empd: an efficient membrane potential driven supervised...

IEEE TRANSACTIONS ON COGNITIVE AND DEVELOPMENTAL SYSTEMS, VOL. 10, NO. 2, JUNE 2018 151

EMPD: An Efficient Membrane PotentialDriven Supervised Learning Algorithm

for Spiking NeuronsMalu Zhang, Hong Qu, Member, IEEE, Ammar Belatreche, Member, IEEE, and Xiurui Xie

Abstract—The brain-inspired spiking neurons, considered asthe third generation of artificial neurons, are more biologicallyplausible and computationally powerful than traditional artificialneurons. One of the fundamental research in spiking neurons is totransform streams of incoming spikes into precisely timed spikes.Due to the inherent complexity of processing spike sequences, theformulation of efficient supervised learning algorithm is difficultand remains an important problem in the research area. Thispaper presents an efficient membrane potential driven (EMPD)supervised learning method capable of training neurons to gen-erate desired sequences of spikes. The learning rule of EMPD iscomposed of two processes: 1) at desired output times, the gradi-ent descent method is implemented to minimize the error functiondefined as the difference between the membrane potential andthe firing threshold and 2) at undesired output time, synapticweights are adjusted to make the membrane potential below thethreshold. For efficiency, at undesired output times, EMPD cal-culates the membrane potential and makes a comparison withfiring threshold only at some special time points when the neuronis most likely to cross the firing threshold. Experimental resultsshow that the proposed EMPD approach has higher learningefficiency and accuracy over the existing learning algorithms.

Index Terms—Membrane potential driven, spiking neurons,supervised learning.

I. INTRODUCTION

TRADITIONAL rate coded artificial neuron modelsencode information through their mean rates of actionpotential firing. However, it is unlikely that rate-based codingcan convey all the information related to a rapid process-ing task, such as color, vision, odor, and sound qualityprocessing [1]–[4]. With the emerging evidence of precisespike-timing neural activities have been observed in manybrain regions, including the retina [5]–[7], the lateral genic-ulate nucleus [8] and the visual cortex [9], the view thatinformation is embedded in the spatiotemporal structure of

Manuscript received June 27, 2016; revised November 21, 2016; acceptedJanuary 7, 2017. Date of publication January 11, 2017; date of current versionJune 8, 2018. This work was supported by the National Science Foundationof China under Grant 61573081, Grant 61273308, and Grant 61370073.

M. Zhang, H. Qu, and X. Xie are with the School of Computer Scienceand Engineering, University of Electronic Science and Technology of China,Chengdu 610054, China (e-mail: [email protected]).

A. Belatreche is with the Department of Computer and Information Science,Faculty of Engineering and Environment, Northumbria University, Newcastleupon Tyne NE1 8ST, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCDS.2017.2651943

spikes rather than in mean firing rates has received increas-ing attention [10], [11]. These findings have led to a new wayof simulating neural networks based on spiking neurons whichencode information by the firing times of spikes [12]–[14]. Therunning mechanism of spiking neurons is more biologicallyplausible and it has been demonstrated that networks of spik-ing neurons can offer significantly computational advantagesover rate encoding scheme [15]–[18]. Although spiking neu-rons show promising capabilities in achieving a performancesimilar to living brains, powerful computing capability, andbroad application prospects are not fully exploited. One of thereasons is that the intrinsic complexity of processing spikesequences might limit the usage of networks of spiking neu-rons. This leads us to the necessity to develop efficient learningalgorithms.

Compared with unsupervised learning, a supervised learn-ing algorithm could improve the learning efficiency with thehelp of an instructor signal. In addition, there is biological evi-dence that the brain performs instruction-based learning [19].The most documented evidence for supervised learning in thecentral nervous system (CNS) comes from the studies on thecerebellum and the cerebellar cortex [20], [21]. Instruction-based or supervised learning algorithms are widely used intraditional artificial neural networks, such as the perceptronlearning rule and the BP algorithm. However, due to theintrinsic complexity of processing spike sequences, the tradi-tional supervised learning methods cannot be directly appliedto the networks of spiking neurons. To train the spiking neu-rons to generate desired sequences of spikes, many supervisedlearning algorithms have been proposed. They can be subdi-vided broadly into two classes: 1) spike-driven methods and2) membrane potential-driven methods.

Spike-driven methods use the desired and actual out-put spikes as the relevant signal for synaptic changes.SpikeProp [22] and the multispike learning algorithm [23] arethe most typical ones of spike-driven learning algorithms. Theyconstruct error functions directly by the difference betweenthe desired and actual output spikes, then update the synapticweights based on gradient descent. ReSuMe [24] is anotherspike-driven method, in which synaptic changes are drivenby the joint effect of two opposite process: 1) strengthen-ing synaptic weights by STDP and 2) weakening the synapticweights by anti-STDP. To enhance the learning performanceof ReSuMe, DL-ReSuMe [25] is proposed to merge thedelay shift approach and ReSuMe-based weight adjustment.

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152 IEEE TRANSACTIONS ON COGNITIVE AND DEVELOPMENTAL SYSTEMS, VOL. 10, NO. 2, JUNE 2018

The biggest difference between PSD [26] and ReSuMe isthat they apply different learning windows. Both ChronotronE-learning rule [27] and the SPAN rule [28] try to minimizethe distance between the desired and actual output spike trains.The distance in Chronotron E-learning rule is defined by theVictor and Purpura metric [29], while in the SPAN rule thedistance is based on a metric similar to the van Rossum met-ric [30]. The common disadvantages of the above mentionedalgorithms are that the learning efficiency and accuracy arerelatively low.

To improve the learning efficiency and accuracy, membranepotential-driven methods emerged recently with the typicalexamples of Tempotron [31], PBSNLR [32], and HTP [33].Compared with spike-driven methods, they take an entirely dif-ferent route in which the relevant signal for synaptic change isthe postsynaptic membrane potential rather than spike times.The Tempotron implements a gradient descent dynamics thatminimizes an error defined as the difference between the max-imum membrane potential and the firing threshold. However,as the Tempotron is designed mainly for pattern recognition,it is unable to produce precise spikes. PBSNLR and HTP per-form a perceptron classification on discretely sampled timepoints of the membrane potential, with the aim to keep mem-brane potential below threshold at undesired spike times andto make sure a threshold crossing at desired spike times [34].As they are based on perceptron learning rule, in theory, ifthe sampled time points of the membrane potential are notlinearly separable, the desired output spike train cannot belearned successfully.

The common disadvantages of these methods are that thelearning efficiency and accuracy are relatively low, whichweakens their ability to solve the real-time and the compli-cated problems. The two disadvantages of the existing methodsprompted us to search for a supervised learning method withhigher learning efficiency and accuracy. Here we present anefficient membrane potential driven (EMPD) learning algo-rithm capable of training neurons to generate desired spikesequences efficiently. At desired output times, to make surea threshold crossing, EMPD employs an error function basedon the voltage difference between output neuron’s membranepotential and the firing threshold, then updates the synap-tic weights by gradient descent rule. On the other hand, thesynaptic weights are adjusted to make the membrane poten-tial below the firing threshold at undesired output time. Unliketraditional supervised learning algorithms of spiking neurons,which require to calculate the membrane potential and makea comparison with the firing threshold continuously. Duringthe undesired output time, EMPD just monitors the membranepotential at some special time points when the neuron is mostlikely to output a wrong spike. This paper is extended from ourpreliminary work [35] by adding more comparative and ana-lytic studies. The experimental results show that the proposedmethod has higher learning accuracy and efficiency over theexisting learning methods, so it will significantly promote theresearches and applications of spiking neurons in the future.

The rest of this paper is organized as follows. In Section II,the spiking neuron model used in this paper is formallydefined. Our learning method is shown in Section III.

(a)

(b)

Fig. 1. Dynamic of the neuron model. (a) Illustration of the spike responsefunction ε(t) with different values of τ . (b) Left: Examples of input patterns.There are two patterns (blue and green) and each spike fired by presynapticneuron is denoted by a dot. Right: Membrane potential traces of the two inputpatterns. Pattern one (in blue) fires two spikes and pattern two does not fire.

In Section IV, some comparison experiments are given toinvestigate the learning performance of the proposed learn-ing method. Discussion and conclusion are presented inSections V and VI, respectively.

II. NEURON MODEL

There are many spiking neuron models that aim to explainthe running mechanism of a biological neuron [12]. The spike-response model (SRM) is a generalization of the integrate-and-fire model and can give a faithful description of biologicalneurons [12], [23], [32]. In this paper, a simplified SRM modelis employed to introduce our method.

The membrane potential of the neuron is represented by avariable u. When there is no spike transmission from the presy-naptic neurons, the variable u is at its resting value, urest = 0.When each spike arrives, a postsynaptic potential (PSP) will beinduced in the neuron. The PSP induced by an incoming spikeis determined by the spike response function ε, defined as

ε(

t − tji)

=⎧⎨⎩

t−tjiτ

exp

(1 − t−t

ji

τ

)if t − tji > 0

0 if t − tji ≤ 0(1)

where tji is the jth spike of presynaptic neuron i, and τ isthe time decay constant determining the shape of the responsefunction [see Fig. 1(a)]. The membrane potential u(t) of theneuron is defined as the sum of the PSPs from the incomingspikes. If the membrane potential reaches the firing threshold,an output spike is triggered. After firing, the membrane poten-tial u(t) resets to the rest potential urest = 0, and stays at theresting level for a time period Ra. After the output spike t fr ,the membrane potential must be calculated by using the inputspikes whose arrival time are after t fr +Ra [23], [32], yielding

u(t) =∑

i

∑j

tji>tfr +Ra

ωiε(

t − tji)

+ urest (2)

ZHANG et al.: EMPD SUPERVISED LEARNING ALGORITHM FOR SPIKING NEURONS 153

where ωi is the synaptic weight of the ith synapse. Thedynamics of the neuron model are illustrated in Fig. 1.

III. EFFICIENT MEMBRANE POTENTIALDRIVEN LEARNING RULE

EMPD divides the running time of a neuron intotwo classes: desired output time Td (Td = {Td(1),Td(2), Td(3), . . .}) and undesired output time NTd. The learn-ing rule of EMPD is composed of two processes: 1) adjustsynaptic weight to make a threshold crossing at Td and2) adjust synaptic weight to keep the membrane potentialbelow threshold at NTd. In the following, we will introducethese two processes, respectively.

A. Learning Rule at TdAt Td, to ensure a spike firing, EMPD implements a gra-

dient descent dynamics that minimizes an error defined asthe difference between the membrane potential and the firingthreshold as

ETd =1

2[u(t) − ϑ]2 if t ∈ Td (3)

where u(t) is the membrane potential and ϑ is the firing thresh-old. During the learning process, as shown in Fig. 2(c) and (d),when we calculate the postsynaptic membrane potential u(t),the tfr in (2) is not the actual but the desired output spiketime. As a wrong output spike (a vanishing spike or a spuriousfiring) can dramatically affect the subsequent membrane poten-tial, which will provide a wrong membrane potential signal forsynaptic change at later times [32], [33].

Then, the weight update rule based on gradient descent atTd is expressed as

�ωi = −β1 ∂ETd∂ωi

(4)

where β1 is the learning rate, and ωi is the synaptic weight ofsynapse i.

B. Learning Rule at NTdTo prevent the neuron emitting a wrong spike at NTd, mem-

brane potential are required to be below the firing thresholdduring NTd period. To achieve this purpose, the existing learn-ing methods calculate the membrane potential and make acomparison with the firing threshold continuously, which isvery time-consuming. EMPD does not monitor the membranepotential continuously, but at some special time points whenthe membrane potentials are most likely to cross the threshold.Here after, we call these special time points as MPs (monitortime points). In the following, we first introduce the ways ofobtaining the MPs, then we describe the learning rule at NTd.

1) Solution of MPs: The membrane potentials are mostlikely to cross the threshold at the extrema points of the func-tion u(t), and the extrema points occur only at stationary pointsand nonderivable points. To get the MPs, we should solve thestationary points and nonderivable points of the function u(t)during the running period of a neuron.

As shown in Fig. 3, assuming that there are m input spikesbetween Td(i) and Td(i + 1), where Td(i) and Td(i + 1) are

(a)

(b)

(c)

(d)

(e)

Fig. 2. Illustration of EMPD learning rule. (a) Input spike pattern, each spikefired by presynaptic neuron is denoted by a dot. (b) Membrane potential traceof a neuron before learning. Actual output spike times and desired output timesare marked by blue bars and red bars, respectively. (c) During the learningprocess, the neuron does not spike at threshold crossing and the membranepotential are forced at the desired times. The dots on membrane potential tracedepict the MPs (monitor time points). (d) Voltage trace of the neuron afterone learning epoch with EMPD. (e) Membrane potential trace after successfullearning. The trained neuron emit spikes at desired times and the membranepotential is below threshold at NTd .

Fig. 3. Illustration of the calculation of MPs. Td(i) and Td(i + 1) are anyof two adjacent desired output times. tn (n = 1, 2, 3, . . .) is the firing time ofa input spike. tn−1 and tn are any of two adjacent time points. There are minput spikes between Td(i) and Td(i + 1).

any of two adjacent desired output times. Here, we illustratethe calculation of MPs within the period [tn−1, tn), where tn−1and tn are two adjacent input spikes. MPs in other periods canbe obtained in the same way. The membrane potential u(t)between tn−1 and tn can be calculated as follows:

u(t) =n−1∑N=1

ωtN ε(t − tN) + urest, t ∈[tn−1, tn

](5)


ωtN represents the weight of the synapse which transmits thetN spike. In the following, tN not only indicates the firing timeof the input spike but also represents the spike itself. Takingthe derivative of μ(t) with respect to t

u′(t) =n−1∑N=1

ωtN

τexp

(1 − t − tN

τ

)(1 − t − tN

τ

). (6)

To get the possible stationary points, we can set

u′(t) = 0 (7)By combining (6) and (7), we get

t =

n−2∑N=1

ωtN

ωtn−1exp

(tN − tn−1

τ

)(tN + τ) + τ + tn−1

n−2∑N=1

ωtN

ωtn−1exp

(tN − tn−1

τ

)+ 1

. (8)

If t ∈ [tn−1, tn), we add it to the set of MPs. As tn−1 isnonderivable point of u(t), it is added into the MPs.

2) Update Rule at NTd: To make the membrane potentialbelow threshold at undesired output time, error function isconstructed as

ENTd = u(t) − ϑ if t ∈ MPs and u(t) ≥ ϑ (9)where u(t) is the membrane potential and ϑ is the firingthreshold.

Applying the gradient descent method to minimize the costlead to the following updating rule:

�ωi = −β2 ∂ENTd∂ωi

. (10)

Fig. 2 illustrates the learning rule of EMPD learning rule.

IV. EXPERIMENT RESULTS

In this section, to investigate the performance of EMPD,some experiments are conducted. In part A, the learningcapability of EMPD is investigated. In part B to part D,we investigate the effect of different factors on the learningperformance. The factors include the spike trains total timeduration (Tt), the number of the synaptic inputs (Ns), the inputspike frequency (Fin), and the output spike frequency (Fo). Inpart E and F, the robustness and memory capacity of EMPDare investigated, respectively. In part G, we investigate theeffect of MPs on the learning performance of EMPD andPBSNLR. Finally, the performance of the proposed EMPDmethod in practical applications is further demonstrated on aclassification task.

A. Learning Sequence of Spikes

In this experiment, a neuron with 400 synaptic inputs istrained to emit a desired sequence of spikes with a length of500 ms. The mean frequency of the input spike trains andthe desired output spike train are set to Fin = 5 Hz andFo = 100 Hz, respectively. The initial synaptic weights areselected as the uniform distribution in the interval [0, 0.025].The learning rates are set as β1 = 0.02 and β2 = 0.005. Fig. 4shows the learning results of EMPD.

(a)

(b)

(c)

(d)

Fig. 4. The learning process and performance of EMPD. (a) Running processof the neuron with the initial synaptic weights. (b) Running process of theneuron with the learned weights. (c) Illustration of the learning process, whichincludes the desired output spike train denoted by ◦ and the actual output spiketrains after each learning epoch denoted by •. (d) Learning accuracy versuslearning epoch.

Fig. 4(a) shows the running process of the spiking neuronwith the initial synaptic weights, in which the actual outputspikes (represented by short blue lines) are very different fromthe desired output spikes (represented by short red lines). Afterlearning with EMPD, as shown in Fig. 4(b), the trained neu-ron output spikes precisely at desired output times. Fig. 4(c)illustrates the evolution of the firing patterns generated by theneuron in consecutive learning epochs. At first, the actual out-put spikes are very different from the desired output spikes.After several learning epochs, the gap gets smaller and smaller.At about 20 epochs the actual output spike train is the same asthe desired one. In order to quantitatively evaluate the learn-ing performance, we use a correlation-based measure C [36](C is assumed 0 for uncorrelated spike trains and 1 for per-fectly matched firing patterns; see the Appendix for details).The measure C plotted as a function of learning epoch isshown in Fig. 4(d), which indicates that the initial value ofC is close to 0.2, and C increases to 1 after about 20 learningepochs.


(a)

(b)

Fig. 5. Comparison of learning performance when the length of spike trainsincrease gradually. (a) Learning accuracy of different methods. (b) Learningefficiency of different methods. Fin = 10 Hz, F0 = 10 Hz, and Ns = 400.

B. Effect of the Spike Trains Total Time Duration (Tt)

In these simulations, a neuron with 400 synaptic inputs istrained to reproduce a desired sequence of spikes. Every inputspike train and the desired output spike train are generatedaccording to a homogeneous Poisson process with rates of10 and 100 Hz, respectively. The length of the desired out-put spike trains varies from 100 to 1000 ms with an intervalof 100 ms. For each Tt value, 20 experiments are carriedout for different input and desired output spike trains. Theexperimental results are shown in Fig. 5.

First, we investigate the learning accuracies of EMPD,PBSNLR, and ReSuMe. For each Tt, the learning is continuedcarried out for a total of 1000 learning epochs and the maxi-mum accuracy C is calculated and is shown in Fig. 5(a). It hasbeen seen that as the length of spike trains gradually increases,the learning accuracy of these three methods decreases. Thelearning accuracy curve of EMPD is significantly higher thanthat of PBSNLR and ReSuMe when the value of Tt variesfrom 100 to 1000 ms. For example, when Tt = 400, thelearning accuracy of EMPD is 1, while the learning accuracyof PBSNLR and ReSuMe are 0.99 and 0.983, respectively.Fig. 5(b) shows the average learning time required to complete1000 learning epochs. The learning time of EMPD, PBSNLR,and ReSuMe increases when the spike train length increasesgradually, however, the learning time of EMPD remains signif-icantly less than that of PBSNLR and ReSuMe counterparts.Therefore, the learning efficiency our method is clearly betterthan that of PBSNLR and ReSuMe.

C. Effect of the Number of the Synaptic Inputs (Ns)

In this part, we investigate the effect of the number of thesynaptic inputs. Every input spike train and the desired outputspike train are generated according to a homogeneous Poissonprocess with rates 10 and 100 Hz, respectively. The length ofthe input and desired output spike train is 300 ms. Ns varies

(a)

(b)

Fig. 6. Comparison of learning performance when the number of the synapticinput increases gradually. (a) Learning accuracy of different methods. (b)Learning efficiency of different methods. Fin = 10 Hz, F0 = 10 Hz, andTt = 300 ms.

from 100 to 500 with an interval of 50. The experimentalresults are shown in Fig. 6.

Fig. 6(a) shows that while the number of synaptic inputs,Ns, gradually increases from 100 to 500 (the x-axis), the learn-ing accuracies of these three methods also increase. However,the proposed EMPD method can quickly reach a very highlearning accuracy within a relatively small number of synap-tic inputs, and the learning accuracy of EMPD remains higherthan that of both PBSNLR and ReSuMe in most cases. Forinstance, when Ns = 250, the learning accuracy of EMPDreaches the highest possible value (C = 1), while the learn-ing accuracies of PBSNLR and ReSuMe are 0.985 and 0.978,respectively. Furthermore, once EMPD reaches the highestpossible accuracy level of 1, it remains steady irrespective offurther increases in the number of synaptic inputs. Fig. 6(b)shows the learning time required to complete 1000 learningepochs for each method. It clearly illustrates that the learn-ing time of EMPD is significantly less than the other twomethods.

D. Effect of Fin, Fo, and TtThe following experiments aim to investigate the effects of

Fin, Fo, and Tt. In these experiments, the number of synap-tic inputs, Ns, is set to 400. Tt varies from 200 to 1000 mswith the interval of 200 ms. F0 varies from 20 to 100 Hzwith the interval of 20. Fin is varied between 10 and 40 Hz inincrements of 10 Hz. The learning is continued for 1000 learn-ing epochs, and the maximum learning accuracy C of EMPD,PBSNLR, and ReSuMe are plotted in Figs. 7–9, respectively.

From Figs. 7–9, EMPD, PBSNLR, and ReSuMe have high-est learning accuracies for the lowest values of Tt and Fo(Tt = 200 ms and Fo = 20 Hz), and there is a trend thatthe performances of these three methods decrease with theincrease of Tt and Fo. When Tt = 1000 ms and F0 = 100 Hz,


Fig. 7. Learning performance of EMPD for different values of the inputspike train frequency Fin, the output spike train frequency Fo, and the lengthof the spike trains Tt .

Fig. 8. Learning performance of PBSNLR for different values of the inputspike train frequency Fin, the output spike train frequency Fo, and the lengthof the spike trains Tt .

EMPD, PBSNLR and ReSuMe reach their lowest values ofperformance. On the other hand, as shown in Figs. 7–9,the area in which EMPD achieve high performance is widerthan that of PBSNLR and ReSuMe. Therefore, the learningaccuracy of EMPD is better than PBSNLR and ReSuMe.

E. Robustness to Noise

In the previous experiments, we only consider the simplecase where the neuron is trained to reproduce target sequencesof spikes under noise-free condition. However, noise widelyexists in the CNS and the neural response can be significantlyaffected by noise [37]–[39]. In order to evaluate the noiserobustness of the proposed EMPD, two noise cases are con-sidered: 1) background noise on the membrane potential and2) presynaptic spike time jitter. In the following experiments, aneuron with 400 synaptic inputs is trained to output the desiredspike train with a length of 200 ms. Every input spike train

Fig. 9. Learning performance of ReSuMe for different values of the inputspike train frequency Fin, the output spike train frequency Fo, and the lengthof the spike trains Tt .

and the desired output spike train are Poisson spike trains withrates Fin = 10 and Fo = 50 Hz, respectively.

1) Training and Recall With Noise on the MembranePotential: In this case, background noise on the membranepotential is considered. The training is performed accord-ing to two scenarios: training under noise-free conditions(deterministic training) and training in the presence of noise(noisy training). After training, the trained neuron is sub-jected to background noise simulated by injecting Gaussianwhite-noise to the neuron membrane potential. The noisemean value is set to 0, and its variance σb is systematicallyincreased in the range of [0.05, 0.5] mV. For each value σb,the similarity between the desired and the observed outputtrain, C, is calculated. The experimental results are shownin Fig. 10.

As shown in Fig. 10(a), the neuron trained under noise-freecondition (noise-free training) is significantly affected by theadded noise, which is reflected in a decrease in accuracy as aresult of noise increase. The measure C of ReSuMe drops moresharply than that of PBSNLR and EMPD. That is, the trainedsynaptic weights obtained by EMPD and PBSNLR are morerobust to membrane potential noise than those of ReSuMe.Furthermore, Fig. 10(b) shows the noise robustness of the dif-ferent methods with noise training. Compared with Fig. 10(a),the anti-noise capability of different methods will be improvedif noise training is performed. In addition, the anti-noise capa-bility of EMPD and PBSNLR are comparable but still strongerthan that of ReSuMe.

2) Training and Recall With Input Spike Time Jitter: Inthis case, the effect of input noise is considered by jittering theinput spike times The jitter intervals are randomly drawn froma Gaussian distribution with mean 0 and variance σj ∈ [0.5, 5]ms. The resulting plots of C are presented in Fig. 11.

Fig. 11 shows that the robustness of EMPD, PBSNLR,and ReSuMe can be improved through noise training. Thenoisy training enables the neuron to reproduce desired spikesmore reliably. On the other hand, the measure C of ReSuMedrops more sharply than that of EMPD and PBSNLR in both


(a)

(b)

Fig. 10. Robustness of the EMPD, PBSNLR, and ReSuMe against back-ground voltage noise. (a) Robustness of the neuron trained under noise-freecondition. (b) Robustness of the neuron trained under noise condition.

(a)

(b)

Fig. 11. Robustness of the EMPD, PBSNLR, and ReSuMe against jitter-ing noise. (a) Robustness of the neuron trained under noise-free condition.(b) Robustness of the neuron trained under noise condition.

noise-free training and noise training. Therefore, the synapticweights obtained by the EMPD and PBSNLR are more robustthan those of ReSuMe against the jittering noise.

Fig. 12. Memory capacity of EMPD, PBSNLR, and ReSuMe with differentnumber of input synapses.

F. Memory Capacity

In this section, the memory capacity of the proposedEMPD rule is investigated. As used for the tempotron [31],the PSD [26], and the SPAN [28], the memory capacity isdescribed in terms of the load factor α which is defined as

α = pn

(11)

where p is the number of input patterns and n is the numberof input synapses.

The p input patterns are randomly generated, where eachpattern contains n spike trains and each train has a singlespike. The patterns are randomly and evenly assigned to cdifferent classes. Here, we choose c = 4 for this experiment.The neuron is trained to emit a single spike at a specified timefor patterns from each category. The target spike times forthe four classes are set to 40, 80, 120, and 160, respectively.An input spike pattern is considered to have been correctlymemorized, if the actual output spike is within 3 ms of the cor-responding target spike. The experiment is repeated on threenetwork architectures with 200, 400, and 600 input synapses,respectively.

Fig. 12 shows the experimental results obtained for differentnumbers of the input synapses. It can be seen from the fig-ure that the number of input synapses has little effect on thememory capacity. In addition, the memory capacity of EMPDand PBSNLR are comparable (α ≈ 0.2) but remain higherthan the capability of ReSuMe (α ≈ 0.1).

G. Effects of MPs

In this section, several experiments are conducted to inves-tigate the effect of using MPs in EMPD and PBSNLR.

1) Effect of Using MPs in EMPD: In the previous experi-ments of EMPD, MPs are selected as the monitor time pointsin NTd. To investigate the effect of using MPs in EMPD, inthe following experiments, the monitor time points in NTd areselected according to three strategies.

Strategy 1: MPs calculated by (8) are selected as themonitor time points.

Strategy 2: The monitor time points are randomly selectedin the total spike train, and the number ofmonitor time points is the same as in strategy 1.

Strategy 3: Every time point is selected as the monitor timepoint.


(a)

(b)

Fig. 13. Effect of MPs in EMPD. (a) Learning accuracy of different strategies.(b) Learning time for one epoch.

In the following experiments, a neuron with 400 synap-tic inputs is trained to output a desired spike train with thelength 400 ms. Every input spike train and the desired out-put spike train are generated according to the homogeneouspoisson process with Fin = 5 and Fo = 100 Hz, respectively.Each experiment is repeated for 20 trials for different input anddesired output pairs. We employ the strategy proposed in [25]where each value of C is replaced by its previous value if thereis a drop in the value compared with its previous one. Thisstrategy results in a smooth C curve. The obtained learningresults are shown in Fig. 13.

Fig. 13(a) shows a comparison between the learning accu-racies obtained for different strategies. It can be seen that bothstrategies 1 and 3 are able to reach the maximum C = 1, whilestrategy 2 can only reach C = 0.9681. Furthermore, strategy 2is unable to reach the maximum value of C = 1 even whenallowed to run for more epochs. Fig. 13(b) shows the averagecomputation time for one learning epoch. The computationtime required by strategy 1 is much less than the time requiredby strategy 3. Fig. 13(b) shows that strategy 2 offers littleadvantage over strategy 1 in terms of learning time. However,the learning accuracy of strategy 2 is much lower than that ofstrategy 1.

2) Effect of MPs in PBSNLR: Experimental results demon-strate that the learning performance of PBSNLR is related tothe selection of negative samples [32]. In this part, we inves-tigate the effect of MPs in PBSNLR. The negative samples ofPBSNLR are selected according to three strategies.

Strategy 1: MPs calculated by (8) are selected as thenegative samples.

Strategy 2: The negative samples are randomly selected inthe total spike train, and the number of negativesamples is the same as in strategy 1.

Strategy 3: Every time point is selected as the negativesample.

As shown in Fig. 14(a), the learning accuracy of strat-egy 1 (PBSNLR combined with MPs) and strategy 3 is almost

(a)

(b)

Fig. 14. Effect of MPs in PBSNLR. (a) Learning accuracy of differentstrategies. (b) Learning time for one epoch.

Fig. 15. General structure and information process of the SNN. It containsthree functional parts: encoding, learning, and recognition. The structure ofthe network is adapted from [40].

the same. However, the learning efficiency of strategy 1 isbetter than that of strategy 3. The computing time requiredstrategy 3 is almost 4 times longer than that of strategy 1.Strategy 2 offers little advantage over strategy 1 in terms ofcomputing time and its learning accuracy is much lower thanthat of strategy 1. The obtained results indicate that MPs canbe used by PBSNLR to improve the learning efficiency andaccuracy.

H. Classification

Spiking neural networks have been successfully used indifferent applications [15], [40]–[45]. Recently, a computa-tional model in which visual information is encoded intoprecisely timed action potentials has been applied in patternrecognition [40]. It consists of three parts: 1) encoding part;2) supervised learning part; and 3) readout part. The structureof the model is illustrated in Fig. 15. In this experiment, weadopt this computational model to evaluate the capability ofthe proposed learning method in practical applications.

In the encoding part, external stimuli are convertedinto spatiotemporal spike patterns. An optical character


Fig. 16. (a) OCR samples. (b) Illustration of the phase encoding schema.Each encoding unit is assigned with an SMO and an input x. The encodingschema is adapted from [26] and [46].

recognition (OCR) task is considered in this experiment whereimages of digits 0–9 are used. Each image has a size of20 × 20 black/white (B/W) pixels. Sample images are shownin Fig. 16(a). In the encoding part, phase encoding methodis used to convert the images into spatiotemporal spike pat-terns [26], [46]. The mechanism of the phase encoding isshown in Fig. 16(b). Each encoding unit consists of a positiveneuron (Pos), a negative neuron (Neg), and an output neuron.Each encoding neuron is assigned to a pixel and a subthresh-old membrane potential oscillation (SMO). The correspondingSMO for the ith encoding neuron is described as

SMOi = A cos(ωt + φi) (12)where A is the magnitude of the SMOs, ω is the phase angularvelocity of the oscillations, and φi is the phase shift of the ithencoding neuron. φi is defined as

φi = φ0 + (i − 1) · �φ (13)where φ0 is the reference initial phase and �φ is the constantphase difference between nearby encoding neurons.

As shown in Fig. 16(b), the Pos and Neg neurons onlyrespond to positive and negative active potentials, respectively.The bottom part of Fig. 16(b) illustrates the dynamics of thePos and Neg neurons. The B/W pixel will cause a down-ward/upward shift from the SMO. The Pos or Neg neuronwill emit a spike if the potential crosses the threshold line(Posthr and Negthr). Through adjustment of parameter A, theamount of shift and the value of the threshold, we set the spiketo occur at the peaks of the oscillation. The firing of eitherthe Pos or the Neg neuron will immediately cause the firingof the output neuron. Therefore, the encoding units will out-put a spike at one phase for a white pixel and another shiftedphase of 180◦ for a black pixel. (For more details about phasecoding, see [26], [46]).

Fig. 17. Phase encoding results of a given image sample. Each dot denotesa spike.

Fig. 18. Learning performance of digit 8. (a) Output spike signals of thelearning neuron corresponding to digit 8. (b) Membrane potential trace of theneuron after learning.

The learning part of the spiking neural network is composedof one layer of 10 spiking neurons, with each learning neuroncorresponding to one category. Each learning neuron is trainedto fire a desired sequence of spikes ([40, 80, 120, 160] ms)when a corresponding pattern is present, and not to spike whenother patterns are presented.

In the recognition part, the relative confidence criterion isused for decision making, where the input pattern will bedecided by one of the neurons that generates the most similarspike train to the target spike train.

1) Learning Performance: After phase coding, differentimages can be converted into corresponding spatiotemporalspike patterns. Fig. 17 demonstrates an encoding result of agiven image sample, in which the output spikes are sparselydistributed over the encoding time window. Experimentalresults shows that the EMPD can successfully accomplish thelearning task with 200 learning epochs. To further illustratethe learning process of the EMPD learning rule, Fig. 18(a)shows the learning performance of digit “8.” The learningneuron corresponding to digit 8 can successfully produce thedesired spike train after about 180 learning epochs. Fig. 18(b)shows the membrane potential trace of the learned neuroncorresponding to digit 8.

2) Robustness to Noise: To study the noise robustnessof the EMPD classification, after learning, the reliability ofthe target recall is tested against two noise cases: 1) back-ground noise on the membrane potential and 2) input jitteringnoise. Figs. 19 and 20 show the classification accuracy of theEMPD against background voltage noise and jittering noise,respectively.

As shown in Figs. 19 and 20, the classification accu-racy of EMPD remains high when the noise level is lowand drops gradually with increasing noise levels. Even whenσb = 0.03 mV and σj = 3 ms, a high classification accuracyof above 90% can still be obtained.


Fig. 19. Robustness of EMPD against the background voltage noise.

Fig. 20. Robustness of EMPD against the jitter noise.

V. CONCLUSION

In this paper, we proposed a new supervised learning algo-rithm for spiking neurons to generate desired sequences ofspikes. In the experiments, the learning accuracy and learn-ing efficiency are obviously higher than that of PBSNLRand ReSuMe. Therefore, EMPD will significantly promote theresearches and applications of spiking neurons in the future.

The experimental results demonstrate that the learning timeof EMPD is shorter than that of PBSNLR. The reason forshorter learning time of EMPD is that, during the undesiredoutput time, EMPD just calculates the membrane potential andmake a comparison with firing threshold at some special timepoints. Another reason is that EMPD is based on gradientdescent, and the size of the weight changes is determinedby learning rate and the difference between the desired andthe actual membrane potential. However, the adjustment ofPBSNLR just relates to learning rate. Therefore, EMPD hashigher flexibility and better efficiency.

Compared to PBSNLR, another advantage of EMPD is thatthe less space complexity. In PBSNLR, all the PSPs inducedby every synapse at the times of all the training samples needto be calculated and stored before training [32]. Assumingthat ts is the time step which used to simulates the continuoustime, Pti is the sum of PSPs induced by all the spikes that havearrived through the ith synapse at t, and P[i][t] is the storagespace for all the PSPs induced by synapse i at time t. Theprocess of storaging PSPs is as follows.

For t = 0 : ts : TtFor i = 1 : 1 : Ns

Calculate the value of PtiEndFor

P[i][t]=Pti.EndFor

It is easy to find that t ∈ [1, Tt/ts] and i ∈ [1, Ns]. Therefore,the storage space of P[i][t] is Ns · Tt/ts. For example, whenTt = 1000, ts = 0.1, and Ns = 400, before learning, PBSNLRneeds 4×106 extra storage units. However, in our experiments,EMPD calculates the PSPs in every learning epoch. Therefore,EMPD does not need to store these quantities, and the spacecomplexity of EMPD is much better than that of PBSNLR.

In future work, we will study how to extend EMPD tothe whole network (three layers or more). It is expected thatsuch an approach would improve the application range andmemory capacity of spiking neurons, at the same time, wouldreduce the size of the networks required to complete particularapplications. Another interesting direction of future work is tosearch for efficient and biological plausibility input and out-put encoding methods for multiple spikes that yield the bestapplication performance.

APPENDIX A

CORRELATION-BASED METRIC

To quantitatively evaluate the learning performance, acorrelation-based measure of spike timing [36] is adopted tomeasure the similar degree between the desired and actual out-put spike trains. The metric is calculated after each learningepoch according to

C =−→vd · −→vo∣∣−→vd

∣∣∣∣−→vo∣∣ (14)

where −→vd and −→vo are vectors representing a convolution (indiscrete time) of desired and actual output spike trains with alow-pass Gaussian filter. −→vd ·−→vo is the inner product, and |−→vd |and |−→vo | are the Euclidean norms of −→vd and −→vo , respectively.

The Gaussian filter function with parameter σ is given by

f (t, σ ) = exp(−t2

2σ 2

). (15)

The measure C equals one for the identical spike trains anddecreases toward zero for loosely correlated spike trains.

APPENDIX B

LEARNING RULE OF PBSNLR ALGORITHM

The following form of PBSNLR algorithm is used in thispaper:

Wnewi =⎧⎨⎩

Woldi + βPti if dt = 1 and at = 0Woldi − βPti if dt = 0 and at = 1Woldi if d

t = at(16)

where β is the learning rate, and Pti is the sum of PSPs inducedby all the spikes that have arrived through the ith synapse at t.dt = 1 (or dt = 0) means t is the desired (or undesired) outputtime. at = 1 (or at = 0) means the neuron has fired (or not fire)a spike at t. In Section IV-H, the negative samples of PBSNLRare selected to three different strategies. In strategy 1, the neg-ative samples are selected as MPs, and every time point beforedesired output time is also added into monitor time points.


APPENDIX C

LEARNING RULES OF RESUME ALGORITHM

ReSuMe [24] is a supervised learning method that aimsto produce desired spike trains in response to the giveninput sequence. In this paper, the following form of ReSuMealgorithm is used:

d

dtωi(t) = [Sd(t) − So(t)]

[α +

∫ ∞0

W(s)Si(t − s)ds]

(17)

where W(s) is the learning window, W(s) = +A+·exp(−s/τ+)for s > 0, W(s) = 0 elsewhere. The parameters values usedin our simulations are α = 0.001, τ+ = 5 ms.

APPENDIX D

SIMULATION INFORMATION

As we simulate spiking neurons in computer, the continuousrunning period of the neurons needs to be discretized. The timeinterval of two successive discrete time points is called timestep. In our simulations, the time step is 0.1 ms and all theexperiments are conducted in the platform of MATLAB.

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Malu Zhang is currently pursuing the Ph.D. degreewith the Department of Computer Science andEngineering, University of Electronic Science andTechnology of China, Chengdu, China.

His current research interests include neural net-works, intelligent computation, deep learning, andoptimization.

Hong Qu (M’09) received the Ph.D. degree incomputer science from the University of ElectronicScience and Technology of China, Chengdu, China,in 2006.

From 2007 to 2008, he was a Post-DoctoralFellow with the Advanced Robotics and IntelligentSystems Laboratory, School of Engineering,University of Guelph, Guelph, ON, Canada. From2014 to 2015, he was a Visiting Scholar with thePotsdam Institute for Climate Impact Research,Potsdam, Germany and the Humboldt University

of Berlin, Berlin, Germany. He is currently a Professor with ComputationalIntelligence Laboratory, School of Computer Science and Engineering,University of Electronic Science and Technology of China. His currentresearch interests include neural networks, machine learning, and big data.

Ammar Belatreche (M’09) received the Ph.D.degree in computer science from Ulster University,Coleraine, U.K.

He is currently a Senior Lecturer of com-puter science with the Department of Computerand Information Sciences, Northumbria University,Newcastle upon Tyne, U.K. He was a Lecturer ofcomputer science with the School of Computingand Intelligent Systems, Ulster University. His cur-rent research interests include bioinspired adaptivesystems, machine learning, pattern recognition,

data analytics, capital market engineering, and image processing andunderstanding.

Dr. Belatreche is a fellow of the Higher Education Academy, an AssociateEditor of Neurocomputing and has served as a Program Committee Memberand a Reviewer for several international conferences and journals.

Xiurui Xie is currently pursuing the Ph.D. degreewith the Department of Computer Science andEngineering, University of Electronic Science andTechnology of China, Chengdu, China.

Her current research interests include neural net-works, intelligent computation, and optimization.

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