiingiing_xiao, yisen wisen.wang, zl #linu edu. cn, qingyanmeng@lind ink. cuhk. edu zongpeng zhang988gmail... com
Abstract
Spiking neural networks (SNNs) are brain-inspired models that enable energy- efficient implementation on ncuromorphic hardware. However, the supervised training of SNNs remains hard problem due to the discontinuity of the spiking neuron model. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks, and surrogate derivatives or compute gradients with respect to the spiking time to deal with the problem. These approaches either accumulate approximation gate information limitedly through existing spikes, and usually require information propagation along time steps with large memory costs and biological implausibility. In this work, we consider feedback spiking neural networks, which are more brain- like, and propose novel training method that does not rely on the exact reverse of the forward computation. First, we show that the average firing rates of SNNs with feedback connections would gradually evolve to an equilibrium state along time, which follows fixed-point equation. Then by viewing the forward computation of feedback SNNs as black-box solver for this equation, and leveraging the implicit differentiation on the equation, can compute the gradient for parameters without considering the exact forward procedure. In this way, the forward and backward procedures are decoupled and therefore the problem of non-differentiable spiking functions is avoided. We also briefly discuss the biological plausibility of implicit differentiation, which only requires computing another equilibrium. Extensive experiments on MNIST, Fashion-MNIST. N-MNIST, CIFAR- 10, and CIFAR-100 demonstrate the superior performance of our method for feedback models with fewer neurons and parameters in a small number of time steps. Our code is available at /tttps::/gtthiii com/ hwmpppuuqq///// IDE-FSNN
only propa-
1 Introduction
Spiking neural networks (SNNs) have gained increasing attention recently due to their inherent energy- efficient computation 32 41 33 8). Inspired by the neurons in the human brain, biologically plausible SNNs transmit spikes between ncurons, enabling event- event-based computation which can be carried out neuromorphic chips with less energy consumption computationally more powerful than artificial neural networks (ANNs) theoretically and are therefore regarded as the third generation of neural network models 24
7 30 33 Meanwhile, SNNs
"Conesponding author.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

---

Despite the advantages, directly supervised training of SNNs remains hard problem, which hampers the real applications of SNNs compared with popular ANNs. The main obstacle lies in the complex spiking neuron model. While backpropagation 35 works well for ANNs, it suffers from the discontinuity of spike generation which is non-differentiable in SNN training. Most recent successful SNN training methods still imitate the backpropagation through time (BPTT) (9) framework by error propagation through the computational graph unfolded along time steps, and they deal with the spiking function by applying surrogate derivatives to approximate the gradients [4] S or by computing the gradients with respect to the spiking time only on the spiking neurons 16 46
|4
38 42
27 16 47
However, these methods either accumulate approximation error along time steps, or suffer from the "dead neuron" problem (38), i... learning would not occur when no neuron spikes. At the same time, BPTT requires memorizing intermediate variables at all time steps and backpropagating along them, which is memory-costing and biologically implausible. So it is necessary to consider training methods other than hackpropagation along computational graphs that fit SNNs better.
On the other hand, most recent SNN models simply imitate the feedforward architectures of ANNS 41 46 47). which ignores the ubiquitous feedback connections in the human brain. Feedback (recurrent) circuits are critical to human's 's vision system for object recognition [15]. Meanwhile, [8] shows that shallow ANNs with recurrence achieve higher functional fidelity of human brains and similarly high performance on large-scale scale vision recognition tasks, compared with deep ANNs. So incorporating feedback connections enables neural networks to be shallower, more efficient, and more brain-like. As for SNNs, feedback was popular in early models like Liquid State Machine (25). which leverages recurrent reservoir layer with weights fixed or trained by unsupervised methods. And compared with the uneconomical cost for ANNS to incorporate feedback connections by unfolding along time, SNNs naturally compute with multiple time steps, which inherently supports feedback connections. Most recent SNN models imitate feedforward architectures because they were once lacking effective training methods and thus they borrow everything from successful ANNs. We focus on another direction, i.c. feedback SNN, which is a natural choice for visual tasks as well.
38 42
In this work, we consider the training of feedback spiking neural networks (FSNN), and propose novel method based on the Implicit Differentiation on the Equilibrium state (IDE). Inspired by recent advances in implicit models 3 4). which treat weight- tied ANNs as solving fixed-poing equilibrium equation and propose alternative implicit models defined by the equation, we derive that when the average inputs converge to equilibrium, the average firing rates of FSNNs would gradually evolve to an equilibrium state along time, which follows a fixed-point equation as well. Then we view the forward computation of FSNN as black-box solver for the fixed-point equation, and borrow the idea of implicit differentiation from implicit models 3 4 to calculate the gradients, which only relies on the equation rather than the exact forward procedure. In this way, gradient calculation is agnostic to the spiking function in SNN, thus avoiding the common difficulties in SNN training. While implicit differentiation may seem too abstract to be computed in the brain, we briefly discuss the biological plausibility and show that it only requires computing another equilibrium along the inverse connections of neurons. Besides, we incorporate the multi-layer structure into the feedback model for better representation ability. Our contributions include:
1. We are the first to theoretically derive the equilibrium states with a fixed-point equation for the average firing rates of FSNNs with the (leaky) integrate and fire model under both continuous and discrete views. According to this, the forward computation of FSNNs can be interpreted as solving a fixed-point equation.
2. We propose a novel training method for FSNNs based on the implicit differentiation on the equilibrium state, which is decoupled from the forward computational graph and avoids SNN training problems, e.g. non-differnntiability and large memory costs. We also discuss the biological plausibility and demonstrate the connection to the Hebbian learning rule.
3. We conduct extensive experiments on MNIST, MASHi N-MNIST, CIFAR-10 10, and CIFAR-100, which demonstrate the superior results of our methods with fewer neurons and parameters in a small number of time steps for both static images and neuromorphic inputs. Especially, our directly trained model can outperform the state-of-the-ar SNN performance on the complex CIFAR-100 dataset with only 30 time steps.
2 Related Work
Training Methods for Spiking Neural Networks. Early works apply biologically inspired method, spike-time dependent plasticity (STDP) 9 to formulate bottom-up unsupervised learning rule, or

---

with limited supervision. Since the rise of successful ANNs, error backpropagation and gradient descent have inspired many methods. One direction is to convert trained ANN to SNN extremely large simulation time steps. The other methods are to directly calculate the gradient and
choose
STDP [22]
[3
34
37
32
44
However, they suffer from conversion errors and
train SNNs. These methods either compute the gradient with respect to spiking times [6, 46). or
leverage a surrogate derivative for discontinuous spiking functions [21 41
5
[4] 38
27 47).
or combine them [16.. However, they suffer from the "dead neuron" problem 388 or accumulated approximation error, and typically require backpropagation along the computational graph to be unfolded by many time steps, which is meeorry-consmmee and biologically implausible. As for SNN with feedback connection, [45] proposes the ST-RSBP method, which backpropagates errors at the spike-train level. They simply approximate the activation function of neuron as linear operation, and require long time steps for satisfactory results. In this work, we propose a new training method that does not rely on the exact reverse of the forward computation, which avoids problems of non-differenniability and large memory costs, and only requires short time steps for superior performance. There are also works trying methods other than BP along computational graphs to train SNNs, such as equilibrium propagation [29]. However, [29] defines a complex computation form rather than common SNN models, and can hardly achieve satisfactory results on the simple MNIST dataset. Instead, our work is based on SNN models applicable on neuromorphic hardware and demonstrates promising results on more complex datasets.
Equilibrium of Neural Networks. The study on the equilibrium of neural networks originates from energy-based based models, e.g. Hopfield Network [2 They view the dynamics iterative procedures of feedhack (recurrent) neural networks as minimizing an energy function, which will converge a minimum of the energy. Based on the energy, several training methods are proposed, including recurrent backpropagation [2] 31 and more recent equilibrium propagation (EP) (36). They heavily rely on the energy function and can hardly achieve competitive results as deep neural networks do. Deep equilibrium models 3 4) on the other hand, are recently proposed models which treat weight-tied deep ANNs as solving fixed-point equilibrium point, and propose alternative implicit models defined by the fixed-point equations rather than energy functions. They express the entire deep network an equilibrium computation and solve and train it by root-finding methods and implicit differentiation, respectively, which achieves superior results. Most of these works are based on ANNs, except that [29] generalizes the EP method to neurons with binary communications. They define a complex neuron computation form and follow the methodology of energy-based EP method to approximate the gradients. Several recent works also study the equilibrium of SNNs (23 26]. They consider equilibrium from the perspective of sol solving a constrained optimization problem, but either do not propose to guide the training based on it or are limited in single-layer structure and simple problems. Differently, we are the first to derive the equilibrium state with a fixed-point equation for average firing rates of FSNNs with common SNN models, and propose to train SNNs by exact gradients through implicit differentiation, which is also scalable to multi-layer structure and deep learning problems.
Preliminaries
3.1 Spiking Neural Network Models
Spiking neurons, different from real-valued ANNs, communicate with each other by binary spike trains along time. Each neuron maintains a membrane potential, which integrates input spike trains, and the neuron would generate an output spike once the membrane potential exceeds threshold. The commonly used integrate and fire (IF) model and leaky integrate and fire (LIF) model describe the dynamics of the membrane potential as:
du R I(1),
IF:
u < Vin
dt
I' du
LIF:
(u Urest) + R I(t),
u < Vin.
dt
where u is the membrane potential, / the input current, VnA is the spiking threshold, and It and Ti are resistance and time constant, respectively. Once reaches VnA at time L/, spike is generated and u is reset to the resting potential u = Wrest, which usually taken as 0. The spike train expressed by the Dirac delta function: s(t) Ens 6(t - (ft-))). We consider the simple current model
3

---

[tt) = Ej wiys((t) - b, where Waj is the weight from neuron j to neuron i, which is the same as ANN. After discretization, the general computation form for the IF and LIF model is described as:
Uq [t + 0.5] = Aua + Σ Wey w
(2)
s; + - H [1 [ + 0.5] Ven),
u, un + Vu 1],
where (x) is the Heaviside step function, i.e. the non-differentiable spiking function, s; [t] the binary spike train of neuron i, and À is 1 for the IF model while À < 1 is leaky term related to the constant Tm and discretization time interval for the LIF model. The constant R, Tm, and time step size are absorbed into the weights Wij and bias b. We use subtraction as the reset operation.
3.2 Implicit Differentiation on the Fixed-Point Equation
We consider a fixed-point equation = (a) parameterized by 8. Let L(a*) denote the objective function with respect to the equilibrium state a", and let go(a) = fo(a) a. The implicit differen-
tiation on the equation satisfies (r @f.(a") 0a" dØ de
Ofe(a") ) 3 Therefore, the differentiation of
C(a") with respect to parameters can be calculated based on implicit differentiation:
OC(a')
al(a*)
0a (J ln-) afo(a")
00
00
where Jm is the inverse Jacobian of evaluated at a". For the proof please refer to [ To solve the inverse Jacobian, follow 3 and solve alternative linear system (J a aL(aa) 0a
0. We can leverage Broyden's method which is second-order quasi- Newton
approach: or we alternatively use fixed-point update scheme x = (Jf xx+ @C(a") since
I = J ln I, and it converges with linear convergence rate long || ln- 1. In this way, gradients for the parameters can be calculated only with the equilibrium state and equation.
4 Proposed IDE Method
In this section, we first derive the equilibrium state of FSNNs under both continuous and discrete views, and demonstrate that FSNNs can be treated solving fixed-point equation. Then introduce how to train the network by the proposed IDE method based on the equation and briefly discuss the biological plausibility. Finally, incorporate the multi-layer structure into the model for more non-linearity and stronger representation ability.
4.1 Derivation of Equilibrium States for Feedback Spiking Neural Networks
4.1.1
Continuous View
We first consider group of spiking neurons with feedback connections. Let u(t) and s(t) denote the membrane potentials and spikes these neurons at time respectively, x(t) denote the inputs, W denote the feedback weight matrix, F denote the weight matrix from inputs these neurons, and b denote constant bias. Under the IF model, the dynamics of membrane potentials are expressed as:
du
Ws(t Ata) + Fx(() + Vu,(())
dt
where Ata is a time delay of feedback connections. and Va, is the threshold. Note that W and F represent linear operations including both fully-connected and convolutional layers. Define the
average firing rates as a(t) So s(r)dr, and the average inputs x(t) 1 So x(r)dr. Then through integration, we have:
1 (t Ats Walt Ata) + FX(t) + at)) = Veh
u(t) 1
(5)

---

5 roughly follows fixed-point update scheme except the existence of u(t). Now we dive into ut).. Since neurons will not spike when the accumulated u(t) is negative, if v;(t) (e Wa(t Atd) + Fx(() + b); < O, ideally neuron generates no spike and its accumu- lated negative term remains in (t). So (t) can be divided as (t( = u, (t) + u (t), where Juz (t) = (i)(v.(t), 0) is the remaining negative term, and u, (t) is the positive one typically bounded in the range between 0 and Va. There could be random error in u, (4) in the context of random arrival spikes rather than the average condition (e.g- the average is O, but large positive input followed by negative one will generate unexpected spike). Despite this, we can still assume u (t) to be bounded by a constant when average inputs converge. By this decomposition, we have the equation with the element- wiis ReLU function (ReLU(x) = max( 0)) and bounded u u+((): (t):
1 V
Atd Wa(1 Ala) + FX (b))))))
1 (t) VAA L
a( ReLU
(6)
With Eq. we can derive that the average firing rate will gradually converge to an equilibrium state. Theorem 1. If the average inputs converge to an equilibrium point x(t) + x", and there exists constant and < such that u; (t) < c, Vi, t and [|W]l2 S <Ven, then the average firing rates of FSNN with continuous IF model in Eq. 66) will converge to an equilibrium point a(t) + a", which
satisfies the fixed-point equation a" Rel.U
(Wa" + Fx' + +x)+
Veh
The proof can be found in Appendix C. Theorem ! rigorously shows the equilibrium state under the IF model, and we can view the forward computation of FSNN solving this fixed-point equation.
As for the LIF model, we can similarly define the weighted average firing rate a(t) = Lú ((t-r)(((())))) ())
Jo <(t <(t-r)dr
and the weighted average inputs (t) fo <(t-r)x(r)dr
Jg <(t-r)dr where k<()) cxp(- Tu. is the response
kernel of the LIF model. this setting, however, there could be random errors caused by ut (t) as its denominator fdr k(t - dd does not go to infinity. We consider it as an approximate solver for the Please refer to Appendix E for details.
equilibrium with random errors, as shown in Proposition [
Proposition If the weighted average inputs converge to an equilibrium point x(t) + x", and there exists constant e and y <. 1 such that |u (t)l S c, Vi, t and W|]2 S VVu, then the weighted average firing rates â(t) of FSNN with continuous LIF model gradually approximate an equilibrium
point a" with bounded random errors, which satisfies a" RRLU (Von (Wa" + Fxx + b)).
Vei
4.1.2 Discrete View
In practice, we will simulate SNNs with discretization. Now consider the computation in (2). With feedback connections, the update equation of membrane potentials under the IF model is:
u(t + 1] = ui4 + Ws[[] + Fx[[] + b VeAs(L + 1],
where we treat the feedback delay in one time step for simplicity. Define the average firing rates
as a[t] & - s(r], the average inputs as x[[] न E' x(7]. and u(0] O, s00] 0. By summation, we have:
1 ( Wall +x(] +b a [1+11= VsA 1+1
ult u + 11] t
(8)
Different from the continuous view, a[t] is bounded in the range of [0, 1]. since there could be at most
t spikes during f time steps. Therefore, u, [f] will maintain both the negative terms and the exceeded
positive ones. Similarly, u; [1] = u, [1] + u, [4], where fu, [4] = min (max (v+(1) - Vin, 0), oz [4] the exceeded term, and u [t) is assumed to be bounded by a constant as previously indicated. Then:
1, x >
1 t
( Wald + FX +))
u u [++++]
a[t+1] = Ven [++1
where (x)) = v, 0 x $1.
Vin
t 6+1
0, x<0
With Eq.19) we can derive the equilibrium state under discrete view.
5

---

Theorem 2. If the average inputs converge to an equilibrium point X - x", and there exists constant e and < such that |u, [t]l < c, Vi, t and [| S VVi, then the average firing rates of FSNN with discrete IF model Eq. 9 will converge an equilibrium point alt] - a", which satisfies the fixed-poin equation a
(Wa'
Fx* +b))
ViA
The proof can be found in Appendix C. And for the LIF model, define the weighted average firing rate 1x1ir then can similarly consider as an approximation solver for the equilibrium state with random errors, as shown in Proposition
alt] Σ [x(]] and the weighted average inputs X|t] Σ Ereg A --== X
2 Please refer to Appendix E for details.
Proposition If the weighted average inputs converge to an equilibrium point 8(4] + x", and there exists constant and y such that | S c, Vi,t and [ S yVu,, then the weighted average firing rates a[t) of FSNN with discrete LIF model gradually approximate equilibrium point a" with bounded random errors, which satisfies a" = Vin (Wa" + Fxx + +b))
4.2 Training of Feedback Spiking Neural Networks
Based on the derivation in Section 4.1 we can view the forward computation of FSNNs as black-box solver for the fixed-point equilibrium equations, with some errors caused by finite time steps or the LIF model. Then we assume the (weighted) average firing rates a [T'] after I time steps approximately follow the equations. We will demonstrate how to train FSNNs and its biological plausibility.
4.2.1 Loss and Gradient Computation
Suppose that we simulate the SNN by I "ime steps. Let a[Z"] denote the final (weighted) average firing rates. We configure a readout layer after these spiking neurons, which performs as a fully-connected classification layer, with the number of outputs class numbers. We assume that these neurons will not spike or reset, and do classification based on the accumulated membrane potential. Then the outputs are equivalent as linear transformation on a(7). i.e. Wo [T]. The loss L is defined on and labels y by commonly used loss functions ((o. y). and we leverage the cross-entropy loss.
Let fo denote the function in fixed-poing equation, e.g. fo
V (Wa" Fxx + +b).
The gradient for parameters can be calculated based on the implicit differentiation as described in Section 3.2 by substituting a* with a[T]. Then parameters be optimized based common gradient descent methods, e.g. SGD 35 and its variants. A pseudocode is presented in Appendix B.
4.2.2 Biological Plausibility of Implicit Differentiation
While implicit differentiation may seem too abstract for information propagation compared with back-
propagation, we will briefly discuss the biological possibility for this calculation and its connection to the Hebbian learning rule [10]. Consider the equilibrium state a" following a" = fe(a", x')
ReLU Via
(Wa' + Fx + bi). Let m fa x*) = H
Ven
(Wa" + Fx + +b)
=
1, x > 0
H ((a'), M (m), w MW, where (xx)
As indicated in Section 3.2
10 x 5 0'
the gradient can be calculated as voL (8))T=
[fa(n"
00
Vin
W
0C 0a
T
and
we can leverage fixed- fixed-point update scheme to solve for 3 =
by iterat-
Vin
ing gh+ = -in W g ac 8 + 3". This can be viewed as computing another equilibrium for these neurons: in this stage, neurons receive the inputs 8a of connections with a mask (M can be viewed as a mask matrix based on the firing condition in the first stage, which may be realized by some inhibition mechanisms) to compute for the equilib- rium 3 If W is symmetric and Vt, = 1, it is similar to the energy-based method equilibrium propagation (6)] to use the same weight as the forward computation for second equilibrium T
ac and they use the inverse directions
computation. Plugging 3 into the gradient and calculating
(0fnia' xx) 09
explicitly, we have:
VwL Vin M3 TT VFC Van x TT It is interesting to find that the change of weight

---

from neuron j to neuron i is proportional to the the equilibrium state of neuron j in the first stage and that of neuron in the second stage, and is related whether neuron i fires in the first stage, because Vw., - = V.,. m aj VF,, Ven m,,3 zj. It is to some extent similar to the locally updated Hebbian learning rule Awi.j 2:2; meaning that neurons wire together if they fire to- gether [00), except that we take average firing rates and some temporal information (two stages) into account. Therefore, implicit differentiation calculation is only related to two equilibrium states by the neurons with a mask possibly realized by inhibition mechanisms, and may correspond to modified locally updated rules, which is more biologically plausible than BPTT with surrogate derivatives for SNNs. Please note that 'biological plausibility" here a brief discussion in the context of the above properties, while there may be other aspects of biological implausibility as well.
4.3 Incorporating Multi-layer Structure into The Feedback Model
The multi-layer structure is commonly adopted in ANNS due to its stronger non-linearity and representation ability. To enhance the non-linearity of the fixed-point equilibrium equation, we propose to incorporate multi-layer structure into FSNNs as well. We configure N subgroups of neurons as different layers, where the inputs have connections to the first layer, the (1 - 1)-th layer has connections to the (-th layer, and the last layer has feedback connections to the first layer. Let u' (t) and s' (t) denote the 1-th layer, x(t) denote the inputs, W denote the feedback connection from the last layer to the first layer, and F' denote the weight from the (1 - 1) layer (or input) to the (-th layer. The discrete update equations of membrane potentials are expressed as:
1]= = Au [4] W's ( F' x[[ + b' Vnn [t + 1],
(10)
( [1 + 1] Au [1] + F ss' [1 + 1] + b V + [1 11,,, 1, N 1.
An illustration figure is presented in Appendix A. With similar definitions of average firing rates a' [f[t] for different layers, and ui[(] = un [4] + u [t] we have the equilibrium state as the following.
Theorem 3. If the average inputs converge to an equilibrium point x(t] - x", and there exists constant e and y < 1 such that juf" [t]] S Vi, 1, and [| " 122|F** l12... F S. <V th then the average firing rates of multi-layer FSNN with discrete model will converge equilibrium points a'[t] + a"", which satisfy the fixed-poin equations a = (fN o fa(a" "). xxx") and a a4+1*=
Ji+ (a'"), where Si (a, x) = (V. (W'a+F'xxt
and fi(a) = Vin (F' (r'a+b)))
The proof can be found in Appendix D. There is also similar proposition for the LIF model, please refer to Appendix E for details. We will do classification based on the (weighted) average firing rate of the last layer and calculate the implicit differentiation for the equation on a [T'] The loss, solution for implicit differentiation, and optimization methods are the same as those in Section 4.2.1
5 Experiments
In this section, we conduct extensive experiments to demonstrate the superior performance of our proposed method, Please refer to Appendix F for implementation details, including restrictions on the spectral norm and batch normalization, as well as training parameters. Since few previous methods leverage feedback architectures, we compare the results of our IDE method for FSNNs with most feedforward SNNs and report network structured" as well as the number of neurons and parameters during computation (calculated according to the papers or released codes) for comparison.
5.1 MNIST and Fashion-MNIST
We first evaluate our method on simple static image datasets including MNIST 19 and Fashion- MNIST [43], and compare the results with other directly trained SNNs (21 41 38 I4 45| 46) or similar ANNs. Inputs are the same images with binary or real values at all time steps, which can be
'The notations are: '64C5' means a convolution with 64 output channels and kernel size 5, 's' after '64C5' means convolution with stride 2 while 'u" after that means a transposed convolution to upscale 2x, 'P2' means average pooling with size 2, '40'' means fully-connected to 400 neurons, and 'F' means feedback layers.
7

---

regarded as imput currents [46]. We leverage single-layer FSNNs, and adopt convolutional layers for MNIST while using fully-con layers for our models achieve comparable or better results with fewer neurons and parameters in relatively small number of time steps, compared with other direct SNN training methods on feedforward or feedback architectures. Especially, our model achieves superior results on Fashion-MNIST with the similar structure in only 5 time steps. The LIF model performs slightly better than the model, probably because it leverages temporal information by encoding weighted average firing rates.
following [45] As shown in Table [
Table 1: Performance on MNIST and Aashion-MNIST Results are based on 5 runs of experiments.
MNIST
Method
Network structure
Time steps
Std
Best
Neurons
Params
BP [2]
20CS-P2-50CSS-P22200
>200
/ 99.31%
33K
518K
STBP 41
1SCS-P2-40CS-P2-300
30
/
99.42%
26K
607K
SLAYER 38
12C5-P2-64CS-PP
300
99.36%:10.05%
99.41%
28K
51K
HM2BP 14
15C5-P2-40C5-P2-300
400
99,42%==0.11 111
99.49%
26K
607K
ST-RSBP
45
15C5-PP--0C5--P2-30
400
99.57% 00.0%% %
99.62%
26K
607K
TSSL-BP
1SC5-P2-40CS-PP-300
99.00%=00002%
99.53%
26K
607K
IDE-IF (ours)
64C5s (F64C5)
30
99.49%=
99.55%
13K
229K
IDE-LIF (ours)
64C5s (F64C5)
30
99.53%=0.04%%
99.59%
13K
229K
Fashion-MNIST
Method
Network structure
Time steps
Std
Best
Neurons
Params
ANN
45
512-512
89.01 %
1.8K
670K
HM2BP
45
400-400
400
88, 88.99%
1.6K
478K
TSSL-BP
46
400-400
89.75%
89.80%
1.6K
478K
ST-RSBP
45
400 (F400)
400
90.00% 14%
90.13%
1.2K
478K
IDE-IF (ours)
400 (F400)
90.04% + 0.09% %
90.14% %
1.2K
478K
IDELLFF (ours)
400 (F400)
90.07 % + 0.10%
90.25 %
1.2K
478K
5.2 N-MNIST
We also evaluate our method on the neuromorphic dataset N-MNIST 28 whose inputs are spikes collected by dynamic vision sensors. We follow the same data pre-possessing as (46) and take time steps, and
view the (weighted) average inputs gradually converge to that at the last time step. demonstrates the comparison results of our models and other directly trained models [14] 38] 46 42 It shows that our method can achieve satisfactory performance neuromorphic data well. Especially, only 30 time steps are required by our method for satisfactory performance.
Table 2
Table 2: Performance on N-MNIST. Results are based on 5 runs of experiments.
Method
Network structure
Time steps
Mea
Best
Neurons
Params
HM2BP 14
400-400
600
98.88%+4..0 0.00
98.88%
3K
1.1M
SLAYER
38
500-500
300
08.89%+0.000
98 98.95%
3K
1.4M
SLAYER
38
12C5-P2-64C55
300
99.20%++.000
02%
99.22%
40K
61K
TSSL-BP
46
12C5-P2-64CS-P2
30
99.23%.10.052 99.28%
40K
61K
STBP w/o NeuNorm
CNN
60
/
99.44%
414K
17.3M
IDE-IF (ours)
64C5s (F64C5)
30
99.30% 10.04%
99.35%
21K
291K
IDE-LIF (ours)
64C5s (F64C5)
30
4..04%
99.47% %
21K
291K
128C3-128C3-P2 128C3-256C3- P2BBC--P-128C1-256C
102
5.3 CIFAR-10 and CIFAR-100
Then we evaluate our method on more complex CIFAR-10 and CIFAR-100 datasets [17] We leverage multi-layer FSNNs with structures modified from AlexNet and CIFARNet proposed in 42). as indicated in the footnote of Table 3 We compare our model with SNNs converted from
ANNs [37 8 32 44 and directly trained SNNs 42 46 20 40). For CIFAR- 100, result of

---

directly trained SNN is reported, so we only compare with the converted ones. Table demonstrates the superior results of our directly trained models with fewer neurons and parameters small number of time steps. Especially, our model can outperform the state-of-the-an SNN performance on CIFAR-100 with only time steps, and achieves 1.59% accuracy improvement when 100 time steps are adopted. Please refer to Appendix G for more comparison results between IF and LIF models.
Table 3: Performance on CIFAR-10 and CIFAR-100 Results are based on 5 runs of experiments.
CIFAR-10
Method
Network structure
Time steps
Std
Best
Neurons
Params
ANN-SNN
g
CIFARNet
400-600
90.61 %
726K
45M
ANN-SNN
37
VGG-16
2500
/
91.55%
311K
15M
ANN-SNN
VGG-16
400-600
92.26%
318K
Hybrid Training
32
VGG-16
100
91.13%
318K
40M
STBP
AlexNet
12
/
85.24%
595K
21M
TSSL-BP 46
AlexNet
5
88.98%-+0.22
89.22%
595K
21M
STBP 42
CIFARNet
12
90.53%
726K
45M
TSSL-BP 46
CIFARNet
5
91.41%
726K
45M
Surrogate gradient
VGG-9
100
90.45%
274K
5.9M
ASF-BP [40]
VGG-7
400
91.35%
>240K
>30M
IDE-LIF (ours)
AlexNet-F
30
91.74%
91.92%
159K
3.7M
IDE-LIF (ours)
AlexNet-F
100
92.03 % +0.07% %
92.15%
159K
3.7M
IDE-LIF (ours)
CIFARNet-F
30
92.08% 02.08%110.144 10.14%
92.23%
232K
11.8M
IDE-LIF (ours)
CIFARNet-F
100
92.52% 10.17%
92.82%
232K
11.8M
AlexNet 5R
AlexNet-1:: 9663366656603338660 384133335883336365833 (196C3u)
CIFARNet (42) 128C3 256C3 $12C3 P2 102433551233-10
CHARRN--F: 210.2-66663-1122--1122220
(F128C3u)
CIFAR-100
Method
Network structure
Time steps
Mta
Best
Neurons
Params
ANN
37
VGG-16
71.22%
311K
15M
ANN-SNN 37
VGG-16
2500
70.77%
311K
15M
ANN-SNN
8
VGG-16
400-600
70.55%
318K
40M
ANN-SNN
VGG.*
300
71.84%
540K
9.7M
IDE-IF (ours)
CIFARNet-F
30
71.56 +.331 %
72.10%
232K
14.8M
IDE-IF (ours)
AlexNet-F
72.02 0.116%
72.23%
159K
5.2M
IDE-IF (ours)
CIFARNet-F
100
73.07 % 10.21%
73.43%
232K
14.8M
5.4 Convergence the Equilibrium
To verify the convergence of FSNNs to equilibrium states, we plot the difference norm on the fixed- point equation at each time step, i... fa(t)) aff]], where fe(x) is the fixed-point equation and a[t] is the (weighted) average firing rate at time step t. Figure demonstrates the convergence of different models. is almost the same among different samples. Since the numerical precision of firing rates is only 1 there would be certain convergence error due to the finite time steps. And for the LIF model, there could be random errors compared with the model, as indicated in Section 4 The results conform to the theorems as the difference norm gradually decreases, i.e. firing rates converge to the equilibrium following the equation. also indicates that exact precision not necessary for satisfactory performance. Networks with fewer neurons converge faster, so smaller number of time steps is needed, explaining why only time steps are enough the Fashion-MNIST experiment. For results
datasets and different time steps, please refer to Appendix G.
5.5 Training Memory Costs
As described in the Introduction, an important advantage of our method is that we can avoid the large memory costs, from which the methods that backpropagate along the computational graph would suffer. To quantify this ease of training, compare the GPU memory costs of our method and the

---

Difference Norm [fftxxx]
14
Difference Norn [[fx)-x]]
IF
Difference Norn [[fx)-x]]]
50
IF First
12
LIF
Ia-Last Layer
10.
LiF-Firtt Layer
LIF Last Layer
6
2-
20:
2-
Time Steps
Steps
Time Steps
(a) MNIST: 64C5s (F64C5)
(b) Fashion-MNIST: 400 (F400)
(c) CIFAR-10: CIFARNet-F
Figure 1: Convergence to the equilibrium of different models random sample in 100 time steps.
representative STBP method A1 42
The architecture and training settings the same, and the well illustrates the smaller memory costs of our method, which also
results are shown in T agnostic time steps. Meanwhile, our method could achieve higher performance.
5.6 Firing Sparsity
As for the efficient neuromorphic computation, the firing rate is an important statistic since the energy consumption is proportional to the number of spikes. We calculate the average firing rate of trained models, and compare the IF and LIF model trained by our IDE method, as well as the LIF model trained by STBP method 41 42 with the same structure. The results in Table demonstrate the firing sparsity of our model, as the average firing rate only around or less than 0.7%. And shows that the LIF model has slightly sparser response compared with the IF model. We note that TSSL-BP [46] also reported the statistics about firing rate. According to their results, their trained model on CIFAR-10 10 has roughly total 9.86% firing rate within 5 time steps. So it is interesting to find that our models have fewer spikes than theirs, even if we have more time steps (30 vs. 5), not to mention that models have fewer neurons. The results also show that the model trained by method has sparser spikes compared with STBP, demonstrating the superiority of our method.
Table Comparison of training memory costs and accuracy between training methods. The model trained on CIFAR-10 with AlexNet-F structure and LIF model.
Table 5:
The average firing rate of trained models. The model trained on CIFAR-10 with AlexNet-F structure and 30 time steps.
Layer
IDE-IF
IDE-LIF
STBP-LIF'
Method
Time steps
Accuracy
GPU memory
Layer
0.0172
0.0166
0.0190
IDE (ours)
30
91.74%8100009
2.8G
Layer
0.0041
0.0039
0.0082
STBP'
87.18%
11G
Layer
0.0025
0.0024
0.0113
IDE (ours)
100
92.03%++0.07%
2.8G
Layer 4
0.0008
0.0008
0.0055
out of memory (x=3G)
Layer 5
0.0200
0.0177
0,0108
STBP"
100
Total
0.0070
0.0066
0,0102
Our implemeniation
implementation
6 Conclusion
In this work, we propose a novel training method for feedback spiking neural networks based on implicit differentiation on the equilibrium state. We first derive the equilibrium states of (weighted) average firing rates for the IF and the LIF models of FSNNs under both continuous and discrete views. Then we propose optimize parameters of FSNNs only based on the implicit differentiation on the underlying fixed- -xeetpointt equation. This enables the backward procedure to be decoupled from the forward computational graph and therefore avoids the common training problems for SNNs. such dim-dfffeentiabiiityy and large memory costs. Meanwhile, we briefly discuss the biological plausibility for the calculation of implicit differentiation, which only requires computing another equilibrium and is related to the locally updated Hebbian learning rule. Extensive experiments demonstrate the superior results of our method and models with fewer neurons and parameters in small number of time steps, and the spikes are sparser in our trained models as well.
10

---

Acknowledgement
Z. Lin was supported by the NSF China (No.s 61625301 and 61731018), NSFC Tianyuan Fund for Mathematics (No. 12026606) and Project 2020BD006 supported by PKU-Baidu Fund. Yisen Wang partially supported by the National Natural Science Foundation of China under Grant 62006153, and Project 2020BD006 supported by PKU-Baidu Fund.
References
[1] Filipp Akopyan, Jun Sawada. Andrew Cassidy, Rodrigo Alvarez-Icaza, John Arthur, Paul Merolla, Nabil Imam, Yutaka Nakamura, Pallab Datta, Gi-Joon Nam, et al. TrueNorth: Design and tool flow of 65 mw million neuron programmable neurosynaptic chip. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 34(10):1537-1557 2015.
[2] L.B Almeida. A leamingg rule for asynchronous perceptrons with feedback in combinatorial environment. In International Conference on Neural Networks, 1987.
[3] Shaojie Bai, Zico Kolter, and Vladlen Koltun. Deep equilibrium models. In Advances in Neural Information Processing Systems, 2019.
[4] Shaojie Bai, Vladlen Koltun, and J Zico Kolter. Multiscale deep equilibrium models. In Advances in Neural Information Processing Systems, 2020.
[5] Guillaume Bellec, Darjan Salaj, Anand Subramoney, Robert Legenstein, and Wolfgang Maass, Long short- memory and leaming-to-leam in networks of spiking neurons. Advances Neural Information Processing Systems, 2018
[6] Sander M Bohte, Joost N Kok, and Han La Poutre. Error-hackrropagationn in temporally encoded networks of spiking neurons. Neurocomputiig. 48(1 (3(1-4):77-3) 2002.
[7] Mike Davies, Narayan Srinivasa, Tsung- Han Lin, Gautham Chinya, Yongqiang Cao, Sri Harsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, et al. Loihi: neuromorphic manycore processor with on-chip learning. IEEE Micro, 38(1):82-99) 09229,997700 2018.
[8] Shikuang Deng and Shi Gu. Optimal conversion of conventional artificial neural networks spiking neural networks. International Conference on Learning Representations, 2021.
[9] Peter U Diehl and Matthew Cook. Unsupervised learning of digit recognition using deeenneed plasticity. Frontiers Computational Neuroscience, 9:99, 2015.
[10] Donald Olding Hebb, The organization of behavior: A neurosyycoloogcal theory. Psychology Press, 2005.
[11] John J Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 79(8):2554-2555 1982.
[12] John J Hopfield. Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences, 81(10))3088 3092, 1984.
[13] Eric Hunsberger and Chris Eliasmith. Spiking deep networks with LIF neurons. arXiv preprint arXiv: axivv:1510008829 2015.
[14] Yingyezhe Jin, Wenrui Zhang, and Peng Li. Hybrid macro/micro level backpropagation for training deep spiking neural networks. In Advances in Neural Information Processing Systems, 2018.
[15] Kohitij Kar, Jonas Kubilius, Kailyn Schmidt, Elias B Issa, and James J DiCarlo. Evidence that recurrent cir- cuits are critical to the ventral stream's 's execution of core object recognition behavior. Nature Neuroscience, 22(6):974 983,993,,999.... 2019.
[16] Jinscok Kim, Kyungsu Kim, and Jac-Joon Kim. Unifying activation-and timing- based learning rules for spiking neural networks. In Advances in Neural Information Processing Systems, 2020.
[17] Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, University of Toronto. 2009.
[18] Jonas Kubilius, Martin Schrimpf. Kohitij Kar, Rishi Rajalingham, Ha Hong, Najib Majaj. Elias Issa, Pouya Bashivan, Jonathan Prescott- Roy. Kailyn Schmidt, et al. Brain-like like object recognition with high high-performing shallow recurrent anns. In Advances in Neural Information Processing Systems, 2019.
11

---

[19] Yann LeCun. Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):278-2224, 1998.
[20] Chankyu Lee, Syed Shakib Sarwar, Priyadarshini Panda, Gopalakrishnan Srinivasan, and Kaushik Roy. Enabling spike- based backpropagation for training deep neural network architectures. Frontiers in Neuro- science, 14, 2020.
[21] Jun Haeng Lee, Tobi Delbruck, and Michael Pfeiffer. Training deep spiking neural networks using backpropagation. Frontiers Neuroscience, 10:508, 2016.
[22] Robert Legenstein, Dejan Pecevski, and Wolfgang Maass. A learning theory for reward-modulated seaeee-iiiinnneeedddee plasticity with application to biofeedback. PLoS Comput Biol, 1(00)=e0001880 2008.
[23] Qianyi Li and Cengiz. Pehlevan. Minimax dynamies of optimally balanced spiking networks of excitatory and inhibitory neurons. In Advances in Neural Information Processing Systems, 2020.
[24] Wolfgang Maass, Networks of spiking neurons: the third generation neural network models. Neural Networks, 1099:165911171, 1997.
[25] Wolfgang Maass, Thomas Natschläger, and Henry Markram, Real-time computing without stable states: new framework for neural computation based on perturbations. Neural Computation, 14(11):2531-2560 2002.
[26]
Allan Mancoo, Sander Keemink, and Christian K Machens. Understanding spiking networks through convex optimization. In Advances in Neural Information Processing Syxtems, 2020.
[27] Emre O Neftci, Hesham Mostafa, and Friedemann Zenke. Surrogate gradient lcarning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks. /EEE Signal Processing Magazine, 36(6):51-63, 2019.
[28] Garrick Orchard, Ajinkya Jayawant, Gregory K Cohen, and Nitish Thakor. Converting static image datasets to spiking neuromorphic datasets using saceades. Frontiers in Neuroscience, 9:437, 2015.
[29] Peter O' O'Connor, Efstratios Gavves, and Max Welling. Training spiking neural network with equilibrium propagation. In International Conference on Artificial Intelligence and Statistics, 2019.
[30] Jing Pei, Lei Deng, Sen Song, Mingguo Zhao, Youhui Zhang, Shuang Wu, Guanrui Wang. Zhe Zou, Zhenzhi Wu, Wei He, al. Towards artificial general intelligence with hybrid Tianjic chip architecture. Nature, 572(7767):106-11
2019.
[31] Fernando Pineda. Generalization of back-propagation recurrent neural networks. Physical Review Letters,
1987.
[32] Nitin Rathi, Gopalakrishnan Srinivasan, Priyadarshini Panda, and Kaushik Roy. Enabling deep spiking neural networks with hybrid conversion and spike timing dependent backpropagation. In International Conference on Learning Representations, 2019.
[33] Kaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. Towards spike-based machine intelligence with neuromorphic computing. Nature, 575(7784):607-617 2019.
[34] Bodo Rueckauer, Iulia-Alexandra Lungu, Yuhuang Hu, Michael Pfeiffer, and Shih-Chii Liu, Conversion of continuous-valued deep networks to efficient event-driven networks for image classification. Frontiers in Neuroscience, 11 11:62,, 2017.
[35] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back- propagating errors. Nature, 32366888):533-536 1986.
[36] Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap hetween energy-hased models and backpropagation. Frontiers in Computational Neumssiencce, 11:24, 2017.
[37] Abhronil Sengupta, Yuting Ye, Robert Wang, Chiao Liu, and Kaushik Roy. Going deeper in spiking neural networks: Vgg and residual architectures. Frontiers in Neuroscience, 13:95, 2019.
[38] Sumit Bam Shrestha and Garrick Orchard. Slayer: spike layer error reassignment in time. In Advances in Neural Information Processing Systems, 2018
[39] Paul J Werbos. Backpropagation through time: what it does and how to do it. Proceedings of the IEEE, 78(10):1550-1555 1990.
12

---

[40] Hao Wu, Yueyi Zhang, Wenming Weng. Yongting Zhang. Zhiwei Xiong, Zheng-Jun Zha, Xiaoyan Sun, and Feng Wu. Training spiking neural networks with accumulated spiking flow. In Proceedings of the AAAI Conference on Artificial Intelligence, 2021.
[41] Yujie Wu, Lei Deng, Guoqi Li, Jun Zhu, and Luping Shi. Spatio-temporal backpropagation for training high-peeformance spiking neural networks. Fromtiers in Neuroscience, 12:331, 2018.
[42] Yujie Wu, Lei Deng. Guoqi Li, Jun Zhu, Yuan Xie, and Luping Shi. Direct training for spiking neural networks: Faster, larger, better. In Proceedings of the AAAI Conference on Artificial Intelligence, 2019.
[43] Han Xiao, Kashif Rasul, and Roland Vollgraf. Dashion-MNIST A novel image dataset for benchmarking machine learning algorithms. arXiv preprint 1708 07747, 2017.
[44] Zhanglu Yan, Jun Zhou, and Weng-Fai Wong. Near lossless transfer learning for spiking neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence. 2021.
[45] Wenrui Zhang and Peng Li. Spike-train level backpropagation for training deep recurrent spiking neural networks. In Advances in Neural Information Processing Systems, 2019.
[46] Wenrui Zhang and Peng Li. Temporal spike sequence learning via backpropagation for deep spiking neural networks. Advances in Neural Information Processing Syxtems, 2020.
[47] Hanle Zheng, Yujie Wu, Lei Deng, Yifan Hu, and Guoqi Li. Going deeper with directly-trained larger spiking neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, 2021.
Checklist
1. For all authors...
(a) Do main claims made the abstract and introduction accurately reflect the paper's contribu- tions and scope? [Yes]
(h) Did you descrihe the limitations of your work? [Yes] See Appendix F.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix F.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including thcorctical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 4
(b) Did you include complete proofs all theoretical results? [Yes] See Appendix A, Appendix B, and Appendix C.
3. If you experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (cither in the supplemental material or as a URL)? (Yes]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix D.
(c) Did you report error bars (c.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, cloud provider)? [Yes] See Appendix D.
If you are using existing assets (e.g., code, data, models) or curating/relcasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] See Section 5
(b) Did you mention the license of the assets? [Yes] See Appendix D.
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] We do not use new assets.
(d) Did you discuss whether and how consent was obtained from people whose data you're 're us- ing/curating? [Yes] See Appendix D.
(e) Did you discuss whether the data you are using/curating contains personally identifiable informa- tion or offensive content? [Yes] See Appendix D.
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, if applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on
participant compensation? [N/A]
13


Reviews:
[{"review_id": "XC0rmV67J4", "reviewer": "Reviewer_W1bo", "summary": "The authors propose a relatively novel way of training spiking networks by considering the equilibrium state of their firing rates in response to input, and training using implicit differentiation, as has recently been explored in deep equilibrium models and implicit layers. They consider leaky and perfect integrate-and-fire neurons, as well as single layers with recurrent feedback, and hierarchical layers with feedback from the last layer to the first. The authors prove that the dynamics do indeed converge to an equilibrium state, and demonstrate the performance of the model on several state-of-the-art datasets.", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The method of using implicit differentiation to train spiking networks appears novel and is very interesting! The description of the model dynamics and implicit differentiation is very clear (for the most part), and the results show that this method is indeed powerful and promising.\n\nHowever, the main criticism of this work that I have is the lack of illustration of the model details. The paper jumps straight to the results, so while the authors show that the model works, no insights are gained beyond that. To aid the reader in understanding, it would be useful to have a couple of schematics illustrating the spiking network architecture for the single layer and hierarchical case. It would also be useful to show an example of the network responding to input over time, and comparing the spikes themselves with the estimates of the average firing rates in equations 6 and 9. This is a crucial part of getting implicit differentiation to work, so it seems important to show an example or two. Also, when thinking about biological plausibility and applications in neuromorphic hardware, the number of spikes fired is important, and it would be interesting if the authors can show some statistics on the average firing rate per neuron for the trained models.\n\nNext, I have two more curiosities that I think are worth elaborating on in the paper a bit. One is about the stability of the equilibrium state during learning. The authors show that given particular conditions, the network will converge to equilibrium, but what’s stopping the network from violating these conditions during training? E.g., does the network ever evolve into limit cycle behavior? Second, while the authors show convergence in Fig 1, I am left wondering how good this convergence needs to be in order to have effective learning. It is perhaps too much for an exhaustive study of results with different number of time steps, but perhaps the authors could discuss where both classification and learning might break down due to insufficient time steps simulated and thus accumulated errors.\n\nLastly, there are a few related works that are worth citing, including Li & Pehlevan 2020 and Mancoo et al. 2020, both of which appeared in neurips last year. These papers describe spiking network models that do evolve to equilibrium states, defined by an implicit function (in those cases the implicit function happens to be a constrained optimization problem). They do not scale them up to deep learning problems, but they have already explored spiking networks evolving to equilibrium states.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors did not include a broader impact statement.", "review_text": "", "extra_scores": {"ethical_concerns": "No concerns", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6h (together with a postdoc)"}, "extra_text": {}}, {"review_id": "IE6DStnsTwb", "reviewer": "Reviewer_RLvt", "summary": "This paper extends prior work [3,4] termed Implicit Differentiation on the Equilibrium state (IDE) to spiking neural networks. The authors derive a learning rule for the weights that is a product of two local equilibrium points in 2 phases. They perform experiments to compare their method with other SOTA SNNs, outperforming some.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The authors work is both an extension of previous work [3,4] and a generalization of equilibrium propagation [34]. The key contribution is the extension to spiking neural networks and finding a seemingly local learning rule for the weights. Of course, key issues in biological plausibility remain, which the authors briefly allude to, namely requiring two phases to compute the equilibria $a^*$ and $\\beta^*$, how the masking would take place, how $\\frac{\\partial \\mathcal{L}}{\\partial a^*}$ would be computed and fed in, etc. These are an intriguing possibilities that require further work. Their network + training outperforms SOTA SNNs in some benchmarks with less neurons / params and time steps. While this work is incremental, the paper could be an important step in improving and incorporating on-chip neuromorphic learning, given the emerging importance of energy-efficient neuromorphic computing and the possibilities this paper raises for online local learning. While the mathematical extensions to spiking neural networks are worked out, these latter aspects of local / bio-plausible / neuromorphic learning are not fleshed out, which greatly limits the applicability of the work. Overall, the work itself is clearly presented with some promising aspects, though raising further questions as above.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2.5"}, "extra_text": {}}, {"review_id": "2VEj0Ugxbr8", "reviewer": "Reviewer_Gsxo", "summary": "This paper presented a training method for feedback spiking neural networks to tackle existing training problems with backpropagation framework. Experiments were conducted on five image datasets, MNIST, Fashion-MNIST, N-MNIST, CIFAR-10 and CIFAR-100, superior performances were claimed.  ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The idea of this paper is novel, training a robust deep SNN is very critical in brain-inspired learning field. From the prespective of equilibrium of NNs is impressive, however, I didn't see more details about the equilibrium of SNN. Based on BP, the authors proposed IDE method to try to train a deep SNN. Experimental results evaluated on MNIST and CIFAR showed that the proposed model could achieve good classification performance on those datasets with fewer time steps.\n\nHowever, there are some missing descriptions. What is the difference when you applied IF or LIF, and which situation did you prefer IF of LIF? In section 3.2, when you choose the fixed-point operation, how can you guarantee the convergence？\n\nThe experiments conducted on MNIST or CIFAR based were not enough when you proposed a BP-based model to train a deep SNN. I prefer to see the results from ImageNet datasets or analysis of energy consumption.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The writing is clear and the motivation is clarified clearly. Besides, the theoretical grounding and experimental evaluation are not sufficient to show their originality and significance. Here are some of the suggestions: \n1)\tI would like to see ablation studies for the proposed training method, the traditional backpropagation framework refers as the baseline. \n2)\tHow to deal with the different types of inputs (e.g., bio-medical signals or speech)? It would be valuable to discuss it and present your solutions in this paper. \nThe citation seems a bit disordered.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "1xbU_m82xAq", "reviewer": "Reviewer_u6bR", "summary": "The paper proposes a new training method for SNNs that relies on decoupling the forward propagation from the backward by viewing the SNNs as solving a fixed-point equilibrium equation and exploiting implicit differentiation results to compute the gradients. The theoretical contribution is to derive the equilibrium states for the fixed-point equations that the SNNs are solving, based on which the implicit differentiation technique can be leveraged. The practical contribution is a training algorithm that avoids the problems of non-differentiability in SNNs and achieves state-of-the-art performance on many datasets. \n", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The paper proposes a new training method for SNNs that relies on decoupling the forward propagation from the backward by viewing the SNNs as solving a fixed-point equilibrium equation and exploiting implicit differentiation results to compute the gradients. The theoretical contribution is to derive the equilibrium states for the fixed-point equations that the SNNs are solving, based on which the implicit differentiation technique can be leveraged. The practical contribution is a training algorithm that avoids the problems of non-differentiability in SNNs and achieves state-of-the-art performance on many datasets. \n \nHighlights:\n \n1. The most important contribution definitely is a training algorithm that does not rely on the exact forward procedure. The SNN literature is full of studies trying to overcome the issue of non-differentiability through pseudo-grad, or large memory requirements using simpler spiking models. There are very few studies that avoid these problems altogether and this is one of them. \n\n2. The authors provide rigorous proof of the equilibrium states that the SNNs converge to under both continuous and discrete dynamics. As a result, one can leverage implicit differentiation models to compute the gradients w.r.t. the fixed point equations, thus avoiding the need for a backward procedure that relies on the forward procedure. \n\n3. Based on 1, the authors propose a training algorithm for SNNs with the assumption that the average firing rates of the SNNs follow the fixed-point equations. The loss and gradients of the loss become straightforward to compute. \nWhile not completely obvious at first, the authors provide a qualitative study of how the learning rule is similar to the Hebbian learning rule, thereby imparting it some degree of biological plausibility. \n\n4. Experiments on many datasets, including both small benchmark datasets and large image datasets, show that the algorithm can obtain state-of-the-art performance with a much less number of network parameters and timesteps. This demonstrates the practical effectiveness of the method.\n \n\n\nConcerns:\n\n1. From equation 10, it doesn’t seem like the network is weight-tied and does not have input injection at each layer like in [1]. As a result, acc. to Theorem 3, the equilibrium point does not directly depend upon the inputs which might affect its stability. That’s not the case in [1] from which the method seems to have been inspired. Can the authors resolve this discrepancy, and conduct additional experiments on weight-tied deep networks?\n\n2. How practical are the assumptions in Theorem 1? In particular, how does the fact that the membrane potentials and network weights need to be bounded affect the initialization of these hyperparameters? This is hinted at in the appendix, but I think it deserves a discussion in the main text. Moreover, why is alpha in equation 25 in appendix D a learnable parameter? \n\n3. Is it fair to characterize the learning algorithm as rate-based? If that is the case, and combined with the fact that the inputs need to be convergent, how does the algorithm exploit the temporal capabilities of the SNNs? Is an event-based version of this algorithm possible?\n \n\n\nMinor Comments:\n\n1. In the introduction, the authors mention that “SNNs naturally compute with multiple time steps, which inherently supports feedback computation”. Yet, authors include a feedback connection in their architecture. Can the authors discuss what is the difference in the feedback information represented by these two mechanisms? In addition, it would be nice to have an ablation study of the network without feedback connections. \n\n2. It seems like one advantage of the method is the decrease in training complexity over other ways of training the networks (such as STBP [2]). This might be due to the fact that backward computation relies on a simpler procedure than following the exact forward propagation. If that is the case, can the authors quantify the ease in training and include a short discussion. \n\n3. It will be more clear if the authors can include algorithm pseudocode for the training algorithm.\n\n[1] Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Deep equilibrium models. In Advances in Neural Information Processing Systems, volume 32, pages 690–701, 2019.\n\n[2] Yujie Wu, Lei Deng, Guoqi Li, Jun Zhu, and Luping Shi. Spatio-temporal backpropagation for training high-performance spiking neural networks. Frontiers in Neuroscience, 12:331, 2018.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The proposed training algorithm for SNN relies on equilibrium states on firing rates for each layer of the network. This limits the SNN for non-stationary inputs (time-series data that changes over time, for example, audio and video) where such equilibrium states may not exist in the network.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "30"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a novel method to train spiking neural networks (SNNs) by considering the equilibrium state of their firing rates. They exploit results on implicit differentiation to compute gradients in the network. They show that the method achieves state-of-the-art performance on several data sets, including CIFAR-10 and CIFAR-100.
The method is based on rigorous proofs about the equilibrium states of an SNN.

All reviewers agree that the paper presents interesting results. The manuscript is well-written and the experimental results are convincing.
Decision: Accept (Spotlight)