Modeling neural networks with lasers
An array of laser diodes and semiconductor amplifiers should be able to physically model the operation of the brain.
Modern electronic programmable digital computers perform less well than the brain at tasks such as spotting your car in a crowded lot from a trunk view, or hearing your name mentioned at a noisy party. While implementation technologies have advanced rapidly since the invention of the programmable digital computer (patent 1936), the basic architecture1–2 has barely changed. The architecture was designed for numerical (exact) computation in science, engineering and accounting. Meanwhile computers are used very differently: primarily as network nodes with real-world interfaces requiring ability to efficiently process inexact data. This mismatch between design and application will constrain progress.
New technologies — optics,1 biological, molecular, quantum — are being investigated for radically new architectures, and advances have been made in artificial intelligence and fuzzy logic to improve handling of real-world inexact data. We believe in learning from the brain because the brain evolved specifically for the tasks we now perform on computers. In the basic brain neuron,3 Figure 1, a dendritic tree collects pulsed signals from other neurons via synapses which convert electric pulse rate to electric current, a larger current producing a higher pulse rate. The summed currents at the cell body charge an equivalent capacitor to a threshold voltage, at which time discharge causes a short pulse. The result is that the relaxation oscillator generates a higher pulse rate for larger sums. Pulse rate modulation provides robust communication against drift and noise. The summing Σ and clipping function S are duplicated in a conventional feed-forward artificial neural network, Figure 2. A typical sigmoid function S(x)1/(1 + e-x) is plotted in Figure 3. Adding a connection from the output, y to an input in Figure 2 creates a dynamic neural network for which typical dynamics may be specified for the ith neuron as a differential equation.
in which the n inputs xj from other neurons are weighted by learned variables cij, added and clipped by S. Also A parameter ρi is added to the weighted sum to control bifurcation.
Because pulse rate can only be positive, biology uses two types of neuron: excitatory and inhibitory. The pulse rate from an excitatory neuron is converted to a positive current in an excitatory synapse and that from an inhibitory neuron is converted to a negative current in an inhibitory synapse. A neural oscillator is formed by cross-coupling excitatory and inhibitory neurons in the commonly used Wilson-Cowan brain model,3–4 Figure 4. The neural oscillator performs a basic Hopf bifurcation function for decision making and communication. In a Hopf bifurcation, the output is switched between zero and a stable oscillation (a limit cycle) by varying ρx and ρy which also determine the amplitude and frequency of the stable oscillation. An oscillation is stable if it recovers from a perturbation from frequency and amplitude.
We show that injecting light with intensity or frequency (corresponding to control ρx) into a laser diode can replicate the Hopf bifurcation and therefore simulate a Wilson-Cowan neural oscillator. Semiconductor optical amplifiers in cross gain modulation can perform the summation and sigmoid clipping.2 Many laser diodes and SOAs on a single chip should physically replicate the neural network model for the brain.
The Wilson-Cowan coupled neuron pair3 may be modeled from Eq. (1),
where a, b, c and d are fixed parameters. Steady state is obtained by setting and . For Hopf bifurcations we select values of ρx and ρy on either side of the bifurcation and fix constants a, b and c. To illustrate convergence to a single stable off-state, we used ρx = 0 and ρy = 20,Figure 5. To illustrate convergence to a stable limit cycle (or stable oscillation) we used ρx = 0 and ρy = 9.6, Figure 6. In this case we started time traces both inside and outside the limit cycle and both converge to the limit cycle (bold closed curve) as shown. Varying ρy from 20 to 9.6 changed the output from a steady off-state to a stable oscillation.
We show that a laser with optical injection can perform an identical Hopf bifurcation to that in a Wilson-Cown neural oscillator and has rate equations:2–5
where E is complex amplitude associated with total number of photons, n is the population inversion, K is the field strength and ω the detuning frequency for the optical injection. We set fixed parameters2α,B, and Γ and use K and ω to transition the Hopf bifurcation. The time trajectory in E, n space is three dimensional because E is complex; therefore we use the real part of E for the plots. To illustrate convergence to a stable off position we used K = 0.25 and ω = -0.3, Figure 7. To illustrate convergence to a stable oscillation, we used K = 0.25 and ω = -0.15, Figure 8 (the converging trace from inside is shown). Figure 7 and Figure 8 show that by varying the frequency of the injection current from ω = -0.3 to -0.15 (we could have changed K instead) we change the output from an off-state to a stable oscillation.
We showed that changing the injection power into a laser diode can model the Hopf bifurcation dynamics of a Wilson-Cowan neural oscillator used to model the basic operation of the brain neuron: switching the output from a stable off-output to a stable oscillation state. In future research, we plan to investigate other bifurcations and show that such optical neurons can implement associative memory and other brain functionality. We also plan to investigate the modification of synaptic gain for Hebbian learning.
I would like to thank Tom Koch and the Center for Optical Technologies at Lehigh University for support.
