Experimental realization of photonic spiking neural network
toward a new neuromorphic computing

1．Background

Information processing in the brain and biological systems is thought to be very efficient and high performance, even though the complete mechanism has not yet been well clarified. Pioneering studies have pointed out that neural information processing is based on spiking signals of neurons making it very different from information processing of the current standard computers (von Neumann type architectures). To clarify the mechanism of neural information processing, a spiking neural network (SNN)^*5, which is an artificial neural network mimicking realistic neurons, has recently attracted much interest. In a SNN, it is important to consider the spiking signals as a point process and simulate the timing and frequency of spike firing, so it is necessary to develop application-specific devices and/or physical implementations.
　Complex and diverse spiking dynamics has been experimentally studied with various nerve cells and analyzed numerically with various mathematical models. A pioneering researcher, A.L. Hodgkin, has demonstrated that the spiking dynamics can be simply classified into two spiking modes (Class-I and Class-II). This is called Hodgkin's classification of spiking modes. In various biological systems, various neurons have been categorized into these two classes of spiking modes. However, it is remained unclear how the two spiking modes are assigned to a role during neural information processing. It is therefore important to study SNN models by considering the diversity of spiking modes.
　We developed an artificial photonic neuron by using degenerate optical parametric oscillators (DOPOs), and applied a nonlinear optical device and measurement feedback technique that have been established to construct a coherent Ising machine, which is a machine for solving combinatorial optimization problems. We used a nonlinear optical device developed in NTT to achieve controllability of spiking modes. A special experimental setup of measurement feedback is implemented for creating correlations between DOPOs to design a network of 240 DOPO neurons with arbitrary connections. As a result, we can develop a photonic neuron with high controllability of spiking modes and the ability to realize arbitrary network structures. Moreover, we observed a novel phenomenon accompanied with synchronization of networked neurons.

2．Results

A DOPO neuron is made from a pair of DOPO pulses as shown in Fig. 1, in which the two DOPOs are connected with opposite signs. The nonlinear effects in a phase-sensitive amplifier (PSA) play an important role to emulate spiking signals. The nonlinear device can be controlled by injecting a pump light and tuning its amplitude. We used this controllable nonlinearity in order to realize tunable spiking modes (see Fig. 2). For a weak pump light amplitude, we observed that spikes with high frequencies are induced by the external bias, and its spiking frequency hardly changes with the amplitude of external bias. This is a characteristic of the Class-II spiking mode within the Hodgkin's classification. For a strong pump amplitude, we observed that spiking frequencies gradually changed from small to large, characterizing the Class-I spiking mode. Without any external bias, we found that a spiking mode gradually changes from Class-II to Class-I with increasing pump light intensity. Note that this simple control with pump lights of spiking modes will be an advantage when applying it to neuromorphic computation.
　We further demonstrated synchronization of DOPO neurons, which is an important concept in SNN systems. We found that spike firing becomes simultaneous when increasing interneuron coupling strength and evaluated an order parameter^*6 that quantitatively characterizes the degree of synchronization. We further observed the following novel phenomenon caused by synchronization (see Fig. 4): A cluster of DOPO neurons changes its spiking modes spontaneously depending on the degree of synchronization. Note that a single DOPO neuron does not exhibit such a spontaneous change in spiking modes, while a group of them has such a function thanks to synchronization. Although spontaneous spiking-mode change is a natural phenomenon caused by synchronization, this "function" controls spiking frequencies to stabilize synchronization similarly to an algorithm on a computer. These findings suggest that a group of DOPO neurons can open the way for a new type of neuromorphic computing.
　In summary, we developed a high-performance photonic artificial neuron by using DOPOs that can arbitrarily control spiking modes and a group of DOPO neurons can change spiking modes spontaneously. The present SNN system can realize arbitrarily network structures. This high controllability can be an advantage when applying DOPO neurons to neuromorphic information processing.

3．Technical points in detail

(1) Implementation of the DOPO neuron

As shown in Fig. 1, a DOPO neuron consists of two DOPO pulses connected to each other, where coupling strengths are set with opposite signs. Mathematical analyses clarified that this coupling creates a flow of energy between two DOPOs and is a key element of spiking dynamics. Another important element is nonlinearity of photons in a PSA.

(2) Network structure of DOPO neurons

As shown in Fig. 5, interneuron couplings were implemented using our measurement feedback technique, which is a hybrid scheme combining optical devices with electronic computing circuits. With this method, intra-neuron coupling (Jvw, Jwv) and inter-neuron coupling (Jij) can be easily controlled to arbitrary design network structures of up to 256 neurons. Figure 3(a) shows a network structure used in this study, in which four clusters are coupled with each other with sparse connections, and each cluster consists of 15 neurons and coupling inside a cluster is completely dense (complete graph). For each cluster, averaged spiking frequency was set to be a different value, suggesting that it was difficult to enable synchronization including all clusters. Interestingly, DOPO neurons controlled spiking modes spontaneously and changed spiking frequencies so that all clusters achieved synchronization even though each cluster had largely different spiking frequencies.

4．Outlook

Our DOPO neuron can control spiking modes by tuning injected pump light. By utilizing the property of DOPO neurons, we will attempt to build a new platform to investigate what role spiking modes play in biological information processing. We also observed a novel phenomenon caused by synchronization, namely, spontaneous controlling of spiking modes. By taking advantage of such cooperative and emerging features of DOPO neurons, we will investigate a new type of neuromorphic computation.

<Figures>

Fig. 1. Schematic of DOPO neuron. Two DOPO pulses (called v- and w-DOPO) are connected, where coupling strengths Jvw and Jwv are set to opposite signs. Inter-neuron coupling between i-th and j-th neurons (Jij) is implemented by two channels (γ,γ'). Couplings Jij can be set for any i and j; thus, any network structure can be designed arbitrarily.

Fig. 2．Spiking signals of single DOPO. (a) Reaction against external bias increasing with time for weak pump amplitude. Spiking signals with high frequencies can be observed from certain time (bias), and spiking frequencies hardly change with time. These features are characteristics of Class-II spiking mode. (b) Same as (a) for strong pump amplitude. First, spiking signals with low frequencies appear, then frequencies become higher, which indicates Class-I spiking mode. (c) Change in spiking frequencies. (left) Frequency as function of external bias with fixed pump at weak pump amplitude. (center) Same as left panel at strong pump amplitude. (right) Frequency as function of pump amplitude without external bias, where spiking mode changes gradually from Class-II to Class-I.

Fig. 3．Synchronization experiments for 60 DOPO neurons. (a) Network structures used in study. (b) Definition of θ, rotation angle of neuron.(c) Dynamical change in θand order parameters in each cluster and all neurons. As coupling strength Jk increases, timing of spike firing synchronizes, which can be quantitatively confirmed from increases in order parameters. Synchronization inside clusters occurs with small Jk, while it seems to be difficult to synchronize in the entire network, which is due to different spiking frequencies of clusters. For most strong Jk (top panels), entire network order parameter reaches maximum, suggesting synchronization of entire network.

Fig. 4. Spontaneous change in spiking modes. Data were extracted from Fig. 3 except for the center-left panel, which shows dynamical change in θ for specific neuron in each cluster. As illustrated in right panel, the center-left panel clearly shows dynamical changes in spiking modes as follows. Before synchronization, linearly-increasing dynamics are characteristics of Class-II. After synchronization, dynamics changes to stepwise, suggesting mode change from Class-II to Class-I. At the proximity of area surrounded with red and blue boxes, Class-I mode changes again to Class-II, where entire-network order parameter increases rapidly. These findings suggest that group of DOPO neurons changes their spiking modes and spiking frequencies to stabilize synchronization.

Fig. 5. Experimental setup of DOPO spiking neural network. 512 DOPO pulses are generated using phase sensitive amplifier with periodically poled lithium-niobate (PPLN) waveguide in 1-km fiber ring cavity. All-to-all coupling of 512 DOPO pulses is implemented with measurement and feedback scheme.

<Glossary>

^(*1)Spiking signal
In biological neurons, point-process like electronic signals with a steep peak structure have been observed, which is called spike. It has been pointed out that these signals transfer information in the brain processing.

^(*2)Degenerate optical parametric oscillator (DOPO)
A DOPO pulse is generated by a phase-sensitive amplifier (PSA) with a nonlinear material in a cavity. Because degenerate parametric amplification is phase sensitive, the optical phase of each DOPO pulse takes only 0 or π.

^(*3)Alan Lloyd Hodgkin
A pioneering researcher who studied biological system based on a mathematical approach and won a Nobel-Prize physiology or medicine at 1963 through collaborated research with Andrew Huxley. Hodgkin proposed that the reaction of realistic neurons can be classified into two classes of spiking modes.

^(*4)Synchronization
A phenomenon induced by the inter-neuron couplings, where timing of spike firings of networked neurons becomes at the same time.

^(*5)Spiking neural network
Artificial neural network that can mimic spiking signals of biological neurons.

^(*6)Order parameter
Physical quantity characterizing synchronization. Large- (small-) order parameters indicate that synchronization (does not) occur.

<Paper publication information>

Title: Collective and synchronous dynamics of photonic spiking neurons

Authors: Takahiro Inagaki, Kensuke Inaba, Timothée Leleu, Toshimori Honjo, Takuya Ikuta,
Koji Enbutsu, Takeshi Umeki, Ryoichi Kasahara, Kazuyuki Aihara, and Hiroki Takesue

Journal: Nature Communications（23rd April 2021）

DOI: 10.1038/s41467-021-22576-4