Implementation of input correlation learning with an optoelectronic dendritic unit

The implementation of machine learning concepts using optoelectronic and photonic components is rapidly advancing. Here, we use the recently introduced notion of optical dendritic structures, which aspires to transfer neurobiological principles to photonics computation. In real neurons, plasticity—the modification of the connectivity between neurons due to their activity—plays a fundamental role in learning. In the current work, we investigate theoretically and experimentally an artificial dendritic structure that implements a modified Hebbian learning model, called input correlation (ICO) learning. The presented optical fiber-based dendritic structure employs the summation of the different optical intensities propagating along the optical dendritic branches and uses Gigahertz-bandwidth modulation via semiconductor optical amplifiers to apply the necessary plasticity rules. In its full deployment, this optoelectronic ICO learning analog can be an efficient hardware platform for ultra-fast control.

1 Introduction

One of the important mechanisms of learning and memory with biological evidence in the brain is synaptic plasticity [1]. Synaptic plasticity can incorporate learning rules based on locally available information, which are well-suited to support bio-inspired physical systems for computing [2]. The majority of synapses are found on dendrites, branch-like extensions of a neuron that receive electrical stimulation from other neurons. Plasticity mechanisms, together with the processing capabilities of dendrites, are considered to play a crucial role in the emergence of human intelligence [3]. New insights into the functionalities of dendritic processing are driving the search for alternative neural network architectures in machine learning [4]. This highlights the importance of exploring different computational approaches and platforms based on dendritic structures [5].

In this context, neuromorphic engineering has drawn inspiration from what is currently known about the functioning of the brain to develop hardware systems with unique computational capabilities, beyond electrical circuits for conventional computing [6, 7]. Recent developments in neuromorphic computing have resulted in various hardware platforms that incorporate brain-inspired concepts [8]. Novel nanoscale electronic devices based on memristive properties [9–14], spintronics [15, 16] or organic electronics [17] have emerged as possible candidates for synapses or neurons in neuromorphic computing systems, enabling ultra-low power consumption. Photonic platforms have also adopted neuromorphic approaches for efficient computing over the last decade [18, 19]. There is a plethora of works that explore diverse photonic platforms for neuromorphic computing, such as optoelectronic or fiber-based with temporal encoding [20–24], free-space optics with spatial encoding [25, 26], or photonic integrated circuits [19, 27, 28]. Some proposals aimed at addressing challenges in analog computing [20, 21, 29], while others aimed at implementing spiking networks with time-dependent plasticity (STDP) [22, 23, 30, 31]. While some of these works were mainly based on proof-of-concept photonic architectures, many works lately focus on the potential of these designs towards scalable, energy-efficient, and robust computing accelerators [32–36].

Moreover, some of the previous hardware systems demonstrate the capability to incorporate the classical Hebbian learning. However, experimental and theoretical studies indicate that the computational power of neurons requires more complex learning rules than the simple “neurons that fire together, wire together” [37]. New insights in neuroscience suggest that neighboring synapses interact with each other and that synaptic plasticity plays a role in implementing this coordination [38]. Furthermore, there is evidence that part of these activity-dependent changes of synaptic strength occurs due to heterosynaptic plasticity, i.e., at inactive synapses [39, 40]. The Input Correlation (ICO) learning rule is a variant of the classical Hebbian learning rule that resembles the dendritic principles of this heterosynaptic plasticity [41, 42]. It is a correlation learning rule, in which the strength (usually called weight) of synaptic connections can be strengthened or weakened based on the correlation between synapses on the same dendrite.

In this work, we draw inspiration from the fields of neuromorphic computing, synaptic plasticity, and dendritic computing to demonstrate for the first time, as a proof-of-concept, a hardware-based ICO learning rule in a reconfigurable and flexible optoelectronic setting. We design and build a fiber-based optoelectronic dendritic unit (ODU), which consists of multiple single-mode fiber pathways, each with a semiconductor optical amplifier, that allows the implementation of fast plasticity rules. These pathways emulate the dendritic branches of biological neurons. The proposed ODU naturally accommodates adaptive mechanisms required for the application of the ICO learning rule. The latter has been demonstrated in the past in hardware systems using electrical signals [41], in robot control [42, 43], and in active noise reduction [44]. With our system, we target an ultra-fast implementation of ICO learning, with information processing and adaptation times of the nanosecond scale, much faster than previous electronic approaches. In this way, computations for which a faster application of ICO learning is required will come into reach.

The manuscript is organized as follows. In Section 2, we present the ICO learning model and the experimental implementation of the ODU. In Section 3, we present the numerical and experimental results of the demonstrated ICO rule. Finally, we discuss the main features of the optoelectronic implementation and the perspectives for this dendritic computing approach.

2 Materials and methods

2.1 Input correlation (ICO) learning rule

The ICO learning rule was first proposed in [41] as a modification of Hebbian differential learning rules, which are variations of the classical Hebbian rule and use derivatives to account for the temporal ordering of neural events. In the ICO rule, the instantaneous change of the presynaptic connection i (Δw_i) is driven by the cross-correlation of two types of presynaptic events rather than between presynaptic and postsynaptic events. In the ICO rule, there are two types of input signals: a “reference” signal u₀, with a synaptic weight (w₀) that is a constant positive value representing a synapse strength that does not change over time, and the “stimulus” signals u_i>0 with their corresponding weights being updated according to the following rule:

$\mathrm{\Delta }{w}_i=\eta u_i{u}_0^{\prime },(1)$

where 0 < η < 1 is the learning rate, u_i is the activation of the presynaptic neuron corresponding to the stimulus signal u_i and $u_0^{\prime }$ is the derivative of the activation of the presynaptic neuron corresponding to the reference signal u₀. The ICO learning rule is an unsupervised neural algorithm that learns temporal separations between individual inputs. Here, synaptic weight w_i increases if the stimulus input u_i activates before the reference input u₀, and decreases if it activates after u₀. This is reminiscent of the principles of heterosynaptic plasticity, where changes in synapse strength are triggered by activity in another pathway. The output of the ICO learning rule is a linear combination of the reference input and the N predictive inputs:

$v=w_0{u}_0+{\displaystyle \sum _{i=1}^N}{w}_i{u}_i(2)$

In the simplest form of the ICO learning rule (Figure 1) we have two raw input signals, the reference x₀ and the stimulus x₁, with a certain delay between them. In this work we consider only one stimulus signal x₁, but the ICO learning rule can be extended and include multiple stimulus inputs. We consider for illustration purposes the sequences of such two neuron spikes, modulated as delta-like functions, x₀(t) = δ(t − T) and x₁(t) = δ(t), which are periodically inserted into the system every 600 time steps (see Figure 2A). If the stimulus signal (x₁) is injected earlier than the reference signal (x₀), T is positive, otherwise, T is negative. In the ICO learning rule, the weight changes only during the temporal overlap between the reference and the stimuli signals, and this change is based on the derivative calculation. Therefore, raw inputs with sharp temporal profiles (e.g., delta-like functions) need to be temporally extended to increase the temporal overlap region. This can be easily implemented by filtering the raw signals and converting them into activation signals u_i. In our case, we use a filter function with a discrete-time response for each input, given by:

$h\left(n\right)=\frac{1}{dc}{e}^{an}\sin \left(dn\right),(3)$

where n is the index of the time steps, a = −πf $/Q,d=\sqrt{{(2\pi f)}^2-a^2}$ and c > 0 is a parameter to tune the amplitude of the filtered signal. The parameter 0 ≤ f < 0.5 is the frequency of the filter normalized to a unit sampling rate. The parameter Q > 0.5 of the filter defines its decay rate.

FIGURE 1

FIGURE 1. Illustration of the ICO learning model. The raw stimulus input x₁ can trigger many different activation signals u_i, which depend on the filters’ properties. The signal x₀ acts as a reference that changes the synaptic weights w_i in an unsupervised manner.

FIGURE 2

FIGURE 2. (A) Two delta-function raw inputs x₁ and x₀ with a time difference of T = 45 steps. (B) Their corresponding filtered activation signals (u₁ and u₀). (C) The corresponding filtered activation signals when considering three different filters for the stimulus signal x₁. The filter parameters are: Q = 0.51 and c =1/(1+0.5 (i −1)), f =0.01/i for the three activation inputs u₁, u₂ and u₃, and c =1, f =0.01 for u₀. (D) The weights for the filtered activation signal of case (B). (E) The weights for the filtered activation signal of case (C).

When using only one filter for each raw input, x₀ and x₁, these are transformed by the filtering process into the activation signals u₀ and u₁ respectively (Figure 2B). We use the same filter for both signals, with Q = 0.51, c = 1, and f = 0.01. Then we apply the ICO rule in Eq. 1 to update the weight associated with the stimulus signal u₁ (Figure 2D). Weight values change only during the time overlap between the increasing and decreasing parts of the filtered activation signals. The increasing part of the reference signal, captured by the first-order derivative, marks also the increasing part of the weight. In general, when the reference signal comes later than the stimulus signal, the weight increases. For example in Figure 2D, the weight goes from an initial value of 1 to eventually a value around 1.1. There is a short region of decrease, around the time steps 70–90, and this is due to the overlap between the final part of the stimulus signal and the decreasing part of the reference signal. The weight w₁ then remains constant in the absence of overlap between u₀ and u₁.

The ICO learning rule can include multiple filtered versions of the original stimulus signal x₁ to increase its range of operation and improve its input resolution. Each new filter is usually designed to expand in time the stimulus signal in a different manner. For instance, a reference signal x₀ appearing at the 200th time step would not interact with activation filtered function u₁ in Figure 2B. We numerically study the case with a bank of 3 filters that are applied to the raw input x₁ (u₁, u₂, u₃) (see Figure 2C). The filter bank increases the temporal range of ICO learning rule operation. Reference inputs appearing up to the 400th time step now interact with at least one filtered version of the stimulus signal. The filter parameters for the stimuli signal are Q = 0.51, c = 1/(1 + 0.5 (i − 1)), and f_i = 0.1/i, where i > 0 is the increment number of the filter bank. In this particular case, the weights (w₁, w₂, w₃) associated with the output of each filter (u₁, u₂, u₃) follow the same trend as in the case of a single filter for the raw stimulus signal (Figure 2E). The filter bank adds expressive power to the ICO rule in resemblance with the way dendritic plasticity adds computational power to a single neuron. Dendrites do not just receive the input information (stimuli) and act as passive receivers; each dendritic branch can apply a different filter to the stimulus signal and then adapt the strength of the contribution of the dendritic branch (for a review see [45]).

2.2 Optoelectronic implementation of the ICO learning rule

In this work, we consider an optoelectronic dendritic unit (ODU) to implement the ICO learning rule. The ODU is an optoelectronic system with individual single-mode fiber optical paths (dendritic branches—DB) that can carry different information. In the context of the ICO learning topology, one DB is assigned to the transmission of the reference information, while the rest of the branches are assigned to the information from the sensory stimuli.

2.2.1 Experimental setup of an optoelectronic dendritic unit

The experimental design of a single-mode fiber (SMF) optoelectronic dendritic unit (ODU) with four dendritic branches, in a coincidence detection architecture, is shown in Figure 3. A 40 mW superluminescent diode (SLD) at 1,550 nm, followed by an optical isolator and a tunable-bandwidth optical filter, provides an incoherent optical carrier to the input of the ODU. The bandwidth of the filter is set to 1.5 nm so that the fiber lengths of the ODU do not introduce considerable chromatic dispersion that distorts the initial pulse shapes. The input information is encoded via a 10 GHz Mach-Zehnder modulator (MZM) biased at its quadrature operating point, using a time-multiplexing approach. Four independent data streams that correspond to the individual filtered stimuli (u₃, u₂, u₁) and reference signals (u₀) are combined into a single data stream in sequential order, with some time separation between them. Time-multiplexing allows the data stream to be electrically generated by a single channel (CH1) of a 9.6 GHz 24 GSa/s arbitrary waveform generator (AWG) and applied in the optical domain using only one MZM element. After encoding, the optical signal is split into four optical paths, via a 1 × 4 optical splitter, representing the four optical dendritic branches. Each path includes a semiconductor optical amplifier (SOA), which can amplify or attenuate the strength g_i of the optical signal. The three DBs (#2, #3, and #4) that carry the experimental stimuli signals, need to be dynamically weighted, following the plasticity rules that apply at any given time. Thus, in these three dendritic branches, the SOAs are operated as modulators (SOAMOD) by using the appropriate circuitry and by modulating their bias current. The supporting modulation rates range from 1.5 MHz up to 2 GHz. The plasticity rules are introduced by the weight data streams (w₃, w₂, and w₁), which are again combined into a single data stream in sequential order via time-multiplexing. Using a second AWG channel (CH2) and after electrical splitting (RFS), these weighted sequences are electrically amplified (RFA) and applied to the SOAMOD devices. In addition, each dendritic branch of the ODU incorporates an SMF spool (SMF-S) of increasing length (1, 1.5, 2, and 2.5 km), so that the time-multiplexed signals encoded at the input are temporally aligned at the coupling stage (4 × 1 optical coupler). By construction, the summation of the optical signals from the different paths is incoherent. An optical filter reduces the amplified spontaneous emission noise–introduced by both the SOA and SOAMOD devices—of the summed optical signal. Finally, the optical signal power is detected by an optical power meter (PM) and its temporal evolution is detected by a 20 GHz AC-coupled photoreceiver with a trans-impedance amplifier (PD). The converted electrical signal is monitored and recorded by a real-time 16 GHz analog bandwidth 40 GSa/s oscilloscope (OSC). The type and operating conditions of the components and instruments are provided in Table 1.

FIGURE 3

FIGURE 3. Optoelectronic dendritic unit (ODU) with four dendritic branches (DB), in a coincidence detection architecture. SLD, Superluminescent diode, OI, Optical isolator, TB-OF, Tunable-bandwidth optical filter, MZM, Mach-Zehnder modulator, 1/4, One-to-four optical splitter, SOA, Semiconductor optical amplifier, SOAMOD, Semiconductor optical amplifier operating as modulator, SMF-S, Single-mode fiber spool, 4/1, Four-to-one optical coupler, OF, Tunable optical filter, 10/90, Optical splitter, PM, Powermeter, PD, Photodetector, OSC, Real-time oscilloscope, AWG, Arbitrary waveform generator, RFA, RF amplifier, RFS, RF splitter.

TABLE 1

TABLE 1. Components, instrumentation and operational parameters used in the ODU experiment. I: Bias current, T: Temperature, λ: Wavelength, Δλ: Wavelength bandwidth, S: Sampling rate.

2.2.2 Time-multiplexing for data encoding

By applying time-multiplexing, we significantly reduce the components that are needed to introduce independent sequences into the different optical dendritic branches. The time-multiplexing of the encoded data, the weighting, and the temporal compensation of the different dendritic branches work as follows. In CH1 of the AWG, we encode sequentially the following input data sets: u₃, u₂, u₁, and finally u₀ (Figure 4). Only when the temporal distance between these sequences matches the physical temporal delay introduced by the length of the ODU’s different DBs, the coincidence of the data sets will happen in predetermined dendritic branches. Specifically, the u₃ sequence, which is encoded first, is assigned in DB #4 which has the longest delay (SMF-S of 2.5 km). u₂ that is encoded second is assigned in DB #3 with the second longest delay (SMF-S of 2 km). u₁ that is encoded third is assigned in DB #2 with the third longest delay (SMF-S of 1.5 km). Finally, u₀ is encoded last and is assigned in DB #1 with the shortest delay (SMF-S of 1 km). As the physical propagation length of the DBs is fixed, the temporal matching is performed at the AWG encoding by setting the appropriate temporal spacing ΔT_uij between the input data sets. The temporal resolution we use for this matching is 100 ps and is defined by the AWG. The length difference of ∼ 500 m between adjacent DBs introduces delays among the optical signals of ∼ 2.45 μs. This delay defines the maximum length of the data sets u_i that can be encoded with time-multiplexing, without overlapping with each other. Eventually, at the summation point - the 4 × 1 optical coupler’s output - the experimental signals $(u_i^E)$ that correspond to the four encoded data sets (u_i, i = 0–3) are temporally aligned. In absence of weighting adaptation and for various optical strengths g_i of the DBs, the corresponding optical signals are summed in the photodetected ODU output v^E:

$v^E\left(t\right)={\displaystyle \sum _{i=0}^3}{u}_i^E\left(t\right)={\displaystyle \sum _{i=0}^3}{g}_i\cdot u_i.(4)$

In Eq. 4, we consider that the encoded signals u_i undergo a linear transformation through the physical ODU system, thus their measured values $u_i^E$ are approximated by a simple multiplication with a gain value. When introducing adaptive weighting in three out of the four dendritic branches, another temporal alignment is required. The time-multiplexed sequences that carry the weight information are uploaded at the encoding stage, in CH2 of the AWG. The weight data sets (w₃, w₂, and w₁) are encoded sequentially. Here, the physical path of the electrical lines and the optical paths are again fixed. Thus, temporal matching is performed via the AWG encoding, by setting the appropriate temporal spacing ΔT_wij between the weight data sets, but also the appropriate temporal shift ΔT_uw with respect to the input signal sequence u (Figure 4). In this way, there is a pre-defined time window in which the optical signals that carry the weighted data sets are temporally aligned and summed at the summation stage (the 4 × 1 optical coupler’s output, Figure 4, red dashed box). Also, in this case, the experimentally applied weights $(w_i^E)$ to the optoelectronic system are approximated by a linear transformation of the encoded weight data sets (w_i). The photodetected ODU output v^E with weighting adaptation is given by:

$v^E\left(t\right)=u_0^E\left(t\right)+{\displaystyle \sum _{i=1}^3}{w}_i^E\left(t\right)\cdot u_i^E\left(t\right)\approx g_0\cdot u_0\left(t\right)+{\displaystyle \sum _{i=1}^3}{w}_i\left(t\right)\cdot g_i\cdot u_i\left(t\right).(5)$

FIGURE 4

FIGURE 4. Schematic diagram of the time-multiplexing encoding and the coincidence detection at the ODU output. The temporal distances between the encoded input u_i (CH1) and weight w_i (CH2) data sets are defined so that there is a predefined time window for the ODU summation stage in which the different, individual signals that propagate along the dendritic branches DB are temporally aligned and summed.

3 Results

3.1 Performance of the ICO learning rule—Numerical results

We show in Figure 5 two cases with different parameter sets for the numerical implementation of the ICO rule. These results illustrate the impact of the time difference between the activation signals of the stimulus (u₁) and the reference (u₀) to the change of the weight w₁ associated to the stimulus branch, and the corresponding output v. In Figure 5A, four pulses are repeated every 600 time steps, under the condition that u₁ comes T = 55 time steps before u₀. For the next three pulses, u₀ is zero, while the last 4 pulses of u₁ come after u₀ with a time difference of T = −55 steps. For this case, a learning rate of η = 0.05 in Eq. 1 is considered. In the case of Figure 5B, the four pulses are repeated every 600 time steps, under the condition that u₁ comes now at T = 20 time steps before u₀. For the next three pulses, u₀ is zero, while the last four u₁ pulses come after u₀ with a shorter time difference of T = −20 time steps. For this case, a slower learning rate η = 0.01 in Eq. 1 is considered. In both cases, the initial plastic weight (w₁) increases when the interval T between the two inputs is positive and decreases when it is negative; the weight remains constant if one of the inputs is zero (Figures 5A, B, middle graphs). The weight increase is linear and depends on the learning rate (η) and the time interval between inputs (T). The output of the ICO learning rule v is given in (Figures 5A, B, lower graphs), for the two cases.

FIGURE 5

FIGURE 5. (A) Top panel: A sequence of two activation inputs u₀ and u₁, with u₀ obeying different conditions: it appears after u₁ with T =55 time steps, it is zero, or it appears before u₁ with T =−55 time steps. Middle panel: The plastic weight w₁. Bottom panel: The ICO learning output v. The learning rate is η =0.05 and w₀=1. (B) Same as A, but with a reduced temporal separation between u₁ and u₀ (T =20 or T =−20 time steps) and smaller learning rate η =0.01.

The values of the stimulus activation inputs play a decisive role in the calculation of their weights. Until now, we have considered input sequences in which the pulse amplitude returns to zero. However, if the input sequences have a DC bias (b), the ICO learning rule implies different weight updates according to Eq. 1. A stimulus input signal with bias (u₁ + b) leads to a different weight update, with respect to an input signal without bias. In contrast, a bias value in the reference input only affects the output signal but not the weight update. This is shown in Figure 6, by considering two cases with minimum amplitude (or DC bias) of b = 2.5 and b = 5. When comparing the above behavior with the case of b = 0 (dashed red line in Figure 6), we conclude that the bias increases the average value of the ICO learning output v and the plastic weight w₁, but the behavior regarding the time difference steps between the inputs remains the same. The reason for which we consider this DC bias is to take into account the characteristics of our experimental demonstration. On the one hand, the light source that we use in our experiment (SLD) is non-monochromatic, with an optical bandwidth set to 1.5 nm, leading to incoherent positive summations only. On the other hand, external optical modulation is performed via an optical MZM, an interferometric device that is designed for coherent light. Thus, when a given wavelength condition matches the maximum extinction ratio obtained at the optical output of the MZM, the neighboring wavelengths do not comply with this interferometric condition. Thus, there will always be significant optical power at the lowest modulation level of our optical input signal, i.e. a DC bias in the optical input.

FIGURE 6

FIGURE 6. (A) The plastic weights w₁, and (B) the ICO learning outputs v, for different DC amplitude bias values b of the activation inputs presented in Figure 5B. The learning rate is η =0.01 and w₀=1.

Next, we numerically study the case with a bank of 3 filters that are applied to the raw input x₁ leading to three different activation signals u₁, u₂ and u₃, following the general ICO learning concept of Figure 1. Here, we consider a total of twenty input time frames, in which raw stimuli and reference pulses are repeated every 600 time steps, with different time distances between them. In the first eight input time frames, x₀ is delayed by T = 45 time steps with respect to x₁; in the next four time frames the reference signal x₀ is zero; in the last eight frames, x₀ is 45 time steps ahead of x₁ (T = −45). The filtered raw inputs result in the activation input sequences u_i of Figure 7A. These filtered inputs are applied to the ICO learning rule with a learning rate of η = 0.01 and a weight for the reference signal equal to w₀ = 1. The ICO learning output v signal is shown in Figure 7B, while the update of the adaptive weights w_i (i > 0) is shown in Figure 7C. The differences among the plastic weights w_i are given by the different input activation amplitudes u_i and the time overlap between the stimuli filtered signals and the reference one.

FIGURE 7

FIGURE 7. Numerical simulations of the ICO rule with one stimulus signal x₁ that feeds three different activation filters (u₁− u₃). (A) Input activation signals (reference and stimuli). (B) Output of the ICO rule. (C) Plastic weights that correspond to the stimuli activation inputs.

3.2 Performance of the ICO learning rule—Experimental results

The filtered inputs (u₀, u₁, u₂, u₃) and weights (w₁, w₂, w₃, are applied in the experimental realization of the ODU, using the time-multiplexing approach explained in Section 2.2.2. We apply the two corresponding sequences u and w to the two channels of the AWG (Figure 3). Appropriate scaling of the electrical modulating sequences is applied. This scaling defines the modulation depth of the optical carriers for each dendritic branch. However, as discussed in the previous section, the incoherent nature of the incoming optical signal sets a non-zero optical signal for the lowest modulation values, introducing a DC bias to the overall experimentally recorded sequence. The SOA and SOAMOD bias currents are tuned to equalize the average optical power of each dendritic branch. As the different DBs experience slightly different optical losses, the provided gain g_i in each path may be slightly different for this optical power equalization. In Figure 8A, we show the experimentally measured filtered input signals $u_i^E$, as they are detected at the output of the ODU, in the absence of the rest of the inputs and plastic weight application. The detected amplitudes of the $u_i^E$ signals become consistent with the numerically computed ones (Figure 7A) by adjusting the gain g_i of each DB, which is incorporated in the amplitude $u_i^E$. The DC bias of each input is not depicted here, as the optoelectronic detection of the ODU is performed with an AC-coupled photoreceiver. In Figure 9, we show a zoom of the measured signals $u_i^E$ for the four DBs, and for the three different u₀ conditions studied in the corresponding encoded sequence. Specifically, in Figure 9A, the measured signal $u_0^E$ (black line) fires after the stimuli filtered input signals ($u_i^E$, i = 1–3); in Figure 9B, $u_0^E$ is zero; and in Figure 9C, $u_0^E$ fires before the stimuli inputs.

FIGURE 8

FIGURE 8. ICO rule applied to the experimental ODU in an open loop configuration. (A) Measured filtered input signals $(u_i^E)$ that propagate along the four DBs before weighting, in absence of weighting and the rest of the inputs. The dashed lines separate the three different reference filtered input signal conditions. (B) Weighted filtered input signals in the three plastic DBs $(w_i^E{u}_i^E)$ and ODU output signal (v^E). The weights $w_i^E$ are obtained from the SOAMODs electro-optic response, after feeding the numerically calculated values w_i. The experimentally measured signals are obtained at the ODU output.

FIGURE 9

FIGURE 9. Detail of Figure 8, focusing on the three different timing conditions of the reference filtered signal $u_0^E$ (black line). In (A) $u_0^E$ fires after the stimuli filtered inputs $(u_1^E-u_3^E)$; in (B) $u_0^E$ is zero; in (C) $u_0^E$ fires before the stimuli filtered inputs $(u_1^E-u_3^E)$. All filtered input signals are monitored experimentally by their physical counterpart $u_i^E$.

In the next step, we apply the pre-calculated weights in the dendritic branches that carry the stimuli information (DB#2-DB#4). In Figure 8B we show the individually measured signals after weighting $(w_i^E{u}_i^E)$, as they are detected at the output of the ODU, in absence of the rest of the inputs. We also show the output summation detected signal v^E, in which three distinct regimes can be identified, in agreement with the numerically computed output of the ICO learning rule of Figure 7. The first eight pulses include a reference signal $(u_0^E)$ that rises in time after the stimuli signals ($u_1^E$, $u_2^E$, and $u_3^E$), as seen in Figure 9A. The experimental ODU output results in an increase of the peak amplitude for these first eight pulses (Figure 8B), every time a reference pulse appears and is lagging the stimuli pulses. The first peak has an amplitude of 103 mV, while the 8th peak’s amplitude is increased to 169 mV. The next four stimuli pulses are not accompanied by any reference pulse (Figure 9B) and, consequently, the output pulses preserve the same amplitude, with only a very slight decrease. The amplitude values vary within the range of 148 mV–156 mV. Finally, the last eight stimuli pulses of $u_1^E$, $u_2^E$, and $u_3^E$ are accompanied by a leading reference signal $(u_0^E)$ (Figure 9C). In the latter case, the ICO learning rule results in a decrease of the peak amplitude at the ODU output, every time a reference pulse appears. The last peak of the sequence is obtained with an amplitude of 50 mV.

The deviation that we find between the numerically computed ICO learning output v (Figure 7) and the corresponding experimental one v^E (Figure 8) is mainly due to the DC bias of the experimental input signals. As we showed in Figure 6, if the pulses do not return to zero, the ODU output will also not return to zero between pulses. This is the case of the experimental implementation, as a “zero” value encoding at the input, results always in non-zero optical power for the optical signals that propagate through the dendritic branches. This DC bias results in the non-return-to-zero pulse behavior in Figure 8B. This effect was not considered in the numerical computation of Figure 7, which were obtained assuming a DC bias b = 0 and no gain (i.e. g = g_i = 1). There, the ODU output exhibits return-to-zero pulses (Figure 7B). In an updated numerical implementation of the ICO learning rule, we consider now a DC bias b = 8 and a gain of g = g_i = 2 for i = 0–3, with these values emulating best the corresponding experimental parameters. By recalculating the ICO learning rule we obtain the output in Figure 10. Now the numerically computed output (Figure 10A, black line) shows a non-return-to-zero pulse behavior, in accordance with the experimentally computed sequence (Figure 10A, red line). The individually weighted signals, for each dendritic branch of the stimuli, show also a non-return-to-zero pulse behavior (Figure 10B) and are also in agreement with the corresponding experimental signals of Figure 8B.

FIGURE 10

FIGURE 10. (A) Comparison between numerical simulations (v, black color) and experimental results (v^E, red color) of the ICO rule’s output, with one stimulus signal x₁ that feeds three activation filters (u₁− u₃). In the numerical model, the gain is g =2 and the bias is b =8. “Det. V” is the abbreviation for “Detected Voltage.” (B) The numerically computed weighted stimulus input signals, for the previous values of gain g and bias.

We can see in Figure 10A that, after adjusting and including both parameters of gain and bias in the numerical computation, we obtain an overall good agreement between the numerical output and the experimentally obtained results at the ODU output. There is only a small divergence between the two signals in the last part of the sequence, due to the faster decay of the experimentally detected output compared to the numerically calculated one. The reason is that the SOAMOD circuitry introduces a non-negligible attenuation in the low-frequency modulation, leading to a stronger decay of the DC-detected signal than numerically expected. This is because the total duration of the signal we process is 1.2 μs (as shown in Figure 10A), which corresponds to a frequency of 833 kHz, slightly below the low cut-off frequency of the SOAMOD circuitry (1.5 MHz).

4 Discussion

The presented optoelectronic ODU and the proof-of-concept implementation of the ICO learning rule, in an open loop configuration and at the nanosecond scale, open the path for future control and adaptation in ultra-fast optoelectronic hardware platforms. Further advances of the presented system can already be envisaged, toward more compact and energy-efficient designs, by transferring the proposed ODU design to photonic integrated platforms, where the fiber-based dendritic branches are substituted by optical waveguides. Here, we made use of incoherent optical signals, which hampers signal subtraction. In a photonic integrated circuit platform, a fully-controllable phase-dependent ODU approach could be considered, while replacing the power-expensive SLD with a coherent laser source. Then, the subtraction of coherent signals would be possible via the appropriate optical phase-sensitive control of the dendritic branches, by exploiting the properties of phase-change materials [46–48]), or graphene ([49]) for phase modulation. A coherent approach would also eliminate the optical bias that appears now in the fiber-based ODU, at the input signal modulation stage. This would allow also an easier parametrization of the system and convergence between the experimental results and the numerically calculated ICO learning performance. Additionally, in the presented ODU, the incorporation of SOAMOD devices offers the flexibility to amplify individually the optical signals in the dendritic branches, apart from the application of the weights. In a photonic integrated platform, the weight adaptation could be introduced by integrated lithium-niobate [50, 51] or microring resonator [52, 53] modulators, in absence of any optical amplification means. Eventually, the optical power budget of the ODU will determine the number of dendritic branches that could be considered without amplification, in energy-efficient designs.

The optoelectronic ODU could also be exploited in a closed-loop configuration for control applications. ICO learning was originally proposed as a model that describes how an agent can learn an anticipatory action to react and avoid a reflex event that is preceded by an earlier alarm signal. As shown by one of the authors [41, 54], the ICO learning rule in a closed-loop approach can be used in various control applications, where the reaction to a sensory stimulus x₁ can be adapted by the ICO learning to predict and eventually suppress the occurrence of a reflex behavior. In such closed-loop feedback control applications, the reference input is an error signal for the learning while the stimulus or sensory inputs x_i>0 will be an early alarm signal. Extending the presented ODU implementation to operate in a closed loop means that the adaptation weights - which in this study were pre-computed and uploaded in the experimental setup - would need to be calculated in real-time from the output signal, for each temporal resolution step and applied to the ODU. Although this approach would still require some conventional computational processing, it would demonstrate an autonomous operation of an ODU that implements the ICO learning rule. Finally, this ODU implementation will serve as a flexible photonic testbed for other synaptic plasticity rules beyond the ICO learning and the exploration of more advanced forms of dendritic computing. This concept could be transferred for dedicated computing tasks into more compact, energy-efficient photonic integrated circuits or memristive platforms.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

SO performed the numerical simulations of the study, in collaboration with CT, FW, and AA. AA designed and implemented the experimental ODU, conducted the experiment and analyzed the corresponding data. SO, MS, CM, and AA drafted the paper. All authors participated in the conceptual design of this study, the discussion of the results, and the final writing of the paper.

Funding

The authors acknowledge the support of the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 899265 (ADOPD), and the fruitful discussions with Robert Legenstein and the rest scientific collaborators that participate in this project.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling editor LP declared a past co-authorship with the authors CRM, AA.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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