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Training Spiking Deep Networks
for Neuromorphic Hardware
Eric Hunsberger
Centre for Theoretical Neuroscience
University of Waterloo
Waterloo, ON N2L 3G1
ehunsber@uwaterloo.ca
Chris Eliasmith
Centre for Theoretical Neuroscience
University of Waterloo
Waterloo, ON N2L 3G1
celiasmith@uwaterloo.ca
Abstract
We describe a method to train spiking deep networks that can be run using leaky
integrate-and-fire (LIF) neurons, achieving state-of-the-art results for spiking LIF
networks on five datasets, including the large ImageNet ILSVRC-2012 bench-
mark. Our method for transforming deep artificial neural networks into spik-
ing networks is scalable and works with a wide range of neural nonlinearities.
We achieve these results by softening the neural response function, such that its
derivative remains bounded, and by training the network with noise to provide
robustness against the variability introduced by spikes. Our analysis shows that
implementations of these networks on neuromorphic hardware will be many times
more power-efficient than the equivalent non-spiking networks on traditional hard-
ware.
1 Introduction
Deep artificial neural networks (ANNs) have recently been very successful at solving image cate-
gorization problems. Early successes with the MNIST database [1] were subsequently tested on the
more difficult but similarly sized CIFAR-10 [2] and Street-view house numbers [3] datasets. Re-
cently, many groups have achieved better results on these small datasets (e.g. [4]), as well as on
larger datasets (e.g. [5]). This work has culminated with the application of deep convolutional neu-
ral networks to ImageNet [6], a very large and challenging dataset with 1.2 million images across
1000 categories.
There has recently been considerable effort to introduce neural “spiking” into deep ANNs [7, 8, 9,
10, 11, 12], such that connected nodes in the network transmit information via instantaneous single
bits (spikes), rather than transmitting real-valued activities. While one goal of this work is to better
understand the brain by trying to reverse engineer it [7], another goal is to build energy-efficient
neuromorphic systems that use a similar spiking communication method, for image categorization
[10, 11, 12] or other applications [13].
In this paper, we present a novel method for translating deep ANNs into spiking networks for im-
plementation on neuromorphic hardware. Unlike previous methods, our method is applicable to
a broad range of neural nonlinearities, allowing for implementation on hardware with idiosyncratic
neuron types (e.g. [14]). We extend our previous results [15] to additional datasets, and most notably
demonstrate that it scales to the large ImageNet dataset. We also perform an analysis demonstrating
that neuromorphic implementations of these networks will be many times more power-efficient than
the equivalent non-spiking networks running on traditional hardware.
1
2 Methods
We first train a network on static images using traditional deep learning techniques; we call this the
ANN. We then take the parameters (weights and biases) from the ANN and use them to connect
spiking neurons, forming the spiking neural network (SNN). A central challenge is to train the ANN
in such a way that it can be transferred into a spiking network, and such that the classification error
of the resulting SNN is minimized.
2.1 Convolutional ANN
We base our network off that of Krizhevsky et al. [6], which won the ImageNet ILSVRC-2012
competition. A smaller variant of the network achieved 11% error on the CIFAR-10 dataset. The
network makes use of a series of generalized convolutional layers, where one such layer is composed
of a set of convolutional weights, followed by a neural nonlinearity, a pooling layer, and finally a
local contrast normalization layer. These generalized convolutional layers are followed by either
locally-connected layers, fully-connected layers, or both, all with a neural nonlinearity. In the case
of the original network, the nonlinearity is a rectified linear (ReLU) function, and pooling layers
perform max-pooling. The details of the network can be found in [6] and code is available1.
To make the ANN transferable to spiking neurons, a number of modifications are necessary. First,
we remove the local response normalization layers. This computation would likely require some
sort of lateral connections between neurons, which are difficult to add in the current framework
since the resulting network would not be feedforward and we are using methods focused on training
feedforward networks.
Second, we changed the pooling layers from max pooling to average pooling. Again, computing max
pooling would likely require lateral connections between neurons, making it difficult to implement
without significant changes to the training methodology. Average pooling, on the other hand, is very
easy to compute in spiking neurons, since it is simply a weighted sum.
The other modifications—using leaky integrate-and-fire neurons and training with noise—are the
main focus of this paper, and are described in detail below.
2.2 Leaky integrate-and-fire neurons
Our network uses a modified leaky integrate-and-fire (LIF) neuron nonlinearity instead of the recti-
fied linear nonlinearity. Past work has kept the rectified linear nonlinearity for the ANN and substi-
tuted in the spiking integrate-and-fire (IF) neuron model in the SNN [11, 10], since the static firing
curve of the IF neuron model is a rectified line. Our motivation for using the LIF neuron model is
that it and it demonstrates that more complex, nonlinear neuron models can be used in such net-
works. Thus, these methods can be extended to the idiosyncratic neuron types employed by some
neuromorphic hardware (e.g. [14]).
The LIF neuron dynamics are given by the equation
τRC ˙v(t) = −v(t) + J(t)(1)
where v(t)is the membrane voltage, ˙v(t)is its derivative with respect to time, J(t)is the input
current, and τRC is the membrane time constant. When the voltage reaches Vth = 1, the neuron
fires a spike, and the voltage is held at zero for a refractory period of τref . Once the refractory
period is finished, the neuron obeys Equation 1 until another spike occurs.
Given a constant input current J(t) = j, we can solve Equation 1 for the time it takes the voltage to
rise from zero to one, and thereby find the steady-state firing rate
r(j) = τref +τRC log 1 + Vth
ρ(j−Vth)−1
(2)
where ρ(x) = max(x, 0).
Theoretically, we should be able to train a deep neural network using Equation 2 as the static non-
linearity and make a reasonable approximation of the network in spiking neurons, assuming that
1https://github.com/akrizhevsky/cuda-convnet2
2
Figure 1: Comparison of LIF and soft LIF response functions. The left panel shows the response
functions themselves. The LIF function has a hard threshold at j=Vth = 1; the soft LIF function
smooths this threshold. The right panel shows the derivatives of the response functions. The hard
LIF function has a discontinuous and unbounded derivative at j= 1; the soft LIF function has a
continuous bounded derivative, making it amenable to use in backpropagation.
the spiking network has a synaptic filter that sufficiently smooths a spike train to give a good ap-
proximation of the firing rate. The LIF steady state firing rate has the particular problem that the
derivative approaches infinity as j→0+, which causes problems when employing backpropagation.
To address this, we added smoothing to the LIF rate equation.
If we replace the hard maximum ρ(x) = max(x, 0) with a softer maximum ρ1(x) = log(1 + ex),
then the LIF neuron loses its hard threshold and the derivative becomes bounded. Further, we can
use the substitution
ρ2(x) = γlog h1 + ex/γ i(3)
to allow us control over the amount of smoothing, where ρ2(x)→max(x, 0) as γ→0. Figure 1
shows the result of this substitution.
2.3 Training with noise
Training neural networks with various types of noise on the inputs is not a new idea. Denoising
autoencoders [16] have been successfully applied to datasets like MNIST, learning more robust
solutions with lower generalization error than their non-noisy counterparts.
In a biological spiking neural network, synapses between neurons perform some measure of filtering
on the spikes, due to the fact that the post-synaptic current induced by the neurotransmitter release
is distributed over time. We employ a similar mechanism in our networks to attenuate some of
the variability introduced by spikes. The α-function α(t)=(t/τs)e−t/τsis a simple second-order
lowpass filter, inspired by biology [17]. We chose this as a synaptic filter for our networks since it
provides better noise reduction than a first-order lowpass filter.
The filtered spike train can be viewed as an estimate of the neuron activity. For example, if the
neuron is firing regularly at 200 Hz, filtering spike train will result in a signal fluctuating around 200
Hz. We can view the neuron output as being 200 Hz, with some additional “noise” around this value.
By training our ANN with some random noise added to the output of each neuron for each training
example, we can simulate the effects of using spikes on the signal received by the post-synaptic
neuron.
Figure 2 shows how the variability of filtered spike trains depends on input current for the LIF
neuron. Since the impulse response of the α-filter has an integral of one, the mean of the filtered
spike trains is equal to the analytical rate of Equation 2. However, the statistics of the filtered signal
vary significantly across the range of input currents. Just above the firing threshold, the distribution
is skewed towards higher firing rates (i.e. the median is below the mean), since spikes are infrequent
so the filtered signal has time to return to near zero between spikes. At higher input currents, on the
3
Figure 2: Variability in filtered spike trains versus input current for the LIF neuron (τRC =
0.02, τref = 0.004). The solid line shows the mean of the filtered spike train (which matches
the analytical rate of Equation 2), the ‘x’-points show the median, the solid error bars show the 25th
and 75th percentiles, and the dotted error bars show the minimum and maximum. The spike train
was filtered with an α-filter with τs= 0.003 s.
other hand, the distribution is skewed towards lower firing rates (i.e. the median is above the mean).
In spite of this, we used a Gaussian distribution to generate the additive noise during training, for
simplicity. We found the average standard deviation to be approximately σ= 10 across all positive
input currents for an α-filter with τs= 0.005. During training, we add Gaussian noise η∼G(0, σ )
to the firing rate r(j)(Equation 2) when j > 0, and add no noise when j≤0.
2.4 Conversion to a spiking network
Finally, we convert the trained ANN to a SNN. The parameters in the spiking network (i.e. weights
and biases) are all identical to that of the ANN. The convolution operation also remains the same,
since convolution can be rewritten as simple connection weights (synapses) wij between pre-
synaptic neuron iand post-synaptic neuron j. (How the brain might learn connection weight pat-
terns, i.e. filters, that are repeated at various points in space, is a much more difficult problem that
we will not address here.) Similarly, the average pooling operation can be written as a simple con-
nection weight matrix, and this matrix can be multiplied by the convolutional weight matrix of the
following layer to get direct connection weights between neurons.2
The only component of the network that changes when moving from the ANN to the SNN is the
neurons themselves. The most significant change is that we replace the soft LIF rate model (Equa-
tion 2) with the LIF spiking model (Equation 1). We remove the additive Gaussian noise used in
training. We also add post-synaptic filters to the neurons, which removes a significant portion of the
high-frequency variation produced by spikes.
3 Results
We tested our methods on five datasets: MNIST [1], SVHN [18], CIFAR-10 and CIFAR-100 [19],
and the large ImageNet ILSVRC-2012 dataset [20]. Our best result for each dataset is shown in
Table 1. Using our methods has allowed us to build spiking networks that perform nearly as well as
their non-spiking counterparts using the same number of neurons. All datasets show minimal loss
in accuracy when transforming from the ANN to the SNN. 3
2For computational efficiency, we actually compute the convolution and pooling separately.
3The ILSVRC-2012 dataset actually shows a marginal increase in accuracy, though this is likely not statisti-
cally significant and could be because the spiking LIF neurons have harder firing thresholds than their soft-LIF
rate counterparts. Also, the CIFAR-100 dataset shows a considerable increase in performance when using soft-
4
Dataset ReLU ANN LIF ANN LIF SNN
MNIST 0.79% 0.84% 0.88%
SVHN 5.65% 5.79% 6.08%
CIFAR-10 16.48% 16.28% 16.46%
CIFAR-100 50.05% 44.35% 44.87%
ILSVRC-2012 45.4% (20.9%)a48.3% (24.1%)a48.2% (23.8%)a
aResults from the first 3072-image test batch.
Table 1: Results for spiking LIF networks (LIF SNN), compared with ReLU ANN and LIF ANN
(both using the same network structure, but with ReLU and LIF rate neurons respectively). The
spiking versions of each network perform almost as well as the rate-based versions. The ILSVRC-
2012 (ImageNet) results show the error for the top result, with the top-5 result in brackets.
Dataset This Paper TN 1-chip TN 8-chip Best Other
MNIST 0.88% (27k) None None 0.88% (22k) [10]
SVHN 6.08% (27k) 3.64% (1M) 2.83% (8M) None
CIFAR-10 16.46% (50k) 17.50% (1M) 12.50% (8M) 22.57% (28k) [11]
CIFAR-100 44.87% (50k) 47.27% (1M) 36.95% (8M) None
ILSVRC-2012 48.2%, 23.8% (493k)aNone None None
aResults from the first 3072-image test batch.
Table 2: Our error rates compared with recent results on the TrueNorth (TN) neuromorphic
chip [12], as well as other best results in the literature. Approximate numbers of neurons are shown
in parentheses. The TrueNorth networks use significantly more neurons than our networks (about
20×more for the 1-chip network and 160×more for the 8-chip network). The first number for
ILSVRC-2012 (ImageNet) indicates the error for the top result, and the second number the more
commonly reported top-5 result.
Table 2 compares our results to the best spiking network results on these datasets in the litera-
ture. The most significant recent results are from [12], who implemented networks for a number of
datasets on both one and eight TrueNorth chips. Their results are impressive, but are difficult to com-
pare with ours since they use between 20 and 160 times more neurons. We surpass a number of their
one-chip results while using an order of magnitude fewer neurons. Furthermore, we demonstrate
that our method scales to the large ILSVRC-2012 dataset, which no other SNN implementation to
date has done. The most significant difference between our results and that of [10] and [11] is that
we use LIF neurons and can generalize to other neuron types, whereas their methods (and those of
[12]) are specific to IF neurons.
We examined our methods in more detail on the CIFAR-10 dataset. This dataset is composed of
60000 32×32 pixel labelled images from ten categories. We used the first 50000 images for training
and the last 10000 for testing, and augmented the dataset by taking random 24 ×24 patches from the
training images and then testing on the center patches from the testing images. This methodology
is similar to Krizhevsky et al. [6], except that they also used multiview testing where the classifier
output is the average output of the classifier run on nine random patches from each testing image
(increasing the accuracy by about 2%).
Table 3 shows the effect of each modification on the network classification error. Rows 1-5 show that
each successive modification required to make the network amenable to running in spiking neurons
adds additional error. Despite the fact that training with noise adds additional error to the ANN,
rows 6-8 of the table show that in the spiking network, training with noise pays off, though training
with too much noise is not advantageous. Specifically, though training with σ= 20 versus σ= 10
decreased the error introduced when switching to spiking neurons, it introduced more error to the
ANN (Network 5), resulting in worse SNN performance (Network 8).
LIF neurons versus ReLUs in the ANN, but this could simply be due to the training hyperparameters chosen,
since these were not optimized in any way.
5
# Modification CIFAR-10 error
0 Original ANN based on Krizhevsky et al. [6] 14.03%
1 Network 0 minus local contrast normalization 14.38%
2 Network 1 minus max pooling 16.70%
3 Network 2 with soft LIF 15.89%
4 Network 3 with training noise (σ= 10) 16.28%
5 Network 3 with training noise (σ= 20) 16.92%
6 Network 3 (σ= 0) in spiking neurons 17.06%
7 Network 4 (σ= 10) in spiking neurons 16.46%
8 Network 5 (σ= 20) in spiking neurons 17.04%
Table 3: Effects of successive modifications to CIFAR-10 error. We first show the original ANN
based on [6], and then the effects of each subsequent modification. Rows 6-8 show the results of
running ANNs 3-5 in spiking neurons, respectively. Row 7 is the best spiking network, using a
moderate amount of training noise.
3.1 Efficiency
Running on standard hardware, spiking networks are considerably less efficient than their ANN
counterparts. This is because ANNs are static, requiring only one forward-pass through the network
to compute the output, whereas SNNs are dynamic, requiring the input to be presented for a number
of time steps and thus a number of forward passes. On hardware that can take full advantage of the
sparsity that spikes provide—that is, neuromorphic hardware—SNNs can be more efficient than the
equivalent ANNs, as we show here.
First, we need to compute the computational efficiency of the original network, specifically the num-
ber of floating-point operations (flops) required to pass one image through the network. There are
two main sources of computation in the image: computing the neurons and computing the connec-
tions.
flops =flops
neuron ×neurons +flops
connection ×connections (4)
Since a rectified linear unit is a simple max function, it requires only one flop to compute
(flops/neuron = 1). Each connection requires two flops, a multiply and an add (flops/connection = 2).
We can determine the number of connections by “unrolling” each convolution, so that the layer is in
the same form as a locally connected layer.
To compute the SNN efficiency on a prospective neuromorphic chip, we begin by identifying the
energy cost of a synaptic event (Esynop) and neuron update (Eupdate), relative to standard hardware.
In consultation with neuromorphic experts, and examining current reports of neuromorphic chips
(e.g. [21]), we assume that each neuron update takes as much energy as 0.25 flops (Eupdate = 0.25),
and each synaptic event takes as much energy as 0.08 flops (Esynop = 0.08). (These numbers could
potentially be much lower for analog chips, e.g. [14].) Then, the total energy used by an SNN to
classify one image is (in units of the energy required by one flop on standard hardware)
ESN N =Esynop
synops
s+Eupdate
updates
s×s
image (5)
For our CIFAR-10 network, we find that on average, the network has rates of 2,693,315,174 syn-
ops/s and 49,536,000 updates/s. This results in EC IF AR−10 = 45,569,843, when each image is
presented for 200 ms. Dividing by the number of flops per image on standard hardware, we find that
the relative efficiency of the CIFAR-10 network is 0.76, that is it is somewhat less efficient.
Equation 5 shows that if we are able to lower the amount of time needed to present each image to
the network, we can lower the energy required to classify the image. Alternatively, we can lower
the number of synaptic events per second by lowering the firing rates of the neurons. Lowering
the number of neuron updates would have little effect on the overall energy consumption since the
synaptic events require the majority of the energy.
To lower the presentation time required for each input while maintaining accuracy, we need to
decrease the synapse time constant as well, so that the information is able to propagate through the
6
Dataset τs[ms] c0[ms] c1[ms] Error Efficiency
CIFAR-10 5 120 200 16.46% 0.76×
CIFAR-10 0 10 80 16.63% 1.64×
CIFAR-10 0 10 60 17.47% 2.04×
MNIST 5 120 200 0.88% 5.94×
MNIST 2 40 100 0.92% 11.98×
MNIST 2 50 60 1.14% 14.42×
MNIST 0 20 60 3.67% 14.42×
ILSVRC-2012 3 140 200 23.80% 1.39×
ILSVRC-2012 0 30 80 25.33% 2.88×
ILSVRC-2012 0 30 60 25.36% 3.51×
Table 4: Estimated efficiency of our networks on neuromorphic hardware, compared with traditional
hardware. For all datasets, there is a tradeoff between accuracy and efficiency, but we find many con-
figurations that are significantly more efficient while sacrificing little in terms of accuracy. τsis the
synapse time constant, c0is the start time of the classification, c1is the end time of the classification
(i.e. the total presentation time for each image).
whole network in the decreased presentation time. Table 4 shows the effect of various alternatives
for the presentation time and synapse time constant on the accuracy and efficiency of the networks
for a number of the datasets.
Table 4 shows that for some datasets (e.g. CIFAR-10 and ILSVRC-2012) the synapses can be com-
pletely removed (τs= 0 ms) without sacrificing much accuracy. Interestingly, this is not the case
with the MNIST network, which requires at least some measure of synapses to function accurately.
We suspect that this is because the MNIST network has much lower firing rates than the other net-
works (average of 9.67 Hz for MNIST, 148 Hz for CIFAR-10, 93.3 Hz for ILSVRC-2012). This
difference in average firing rates is also why the MNIST network is significantly more efficient than
the other networks.
It is important to tune the classification time, both in terms of the total length of time each example
is shown for (c1), and when classification begins (c0). The optimal values for these parameters are
very dependent on the network, both in terms of the number of layers, firing rates, and synapse time
constants. Figure 3 shows how the classification time affects accuracy for various networks.
Given that the CIFAR-10 network performs almost as well with no synapses as with synapses, one
may question whether noise is required during training at all. We retrained the CIFAR-10 network
with no noise and ran with no synapses, but could not achieve accuracy better than 18.06%. This
suggests that noise is still beneficial during training.
4 Discussion
Our results show that it is possible to train accurate deep convolutional networks for image clas-
sification without adding neurons, while using more complex nonlinear neuron types—specifically
the LIF neuron—as opposed to the traditional rectified-linear or sigmoid neurons. We have shown
that networks can be run in spiking neurons, and training with noise decreases the amount of error
introduced when running in spiking versus rate neurons. These networks can be significantly more
energy-efficient than traditional ANNs when run on specialized neuromorphic hardware.
The first main contribution of this paper is to demonstrate that state-of-the-art spiking deep networks
can be trained with LIF neurons, while maintaining high levels of classification accuracy. For exam-
ple, we have described the first large-scale SNN able to provide good results on ImageNet. Notably,
all other state-of-the-art methods use integrate-and-fire (IF) neurons [11, 10, 12], which are straight-
forward to fit to the rectified linear units commonly used in deep convolutional networks. We show
that there is minimal drop in accuracy when converting from ANN to SNN. We also examine how
classification time affects accuracy and energy-efficiency, and find that networks can be made quite
efficient with minimal loss in accuracy.
7
CIFAR-10 (τs= 5 ms) CIFAR-10 (τs= 0 ms)
MNIST (τs= 2 ms) ILSVRC-2012 (τs= 0 ms)
Figure 3: Effects of classification time on accuracy. Individual traces show different starting classi-
fication times (c0), and the x-axis the end classification time (c1).
By smoothing the LIF response function so that its derivative remains bounded, we are able to use
this more complex and nonlinear neuron with a standard convolutional network trained by back-
propagation. Our smoothing method is extensible to other neuron types, allowing for networks to be
trained for neuromorphic hardware with idiosyncratic neuron types (e.g. [14]). We found that there
was very little error introduced by switching from the soft response function to the hard response
function with LIF neurons for the amount of smoothing that we used. However, for neurons with
harsh discontinuities that require more smoothing, it may be necessary to slowly relax the smoothing
over the course of the training so that, by the end of the training, the smooth response function is
arbitrarily close to the hard response function.
The second main contribution of this paper is to demonstrate that training with noise on neuron
outputs can decrease the error introduced when transitioning to spiking neurons. The error decreased
by 0.6% overall on the CIFAR-10 network, despite the fact that the ANN trained without noise
performs better. This is because noise on the output of the neuron simulates the variability that a
spiking network encounters when filtering a spike train. There is a tradeoff between training with
too little noise, which makes the SNN less accurate, and too much noise, which makes the initially
trained ANN less accurate.
These methods provide new avenues for translating traditional ANNs to spike-based neuromorphic
hardware. We have provided some evidence that such implementations can be significantly more
energy-efficient than their ANN counterparts. While our analyses only consider static image classi-
fication, we expect that the real efficiency of SNNs will become apparent when dealing with dynamic
inputs (e.g. video). This is because SNNs are inherently dynamic, and take a number of simulation
steps to process each image. This makes them best suited to processing dynamic sequences, where
adjacent frames in the video sequence are similar to one another, and the network does not have to
take time to constantly “reset” after sudden changes in the input.
Future work includes experimenting with lowering firing rates for greater energy-efficiency. This
could be done by changing the neuron refractory period τref to limit the firing below a particular
8
rate, optimizing for both accuracy and low rates, using adapting neurons, or adding lateral inhibition
in the convolutional layers. Other future work includes implementing max-pooling and local contrast
normalization layers in spiking networks. Networks could also be trained offline as described here
and then fine-tuned online using an STDP rule [22, 23] to help further reduce errors associated
with converting from rate-based to spike-based networks, while avoiding difficulties with training a
network in spiking neurons from scratch.
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