Introduction. Photonic systems that display neuromimetic phenomena have been the subject of intense study in recent years [1]. Photonic recurrent neural networks [2] and photonic reservoir computing systems [3,4] have been particularly successful. The realization that photonic systems can also be used to mimic the dynamic spiking behavior of neurons [5,6] has led to intensive studies of excitable laser systems [7]. An optically injected laser system is particularly attractive as it provides sufficient nonlinearity to allow for excitable behavior while remaining amenable to experiment and theoretical analysis. What is more, it yields pulse trains with periods and pulse lengths orders of magnitude shorter than those of biological neurons while simultaneously allowing for excellent deterministic control.

Type I excitability has been observed across a multitude of device types in this configuration including edge emitting quantum well [8–10] and quantum dot (QD) [9,11,12] devices, vertical cavity surface emitting lasers (VCSELs) [13], and semiconductor ring lasers [14]. Generalizations of the Adler mechanism allowing multipulse excitability have also been observed [9,15,16]. Optically injected QD lasers have been a particularly bountiful source of excitable regimes. As well as Type I excitability, Fitzhugh–Nagumo like Type II excitability and associated canard phenomena have also been observed [17,18] unlike with optically injected conventional semiconductor lasers.

In this work we focus on dual-state QD devices. QD lasers can emit from either the first excited state (ES) or the ground state (GS), or indeed simultaneously from both [19–21]. By biasing the device so that it emits from the ES only and injecting the GS with an external laser, the ES can be made to turn off, with lasing from the GS obtained instead. A dual-state, antiphase, excitable regime is found at low injection strengths, comprised of GS dropouts and corresponding antiphase short ES pulses [22].

With quantum-well-based devices and with single-state QD devices, excitable pulses have been excited deterministically via several different types of perturbations. These include incoherent pulse triggering [23] and, most importantly for the work here, phase perturbations [24]. The direction of the phase perturbations is crucially important and depends on the sign of the detuning (the frequency of the primary laser minus that of the secondary laser). For negative detuning, the perturbations must be in the anticlockwise direction and there is a threshold perturbation amplitude which must be exceeded so as to excite a pulse. For perturbations below this threshold no excitable response is generated. However, in the dual-state system, both directions can be used to excite a pulse and there are two thresholds found—one for each perturbation direction [25]. There is also an asymmetry in the threshold values, with clockwise perturbations typically requiring larger amplitudes than their anticlockwise counterparts [25].

In real biological systems, rather than responses to individual perturbations, systems of neurons instead receive many perturbations which are then summed—integrated—together to find the response of the system. One of the prototype models for the response to integrated perturbations is the so-called integrate-and-fire mechanism. In this case, two (or more) sub-threshold perturbations are integrated together to yield an effective supra-threshold perturbation and a resulting pulse. A more realistic extension is the leaky integrate-and-fire (LIF) mechanism [26]. In this case, the time between the perturbations is crucial. The second perturbation must arrive before the effect of the first has leaked away like a discharging capacitor. Such a dynamic has been previously displayed in a number of neuromorphic photonic systems including those in Refs. [27,28]. It has also been observed in Ref. [29] where some deviations from pure integration phenomena were obtained due to the laser relaxation oscillations.

In contrast to excitatory/firing mechanisms, inhibitory mechanisms are also extremely important in neural networks [30,31]. Inhibitory mechanisms with optically injected VCSELs have been investigated in Ref. [32] where inhibition of a periodic spiking was reported by changing the injection strength and in Ref. [33] where the injection of optical pulses could inhibit spike creation via polarization mode competition. Our inhibition mechanism is quite different. We show that an integrate-and-inhibit phenomenon can be obtained where multiple above-threshold perturbations can be integrated together to yield an effective below-threshold perturbation. Such an integrate-and-inhibit phenomenon is heretofore unobserved with photonic artificial neurons.

The demonstration of both phenomena with the same system—an optically injected dual-state QD laser neuron—suggests great potential for novel spike processing functionality.

Experimental setup. Figure 1 shows a schematic of the experiment. The QD laser used is a 300 $\mu$m long InAs/GaAs-based discrete mode laser similar to that used in Ref. [34]. It lases from the ES when in the free-running configuration. The tunable laser is a commercially available device, tunable in steps of 0.1 pm with a linewidth of approximately 100 kHz. The phase perturbations are of the same form as those used in Ref. [25]. A square voltage pulse is sent from a pulse generator to a phase modulator, quickly changing the relative phase of the primary and secondary lasers. The signal from the pulse generator is split with a 50/50 power splitter to give the two perturbations that will be integrated. Different length, high-speed RF cables are used for coarse adjustments to the time between the perturbations and a variable electrical delay line is used for fine adjustments. There is a clockwise perturbation during the rise of each square pulse, and an anticlockwise perturbation during the fall of each pulse. The rise time of $\sim$200 ps is much faster than the fall time of $\sim$470 ps meaning that the clockwise perturbations are much faster and approximate idealized discrete perturbations better than the anticlockwise perturbations. The duration of the pulse is set to 5 ns, much longer than the intrinsic time scales so that the system can be taken to have relaxed to steady state by the end of each square. We use the phase resolving technique [10] to measure the perturbation amplitude in radians.

Leaky integrate and fire. Figure 2 shows efficiency curves for both perturbation directions. For a noise-free system the efficiency curves would be step functions with the step at the threshold strength. However, due to the unavoidable presence of noise, the step is elongated. The threshold is then typically defined as halfway along the elongated section (i.e., halfway between the 0% and 100% efficiencies). There is a clear threshold in each direction as seen in the figure with $-5.6$ rad for clockwise perturbations and $5.2$ rad for anticlockwise perturbations.

 

Fig. 1. Schematic of the experiment. SL, secondary QD laser; PL, primary laser; MOD, LiNbO$_3$ phase modulator; PG, pulse generator; VDL, variable electrical delay line; PC, polarization controller; filter, splits GS (red) and ES (blue) light; OSC, oscilloscope; red lines, GS light; blue lines, ES light; purple lines, GS and ES light; black lines, high-speed electrical cables; $3 \times 3$, electric field phase resolving measurement of the QD laser relative to the PL.

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Fig. 2. Efficiency curves for both perturbation directions. The clockwise threshold is at $-5.6$ rad (green line). The anticlockwise threshold is at $5.2$ rad (back line).

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We first consider the conventional LIF mechanism. In LIF networks, two or more sub-threshold perturbations can be integrated together to yield an effective above-threshold perturbation and a pulse excitation but the time between the perturbations must be short. That is, the second perturbation must arrive before the effect of the first has “leaked" away. The basic LIF mechanism can be investigated with two perturbations where each individual perturbation is sub-threshold. Due to the nature of our perturbations, both the anticlockwise and clockwise perturbations have the same strength. For integrate and fire we need two sub-threshold perturbations that together can exceed the threshold. (The control parameters for the experiment were chosen with this in mind which is why the perturbation thresholds are both relatively high.)

We set our perturbation strength to 3 rad at which strength single perturbations do not yield any pulses. When the time between the two pulses is zero (or as close to zero as can be achieved, given experimental limitations) there is 100% efficiency in pulse excitations for both directions. On gradually increasing the time between the two perturbations we find successful integrate-and-fire behavior up to 320 ps for the anticlockwise perturbations and up to 120 ps for the clockwise perturbations. The shorter time for the clockwise case is not surprising since the threshold for clockwise pulses is higher and so the integration must be done in a shorter time to reach the threshold. Figure 3 shows two examples of anticlockwise perturbation integration. In Fig. 3(a) the perturbations arrive together and successfully trigger a pulse, while in Fig. 3(b) the time between the perturbations is approximately 320 ps and the integration does not yield an excitable response. Figures 3(c) and 3(d) show the total phase perturbation in each case. Both show the same overall amplitude, and the only change comes from the introduction of the delay in Fig. 3(d). Figure 4 shows the corresponding results for the clockwise case. Figure 4(a) shows a successful excitation via integration. Figure 4(b) shows a failure to excite a pulse when the separation between the perturbations is 120 ps. Figures 4(c) and 4(d) show the total phase perturbation and, as with the anticlockwise case, the only change is the delay introduced in Fig. 4(d).

 

Fig. 3. (a) GS (red) and ES (blue) intensities for the successful triggering of an excitable response via the integration of two anticlockwise perturbations with approximately zero time separation. The integration leads to a successful GS dropout and the accompanying ES pulse. (b) Situation for an unsuccessful triggering with approximately 320 ps perturbation separation. No dropout-pulse pair is obtained. (c) Effective perturbation amplitude from the integration of the anticlockwise perturbations in the zero separation case. (d) Same but for the 320 ps separation case.

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Fig. 4. (a) GS (red) and ES (blue) intensities for the successful triggering of an excitable response via the integration of two clockwise perturbations with approximately zero time separation. The integration leads to a successful GS dropout and the accompanying ES pulse. (b) Situation for an unsuccessful triggering with approximately 120 ps perturbation separation. No dropout-pulse pair is obtained. (c) Effective perturbation amplitude from the integration of the anticlockwise perturbations in the zero separation case. (d) Same but for the 120 ps separation case.

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We note that where the pulse triggering is unsuccessful there is a response in the GS intensity albeit smaller than the full excitable response. However, the ES intensity is completely unchanged as can be seen in Figs. 3(b) and 4(b). Thus, by viewing the ES spikes as the computationally important part of the excitable trajectory, there is a true all or none response.

Leaky integrate and inhibit. We now consider the contrasting integrate-and-inhibit phenomenon. In this case, we consider the response to the integration of two supra-threshold perturbations. In the LIF case we deliberately set the detuning so that the threshold for excitability was quite large so that both directions could be analyzed with the same voltage pulses. However, in particular for the clockwise case, this is very close to the limit of our available perturbation strength. Thus, we change the detuning here, making its magnitude as large as possible without generating stochastic excitations. This decreases the excitability threshold perturbation strengths. For 100% efficiency in the clockwise direction in this case, perturbations exceeding 4.2 rad are needed with 2.7 rad needed for 100% efficiency in the anticlockwise direction. We apply perturbations of magnitude 5.5 rad (close to the maximum possible with our pulse generator) and find 100% efficiency for both directions as expected.

We now apply two perturbations to the system and vary the time between them. We find that we can completely inhibit the excitable response for the clockwise case. In Fig. 5 we show the inhibition of the response with a delay of 0.22 ns between the two perturbations. This complete inhibition is observed as the delay is increased up to 0.47 ns. For delay times between 0.47 ns and approximately 1 ns we find that noise is a major influence with complete pulse inhibition lost and instead a mixture of behaviors obtained: pulse inhibition, partial pulse inhibition, and pulse generation, sometimes even for the same control parameters. This pronounced influence of noise on the system is due largely to the finite rise time where the system can evolve during the rise time itself. For faster perturbations, the influence of noise would be much less pronounced. We interpret the phenomenon as follows. The magnitude of our perturbations is 5.5 rad. By integrating two perturbations, one might conclude that this is an effective perturbation of approximately 11 rad. A 2$\pi$ perturbation would of course put the system back on the stable point if done instantaneously. Modulo this 2$\pi$ phase periodicity, our two perturbations integrate to approximately 4.72 rad. This is in principle above the threshold, but because of the finite time associated with the perturbations, it is effectively less than this and complete inhibition can be obtained when the effective strength (modulo 2$\pi$) drops below the threshold.

 

Fig. 5. Timetrace showing that a second clockwise perturbation can inhibit the first perturbation if it arrives close enough in time. In this case the perturbations are separated by 0.22 ns.

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We do not observe the phenomenon for the anticlockwise perturbations due to the slow fall time of our voltage pulse. This slow time allows the system to evolve significantly during the perturbations and could potentially be compensated by applying a larger perturbation. However, we are close to the maximum strength of our pulse generator and so cannot go higher. We believe that the phenomenon would also be observed with faster anticlockwise perturbations of sufficient strength in this QD system. Thus, the lack of observation is purely an issue of equipment and with faster generators or indeed an arbitrary waveform generator, clear integrate-and-inhibit behavior would be obtained. Nonetheless, our experimental results show a definitive integrate-and-inhibit phenomenon for above-threshold perturbations, which is a completely new feature in neuromorphic optical injection. This is reminiscent of pre-pulse inhibition (PPI) [35], an important feature in several neuronal systems and phenomena such as the startle reflex and schizophrenia [36]. With PPI, the response to a supra-threshold perturbation can be inhibited by sending an earlier perturbation in advance with a sufficiently short delay between the two, that is, via an inhibitory integration. The real-life manifestation of PPI is naturally much more complicated and involved, but we believe that the functional similarity is worthy of notice.

Conclusion. Optically injected dual-state QD lasers exhibit a multitude of neuronal phenomena and we focus on excitability in this work. Deterministic excitable events consisting of GS dropouts and accompanying antiphase ES pulses can be generated via a LIF process with delays up to 120 ps between clockwise perturbations and 320 ps between anticlockwise perturbations. Intriguingly the same system can be used to display an integrate-and-inhibit phenomenon where the integration of two above-threshold perturbations can block pulse excitation in the system response with delays up to 0.47 ns between the perturbations. In the integrate-and-fire case multiple below-threshold perturbations are integrated to provide an effective above-threshold perturbation. In the integrate-and-inhibit case, multiple above-threshold perturbations are integrated to provide an effective below-threshold perturbation due to the $2\pi$ periodicity of the phase. All of the perturbations used are of the same form: perturbations of the relative phase between the primary and secondary laser.

There is a true all or none response in the system when the ES pulses are viewed as the computationally important part of the response. This strongly suggests that QD laser-based artificial neurons can bring novel spike processing functionalities in neuromorphic photonic systems.