As the improvement of traditional neural networks has gradually approached an upper limit, research focuses on neural networks with more biological reality. The spiking neural networks (SNNs), normally known as the third generation of artificial neural networks (ANNs), are more biologically plausible than previous ANNs [1] and have attracted more and more attention in recent decades [2–6]. Spikes transmitted in the biological neural networks enable the network to capture the rich dynamics of neurons and to integrate different information dimensions [3]. However, the information representation and processing manner has become a controversial issue and remains very challenging.

Rate coding is widely used in traditional SNNs; however, there is biological evidence that the precise timing of spikes also conveys information in nervous systems [7–9]. The precise timing of spikes enables higher information encoding capacity and lower power consumption [10–12], which are extremely important in information processing of the human brain. However, the exact learning mechanism still remains an open problem [13]. It has been shown that in the cerebellum and the cerebellar cortex, there exist signals that act like an instructor that helps the processing of information [14,15]. Several supervised learning algorithms have been proposed upon which specific problems that are tightly related to neural processing such as spike sequence learning and pattern recognition have been solved successfully [16–20]. Remote supervised method (ReSuMe) is one of the supervised learning algorithms originally derived from the well-known Widrow-Hoff rule [17]. Based on photonic spike-timing-dependent plasticity (STDP) and anti-STDP rules, the synaptic weights can be adjusted to train the output neuron to fire spikes at the desired time.

Time-delayed transmission is an intrinsic feature in neural networks. Biological evidence shows that the transmission velocities in the nervous system can be modulated [21,22], for example, by changing the length and thickness of dendrites and axons [23]. The adjustability of both delay and weight of a synapse is referred to as synaptic plasticity. Delay plasticity has also been found to be helpful for the neuron in changing its firing behavior and synchronization, and it helps to understand the process of learning [24,25]. Delay selection and delay shift are two basic approaches incorporated in delay learning works [26,27]. To be specific, delay selection is to strengthen the weight of an optimal synapse among multiple subconnections, and delay shift is a more biologically plausible method training neurons to fire with coincident input spikes and with constant weight. Recently, researchers found that combined adjustment of delay and weight enhances the performance of an SNN [28–32]. In 2015, a delay learning remote supervised method (DL-ReSuMe) for spiking neurons was proposed to merge the delay shift approach and weight adjustment based on ReSuMe, by which the learning accuracy and learning speed are enhanced [30]. In 2018, Taherkhani et al. proposed to appropriately train both weights and delays of excitatory and inhibitory neurons in a multilayer SNN to fire multiple desired spikes. Experimental evaluation on benchmark data sets shows that higher classification accuracy than single layer and a similar multilayer SNN can be achieved by the proposed method [31]. In 2020, Zhang et al. investigated the synaptic delay plasticity and proposed a novel learning method, where two representative supervised learning methods, ReSuMe and the perceptron based spiking neuron learning rule (PBSNLR), were studied and found to outperform the traditional synaptic weight learning methods [32].

For the sake of emulating realistic biological behaviors, SNN hardware realizations are designed to seek ultralow power consumption [5]. Devices for the implementation of basic elements of SNN, namely, artificial spiking neurons and synapses, have been achieved via complementary metal-oxide-semiconductors (CMOS), transistors, and the emerging nonvolatile memory technologies [33–38], etc. Photonic neuromorphic systems have attracted attention for being a potential candidate in applications of ultrafast processing. Despite its similarity with biological neurons [39], the semiconductor laser also exceeds its electronic counterpart in its ultrafast response and low power consumption. Numerous studies on photonic synapses and photonic neurons [40–53] have laid a solid foundation for significant progress in photonic neuromorphic computing based on both software and hardware implementations [54–59]. In 2019, an all-optical SNN with self-learning capacity was physically implemented on a nanophotonic chip, which is capable of supervised and unsupervised learning [55]. In 2020, we proposed to solve XOR in an all-optical neuromorphic system with inhibitory dynamics of a single photonic spiking neuron based on vertical-cavity surface-emitting lasers (VCSELs) with an embedding saturable absorber (VCSEL-SA) [58], and an all-optical spiking neural network based on VCSELs was also proposed for supervised learning and pattern classification [59]. However, as far as we know, delay learning has not yet been applied in photonic SNNs.

In this work, we propose to incorporate delay learning with the traditional algorithm based on an optical SNN. First, we propose a modified algorithm that combines delay learning with ReSuMe in a photonic SNN, which adjusts synaptic weight and delay simultaneously. By implementing a spike sequence learning task, better performance and learning efficiency of the proposed algorithm than that of its weight-based counterpart are verified. Then, the proposed algorithm is also implemented for classification, where two benchmarks, the Iris data set and the breast cancer data set, are adopted. By applying the delay-weight (DW)-based algorithm, the testing accuracy for both benchmarks is significantly improved (reaching 92%).

The photonic neurons and synapses are the basic elements in a photonic SNN. VCSEL-SA can mimic spiking dynamics of a biological neuron [51] and VCSEL that works below threshold value can serve as a vertical-cavity semiconductor optical amplifier (VCSOA) that is able to perform the STDP function [47], which provides possibilities of large-scale integration and low power consumption in a photonic SNN. In this work, the spiking dynamics are implemented via the excitable VCSEL-SA neurons. The rate equations of a VCSEL-SA are written as follows [57]: S˙i,o=Γaga(na−n0a)Si,o+Γsgs(ns−n0s)Si,o−Si,o/τph+βBrna2,(1)$$\begin{gathered} na·=−Γaga(na−n0a)(S−Φpre,i−Φpost,o)−na/τa+Ia/(eVa), \\ Φpre,i=keiτphλiPei(τi,Δτ)/(hcVa), \\ Φpost,o=∑i=1nωiλiτphPi(t−di)/(hcVa), \end{gathered}$$(2)ns·=−Γsgs(ns−n0s)Si,o−ns/τs+Is/eVs,(3)where i = 1, 2, ..., n is the number of the PREs, and o denotes the POST. The subscripts a and s represent the gain and absorber regions, respectively. Si,o(t) stands for the photon density in the cavity of the PREs and POST. Γa is the gain region confinement factor. ga is the gain region differential gain/loss. na (ns) is the carrier density in the gain (absorber) region.

The term Φpre,i in Eq. (2) describes the pre-coded square-wave pulses injected as external stimulus into PREs, and Φpost,o represents the weighted sum of all of the pre-synaptic spikes fed into the POST. di is the adjustable transmission delay from PREi to the PSOT. kei, τi, and Δτ denote the strength, the central timing, and the temporal duration of the pulse, respectively. Δτi=τi−τi−1 is the time interval between two adjacent input pulses. ωi is the coupling weight between the PRE_i and the POST that can be tuned according to the supervised training method. The output power of PREs and POST can be calculated by Pi,o(t)≈ηcΓaSi,o(t)Vahc/(τphλi,o). In practice, the ωi is calculated as an initial weight ω0 multiplying a constant coefficient ωf=ηcΓahc/(τphλi,o) to match the optical system. The remaining parameters are the same for all neurons as in Ref. [56]. The rate equations are numerically solved by using the fourth-order Runge-Kutta method.

ti≤td and ti≤to in Hebbian terms represent that this rule only modifies inputs that contribute to the neuron state before the desired or actual output firing time but neglect those inputs that fire afterward, which leads to better performance of the proposed DW-based algorithm.

The term Did (Dio) is the distance between ti and td (to). The delay adjustment is based on the distance between input spikes and output spikes (target spikes), in a way similar to that of synaptic weight. Note that both Δω and Δd approach 0 if the POST fires at the desired times, which ensures the convergence of the proposed algorithm. In addition, the synaptic weights (delays) are adjusted with a learning rate ηf(ηd) and within a learning window Tω(Td). The weight and delay of the i-th synapse are adjusted only if the input spike distance Did is less than the learning window. Finally, the weight and delay of the i-th synapse are updated by ωi(x+1)=ωi(x)+ηωΔωi,(8)di(x+1)=di(x)+ηdΔdi,(9)where the term x denotes the training cycle. In general, ηd=0.5 contributes to better performance, while other parameters should be carefully selected according to different tasks.

However, note that if the desired output contains more than one spike, the delay learning window Td has to be limited within the range of the minimum inter-spike interval (ISI) to maintain stable performance. Since some of the input spikes are shifted toward a certain target spike, there is an additional consideration in the DW-ReSuMe that the injection power of the output neuron should be limited to protect devices of photonic neurons and to ensure spiking dynamics.

The Fisher’s Iris flower data set (Fisher, 1936) is a classic benchmark of pattern recognition that contains three classes of Iris flowers with a total of 150 case entries [31]. One of the classes is linearly separable from the other two, while the other two classes are linearly inseparable. Four measurements are used to describe and differentiate the three classes, and each measurement is directly encoded into single spike firing at different times and is linearly rescaled into the interval of [5 ns, 10 ns]. The network comprises 4 PREs and a single POST. The encoding spikes are fed into the 4 PREs of the SNN via synapses with adjustable delay and weight. The initial delays are 2 ns for all of the four input neurons, and the initial weight ω0 for each synapse is randomly selected as a constant coefficient (which is 0.1 here) multiplied by a random number from [0,1]. In this case, nearly all of the PRE neurons are capable of generating just a single spike in response of each input pattern. The weight learning rate ηω is 0.01 and decays by half every 20 training epochs to enhance the convergence of learning. The output of the network is represented by the precise spiking time of the POST, which fires a desired spike at different times for different classes. Here, the target spikes for the three classes are 8 ns, 9 ns, and 10 ns, respectively. If the output spike locates within 40% the interval of target spikes around a certain td, the input entry is classified into the corresponding class.

DW-based learning has shown excellent performance in single-layer networks. However, the real neural networks are usually hierarchical, and synaptic weights and delays can be modulated based on different biological mechanisms, which form the foundations of complex brain functions. It is quite interesting and challenging to consider how to effectively introduce delay adjustment into deep learning networks.

This paper proposed a supervised DW learning method in an optical SNN. Based on precise timing of spikes, delay learning trains coherent inputs in the input layer of an SNN via shifting the synaptic delays according to the desired and actual output timing. The proposed DW-ReSuMe is applied to spike sequence learning and two classification benchmarks, the Iris data set and breast cancer data set, successfully. The performance of the SNN is significantly improved compared with its weight-based counterpart. By introducing time-delay learning in a photonic SNN, fewer optical neurons are required to solve different tasks, which is significant to photonic neuromorphic computing. The results also suggest that synaptic delay and weight may be a combined learning mechanism in real biological neural networks, which deserves deeper investigation.