Collective and synchronous dynamics of photonic spiking neurons

Nonlinear dynamics of spiking neural networks have recently attracted much interest as an approach to understand possible information processing in the brain and apply it to artificial intelligence. Since information can be processed by collective spiking dynamics of neurons, the fine control of spiking dynamics is desirable for neuromorphic devices. Here we show that photonic spiking neurons implemented with paired nonlinear optical oscillators can be controlled to generate two modes of bio-realistic spiking dynamics by changing optical-pump amplitude. When the photonic neurons are coupled in a network, the interaction between them induces an effective change in the pump amplitude depending on the order parameter that characterizes synchronization. The experimental results show that the effective change causes spontaneous modification of the spiking modes and firing rates of clustered neurons, and such collective dynamics can be utilized to realize efficient heuristics for solving NP-hard combinatorial optimization problems.


Supplementary Note 1 -Numerical simulation
The dynamics of a single DOPO neuron were investigated by numerical simulations based on the following coupled ordinary differential equations (ODEs): where variables and are amplitudes of DOPOs called -and -DOPO, * and * are their complex conjugates, and Re( ) and Im( ) are in-phase and quadrature-phase components of DOPO amplitudes for = , .
Supplementary Figure 1 clearly shows that the two nullclines become tangential for large ̃, suggesting that the saddle-node bifurcation on the limit cycle (SNLC) emerges. 6 Supplementary Figure 1 Numerical simulation of DOPO neuron dynamics.
Time series of DOPO amplitudes, and the trajectories (red) in the v-w space and the nullclines, which are curves satisfying dv/dt=0 (cyan) and dw/dt=0 (blue). For comparison, trajectories calculated without noise terms are also shown (black), which correspond to the orbit of the limit cycles. Supplementary
Although this parameter is zero ( = 0) and is uncontrollable in the present experiments, it helps us to discuss the change of types of bifurcation. It should be noted that some physical systems can be described by the analogical equation to the above with = 1, and thus this parameter makes it possible to clarify the difference between the bifurcation mechanisms of different physical systems.
Hereafter, it is assumed that < 0, and only the case of < 0 and > 0 is considered. The opposite-sign case is straightforward under the transformation of → − . And, is rescaled as √ , and then the above equations can then be rewritten by using = + as which can be rewritten with = √ and = 2 + 2 / as From the above expressions, it can be easily understand that 0 corresponds to the angular frequency of the oscillator at = 1, or at the limit of → 0 with → +0, where a simple oscillator with = 0 appears.
It can thus be expected that the AH bifurcation should occur near these two limits. For example, for α = = 1, the normal form of the AH bifurcation is obtained as = 0 + − | | 2 . From another viewpoint, two separable variables ( and ) can be obtained as = 0 and = 2 − 2 2 .
However, for = 0 and > 0, the connection between and induces change of bifurcations from AH to SNLC types. Roughly speaking, the normal form of the saddle-node bifurcation can be derived too; for example, at α = 1 and = 0 the form = 0 + 4 sin 4 can be reduced to . This point will be discussed again in Supplementary Notes 3.
It is thus clear that parameters and play important roles in the mechanism of the bifurcations. , and angular frequency , corresponding to firing rate, is given as a function of as follows: Imposing the limit of → +0 gives basic frequency 0 as mentioned above, suggesting that the AH bifurcation (class II neuron) can be found near ∼ 0.
At = ≡ √8 0 , the frequency gradually reaches zero, suggesting the SNLC bifurcation (the class I neuron). Point is exactly the same as the analytically obtained expression of the bifurcation point as mentioned below.
The period of the saddle node bifurcations is then obtained as is equivalent with the above expression near ∼ , is obtianed. Note where Ψ = The following standard Kuramoto model 4,5 is considered to explain one of the essential points of the synchronization of the DOPO neurons: where node-dependent is assumed to be a natural distribution with a variance σ ω . In this well-known model, the synchronization can be understood from the analogy to the phase transition with order parameter As discussed later, it should be noted that the frequencies  With the assistance of the selective spin-flip mechanism discussed above, the DOPO-SNN showed better performance than the CIM in terms of the time-to-solution for this hard instance. It is useful for comparison to consider the time-to-solution with 99% success probability which is given as t99 = T log(10.99) / log(1Ps), where T is the computation time and Ps is the success probability 6 . CIM and the proposed spiking dynamics have t99 = 2867 ms and 692 ms, respectively. Thus, the time-to-solution with 99% success probability is reduced by a factor of four by introducing the spiking dynamics. It should be noted that the time scale of spiking frequency, 0 , restricts the circulation steps ( ) of the DOPO-SNN scheme; i.e., ω 0 ≫ 1.
Namely, a sufficient number of spikes is required during all circulations.
Here, 0 ∼1/20 (with a unit of [spikes/steps]) with =3500 was used. This 0 is determined from experimental parameter . Note that an important parameter regarding the Ising problem solver (see Figure 4 in the main text) is the ratio between spiking frequency ( ) and Ising coupling ( ), corresponding to ̃ (= / 0 ). Controlling parameter ̃ is now strictly limited due to the performance of the FPGA; namely, and should be 8 bit integers. It is expected that the performance of the FPGA can be improved in the future, and furthermore, v-w coupling can be replaced by Ising energy direct optical coupling; and then, the ratio of and can be tuned widely.
We thus believe that the current limitation on number of circulation steps can be relaxed in the future work.