Introduction

Spiking neural networks (SNNs) in biology encode information through timed events or spikes, utilizing distributed, sparse, and robust coding schemes. Neuromorphic systems are designed to implement artificial neural networks with individual neurons in hardware, aiming for enhanced speed, latency, and energy efficiency1. Here, we introduce a photonic foundry-compatible, light-based silicon-on-insulator excitable neuron. We discuss how technological advancements inform and guide its design. Photonic interconnects known for their low loss, low latency, and high bandwidth, are increasingly utilized in neuromorphic systems. They enable high throughput by parallelizing channels through wavelength-division multiplexing (WDM)2,3, facilitating easier and denser routing as multiple signals traverse the same waveguide. In a neural network, each individual neuron inputs the outputs of other neurons and weighs their importance through a linear weighted addition operation implemented in the interconnects between neurons. Although a single neuron offers limited computational utility; a large network of neurons is often necessary to perform a task, as the computation is distributed over the whole network4. Consequently, many neurons are required with interconnects capable of transmitting large amounts of information across the network5. Utilizing WDM is critical to increasing data transfer, and silicon photonics technology is promising due to its silicon foundry compatibility and potential for large-scale neural systems.

Neurons also require a nonlinear component for decision-making. For SNNs, this component is a nonlinear dynamical system that depends on the timing and strength of the inputs. These inputs determine the behavior of a spiking neuron’s output over time, whether it spikes or remains at rest. One approach to implement the nonlinearity of a neuron is to use nonlinear optics, such as in all-optical designs6,7,8,9. However, the weak interaction between photons presents a challenge, necessitating high optical powers. All-optical designs with WDM are particularly challenging, often because these designs rely on resonators to enhance the otherwise weak nonlinear behavior10,11. These optical resonators are only highly nonlinear near their resonance wavelengths, which restricts the inputs of the neuron to on-resonance wavelengths. Additionally, all-optical designs do not regenerate their output light. The output will be a mixture of wavelengths depending on the inputs which complicates how they can be weighted by subsequent neurons. This makes their integration into a WDM architecture using multiple different wavelengths problematic.

Another approach involves converting the optical signal into an electrical signal and performing the nonlinear transform electronically where nonlinear phenomena are more accessible. The electrical signal is then sent to either a laser or a modulator to transition back into the optical domain for processing by subsequent neurons. This conversion from electrical to optical signals often exhibits inherent nonlinearity. These are known as optical-electrical-optical (OEO) designs12. Such designs facilitate scalability, a crucial attribute for developing larger neural networks. They are also more WDM compatible since germanium photodetectors on silicon offer responsivities relatively independent of wavelength within the C-band. Furthermore, OEO neurons are more amenable to cascading: the first neuron needs to be capable of driving the second neuron and so on. This requires signal gain, which can be provided by lasers. However, integrating lasers with silicon chips, a long-standing goal in photonics has been impeded by challenges related to interconnection and material compatibility issues13. In contrast, modulators are commonly offered devices by foundries and are guaranteed to work with their process. They can also provide signal gain via a regenerative process12, but not always14. Moreover, a photodetector-modulator setup can allow the output optical wavelength to be arbitrary and independent of the input optical wavelengths. Ring modulators can only modulate light near their resonance, but many WDM signals can be independently modulated with rings of varied resonance wavelengths. The primary drawback of OEO designs is the inherent speed limitation imposed by photodetection and the transmission line between the photodetector and modulator, which needs careful design15. Despite this, the potential for scalable analog and spiking neuron designs has sparked considerable interest16.

Here, we propose a monolithic silicon-on-insulator resonator spiking neuron incorporating a modulator and optical feedback. This device is WDM compatible, which is critical for scalability3. It benefits from regenerative signal gain, allowing its output optical wavelength to be independent of the input wavelength, which is critical for cascadability. These benefits have been discussed before in ref. 12, but not for spiking neurons. The proposed spiking neuron is fully compatible with all mainstream silicon photonic platforms, and does not require any post-processing steps. Unlike prior photonic SNN designs, our proposed model does not require the use of any transistors17, resonant tunneling photodiodes18, 2D or exotic materials19,20, or lasers21,22. This simplification enables its implementation in common multi-project-wafer runs. The components of our design can be tuned to generate spike rates in the gigahertz range, showcasing its potential for high-speed applications in adaptive control23, learning24, and cognitive processing of the radio frequency spectrum25.

Methods

A simplified mechanism of resonant spiking

Resonator spiking neurons are especially excitable when input pulses arrive at their characteristic resonant spike rate26,27. This means they can more easily ignore noise or other off-resonance signals. In this way, they can be also used for coincidence detection. If input pulses come at the same time or at some multiple of the resonant period, the neuron has a better opportunity to spike. Whereas, input pulses arriving off the resonant period will destructively interfere and prevent spiking. Resonator neurons are Class 2 excitable, so the rate of spikes at the output is roughly constant regardless of the input. Additionally, sufficiently strong negative inputs can also induce spiking. Some Class 2 neurons can also act as memory elements if they are bistable. This requires a rest state and a spiking state to coexist in the same parameter space. The transition from rest to spiking (or vice versa) is due to a bifurcation. A bifurcation is a sudden qualitative shift in the behavior of a dynamical system under the smooth variation of a parameter28. There are many different types of bifurcations, but we are concerned with three different types for this article. The first is a supercritical Hopf bifurcation. This is the conversion of a stable fixed point to a stable limit cycle (or vice versa). A fixed point (also known as an equilibrium point) is a point where a steady-state solution of the system exists. Fixed points can be stable or unstable; they will attract or repel nearby trajectories, respectively. A limit cycle is an isolated closed orbit in the phase space. Similar to fixed points, stable and unstable limit cycles exist. The supercritical Hopf bifurcation transforms the dynamical system so that trajectories are attracted to an oscillating state instead of a stable rest state. A subcritical Hopf bifurcation occurs when an unstable limit cycle collapses onto a stable fixed point to form an unstable fixed point. This forces nearby trajectories that were originally attracted to the stable fixed point to be suddenly repelled away from the formerly stable resting state. Often, the unstable limit cycle is surrounded by a stable limit cycle, so the net effect of the subcritical Hopf bifurcation is to suddenly shift from a resting state to a spiking state. Note the presence of two separate stable states is called bistability. The third type of bifurcation we are interested in is called a fold bifurcation of cycles. It occurs when an unstable and stable limit cycle appears from nothing due to the variation of a parameter. In the spiking neuron device we propose, all three of these bifurcations appear in the parameter space investigated.

This spiking neuron is an electro-optic device integrated on a silicon photonics platform, which takes its inputs encoded in the optical input power. The inputs are detected and converted to electrical currents by germanium photodiodes. These electrical currents pass through an electrical circuit (Fig. 1) which drives an optical modulator. The optical modulator is a ring resonator with a PN junction formed in the ring waveguides. An important note is the ring resonance is unrelated to the resonant behavior of the spiking. That is to say, the natural spike rate of the neuron is unrelated to the round-trip lifetime or the resonant wavelength of the microring. The spike rate depends on the inductance, capacitance, and resistance of the circuit elements and the pump power. The resonance behavior of the microring is dependent on the radius, propagation constant, coupling efficiency, and round-trip optical loss. The voltage across the PN junction determines the free-carrier concentration in the waveguides. Via the free-carrier electro-optical effect, the effective refractive index and absorption of the ring waveguide changes, and this is used to shift the optical resonant peak. The modulator is optically pumped with light of a wavelength just off the Lorentzian resonant peak of the modulator. If the modulator is off-resonance with the pump light, the majority of the light passes to the THRU port. The THRU port is also the output of the neuron in this system. If the modulator is on-resonance with the pump light, the majority of the light passes to the DROP port. The DROP port is fed back into another germanium photodiode to form a current source dependent on the voltage across the modulator. This provides the positive nonlinear feedback connection that is fundamentally required for spiking. The voltage dependence of the feedback current takes roughly the shape of a Lorentzian function since the transmission of light through a microring modulator can be approximated as a Lorentzian function of the optical wavelength. By adjusting the voltage applied to the modulator, the current feedback from the DROP port appears as a Lorentzian function as the ring resonator is pushed across its resonance.

Fig. 1: Resonant spiking device circuit diagram.
figure 1

Excitatory/inhibitory inputs enter in the photodiode labeled E/I. The optical feedback from the drop port of the modulator goes into the feedback photodiode labeled F. The photocurrents are accepted by the input net u and passed to the electrical resonator circuit. The inductors and capacitors in the dashed boxes are biasing components, unrelated to the spiking dynamics. The microring modulator (boxed by the dot dash) has two bus waveguides and a ring waveguide that is flanked by p and n-doped silicon. Its equivalent electrical diagram is also shown in the other dot-dash box. The voltage across the junction capacitor is vC and the current passing through the resonator inductor is IL. Though not explicitly simulated in this work, this design is compatible with WDM-based weighting designs.

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The device proposed includes biasing circuitry, parasitics, time delays, and losses in the optical waveguides. While these details are necessary for verifying the functionality of the system, they are unrelated to the spiking dynamics in which we are interested. For explanation purposes, we first discuss simplified circuit equations of the device. In general, two differential equations are required for limit cycles to be possible. The following two equations have been non-dimensionalized in time. From the 30nH inductor in Fig. 1 using Kirchoff’s Voltage Law, we can obtain

$$\frac{\partial w}{\partial t}=\frac{1}{\tau }(v-bw+a),$$
(1)

and from the shunt capacitance of the modulator using Kirchoff’s Current Law, we can also get

$$\frac{\partial v}{\partial t}=-v+L(v)-w+{R}_{1}{I}_{{\rm{input}}},$$
(2)

where L(v) is the Lorentzian and nonlinear part of the equations.

$$L(v)=\frac{{R}_{1}\eta {P}_{{\rm{pump}}}\Delta {V}^{2}}{{\left(v-{V}_{{\rm{peak}}}\right)}^{2}+\Delta {V}^{2}}$$
(3)

The Lorentzian has parameters Vpeak to set the peak position, ΔV to set the half-width half-max, η to set the photodiode responsivity, and Ppump to set the optical pump power. To return the equations to their dimensionalized form, \(a=\frac{{V}_{{\rm{bias}}}{R}_{1}}{{R}_{1}| | {R}_{L}}\), \(\tau =\frac{L}{{R}_{1}^{2}C}\), w = ILR1, v = vC − a, \(t=\frac{t^{\prime} }{{R}_{1}C}\), \(b=\frac{{R}_{L}}{{R}_{1}}\). The modulator is operated just before the cut-in voltage, so its shunt resistance (~2k Ω) is still much greater that its series resistance (~38 Ω). This series resistance is small and is ignored in the non-dimensionalized equations, but it is accounted for in the VerilogA simulations that follow in the “Results” section. The junction capacitance is estimated to be 500 fF. These equations were designed to have similar characteristics to the Fitzhugh–Nagumo spiking equations29,30.

The dynamics of the device can be analyzed by obtaining the nullcline curves (i.e., when \(\frac{\partial v}{\partial t}\) and \(\frac{\partial w}{\partial t}\) equal zero). These are graphed in Fig. 2. Depending on the input currents and the optical pump power, the stable fixed point can become unstable via a subcritical Hopf bifurcation causing the trajectories to attract toward the stable spiking limit cycle. Note that the output of the neuron is the optical power leaving the THRU port, so while most spiking neuron devices would take v as the output, our output is actually closer to

$${P}_{{\rm{output}}}={P}_{{\rm{pump}}}(1-L(v)).$$
(4)

This becomes relevant later when discussing the shape of the output spikes.

Fig. 2: Phase portrait of the resonant spiking system.
figure 2

The nullclines are shown in black. The non-monotonic “n-shaped" nullcline is critical for spiking behavior. Stable trajectories are shown in solid dark blue. Here both a stable fixed point and stable limit cycle are present. They are separated by an unstable limit cycle approximated as the dashed curve. Light blue arrows are sudden perturbations on input light that take the system off the steady state. Note both positive and negative perturbations can excite the system. The red cross indicates a perturbation timed incorrectly and will not allow the system to move back to rest. However, the perturbation with the green check mark can return the system to rest. Parameters are τ = 0.0625, b = 0.05, a = 0.8, R1 = 1, η = 1, Pdrop = 0.625, ΔV = 0.0625, Vpeak = 0.925.

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Note that the inductor, resistors, and optical pump values have a direct impact on the resonant spike rate of the neuron. We want a high spike rate to increase the maximum operation speed. Ideally, we want the device to be high speed and have low power consumption. These two constraints end up being competitive. If we decrease the shunt resistance of the modulator the speed increases. However, the amount of optical input power required to drive the modulator increases. We have chosen a shunt resistance here so that operation in the GHz regime is possible and the required pump powers are in the mW range. In the above discussion, we have simplified the physics to the bare necessities for demonstrating spiking. In the next section, we discuss a more realistic system that emulates past measurements of real silicon photonic devices31 and includes the effects of parasitic and bias circuitry.

Results

Demonstration of behavior

This work utilizes the tools created in past work in simulated analog opto-electronic neuromorphic devices using VerilogA31. VerilogA is compatible with SPICE models and Spectre simulators, which makes it critical for co-simulating photonic and electronic effects simultaneously. This is especially vital for this work, since the nonlinearity is fundamentally opto-electronic. In particular, the modulator model used in this work is described much more deeply in refs. 15,31,32. The results in what follows come from time transient simulations using the Runge–Kutta method.

The timing and strength of the input pulses to the resonator neuron greatly changes the output pulse response. In Fig. 3, we send in one square pulse with a pulse height and width of 250 μW and 200 ps. This pushes the system off its stable fixed point, and the system gradually spirals back to stability. If we input two pulses spaced at half a resonant period apart, the second pulse stops the spiking resonator from oscillating by destructive interference and further prevents the system from passing the threshold. The two pulses work against each other by perturbing the system toward and then away from the unstable limit cycle threshold. However, if we input two pulses spaced a resonant period apart, the output signal is significantly enhanced since the system is pushed past the threshold. Note that these two pairs of pulses have the exact same total pulse energy, but wildly different output responses. This is how these resonant spiking neurons can encode the timing of input pulses in the output response of the system. This is useful for coincident detection and potentially learning tasks that incorporate timing-based information, such as spike timing-dependent plasticity.

Fig. 3: Time trace of input optical pulses in the resonant spiking device.
figure 3

Input optical pulses are shown in red and output optical pulses are shown in blue at 2 and 3 mW pump power. The green check mark indicates the pulse changes the trajectory to the other stable state. The pulse indicated by the red cross is unable to do so due to its timing.

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The qualitative behavior of the neuron is different depending on the pump power. At zero input power, the 2 mW pumped neuron is monostable and will eventually go to the rest state if previously excited. However, for the 3 mW pumped neuron, bistability exists even when the input power is zero. This means the system can remain in the spiking state even in the absence of input power. Returning the 3 mW pumped system to the rest state requires a properly timed pulse. The pulse denoted with a red cross is incorrectly timed, and they are incapable of switching to the rest state. The pulse denoted with the green check mark is timed to push the system back to the rest state domain. Note that both of these pulses have the same energy and polarity, yet have completely different outcomes.

Another detail for resonator neurons is they can be excited past threshold by sufficiently strong inhibitory spikes. This is because the threshold is an unstable limit cycle that surrounds the stable fixed point. As long as the threshold is past, whether positively or negatively, the trajectories will tend toward the stable limit cycle surrounding the unstable limit cycle. This is especially important for our device since the spikes it outputs are negative relative to the DC operating point.

We have shown that the resonator neuron is more excitable when pulses arrive at its resonant period. However, under a large enough constant excitation, the spiking neuron can still output spikes. In Fig. 4, we pump the microring with 2 mW of optical power and a step pulse is inputted at 350 μW pulse height with a width far longer than the resonant period. Thus, the resonator circuit sees this input as effectively a DC input after the initial sharp step in input power. The sharp increase pushes the trajectory off the stable state, and surpasses the unstable limit cycle. The only available attractor is the surrounding stable limit cycle. Thus, the system self-pulsates. When the input power is brought to 340 μW, the self-pulsation continues because the system remains in the bistable regime where the spiking limit cycle exists. The input is turned off and the spiking neuron is allowed to settle and reset. We then apply another excitation of 340 μW. This is enough to push the system far away from its stable fixed point, but the system eventually decays back to rest because the excitation was not strong enough to push past the unstable limit cycle threshold. Note that the “rest state” at 340 μW is at a different position than at 0 μW. This is still the same stable fixed point, its position has just slightly shifted due to the change in the input. Now, if the input is slightly raised to 350 μW, not much happens besides the stable fixed point slightly shifting again because the perturbation is too small to exceed the threshold. Both the resting and the self-pulsation state can exist at 340 μW input due to the bistability of the system. However, a sudden excitation of 340 μW was not enough to switch states. By dropping to zero input again, the system returns to the monostable state space. Thus, the 2 mW pumped neuron can exhibit bistability but only at input powers >330 μW and <1.3 mW.

Fig. 4: Self-pulsation behavior from a constant inputs.
figure 4

The red input pulses are effectively DC with respect to the resonator circuit, but occur over a short enough period of time to be considered AC for the biasing circuit. The blue trace shows the output of the neuron, and the yellow trace indicates the feedback signal that comes from the microring DROP port. The dashed box zooms in to show the shape of the spikes exiting the neuron and entering the feedback diode from the drop port during self-pulsation.

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To examine the class of excitability of the spiking neuron, we ramp the input power smoothly. Class 1 excitable systems see the spike rate gradually increase from zero as system is pushed past the threshold. Class 2 excitable systems see a sudden discontinuous shift in the spike rate from zero to a nonzero value. They tend to not shift in spike rate much. The results of Fig. 5 indicate that this device is a Class 2 excitable system. The shift from 0 to  ~1.1 GHz spike rate is due to a subcritical Hopf bifurcation, as evidenced by the sudden shift from rest to the spiking state. When the input power is ramped high enough, eventually the stable limit cycle disappears via a supercritical Hopf bifurcation. This effectively represents the maximum input power the spiking neuron can accept before it stops behaving like a spiking neuron. One clue that identifies this as a supercritical Hopf bifurcation is that the transformation is reversible, so that when the input power is ramped back down, oscillations begin again. The bistability of the system is very visible in Fig. 5 because the spiking does not stop even when the input power is dropped past the subcritical Hopf bifurcation again. Continuing to ramp the input downwards, the system returns to the rest state because of a fold bifurcation of cycles, where the unstable and stable limit cycles collide and destroy each other. This leaves only the rest state remaining so the system falls back to this state. Overall, the rate of spikes does not shift drastically when in the spiking state. It only varies within 100 MHz of the base rate.

Fig. 5: Continuous wave (CW) behavior of the resonant spiking system.
figure 5

The input power is slowly and smoothly ramped up and the spike rate is measured. Most of the parameter space is monostable except for the shaded region which is bistable. The pump power is 2 mW.

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Next, we demonstrate the frequency preference of the spiking neuron explicitly. We feed the spiking neuron square wave inputs of varying amplitude and frequency and plot the modulation amplitude (Fig. 6). We see the intended spiking resonant behavior in the amplitude response of the output; the modulation amplitude of the output is greatly increased if excited on-resonance with the natural spike rate of the neuron. However, there is still a certain amplitude required to pass the threshold and output a large spike. Note that the spiking resonance peak broadens for larger amplitudes. This reflects the fact that strong enough off-resonance inputs can still induce spiking, but they require larger input amplitudes. An extreme example of this is the self-pulsation behavior displayed earlier. There does appear to be a very slight distortion of the resonance peak with increasing amplitude, causing the true peak to slightly shift. This was also observed in Fig. 5, as the spike rate slightly drops with increasing continuous wave (CW) power at the input. The position of the spiking resonance peak is determined by the electrical circuit components and the optical pump power. In this way, the peak can be shifted to a different frequency by using a different inductor or capacitance value. However, since the capacitance is a property of the PN junction in the ring modulator itself, it is difficult to change without altering the optical properties.

Fig. 6: Resonant behavior of the spiking system.
figure 6

A square wave with varying frequency and amplitude is inputted into the spiking neuron and the modulation amplitude is measured. The amplitude variation is represented by a color gradient. The frequency response of two separate spiking neurons with different resonator inductances and pump powers are shown.

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In order to eventually build neural networks from this device, it needs to be cascadable. The output of one spiking neuron must be capable of driving another spiking neuron. If this is impossible, then the signal will eventually die out as it passes through layer by layer. As an example, the off-resonance signal with 200 μW amplitude at 3 GHz in Fig. 6 will only yield a modulation amplitude of roughly 100 μW. This will excite the next neuron even less, so this off-resonance signal will die out as it passes through multiple layers. This is useful because it means noise or other undesired signals will be ignored by the system. However, for an on-resonance square wave signal with an amplitude of 200 μW, the neuron modulates the pump light with an amplitude of 1100 μW. This effective increase in signal magnitude is due to the regenerative signal gain, and allows for continual cascading of the neurons so that signals can pass through any number of layers. Low-frequency signals, such as the quiescent power, will pass through the bias inductor while higher frequency signals will excite the resonator circuit. This means the resonator circuit responds to spike inputs without being affected by changes in quiescent powers. Spike events are represented by negative deflections in output optical power. Sending the output of an initial neuron to the inhibitory photodetectors of a subsequent neuron results in a net excitatory effect. For demonstration purposes, we can cascade ten spiking neurons directly after each other, feeding the outputs into negative inputs, and show that output power is still capable of driving subsequent neurons (see Fig. 7). Note this analysis ignores any insertion loss or intentional linear weighting that would normally be present between neurons. The insertion loss will depend heavily on the weighting architecture and number of weights and is not the focus of this paper. However, we think it is fair to say in an ideal weight bank, it is small or nonexistent.

Fig. 7: A demonstration of cascadability.
figure 7

Ten spiking neurons are cascaded in series. The first positive square wave input (red line) is fed into the positive input photodiode of the first neuron. The output (blue line) of the first neuron is fed into the inhibitory photodiode of the subsequent neuron. This connection scheme is repeated for the rest of the neurons.

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Discussion

The previous results indicate that the proposed device functions as intended as a resonator spiking neuron. The ease of fabrication of this device is one of its main benefits. It is made of devices commonly found in many multi-project wafer runs provided by silicon photonic chip foundries and does not require the use of any transistors or lasers to implement the dynamics. Its reliance on the electro-optical nonlinearity has some optical power consumption benefits compared to purely nonlinear optical designs. All-optical designs often either require large optical pump powers or high-quality factor cavities to induce their nonlinearity. High-quality factor cavities help lower the power required for excitability, but they simultaneously limit the speed of the device by increasing the photon lifetime. This system intentionally avoids nonlinear optical behavior. However, this system requires that the photon lifetime of the modulator be effectively instantaneous compared to the electrical resonator dynamics. This limits the maximum quality factor of the microring modulator. It is the capacitance of the microring modulator that primarily limits the speed of this spiking neuron. In principle, other faster modulators could produce faster neurons, but eventually the bandwidth of the photodetector and the photon lifetime will also matter. Lower modulator capacitance could allow for a smaller inductor, which will take up a large portion of the footprint of this device. Most of the power consumption for this device comes from the DC biasing of the modulator at 0.8 V. The voltage drop across the 50 Ω resistor contributes 12.8 mW static power usage. In principle, larger resistors could allow for lower power usage, but they will also increase the electrical time constants. There is a fundamental trade-off between power consumption and speed in this case. The choice to use a modulator over a laser has its own set of issues. The output of the modulator always has some non-negligible static quiescent output power. This static output can be ignored with careful design of the biasing circuitry, but it does complicate the design. The biasing inductor will also put a lower limit on how fast the spiking neuron can operate, as signals that last too long will pass through the biasing inductor instead of interacting with the spiking resonator circuit. Additionally, if the weights of the neuron are changing quickly, such as in training, the quiescent power could be directed to the spiking resonator circuit and could disrupt proper operation. It is often said that spiking systems are energy efficient because they primarily only consume power when they spike, which is not the case for this design. The modulator always requires optical pump powers and spiking has little effect on the power usage needed to bias the neuron. The modulator merely controls whether the power is switched between acting as a feedback or an output signal.

Conclusion

Here, we have proposed a monolithic resonator opto-electronic spiking neuron. Its nonlinear behavior comes from the optical feedback of an electrically tunable PN junction microring modulator neuron. The outputs of this neuron strongly depend on the timing and strength of input pulses. We have demonstrated bistable, self-pulsation behavior characteristic of a subcritical Hopf bifurcation. We have also shown that fold and supercritical Hopf bifurcations can also occur under different input powers. It is Class 2 excitable and is most excitable at the resonant spike rate of the neuron. This device is cascadable as it can utilize regenerative signal gain. Additionally, its ability to accommodate WDM multiplexing schemes at GHz speeds is a significant advantage for scaling to high-speed large neural networks.