Nonlinear dynamics in neuromorphic photonic networks: physical simulation in Verilog-A

TL;DR

The paper addresses the challenge of accurately simulating neuromorphic photonic networks that couple analog electronics with multi-channel photonics to predict on-chip dynamics. It adopts Verilog-A photonic neuron models, incorporating parasitics, to study continuous-time recurrent neural network ($CTRNN$) dynamics, exploring cascadability, bistability, Hopf bifurcations, and winner-take-all behavior in single- and two-neuron circuits. The key finding is that parasitics yield a topological equivalence rather than a strict isomorphism to the abstract $CTRNN$ model, with qualitative alignment but quantitative deviations, highlighting the need for physical simulation in design. The work establishes a path toward predictive, large-scale silicon photonic recurrent networks and motivates integration with packaging, CMOS control, and CAD tools to close the design loop for photonic neuromorphic hardware.

Abstract

Advances in silicon photonics technology have enabled the field of neuromorphic photonics, where analog neuron-like processing elements are implemented in silicon photonics technology. Accurate and scalable simulation tools for photonic integrated circuits are critical for designing neuromorphic photonic circuits. This is especially important when designing networks with recurrent connections, where the dynamics of the system may give rise to unstable and oscillatory solutions which need to be accurately modelled. These tools must simultaneously simulate the analog electronics and the multi-channel (wavelength-division-multiplexed) photonics contained in a photonic neuron to accurately predict on-chip behaviour. In this paper, we utilize a Verilog-A model of the photonic neural network to investigate the dynamics of recurrent integrated circuits. We begin by reviewing the theory of continuous-time recurrent neural networks as dynamical systems and the relation of these dynamics to important physical features of photonic neurons such as cascadability. We then present the neural dynamics of systems of one and two neurons in the simulated Verilog-A circuit, which are compared to the expected dynamics of the abstract CTRNN model. Due to the presence of parasitic circuit elements in the Verilog-A simulation, it is seen that there is a topological equivalence, but not an exact isomorphism, between the theoretical model and the simulated model. The implications of these discrepancies for the design of neuromorphic photonic circuits are discussed. Our findings pave the way for the practical implementation of large-scale silicon photonic recurrent neural networks.

Figures (9)

• Figure 1: Schematic of a silicon photonic neuron with $N$ input channels. Each input signal $x_i$ is amplitude modulated onto an optical carrier with wavelength $\lambda_i$ (modulation not depicted) and then wavelength-division multiplexed (WDM) onto a single waveguide. Synapses are implemented with a bank of tunable add-drop microring filters. The labels $I_i$ represent current sources applied to in-ring heaters used to tune each filter and give its corresponding channel weight $W_i$. The photocurrent $I_{ph}$ generated by the balanced photodetector configuration is the result of the weighted sum across all the optical channels. Each neuron has electrical connections $\pm V_{PD}$ to bias the photodetectors. A PN junction-integrated microring modulator is used to apply a nonlinear activation function $\sigma_L$ to the weighted sum, modulating a signal $y_L$ onto a new optical carrier (here of wavelength $\lambda_L$, where $L\in{\lbrack 1, N \rbrack}$). The modulator is biased using a current source with current $I_{bias}$ to achieve a bias voltage $V_b=I_{bias} R_b$. The connection points show where bond pads introduce parasitics into the integrated circuit.
• Figure 2: Two-channel addition and subtraction, and nonlinear activation functions in a simulated photonic modulator neuron. (a) Schematic of Verilog-A simulation and equivalent conceptual schematic. (b) Input and output optical signals for various circuit configurations. $A$ and $B$ are input signals at $\lambda_1$ and $\lambda_2$. Output signals on $\lambda_3$ demonstrate four different operations applied to the inputs, achieved by programming the weights and activation function. All signals have a high-pass filter with a cutoff frequency of 400 applied to remove the DC component and are scaled to emphasize the qualitative aspects of the results. (c) Linear and quadratic activation functions obtained by plotting the output power of the neuron against the voltage of the junction. Circular markers are used for $A+B$ and $(A+B)^2$ while square markers used for $A-B$ and $(A-B)^2$. The black lines show fits to the data using a linear and quadratic functions.
• Figure 3: Bistability in single self-afferent neuron. (a) Schematic of simulated recurrent circuit and equivalent conceptual schematic. (b,c,d) Relationship between neuron bias $b(t)$ and output $y(t)$ for three different bias signals. The output $y$ on $\lambda_1$ when $W_F=0$ is shown in green, while the red and blue traces show the rising and falling edges of $y$ when $W_F=+1$. The $y$ vs. $b$ plots on the left shows the signals along with the system's nullcline (grey trace). The transient plots on the right shows the same signals along with the bias waveform (black trace). Optical powers are normalized using $y(t)=P(t)/P_{pump}$, the bias current is normalized using $b(t)=(I(t)-I_{min})/(I_{max}-I_{min})$.
• Figure 4: (a) Conceptual representation of the two-neuron circuit showing the weight values. (b, c) Transient response of neuron outputs for $W_F=0$ and $W_F=1$, respectively. The outputs $y_i$ are given by the normalized output power from each modulator. (d) 3D bifurcation diagram of the Hopf system. The stability transitions from a stable fixed point to a stable limit cycle as $W_F$ is varied. Black shadows show the projections of simulated data onto the $y_i-W_F$ axes. The projection on the $y_1-y_2$ axes shows both the expected dynamics as a vector field, and the slice of the simulated data where $W_F=1$.
• Figure 5: (a) Conceptual circuit diagram of winner-take-all simulation showing programmed weights and output signals. (b) Transient input and output signals for each of the two neurons. (c) State-space plot of outputs, where the colour of the marker corresponds to the neuron with the greater input (the expected winner). The vector field depicts the expected dynamics in the absence of any input signal.