# Abstracts

## Large scale spatio-temporal networks of nonlinear oscillators for neuromorphic computing

## Resonator and multipulse property of a neuromimetic excitable laser system

Excitability is a property that is shared by many different systems, from biology to chemistry to laser systems. It is defined as the possibility to respond to an external perturbation by either producing an excitable response (e.g. a spike) if the perturbation is over a certain threshold, or otherwise relaxing to its stable state. The feasibility of producing an excitable event using a laser with injected signal was already suggested several years ago [1], where the phase dynamics of these events can be basically described by the Adler equation. An observation of such events was realized in [2], while a control of their generation was obtained in [3].

Our work is also focused on the generation of excitable pulses using a laser system with optical injection, where the perturbation used to generate the excitable events consists of two narrow pulses of a duration of 0.12 ns, separated by a variable time-delay, from a minimum of 0.08 ns to a maximum of 1.05 ns. What we unexpectedly find is that there is a particular delay, around 0.10 ns, where there is a higher efficiency of generating a single response. This means that the system has a preferred frequency to which it will respond: this is called a resonator property. We also observed multipulse excitability, that is, the possibility of producing two or more responses for a single perturbation. Both of these properties cannot be explained by the simple Adler model.

Similar features of resonator property, multipulse excitability and refractory period [4] can already be found in the case of neurons, which are the most typical example of excitable system. Guided by this observation we started to investigate the connections between these two words, the laser word and the neuroscience word. Using tools from Singolar Pertubation theory we study the theoretical model of a class B laser, and we try to describe the behavior of our system going beyond the Adler equation. We also investigate the connections between our system and the dynamics of neuronal models that better fit our physical data. The long term goal we have in mind is to exploit the neuron-like properties of our system for optical data processing and to provide possible insight about complex solitons interactions in forced oscillatory media [5].

Coullet, P., D. Daboussy, and J. R. Tredicce. “Optical Excitable Waves”. Physical Review E 58, no. 5 (1998)

Goulding, D., S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet. “Excitability in a Quantum Dot Semiconductor Laser with Optical Injection”, Physical Review Letters 98, no. 15 (2007)

Turconi, M., B. Garbin, M. Feyereisen, M. Giudici, and S. Barland. “Control of Excitable Pulses in an Injection-Locked Semiconductor Laser”, Physical Review E 88, no. 2 (2013)

Garbin, B., A. Dolcemascolo, F. Prati, J. Javaloyes, G. Tissoni, and S. Barland. “Refractory Period of an Excitable Semiconductor Laser with Optical Injection”, Physical Review E 95, no. 1 (2017)

Garbin, Bruno, Julien Javaloyes, Giovanna Tissoni, and Stephane Barland. “Topological Solitons as Addressable Phase Bits in a Driven Laser”, Nature Communications 6 (2015)

## Modes of oscillation of a neural field model

We investigate the modes of oscillation of a large network of spiking neurons arranged in a one dimensional ring. Perturbations of the equilibrium state with a particular wave number produce standing waves with a specific frequency, in a similar fashion as in a tense string. In the neuronal network, the equilibrium state corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, reflecting episodes of spike synchrony of the spatially distributed neuronal ensembles and excitatory-inhibitory spatial interactions. In the thermodynamic limit, a low-dimensional neural field model describing the macroscopic dynamics of the ring network is exactly derived. This allows us to analytically obtain the spectrum of the normal modes of oscillation of the network. We find that the frequency of each mode only depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity; the decay rate, instead, is exactly the same for all oscillation modes. We finally show that the spectrum of spatially inhomogeneous solutions has a continuous part off the real axis, indicating that similar oscillatory modes operate in neural bump states as well.

E. Montbrió, D. Pazó and A. Roxin, *Macroscopic Description for Networks of Spiking Neurons* Phys Rev X. 06/2015 5(2).

## Correlation processing by neurons with active dendrites

## A simple spiking neuron model based on stochastic STDP

Synaptic plasticity refers to a change of neuron's morphology, sensitivity and reactivity. Such a plastic behaviour is thought to be at the basis of memory formation which makes it particularly interesting. Nowadays, popular plasticity models are based on Spike-Timing-Dependent-Plasticity (STDP) rules. In STDP models spikes timings are central, but current works showed that firing rate, membrane potential, neuromodulators and many other factors affect synaptic plasticity. These numerous perturbations partially explain the stochastic behaviour of neurons and plasticity. Another important remark about plasticity is the presence of different time scales. Indeed, long term plasticity time scale ranges from minutes to more than one hour. On the other hand, a spike lasts for 1 millisecond. Thus, there is a need to understand how to bridge this time scale gap at the synapse level and how it interplays with noise.

Here, we would like to present a new model of stochastic plasticity in networks of spiking neurons described by 2 states Markov chains. The non-plastic network is rich enough to be realistic: it reproduces phenomena which have been widely observed by biologists. For example, spontaneous oscillations, bi-stability and different time scales. In addition, it is simple enough to be mathematically analyzed and numerically simulated. The most original point of our study concerns the introduction of a new STDP rule which we implement in the well-known stochastic Wilson-Cowan model of spiking neurons. More precisely, because of the plasticity rule, our model is a piecewise deterministic Markov process. In the context of long term plasticity, synaptic weights dynamic is much slower than the network one, a time scale analysis enables us to remove the neurons dynamics from the equations. Indeed, this allows us to derive an equation, for the slow weight dynamic alone, in which neurons dynamics are replaced by their stationary distributions. Thereby, we don't need to simulate the dynamics of thousands fast neurons any more and we get an equation much simpler to analyze. We then discuss the implications of such derivation for learning and adaptation in neural networks.

## Activation, Inhibition and Propagation of Spiking Regimes in Vertical Cavity Surface Emitting Lasers

Photonic techniques emulating the brain powerful computational capabilities are the subject of increasing research interest as these offer excellent prospects for ultrafast neuro-inspired information processing systems going beyond classical digital modules. One of these approaches considers the use of semiconductor lasers, as these devices can undergo a rich variety of dynamical responses similar to those observed in neurons; yet, remarkably these are obtained at speeds up to 9 orders of magnitude faster than the millisecond timescales of biological neurons. Amongst semiconductor lasers, Vertical Cavity Surface Emitting Lasers (VCSELs) are ideal for use in neuromorphic photonics as they possess important inherent advantages, e.g. low fabrication costs, ease to integrate in 2- and 3-Dimensional arrays, high coupling efficiency to optical fibres, etc.

In this talk, we will review our recent progress on the achievement of controllable and reproducible spiking patterns in VCSELs with ultrafast speed resolution. Specifically, we will show that a wide variety of spiking regimes, e.g. single and multiple spiking and bursting patterns can be controllably activated and inhibited in these devices in response to external perturbations. Additionally, we will introduce our recent results demonstrating the successful communication of spiking photonic signals between two interconnected VCSELs. Moreover, the activation, inhibition and propagation of the aforementioned spiking regimes are all obtained at sub-nanosecond speeds, thus offering high potentials for novel ultrafast non-traditional information processing capabilities with these laser sources. Also, our results, obtained with off-the-shelf inexpensive components operating at the most relevant wavelengths in present optical fibre networks (1300 and 1550 nm), make our approach fully compatible with optical communication technologies.

In summary, the reproducible and controllable activation, suppression and propagation of spiking photonic signals at high speeds in VCSELs operating at telecom wavelengths offer great potential for the use of these devices in excitatory and inhibitory photonic neuronal models for future neuromorphic photonic information processing systems.

## Regenerative memory in time-delayed neuromorphic photonic resonators

## Slow passage to two-tone bursting

## Delay-based Reservoir Computing: a meeting point for nonlinear dynamics, signal theory, and brain-inspired computing

Nonlinear delay dynamics have attracted lots of interest from their autonomous operation, being capable for various, complex and beautiful behaviors, from period doubling cascade to high dimensional chaos among others. They were also recognized to be relevant models for many practical situations, from physiology to optics through mechanics. In a more applied perspective as for example in photonic, they were used to develop novel encryption schemes that are hiding bit streams into chaotic waveform, decryption being achieved after chaos synchronization ; One also used their extremely long temporal memory provided by large delay to dramatically decrease the phase diffusion constant in microwave oscillators, thus providing high performance (ultra-low phase noise) microwave optoelectronic oscillators for Radar applications. More recently the high dimensional phase space of delay systems and their known analogy with spatio-temporal dynamics, have been used as a technologically feasible and efficient way to emulate a network of neurons, through which novel brain-inspired computing (Reservoir Computing) concepts can be implemented.

In this contribution, we will particularly emphasize on how nonlinear delay dynamics, moreover photonic ones, can be regarded as a kind of recurrent neural network with which spatio-temporal information processing is fully expanded in the time domain through basic signal theory principles. We will show how this description can be used in programming physically classification tasks performed with the bandwidth of Telecom devices, thus reaching record speed up to 1 million words recognized per second with a dedicated photonic hardware.

## How noise shapes the spontaneous activity and signal transmission properties of neurons

In my talk I discuss seemingly complex spontaneous spike statistics of some sensory neurons, that arises due to correlated fluctuations and adaptation mechanisms. I sketch novel analytical methods to deal with the associated non-Markovian first-passage-time problems. I furthermore show that such correlated (colored) noise also emerges in an autonomous fashion in recurrent neural networks and leads to similar features in the spike statistics of cortical cells. Fluctuations do not only lead to interesting spontaneous neural activity but also shape the way in which neurons encode information about time-dependent stimuli. I elucidate in particular mechanisms of information filtering in single neurons and neural populations.

**References:**

B. Dummer, S. Wieland, and B. Lindner Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity Front. Comp. Neurosci. 8 , 104 (2014)

T. Schwalger, F. Droste, and B. Lindner Statistical structure of neural spiking under non-Poissonian or other non-white stimulation J. Comp. Neurosci. 39 , 29 (2015)

S. Wieland, D. Bernardi, T. Schwalger, and B. Lindner Slow fluctuations in recurrent networks of spiking neurons Phys. Rev. E 92 040901(R) (2015)

B. Lindner Mechanisms of information filtering in neural systems IEEE Transactions on Molecular, Biological, and Multi-Scale Communications 2 5 (2016)

J. Doose, G. Doron, M. Brecht, and B. Lindner Noisy juxtacellular stimulation in vivo leads to reliable spiking and reveals high-frequency coding in single neurons J. Neurosci. 36 11120 (2016)

J. Grewe, A. Kruscha, B. Lindner, and J. Benda Synchronous Spikes are Necessary but not Sufficient for a Synchrony Code PNAS 114 E1977 (2017)

## How much is it possible to perturb a mean-field interaction ?

A convenient framework for the study of synchronization of interacting individuals is to consider systems of exchangeable particles interacting on the complete graph. This assumption allows for a rigorous analysis of the behavior of the system in the limit of large population, at least on a bounded time interval. From a practical point of view (e.g. simulations), a crucial question is how to derive from this formalism the behavior of the system on a longer time scale (that is dependent on the size of the population). I will first discuss this issue on the particular case of the Kuramoto model. And what if the graph of interaction is no longer the complete graph? Informally, in a situation where each particle has "enough" neighbors to interact with, it is natural to expect a similar mean-field behavior. The second point of the talk will be to show that this statement holds for a large class of (possibly inhomogeneous and sparse) random graphs of interaction, at least on a bounded time interval.

This is joint works with Christophe Poquet (Lyon 1) and Giambattista Giacomin and Sylvain Delattre (Paris 7).

## Dynamics and stochastic phenomena in an excitable laser with long delayed feedback

## Subthreshold signal encoding and transmission in coupled FitzHugh-Nagumo neurons

We study numerically the dynamics of two mutually coupled neurons using the well-known stochastic FitzHugh-Nagumo (FHN) model. We analyze how the coupling parameters affect the detection and transmission of a periodic weak signal that is applied to only one of the neurons. In a recent work [1] it was shown that, in a single FHN neuron, the interplay of noise and periodic subthreshold modulation induced the emergence of relative temporal ordering in the timing of the spikes. Different types of relative temporal order were found, in the form of preferred and infrequent ordinal patterns [2] that depended on both, the strength of the noise and the period of the input signal. A resonance-like behavior was also found, as the probability of the preferred (infrequent) pattern was maximum (minimum) for certain periods of the input signal and noise strengths. Here we analyze under which conditions the coupling to a second neuron, which is assumed to be linear and instantaneous, can further enhance the temporal ordering in the spike sequence of the first neuron, improving the encoding of the external signal. As in [1] we apply the symbolic method of ordinal analysis [2] to the output sequence of inter-spike intervals (ISIs). We find that for certain periods and amplitudes of the external signal, the coupling to the second neuron changes the preferred (and also the infrequent) ordinal patterns. A detailed study of how the ordinal probabilities vary with the coupling strength is performed. In a second step, we consider the situation in which the external signal is applied to both neurons. We discuss under which conditions mutual coupling enhances (or degrades) the encoding of the signal in the neuronal spike sequences.

**Acknowledgments:** This work was supported in part by Spanish MINECO/FEDER (FIS2015-66503-
C3-2-P) and ICREA ACADEMIA, Generalitat de Catalunya.

[1] J. A. Reinoso, M. C. Torrent and C. Masoller, Emergence of spike correlations in periodically forced excitable systems, Phys. Rev. E. 94, 032218 (2016).

[2] C. Bandt and B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88, 174102 (2002).

## Are the spikes emitted by a semiconductor laser with feedback similar to neuronal spikes?

In semiconductor lasers optical feedback induces a wide range of dynamical regimes. In this talk I will focus on the low frequency fluctuations (LFFs) regime, in which the laser emits optical spikes that resemble the spikes of biological neurons. In this regime, semiconductor lasers have potential to act as ultra-fast photonic neurons, which can be building blocks of novel information processing systems [1], inspired in the way biological neurons process information. In order to use the laser as a basic, neuro-inspired information processing unit, we first need to understand how an input signal can be encoded in the output sequence of optical spikes [2-4], and then compare with neuronal encoding. Single biological neurons may encode information in the spike rate (“rate coding”) or in the relative timing of the spikes (“temporal coding”). In a recent work [5] we have analyzed the well-known FitzHugh-Nagumo (FHN) single-neuron model and we have shown that the interplay of noise and a weak (subthreshold) periodic input signal induced temporal ordering in the timing of the spikes (i.e., induced temporal correlations among several inter-spike-intervals, ISIs), in the form of more/less frequently expressed patterns, which depend of the period of the input signal. Our results suggested that single neurons in noisy environments encode information about the period of a subthreshold periodic input in the relative timing of the spikes.

In this talk I will present an experimental study of the laser dynamics in the LFF regime under weak pump current sinusoidal modulation and I will compare the statistical properties of optical spikes with those of synthetic neuronal spikes [5]. In spite of the fact that the laser with feedback is a high dimensional system (due to the feedback delay time) while the FHN model is a low dimensional system, in general a good qualitative agreement is found; however some relevant differences will also be discussed.

[1] P. R. Prucnal, B. J. Shastri, T. F. de Lima, M. A. Nahmias, and A. N. Tait, “Recent progress in semiconductor excitable lasers for photonic spike processing”, Advances in Optics and Photonics 8, 228 (2016).

[2] T. Sorrentino, C. Quintero-Quiroz, A. Aragoneses, M. C. Torrent, and C. Masoller, “The effects of periodic forcing on the temporally correlated spikes of a semiconductor laser with feedback”, Optics Express 23, 5571 (2015).

[3] C. Quintero-Quiroz, J. Tiana-Alsina, J. Roma, M. C. Torrent, and C. Masoller, “Characterizing how complex optical signals emerge from noisy intensity fluctuations”, Sci. Rep. 6 37510 (2016).

[4] A. Aragoneses, S. Perrone, T. Sorrentino, M. C. Torrent and C. Masoller, "Unveiling the complex organization of recurrent patterns in spiking dynamical systems", Sci. Rep. 4, 4696 (2014).

[5] J. A. Reinoso, M. C. Torrent and C. Masoller, “Emergence of spike correlations in periodically forced excitable systems”, Phys. Rev. E. 94, 032218 (2016).

## Excitable knots

There are a range of chemical, physical and biological excitable media that support spiral wave vortices. Examples include the Belousov-Zhabotinsky redox reaction, the chemotaxis of slime mould and action potentials in cardiac tissue. Usually spatio-temporal three-dimensional interacting vortex strings, e.g. superfluid vortices [Kleckner16], are prone to reconnection and untying. However, it appears that there are types of reaction-diffusion equations - for example the so-called FitzHugh-Nagumo equations which constitute the simplest mathematical model of cardiac tissue as an excitable medium - that may give rise to strong short-ranged repulsive interactions between vortex strings. The latter may lead to a preservation of topology of vortex strings [Winfree84,Sutcliffe03]. We use a Biot-Savart construction to initialize a given knot as a vortex string in the FitzHugh-Nagumo equations. We show, that the evolution can be used to untangle knotted vortex strings [Maucher16]. Furthermore, the evolution yields a well-defined minimal length for a range of knots that is comparable to the ropelength of ideal knots [Maucher17]. We highlight the role of the medium boundary in stabilizing the length of the knot and discuss the implications beyond torus knots. By applying Moffatt’s test we show that there is not a unique attractor within a given knot topology.

[Kleckner16] D. Kleckner, L. H. Kauffman, W. T. M. Irvine, *Nature Physics* **12**, 650 (2016)

[Winfree84] A. T. Winfree and S. H.
Strogatz, *Nature* **311**, 611 (1984)

[Sutcliffe03] P. Sutcliffe, A.T. Winfree *Phys. Rev. E* **68**,
016218 (2003)

[Maucher16] F. Maucher, P. Sutcliffe, *Phys. Rev. Lett.*
**116**, 178101 (2016)

[Maucher17] F. Maucher, P. Sutcliffe,
*(submitted)* (2017)

## Death and rebirth of neural activity in sparse inhibitory networks

In this presentation, we clarify the mechanisms underlying a general phenomenon present in pulse-coupled heterogeneous inhibitory networks: inhibition can induce not only suppression of the neural activity, as expected, but it can also promote neural reactivation. In particular, for globally coupled systems, the number of firing neurons monotonically reduces upon increasing the strength of inhibition (neurons' death). However, the random pruning of the connections is able to reverse the action of inhibition, i.e. in a sparse network a sufficiently strong synaptic strength can surprisingly promote, rather than depress, the activity of the neurons (neurons' rebirth). Thus the number of firing neurons reveals a minimum at some intermediate synaptic strength. We show that this minimum signals a transition from a regime dominated by the neurons with higher firing activity to a phase where all neurons are effectively sub-threshold and their irregular firing is driven by current fluctuations. We explain the origin of the transition by deriving an analytic mean field formulation of the problem able to provide the fraction of active neurons as well as the first two moments of their firing statistics. The introduction of a synaptic time scale does not modify the main aspects of the reported phenomenon. However, for sufficiently slow synapses the transition becomes dramatic, the system passes from a perfectly regular evolution to an irregular bursting dynamics. In this latter regime the model provides predictions consistent with experimental findings for a specific class of neurons, namely the medium spiny neurons in the striatum.

**Reference:**

D Angulo-Garcia, S Luccioli, S Olmi, A Torcini, "Death and rebirth of neural activity in sparse inhibitory networks", New Journal of Physics (2017) doi.org/10.1088/1367-2

## Synchronization phenomena in coupled excitable lasers

## Detection of dependence between neurons and synchronization

## Exact macroscopic equations for globally coupled heterogeneous neurons

Many phenomena in the natural and biological sciences reflect the collective behavior of large numbers of interacting dynamical units. Insight into such phenomena can therefore by gleaned by studying idealized systems in which the dynamics of the individual units are taken to be as simple as possible, while still retaining the most salient features for the physical system of interest. A well-known example is the Kuramoto model, a canonical example of collective synchronization. In the field of neuroscience, the individual unit is the neuron and the collective behavior of interest is often the response of the neuronal network to external inputs. Here, as opposed to the Kuramoto model, the parameter order of interest is not necessarily the degree of synchronization, but rather the average number of electrical impulses or “spikes” generated by the network per unit of time, known as the “firing rate”.

Here I will discuss the derivation of an exact mean-field model for a large network of coupled, heterogeneous model neurons [Montbrio15]. The neural model, that is the individual element, is the “quadratic integrate-and-fire” neuron, a canonical model which approximates how spikes are emitted in a large class of more realistic neuronal models. We find that the relevant macroscopic variables are the mean firing rate as well as the mean voltage drop across the cellular membrane of neurons in the network; the collective dynamics are described by a system of two coupled ODEs. Interestingly, we can show that there is a deep connection between this neuronal description of collective behavior, and the more traditional Kuramoto order parameter which measures collective synchronization. Specifically, the conformal transformations which map between the unit circle and the positive half plane allow us to switch between these two descriptions and tie our work to the so-called Ott-Antonsen ansatz [Ott08], which provides an exact solution to the Kuramoto model.

[Montbrio15] E. Montbrió, D. Pazó, and A. Roxin. Macroscopic description for networks of spiking neurons. , 5, 2015.

[Ott08] E. Ott and T. M. Antonsen. Low-dimensional behavior of large systems of globally coupled oscillators. , 18:037113, 2008.

## How fast is the self-pulsation in a laser diode with optical feedback?

When subject to optical feedback, a laser diode exhibits a large variety of nonlinear dynamics including self-pulsation, period-doubling bifurcations, intermittent pulsating dynamics and chaos. In the past forty years these dynamics have been scrutinized both theoretically and experimentally, paving the way towards innovative applications such as in chaos-based cryptography, physical sources of entropy, or, recently, neuromorphic computing [1].

A crucial question is how fast can self-pulsation occur. As in any system with time-delayed feedback, competition may occur between the natural time-scale of the damped nonlinear oscillator and the time-delay. Within the framework of a rate equation model one can easily demonstrate that when increasing the amount of optical feedback, a laser diode bifurcates from a steady-state to self-pulsation (Hopf bifurcation) with a frequency either close to the relaxation oscillation frequency of the free-running laser diode (internal, natural frequency of the nonlinear damped oscillator), or, close to the inverse of the time-delay [2]. Integrating a passive feedback section into an active laser medium therefore leads to a short time-delay enabling several tens of GHz self-pulsation from a laser diode [3].

A slightly different situation was analyzed in the early 1980s, in which the optical feedback does not occur from a conventional mirror but from a phase-conjugate mirror. In that situation, summarized as "phase-conjugate feedback (PCF)", self-pulsation at frequencies being harmonic of the inverse of time-delay have been theoretically predicted [4]. Recently we have found these dynamics experimentally [5] and analyzed how they get stabilized and bifurcate when varying the feedback strength and/or time-delay [6]. We will summarize our latest conclusions on the PCF configuration and shall highlight the important role played by the filtering effect in the phase-conjugate mirror on the high-speed self-pulsating dynamics [7]. For a relatively long nonlinear mirror, the dynamics get close to those observed in a laser diode with optical injection [8].

[1] M. Sciamanna, K.A. Shore, Nat. Photonics 9, 151-162 (2015)

[2] Th. Erneux, A. Gavrielides, M. Sciamanna, Phys. Rev. A 66, 033809 (2002)

[3] O. Ushakov, S. Bauer, O. Brox, H.J. Wünsche, F. Henneberger, Phys. Rev. Lett. 92, 043902 (2004)

[4] G.P. Agrawal, J.T. Klaus, Opt. Lett. 16, 1325-1327 (1991)

[5] A. Karsaklian Dal Bosco, D. Wolfersberger, M. Sciamanna, Appl. Phys. Lett. 105, 081101 (2014)

[6] E. Mercier, C.H. Uy, L. Weicker, M. Virte, D. Wolfersberger, M. Sciamanna, Phys. Rev. A 94, 061803 (2016)

[7] E. Mercier, L. Weicker, D. Wolfersberger, D.M. Kane, M. Sciamanna, Opt. Lett. 42, 306-309 (2017)

[8] L. Weicker, T. Erneux, D. Wolfersberger, M. Sciamanna, Phys. Rev. E 92, 022906 (2015)

## Machine learning for signal processing

We have proposed the use of sparse identification method for optical systems, or SINO. This new approach determines the optimum number of variables in the transmission system required for adaptive mitigation of effects (nonlinearities in fibre optic cable) that limit the throughput of standard optical fibre. Demand for data is high with today’s online culture and the introduction of 8K TV, the Internet of Things and the ever-increasing use of streaming services mean that this demand could outstrip network capacity. Novel techniques, such as SINO, could help to future-proof our broadband infrastructure. The SINO method is significantly less complex than other similar compensation techniques. This bodes well for future commercial deployment. SINO is particularly useful for flexible smart-grid networks, as it does not require a knowledge of system parameters and is scalable to difference power levels. Such networks are more sustainable and more reliable, considering the needs of modern society.

**Reference:**

Sorokina M., Sygletos S. & Turitsyn S., Sparse identification for nonlinear optical communication systems: SINO method, Opt. Express 24, 30433-30443 (2016).

## Reservoir computing using integrated semiconductor lasers with delayed feedback

The concept of delay-based reservoir computing, using only a single nonlinear node with delayed feedback instead of a very big random network, was introduced some years ago as a means of limiting hardware complexity in photonic systems. In essence, the idea of delay line reservoir computing constitutes an exchange between space and time: what is normally done spatially with many nonlinear nodes, is now done in a single node that is multiplexed in time. A virtual network configuration is created through the inertia of the system. After the first mixed analog/digital implementations in electronics, high processing speeds have been demonstrated based on the transient response to optical data injection in nonlinear optical systems such as semiconductor lasers. While previous efforts have focused on signal bandwidths limited by the semiconductor laser’s relaxation oscillation frequency, we have demonstrated numerically that the much faster optical phase response makes significantly higher processing speeds attainable. Using complex nonlinear interactions between optical field and material, we demonstrate that it is possible to reduce the delay time to within lengths that can be integrated on chip. Relying on these unique nonlinear optics properties, we show that multiple optical modes can be used to either process in parallel several independent computational tasks or use them in conjunction to work on the same problem. We will illustrate our approach on a single-longitudinal mode semiconductor ring laser with optical feedback and on a semiconductor laser with multiple longitudinal modes. Finally, we have further developed the structure of delay-based reservoir computing to accommodate a cascade of several reservoir computers. In electronics, we have already demonstrated that such a cascade can perform several complex audio-processing tasks without the need for any audio-related pre-processing. With integrated delay-based reservoir computers, such a scheme could be achieved in optics.