1 Introduction
Artificial Neural Networks (ANNs), especially Deep Neural Networks have become the goto tool for many machine learning tasks. ANNs achieve state of the art performance in applications ranging from image classification and object recognition, to object tracking, signal processing, natural language processing, self driving cars, health care diagnostics, and many more. In the currently popular second generation of ANNs, backpropagation of error signal to the neurons in preceding layer is the key to their learning prowess.
However, ANNs generally require powerful GPUs and computing clusters to crunch their inputs into useful outputs. Therefore, in scenarios where power consumption is constrained, on site use of ANNs may not be a viable option. On the other hand, biologically inspired spiking neurons have long shown great theoretical potential as efficient computational units Maass1996 ; Maass1996a ; Maass1997a and recent advances in Spiking Neural Network (SNN) hardware Merolla2014 ; Furber2014 ; davies2018loihi have renewed research interest in this area.
SNNs are similar to ANNs in terms of network topology, but differ in the choice of neuron model. Spiking neurons have memory and use a nondifferentiable spiking neuron model (spike function
) while ANNs typically have no memory and model each neuron using a continuously differentiable activation function. Since the spike function is nondifferentiable, the backpropagation mechanism used to train ANNs cannot be directly applied.
Nevertheless, a handful of supervised learning algorithms for SNNs have been proposed previously. The majority of them are designed for a single neuron
Ponulak2010 ; MOHEMMED2012 ; GuetigSompolinsky2006 , but a few have proposed methods to work around the nondifferentiable spike function and backpropagate error through multiple layers Bohte2002a ; Shrestha2017 ; Panda2016 ; Lee2016 ; Zenke2017 . Event based methods such as SpikePropBohte2002a and EvSpikePropShrestha2017 have the derivative term defined only around the firing time, whereas Panda2016 ; Lee2016 ; Zenke2017 ignore the temporal effect of spike signal. In Section 3.1 we describe the strengths and weaknesses of these approaches in more detail.The main contribution of this paper is a general method of error backpropagation for SNNs (Section 3) which we call Spike LAYer Error Reassignment (SLAYER). SLAYER distributes the credit of error back through the SNN layers, much like the traditional backprop algorithm distributes error back through an ANN’s layers. However, unlike backprop, SLAYER also distributes the credit of error back in time because a spiking neuron’s current state depends on its previous states (and therefore, on the previous states of its input neurons). SLAYER can simultaneously learn both synaptic weights and axonal delays, which only a few previous works have attempted schrauwen2004extending ; TaherkhaniBelatrecheLiEtAl2015 .
We have developed and released^{1}^{1}1The code for SLAYER learning framework is publicly available at: https://bitbucket.org/bamsumit/slayer a CUDA accelerated framework to train SNNs using SLAYER. We demonstrate SLAYER achieving state of the art accuracy for an SNN on neuromorphic datasets (Section 4) for visual digit recognition, action recognition, and spoken digit recognition.
The rest of the paper is organized as follows. We start by introducing notation for a general model of a spiking neuron and extending it to a multilayer SNN in Section 2. Then, in Section 3 we discuss previously published methods for learning SNN parameters before deriving the SLAYER backpropagation formulae. In Section 4, we demonstrate the effectiveness of SLAYER on different benchmark datasets before concluding in Section 5.
2 Spiking Neural Network: Background
An SNN is a type of ANN that uses more biologically realistic spiking neurons, as its computational units. In this Section we introduce a model for a spiking neuron before extending the formulation to a multilayer network of spiking neurons (SNN).
2.1 Spiking Neuron Model
Spiking neurons obtain their name from the fact that they only communicate using voltage spikes. All inputs and outputs to the neuron are in the form of spikes, but the neuron maintains an internal state over time. In this paper, we will use a simple yet versatile spiking neuron model known as the Spike Response Model (SRM)Gerstner1995 , described below.
Consider an input spike train to a neuron, . Here is the time of the spike of the input. In SRM, the incoming spikes are converted into a spike response signal, , by convolving with a spike response kernel . This can be written as . Similarly, the refractory response of a neuron is represented as , where is the refractory kernel and is the neuron’s output spike train.
Each spike response signal is scaled by a synaptic weight to generate a Post Synaptic Potential (PSP). The neuron’s state (membrane potential), , is simply the sum of all PSPs and refractory responses
(1) 
An output spike is generated whenever reaches a predefined threshold . More formally, the spike function is defined as
(2) 
Unlike the activation functions used in nonspiking ANNs, the derivative of the spike function is undefined which is a major obstacle for backpropagating error from output to input for SNNs. Also, note that the effect of an input spike is distributed in future via the spike response kernels which is the reason for temporal dependency in the spiking neuron.
The above formulation can be extended to include axonal delays by redefining the spike response kernel as , where is the axonal delay.^{2}^{2}2 Synaptic delay can also be modelled in similar manner. Here, we only consider axonal delay for simplicity.
2.2 SNN Model
Here we describe a feedforward neural network architecture with layers. This formulation applies to fully connected, convolutional as well as pooling layer. For details, refer to supplementary material. Consider a layer with neurons, weights and axonal delays . Then the network forward propagation is as described below.
(3)  
(4)  
(5) 
Also note that the inputs, , and outputs, , are spike trains rather than numeric values.
3 Backpropagation in SNN
In this Section, we first discuss prior works on learning in SNNs before presenting the details of error backpropagation using SLAYER.
3.1 Existing Methods
Previous works which use learning to configure a deep SNN (multiple hidden layers) can be grouped into three main categories. The first category uses an ANN to train an equivalent shadow network. The other two categories train directly on the SNN but differ in how they approximate the derivative of the spike function.
The first category leverages learning methods for conventional ANNs by training an ANN and converting it to an SNNEsser2015 ; Hunsberger2015 ; Esser2016 ; OConnor2013 ; Liu2017 ; Diehl2015 ; Diehl2016a ; Rueckauer2017 with some loss of accuracy. There are different approaches to overcome the loss of accuracy such as introducing extra constraints on neuron firing rateDiehl2015 , scaling the weightsDiehl2015 ; Diehl2016a ; Rueckauer2017 , constraining the network parametersEsser2016 , formulating an equivalent transfer function for a spiking neuronHunsberger2015 ; Esser2016 ; OConnor2013 ; Liu2017 , adding noise in the modelOConnor2013 ; Liu2017 , using probabilistic weightsEsser2015 and so on.
The second category keeps track of the membrane potential of spiking neurons only at spike times and backpropagates errors based only on membrane potentials at spike times. Examples include SpikePropBohte2002a and its derivativesmckennoch2006fast ; Shrestha2017
. These methods are prone to the “dead neuron” problem: when no neurons spike, no learning occurs. Heuristic measures are required to revive the network from such a condition.
The third category of methods backpropagates errors based on the membrane potential of a spiking neuron at a single time step only. Different methods are used to approximate the derivative of the spike function. Panda et al.Panda2016
use an expression similar to that of a multilayer perceptron system, Lee et al.
Lee2016 use small signal approximation at the spike times, and Zenke et al.Zenke2017 simply propose a function to serve as the derivative. All these methods ignore the temporal dependency between spikes. They credit the error at a given time step to the input signals at that time step only, thus neglecting the effect of earlier spike inputs.3.2 Backpropagation using SLAYER
In this Section we describe the Loss Function (Section
3.2.1), how error is assigned to previous timepoints (Section 3.2.2), and how the derivative of the spike function is approximated (Section 3.2.4).3.2.1 The Loss Function
Consider a loss function for the network in time interval , defined as
(6) 
where is the target spike train, is the loss at time instance and is the error signal at the final layer. For brevity we will write the error signal as from here on.
To learn a target spike train an error signal of the form
(7) 
is a suitable choice. This loss function is similar to the vanRossum distancedauwels2008similarity .
For classification tasks, a decision is typically made based on the number of output spikes during an interval rather than the precise timing of the spikes. To handle such cases, the error signal during the interval can be defined as
(8) 
and zero outside the interval . Here we only need to define the number of desired spikes during the interval (the second integral term). The actual spike train need not be defined.
3.2.2 Temporal Dependencies to History
In the mapping from input spikes, , to membrane potential, , temporal dependencies are introduced due to spike response kernel which distributes the effect of input spikes into future time values i.e. the signal is dependent on current as well as past values of inputs . Step based learning approachesLee2016 ; Panda2016 ; Zenke2017 ignore this temporal dependency and only use signal values at the current time instance. Below we describe how SLAYER accounts for this temporal dependency. Full details of the derivation are provided in the supplementary material.
Let us, for the time being, discretize the system with a sampling time such that and use to denote the total number of samples in the period . The signal values and have a contribution to future network losses at samples . Taking into account the temporal dependency, the gradient term is given by
(9) 
The backpropagation estimate of error in layer
is then(10)  
(11) 
Here represents elementwise correlation operation in time. The summation from to assigns the credit of all the network losses in a future time to the neuron at current time. Note that at the output layer, for which results in . This is in agreement with the definition of output layer error in (6).
Similarly for axonal delay with , one can derive the delay gradient as follows.
(12) 
3.2.3 The Derivative of the Spike Function
The derivative of the spike function is always a problem for supervised learning in a multilayer SNN. In Section 3.1, we discussed how prior works handle the derivative. Below we describe how SLAYER deals with the spike function derivative.
Consider the state of a spiking neuron at time . The neuron can either be in spiking state () or nonspiking state (). Now consider a perturbation in the membrane potential by an amount for .
A neuron in the nonspiking state will switch to the spiking state when perturbed by if . Similarly, a neuron in the spiking state will switch to the nonspiking state when perturbed by if . In both the cases, when there is a change in spiking state of the neuron when . Fig. 1(a) shows these transitions. Therefore,
(13) 
This formulation is still problematic because of Diracdelta function. However, we can see that the derivative term is biased towards zero as increases. A good estimate of the derivative term can be made using the probability of a change in spiking state.
If we denote the probability density function as , then the probability of spiking state change in an infinitesimal time window of width around and a small perturbation as can be written as . Now, the expected value of can be written as
(14) 
The derivative of spike function represents the Probability Density Function (PDF) for change of state of a spiking neuron. For a completely deterministic spiking neuron model, it is a sum of impulses at spike times, which is equivalent to the spike train . Nevertheless, we can relax the deterministic nature of spiking neuron and use the stochastic spiking neuron approximation for backpropagating errors.
The function must be high when is close to and must decrease as it moves further away. An example PDF is shown in Figure 1(b). A good formulation of this function is the spike escape rate function Gerstner2002 ; Jolivet2003 which is usually represented by an exponentially decaying function of
(15) 
Zenke et al.Zenke2017
use the negative portion of a fast sigmoid function to represent the derivative term, which is also a suitable candidate for
.3.2.4 The SLAYER Backpropagation Pipeline
Now, applying the limit for (9) (10) and (11) and using the expectation value of , we arrive at the SLAYER backpropagation pipeline.
(16)  
(17)  
(18)  
(19) 
The gradients with respect to weights and delays are given by (18) and (19
). It is straightforward to use any of the optimization techniques from simple gradient descent method to adaptive methods such as RmsProp, ADAM, and NADAM to drive the network towards convergence.
4 Experiments and Results
In this Section we will present different experiments conducted and results on them to evaluate the performance of SLAYER. First, we train an SNN to produce a fixed Poisson spike train pattern in response to a given set of Poisson spike inputs. We use this simple example to show how SLAYER works. Afterwards we present results of classification tasks performed on both spiking datasets and nonspiking datasets converted to spikes.
Simulating an SNN is a time consuming process due to the additional temporal dimension of signals. An efficient simulation framework is key to enabling training on practical spiking datasets. We use our CUDA accelerated SNN deep learning framework for SLAYER to perform all the simulations for which results are presented in this paper. All the accuracy values reported for SLAYER are averaged over 5 different independent trials. In our experiments, we use spike response kernels of the form
and . Here, is the Heaviside step function. SLAYER, however, is independent of the choice of the kernels.Throughout this paper, we will use the following notation to indicate the SNN architecture.
Layers are separated by 
and
spatial dimensions are separated by x
.
A convolution layer is represented by c
and
an aggregation layer is represented by a
.
For example 34x34x28c52a5o
represents a 4 layer SNN with 32342 input, followed by
8 convolution filters (55), followed by 22 aggregation layer
and finally a dense layer connected to 5 output neurons.
4.1 Poisson Spike Train
This is a simple experiment to help understand the learning process in SLAYER. A Poisson spike train was generated for 250 different inputs over an interval of
ms. Similarly a target spike train was generated using a Poisson distribution. The task is to learn to fire the desired spike train for the random spike inputs using an SNN with 25 hidden neurons.
The learning plots are shown in Figure 2. From the learning spike raster, we can see that initially there are output spikes distributed at random times (Figure 2(a) bottom). As learning progresses, the unwanted spikes are suppressed and the spikes near the desired spike train are reinforced. The learning finally converges to the desired spike train at the 739 epoch. The learning snapshot at epoch 20 (Figure 2(b)), shows how the error signal is constructed. The spike raster for input, hidden and output layer is shown at the bottom, The blue plots show the respective signals for output layer. The conversion from error signal, , to delta signal, , shows that the error credit assigned depends on the membrane potential value . Note the temporal credit assignment of error. A nonzero value of results in nonzero values of at earlier points in time, even if the error signal, , was zero at those times. Similar observations can be made for hidden layer signals. Out of 25 hidden layer signals, one is highlighted in black and rest are shown faded.
Dataset  Method  Architecture  Accuracy 
MNIST  Lee et al.Lee2016 
28x2880010 

Rueckauer et al.Rueckauer2017  SNN converted from standard ANN  
SLAYER 
28x2812c52a64c52a10o 

NMNIST  Lee et al.Lee2016 
34x34x280010 

SKIMCohen2016 
34x34x21000010 

DARTRamesh2017  DART feature descriptor  
SLAYER 
34x34x250050010 

SLAYER 
34x34x212c52a64c52a10o 

DVS Gesture  TrueNorthAmir2017  SNN (16 layers)  
SLAYER  SNN (8 layers)  
TIDIGITS  SOMSNNWu2016  MFCCSOMSNN  
Tavanaei et al.Tavanaei2017  Spiking CNN and HMM  
SLAYER  MFCCSOM, 48450050011 
4.2 MNIST Digit Classification
MNIST is a popular machine learning dataset. The task is to classify an image containing a single digit. This dataset is a standard benchmark to test the performance of a learning algorithm. Since SLAYER is a spike based learning algorithm, the images are converted into spike trains spanning
ms using Generalized Integrate and Fire Model of neuronPillow2005 . Standard split of 60,000 training samples and 10,000 testing samples was used with no data augmentation. For classification, we use the spike counting strategy. During training, we specify a target of 20 spikes for the true neuron and 5 spikes for each false neuron over the ms period. During testing, the output class is the class which generates the highest spike count.The classification accuracy of SLAYER for MNIST classification is listed in Table 1 along with other SNN based approaches. We achieve testing accuracy of on the network which is the best result for completely SNN based learning. Although this accuracy does not fare well with state of the art deep learning methods, for an SNN based approach it is a commendable result.
4.3 NMNIST Digit Classification
The NMNIST datasetOrchard2015 consists of MNIST images converted into a spiking dataset using a Dynamic Vision Sensor (DVS) moving on a pantilt unit. Each dataset sample is ms long, and 3434 pixels big, containing both ‘on’ and ‘off’ spikes. This dataset is harder than MNIST because one has to deal with saccadic motion. For NMNIST training, we use a target of 10 spikes for each false class neuron and 60 spikes for the true class neuron. The output class is the one with greater spike count. The training and testing separation is the same as the standard MNIST split of 60,000 training samples and 10,000 testing samples. The NMNIST data was not stabilized before feeding to the network.
The results on NMNIST classification listed in Table 1 show that SLAYER learning surpasses the current reported state of the art result on NMNIST dataset by Lee et. al.Lee2016
with a comparable number of neurons.
However, the CNN architecture trained with SLAYER achieves the best result.
An SNN with architecture 34x34x250050010
was also trained with no delay learning applied.
All the axonal delays were set to zero.
It resulted in testing accuracy of , an increase in accuracy by .
4.4 DVS Gesture Classification
The DVS GestureAmir2017 dataset consists of recordings of 29 different individuals performing 10 different actions such as clapping, hand wave etc. The actions are recorded using a DVS camera under three different lighting conditions. The problem is to classify the action sequence video into an action label. For training we set a target spike count of 30 for false class neurons and 180 for the true class neuron. Samples from the first 23 subjects were used for training and last 6 subjects were used for testing.
The results for DVS Gesture classification are listed in Table 1. SLAYER achieves a very good testing accuracy of on average. In SLAYER training as well as testing, only the first s out of s of action video for each class were used to classify the actions. For speed reasons, the SNN was simulated with a temporal resolution of ms. Despite these shortcomings, the accuracy results are excellent, surpassing the testing accuracy of TrueNorth trained with EEDNAmir2017 . With output filtering, the TrueNorth accuracy can be increased to . Nevertheless, SLAYER is able to classify with a significantly less number of neurons and layers. The TrueNorth approach uses additional neurons before the CNN classifier for preprocessing, whereas in SLAYER, the spike data from the DVS is directly fed into the classifier.
4.5 TIDIGITS Classification
TIDIGITSLeonard1993
is an audio classification dataset containing audio signals corresponding to digit utterances from ‘zero’ to ‘nine’ and ‘oh’. In this paper, we use audio data converted to spikes using the MFCC transform followed by a Self Organizing Map (SOM) as described in
Wu2016 .For training, we specify a target of 5 spikes for false classes and 20 spikes for the true class. The dataset was split into 3950 training samples and 1000 testing samples.
The results for TIDIGITS classification are listed in Table 1. SLAYER significantly improves upon the testing accuracy results of SNN based approach using SOMSNNWu2016 . However, the best reported accuracy for TIDIGITS classification AbdollahiLiu2011 is using MFCC and HMMGMM approach (non spiking). The accuracy of SLAYER, however, is still competitive at .
5 Discussion
We have proposed a new error backpropagation for SNNs which properly considers the temporal dependency between input and output signals of a spiking neuron, handles the nondifferentiable nature of the spike function, and is not prone to the dead neuron problem. The result is SLAYER, a learning algorithm for learning both weight and axonal delay parameters in an SNN.
We have demonstrated SLAYER’s effectiveness in achieving state of the art accuracy for an SNN on spoken digit and visual digit recognition as well as visual action recognition.
During training, we require both true and false neurons to fire, but specify a much higher spike count target for the true class neuron. This approach prevents neurons from going dormant and they easily learn to fire more frequently again when required.
We believe that SLAYER is an important contribution towards efforts to implement backpropagation in an SNN. The development of a CUDA accelerated learning framework for SLAYER was instrumental in tackling bigger datasets in SNN domain, although they are still not big when compared to the huge datasets tackled by conventional (nonspiking) deep learning.
Neuromorphic hardware such as TrueNorthMerolla2014 , SpiNNakerFurber2014 , Intel Loihidavies2018loihi show the potential of implementing large spiking neural networks in an extremely low power chip. These chips usually do not have learning mechanism, or have a primitive learning mechanism built into them. Learning must typically be done offline. SLAYER has good potential to serve as an offline training system to configure a network before deploying it to a chip.
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