Ultrafast neural sampling with spiking nanolasers

TL;DR

This theoretical work shows how networks of spiking photonic crystal nanolasers can be trained to perform Bayesian inference through sampling from multivariate probability distributions.

Abstract

Owing to their significant advantages in terms of bandwidth, power efficiency, and latency, optical neuromorphic systems have arisen as interesting alternatives to digital electronic devices. Recently, photonic crystal nanolasers with excitable behavior were first demonstrated. Depending on the pumping strength, they emit short optical pulses -- spikes -- at various intervals on a nanosecond timescale. In this theoretical work, we show how networks of such photonic spiking neurons can be used for Bayesian inference through sampling from learned probability distributions. We provide a detailed derivation of translation rules from conventional sampling networks, such as Boltzmann machines, to photonic spiking networks and demonstrate their functionality across a range of generative tasks. Finally, we provide estimates of processing speed and power consumption, for which we expect improvements of several orders of magnitude over current state-of-the-art neuromorphic systems.

Figures (8)

• Figure 1: Spiking nanolasers. (a) Schematic of a PSN coupled to a silicon waveguide seen from above (left) and the midsection of the structure (right). Here, the laser is a nanobeam photonic crystal; its modes are standing-wave and optical spikes are coupled out to the waveguide in both directions. (b) Optical spike generation. Dashed lines separate sections with different scaling of the time axis.
• Figure 2: PSNs emulate LIFS neurons. (a) Membrane potential of a LIFS neuron. Shading represents the active state after a spike emission. Below: properties of a PSN far from (blue, $\gamma_{\text{p}} = 0.955\gamma_{\text{p}}^{\text{thr}}$) and close to (orange, $\gamma_{\text{p}} = 0.995\gamma_{\text{p}}^{\text{thr}}$) the spiking threshold. (b) Gain time traces. After a spike is emitted, the gain is reduced drastically, and the PSN is considered active for a time shown with shading. (c) Gain time traces between spike emissions. (d) Inter-spike interval histogram. (e) Autocorrelation of the gain. For each spike at $t_{\text{s}}$, the interval $(t_{\text{s}}- 0.5\tau; t_{\text{s}} + \tau)$ was omitted from the analysis to exclude the highly nonlinear regime dynamics of the spiking process. The case far from the threshold is fitted with an Ornstein-Uhlenbeck process, as also obeyed by the free membrane potential of LIFS neurons. (f) Histograms of gain values between spike emissions and fit with normal distributions (black lines).
• Figure 3: Setup of a PSN network. (a) Possible implementation of a PSN with a photodetector and a nanolaser for incoherent excitation. An optical signal $s^{(i)}(t)$ composed of optical spikes $s_{j}^{(i)}(t)$ at various wavelengths $\lambda_{j}$ (shown with various colors) is absorbed by a photodetector. Limited electrical circuit bandwidth leads to filtering of the spike, resulting in an almost alpha-shaped PSP. The sign of $I_{\text{exc}}$ impacts the emission of a spike. (b) Change of the nanolaser gain upon a reception of a spike (i.e. a PSP). The blue line shows an alpha-shaped fit. (c) Possible interconnection of nanolasers using a $N\times 2N$ photonic matrix that acts as a tunable broadband scatterer. Here, $2N$ outputs are required for balanced photodetectors that provide positive and negative connections between PSNs. (d) Same, with a waveguide crossbar array and microrings as weighting elements. Here, each wavelength is selectively weighted, making tuning easier than in (c). (e) Gain time trace of a PSN affected by incoming spikes (vertical lines) from two other PSNs: one is connected with a positive weight (excitatory, green) and the other with a negative one (inhibitory, orange).
• Figure 4: Translation of Boltzmann parameters to PSN parameters. (a) Spiking nanolaser activation function (crosses) with a logistic function fit (line). (b) Impact of connection weight on activation of a receiving neuron. Lines and crosses show activation of a BM neuron and a nanolaser, respectively. Each color corresponds to a receiving neuron bias $b_1 = -3, -2, \dots 3$ increasing along the arrow.
• Figure 5: Sampling from random Boltzmann distributions with PSN networks. (a) Optimization of the photodiode timescale: $\tau_\text{U}$ is swept and sampling performance computed for PSN networks with maximum weights of $0.6$ (blue), $1.2$ (orange) and $2.4$ (green). The solid black line is the mean $D_\text{KL}$ for each $\tau_\text{U}$. The dashed line shows $\tau_\text{U} = 0.37\tau$. (b) Convergence of sampling from $10$ random Boltzmann distributions with weights up to $0.6$, $1.2$ and $2.4$ from left to right. (c) Spike raster during sampling from a Boltzmann distribution with weights up to $2.4$. (d) Sampling result for a Boltzmann distribution in (c). Gray: analytical distribution, green: sampling result. (e) Sampling from conditional distributions. From left to right: $p(z_{1345} | z_{2} = [1])$, $p(z_{245} | z_{13} = [1,0])$ and $p(z_{12} | z_{345} = [1,1,1])$. Colors match (d).