A Leaky Integrate-and-Fire Laser Neuron for Ultrafast Cognitive Computing 用于超快认知计算的LIF激光神经元

Abstract 摘要

We propose an original design for a neuron-inspired（神经元） photonic（光子） computational primitive（原始） for a large-scale, ultrafast cognitive（认知） computing platform.

我们提出了一个用于大规模超快认知的神经元启发光子计算原始计算平台。

The laser（激光） exhibits（展示） excitability（刺激性） and behaves（行为）analogously（类似） to a leaky integrate-and-fire (LIF) （泄露整合放电）neuron.

激光表现出兴奋性并且行为类似于泄露整合放电神经元。

This model is both fast and scalable（延展）, operating up to a billion times faster than a biological equivalent（等量物） and is realizable（实现） in a compact（紧凑，严谨，合同）,vertical-cavity surface-emitting laser (VCSEL).

这个模型既快速又可扩展，运行速度比生物等效物快十亿倍，并且可在在协议下实现，垂直腔面发射激光器（VCSEL）。

We show that—under a certain set of conditions—the rate（速率） equations （方程）governing（控制）a laser with an embedded（嵌入） saturable（饱和） absorber （吸收）reduces to the behavior of LIF neurons.

我们表明，在一定条件下，速率方程控制具有嵌入式可饱和吸收体的激光器减少了LIF神经元的行为。

We simulate（仿真） the laser using realistic（实际） rate equations governing a VCSEL cavity, and show behavior representative（代表） of cortical（皮质） spiking （尖峰）algorithms（算法） simulated in small circuits of excitable lasers.

我们使用实际速率方程控制VCSEL模拟激光，并展现了代表用小型可激发激光线路模拟皮质尖峰算法的行为。

Pairing（配对） this technology with ultrafast, neural learning algorithms would open up a new domain of processing.

将这项技术与超快的神经学习算法配对将开辟一个新的处理领域。

I. INTRODUCTION(导言)

1. In an effort to break the limitations（局限性） inherent（固有） in traditional von Neumann（冯.诺依曼） architectures（结构）, some recent projects in computing have sought（seek 寻求） more effective signal processing techniques by leveraging（借用） the underlying（底层，基础） physics of devices.

为了打破传统冯·诺依曼架构中固有的局限性，最近的一些计算项目通过利用设备的基础物理学来寻求更有效的信号处理技术。

Cognitive computing platforms inspired by biological neural networks could solve unconventional(非传统的) computing problems and outperform（比..表现的更好） current technology in both power efficiency and complexity.

受生物神经网络启发的认知计算平台可以解决非常规计算问题，并在功效和复杂性方面优于当前技术。

These novel（新颖） systems rely on an alternative（另一个，可选择的） set of computational principles, including hybrid（混血） analog-digital（模拟-数字） signal representations（表示）, collocation of memory and processing, unsupervised（无监督） learning, and distributed representations of information.

这些新颖的系统依赖于另一组计算原理，包括混合模拟 - 数字信号表示，存储器和处理器的搭配，无监督学习以及信息的分布式表示。

Fig. 1. Spiking neural networks encode information as events in time rather than bits. Because the time at which a spike occurs is analog while its amplitude（幅度） is digital, the signals use a mixed signal or hybrid-encoding scheme.

图一尖峰神经网络将信息编码为事件的时间而不是比特。因为尖峰发生的时间是模拟的，而其幅度是数字的，所以信号使用混合信号或混合编码方案。

2.On the cellular（细胞） level, the brain encodes information as events or spikes in time , a hybrid signal with both analog and digital properties（特性） as illustrated（插图） in Fig. 1.

在细胞水平上，大脑将信息编码为事件或时间峰值，具有模拟和数字特性的混合信号，如图1所示。

This encoding scheme（方案） is equivalent（等同） to analog pulse position modulation（调制） (PPM) in optics（光学）,which has been utilized（利用） in various applications including the implementation（实现） of robust chaotic communication （鲁棒混沌通信）and power efficient channel coding .

该编码方案等同于光学中的模拟脉冲位置调制（PPM），其已经在各种应用中使用，包括鲁棒混沌通信和功率有效信道编码的实现。

Spike processing has evolved（发展） in biological (nervous systems) and engineered (neuromorphic（神经形态） analog VLSI（超大规模集成电路Very Large Scale Integration）) systems as a means to exploit（利用） the efficiency of analog signals while overcoming the problem of noise accumulation（累积） inherent （固有）in analog computation .

尖峰处理已经在生物（神经系统）和工程（神经形态模拟VLSI）系统中发展，作为利用模拟信号效率同时克服模拟计算中固有的噪声累积问题的手段。

Various technologies have emulated（仿真） spike neural networks in electronics,including IBM’s neuro synaptic（突触）core as part of DARPA’s（美国国防高级研究计划局（Defense Advanced Research Projects Agency）） SyNAPSE （Systems of Neuromorphic Adaptive Plastic Scalable Electronics自适应可塑扩展电子神经系统）program and Neurogrid as part of Stanford’s Brains in Silicon program.

各种技术模拟了电子学中的尖峰神经网络，包括作为美国国防高级研究计划局自适应可塑扩展电子神经系统计划的一部分的IBM神经突触核心和作为斯坦福大学硅脑计划的一部分的Neurogrid。

Although these architectures have garnered success in various applications, they aim to target biological time scales rather than exceed（超越） them.

虽然这些架构在各种应用中取得了成功，但它们的目标是针对生物时间尺度而不是超过它们。

Microelectronic neural networks that are both fast and highly interconnected are subject to(受制于) a fundamental bandwidth fan-in（fan扇） tradeoff.

快速且高度互连的微电子神经网络受到基本带宽扇入权衡的影响。

3.Photonic platforms offer an alternative approach to microelectronics.

光子平台为微电子学提供了另一种方法。

The high speeds, high bandwidth, and low crosstalk achievable in photonics are very well suited for an ultrafast spike-based information scheme.

高速，高带宽和低串扰在光子学中可实现的非常适合超快速基于尖峰的信息方案。

Because of this, photonic spike processors could access a computational domain that is inaccessible（难得到的） by other technologies.

因此，光子尖峰处理器可以访问其他技术无法访问的计算域。

This domain, which we describe as ultrafast cognitive computing, represents an unexplored processing paradigm（范例） that could have a wide range of applications in adaptive（自适应） control, learning, perception（感觉）, motion（运动） control, sensory（感觉） processing (vision systems, auditory（听觉） processors, and the olfactory（嗅觉） system), autonomous robotics, and cognitive processing of the radio（无线电） frequency spectrum（波谱）.

这个领域，我们称之为超快认知计算，代表了一种未开发的处理范例，可以在自适应控制，学习，感知，运动控制，感觉处理（视觉系统，听觉处理器和嗅觉系统）中具有广泛的应用，自主机器人，以及无线电频谱的认知处理。

4.There has been a growing interest in photonic spike processing which has spawned（催生） a rich search for an appropriate computational primitive.

人们越来越关注光子尖峰处理，它已经产生了对适当计算原语的丰富搜索。

The first category includes those based on discrete, fiber components .

第一类包括基于分立光纤组件。

However, the platform’s reliance（依赖） on nonlinear fibers and other similar technologies have made demonstrations（示范） bulky（庞大） (on the order of meters), complex,and power-hungry (hundreds of watts（瓦特）).

然而，该平台对非线性光纤和其他类似技术的依赖使得演示体积庞大（大约几米），复杂且耗电量大（数百瓦）。

The platform is simply unscalable beyond a few neurons.

该平台在几个神经元之外是不可扩展的。

Integrated lasers, in contrast,
are physically compact（紧凑） and are capable of using feedback rather
than feedforward dynamics（动态） to radically（根本） enhance nonlinearity.

相比之下，集成激光器在物理上非常紧凑，能够使用反馈而不是前馈动态来从根本上增强非线性。

Feedback allows for the emergence（情况） of more complex behaviors,including bistability（双稳态）, the formation of attractors, and excitability.

反馈允许出现更复杂的行为，包括双稳态，吸引子的形成和兴奋性。

5.Excitability is a dynamical system property that underlies（背后） allor-none responses.

兴奋性是一种动态系统属性，是所有或没有响应的基础。

Its occurrence in a variety of different lasing systems has received considerable interest .

它在各种不同激光系统中的出现引起了人们的极大兴趣.

Excitability is also a critical(关键) property of biological spiking neurons.

兴奋性也是生物尖峰神经元的关键特性。

More recently, several excitable lasers have demonstrated（证明） biological-like spiking features.

最近，几种可激发的激光器已经证明了类似生物的尖峰特征。

One proposal suggests using excitability in semiconductor lasers based on weakly
broken Z2 symmetry（对称性） close to a Takens–Bogdanov bifurcation（分支）,yet another suggests using emergent（紧急） biological features from polarization（偏振） switching in a vertical-cavity surface-emitting laser (VCSEL).

一项提议建议利用基于接近Takens-Bogdanov分岔的弱破坏Z2对称性的半导体激光器中的兴奋性，而另一个建议在垂直腔面发射激光器（VCSEL）中使用来自偏振切换的紧急生物特征。

However, these models have yet to demonstrate(演示) some key properties of spiking neurons: the ability to perform computations without information degradation（降解）, clean-up noise,or implement algorithms.

然而，这些模型尚未展示尖峰神经元的一些关键属性：能够在没有信息降级的情况下执行计算，清除噪声或实现算法。

6.In this paper, we show for the first time that a photonic computational primitive based on an integrated, excitable laser with an embedded（嵌入式） saturable （饱和）absorber（吸收器） (SA) behaves analogously（类似） to a leaky integrate-and-fire (LIF) neuron.

在本文中，我们首次展示了基于具有嵌入式可饱和吸收器（SA）的集成可激发激光器的光子计算原语，其行为类似于泄漏的积分 - 激发（LIF）神经元。

The LIF model is one of the most ubiquitous（普及） models in computational neuroscience and is the simplest known model for spike processing [25]. We also show that our laser neuron can be employed to carry out cortical（皮质） algorithms through several small circuit demonstrations. Emulating this model in a scalable device represents the first step in building an ultrafast cognitive computing platform.

LIF模型是计算神经科学中最普遍存在的模型之一，也是最简单的尖峰加工模型[25]。我们还表明，我们的激光神经元可用于通过几个小电路演示来执行皮质算法。在可扩展设备中模拟此模型代表了构建超快认知计算平台的第一步。

II. LASER NEURON—THEORETICAL FOUNDATIONS（激光神经元理论基础）

Our device is based upon a well-studied （充分研究）and paradigmatic（范式） example of a hybrid computational primitive: the spiking neuron.In this section, we briefly review the spiking neuron model and reveal the analogy（对比） between the LIF model and our own.

我们的设备基于混合计算原语的充分研究和范例：尖峰神经元。在本节中，我们简要回顾尖峰神经元模型，并揭示LIF模型与我们自己的模型之间的类比。

A. Spiking Neuron Model

Studies of morphology(形态学) and physiology(生理) have pinpointed(精确定位的) the LIF model as an effective spiking model to describe a variety of different biologically observed phenomena(现象) [26]. From the standpoint（立场） of computability（可计算） and complexity theory, LIF neurons are powerful computational primitives that are capable of simulating both Turing machines and traditional sigmoidal（s型） neural networks [27]. Signals are ideally represented by series of delta functions: inputs and outputs take the form
x(t) = Σnj =1δ(t − τj ) for spike times τj . Individual（个人） units perform a small set of basic operations (delaying, weighting, spatial（空间的） summation, temporal （时间）integration, and thresholding) that are integrated into a single device capable of implementing a variety of processing tasks, including binary classification, adaptive feedback, and temporal logic.

形态学和生理学研究已经将LIF模型确定为一种有效的尖峰模型，用于描述各种不同的生物学观察现象[26]。从可计算性和复杂性理论的角度来看，LIF神经元是强大的计算原语，能够模拟图灵机和传统的S形神经网络[27]。信号理想地由一系列delta函数表示：输入和输出采用形式x（t）=Σnj=1δ（t-τj）用于尖峰时间τj。各个单元执行一小组基本操作（延迟，加权，空间求和，时间积分和阈值处理），这些操作被集成到能够实现各种处理任务的单个设备中，包括二进制分类，自适应反馈和时间逻辑。

The basic biological structure of an LIF neuron is depicted（描绘） in Fig. 2(a). It consists of a dendritic（树突） tree that collects and sums inputs from other neurons, a soma（体细胞） that acts as a lowpass filter （低通滤波器）and integrates the signals over time, and an axon（轴突） that carries an action potential（电位）, or spike, when the integrated signal exceeds（超过） a threshold. Neurons are connected to each other via synapses（突触）, or extracellular（细胞外） gaps（间隙）, across which chemical signals are transmitted（发送）. The axon, dendrite, and synapse all play an important role in the weighting and delaying of spike signals.

LIF神经元的基本生物结构如图2（a）所示。它由一个树枝状树组成，它收集和汇总来自其他神经元的输入，一个作为低通滤波器的体细胞，随时间积分信号，以及当集成信号超过阈值时携带动作电位或尖峰的轴突。神经元通过突触或细胞外间隙相互连接，化学信号通过突触传递。轴突，树突和突触都在加权和延迟尖峰信号中起重要作用。

According to the standard LIF model, neurons are treated as electrical devices. The membrane（膜） potential Vm(t), the voltage difference across their membrane, acts as the primary internal (activation) state variable. Ions（离子） that flow across the membrane experience a resistance R = Rm and capacitance C = Cm associated with the membrane. The soma（体细胞） is effectively a firstorder（一阶） low-pass filter, or a leaky integrator, with the integration time constant τm = RmCm that determines the exponential （指数）decay（衰变）rate of the impulse response function. The leakage current through Rm drives the membrane voltage Vm(t) to 0, but an active membrane pumping（泵） current counteracts（抵消） it and maintains a resting membrane voltage at a value of Vm(t) = VL .

根据标准LIF模型，神经元被视为电子设备。膜电位Vm（t）（其膜上的电压差）充当主要内部（激活）状态变量。流过膜的离子经历电阻R = Rm和与膜相关的电容C = Cm。该体细胞实际上是一阶低通滤波器或漏泄积分器，其积分时间常数τm= RmCm确定脉冲响应函数的指数衰减率。通过Rm的泄漏电流将膜电压Vm（t）驱动为0，但是有源膜泵浦电流抵消它并且将静止膜电压维持在Vm（t）= VL的值。

Fig. 2. (a) Illustration and (b) functional description of a leaky integrate-and fire
neuron. Weighted and delayed input signals are summed into the input
current Iapp (t), which travel to the soma and perturb（干扰） the internal state variable,the voltage V . Since V is hysteric（过度狂烈的）, the soma performs integration and then applies a threshold to make a spike or no-spike decision. After a spike is released, the voltage V is reset to a value Vreset . The resulting spike is sent to other neurons in the network.

图2.（a）图示和（b）漏泄积分和放电神经元的功能描述。加权和延迟的输入信号被加到输入电流Iapp（t）中，输入电流Iapp（t）传递到体细胞并扰乱内部状态变量电压V.由于V是歇斯底里的，因此体细胞执行积分，然后应用阈值来进行尖峰或无尖峰决定。在释放尖峰之后，电压V被重置为值Vreset。产生的尖峰被发送到网络中的其他神经元。

Fig. 2(b) shows the standard LIF neuron model [27]. A neuron has: 1) N inputs which represent induced(感应) currents in input synapses xj (t) that are continuous time series consisting either of spikes or continuous analog values; 2) an internal activation state Vm(t); and 3) a single output state y(t). Each input is independently weighted by ωj , which can be positive or negative,and delayed by τj resulting in a time series that is spatially（空间地） summed (summed pointwise逐点求和). This aggregate（合计） input induces an electrical current, Iapp(t) = Σnj=1ωjxj (t − τj ) between adjacent（相邻） neurons.

图2（b）显示了标准的LIF神经元模型[27]。神经元具有：1）N个输入表示输入突触中的感应电流，xj（t）是连续时间序列，包括尖峰或连续模拟值；2）内部激活状态Vm（t）; 3）单个输出状态y（t）。每个输入由ωj独立加权，ωj可以是正的或负的，并且延迟τj，从而得到空间求和的时间序列（逐点求和）。该聚合输入在相邻神经元之间感应出电流Iapp（t）=Σnj=1ωjxj（t-τj）。

Fig. 3. An illustration of spiking dynamics in an LIF neuron. Spikes arriving from inputs xj (t) that are inhibitory(抑制) (red arrows) reduce the voltage V (t),while those that are excitatory (green arrows) increase V (t). Enough excitatory activity pushes V (t) above Vthresh , releasing a delta function spike in y(t), followed by a refractory（抵抗刺激） period during which V (t) recovers to its resting potential VL .

图3. LIF神经元中尖峰动力学的图示。输入具有抑制性（红色箭头）xj（t）的的尖峰降低电压V（t），而那些兴奋的尖峰（绿色箭头）增加V（t）。足够的兴奋活动将V（t）推高到Vthresh以上，释放y（t）中的δ函数峰值，然后是不应期，在此期间V（t）恢复到其静止电位VL。

The weights wj and delays τj determine the dynamics of network,providing a way of programming a neuromorphic system.The parameters internal to the behavior of individual neurons include the resting potential VL and the membrane time constant
τm. There are three influences on Vm(t)—passive leakage of current, an active pumping current, and external inputs generating time-varying membrane conductance changes. Including a set of digital conditions, we arrive at a typical LIF model for an individual neuron:

权重wj和延迟τj决定了网络的动态，提供了一种编程神经形态系统的方法。个体神经元行为内部的参数包括静息电位VL和膜时间常数τm。 Vm（t） - 电流的无源泄漏，有源泵浦电流和产生时变膜电导变化的外部输入有三种影响。包括一组数字条件，我们得到了一个典型的单个神经元LIF模型：

The dynamics of an LIF neuron are illustrated in Fig. 3. If Vm(t) ≥ Vthresh , then the neuron outputs a spike which takes the form y(t) = δ(t − tf ), where tf is the time of spike firing, and Vm(t) is set to Vreset . This is followed by a relatively refractory
period, during which Vm(t) recovers from Vreset to the resting potential VL in which is difficult, but possible to induce the firing of a spike. 1Consequently, the output of the neuron consists of a continuous time series comprised(包括) of spikes taking the form y(t) = Σiδ(t − ti) for spike firing times ti .

LIF神经元的动力学如图3所示。如果Vm（t）≥Vthresh，那么神经元输出一个尖峰，其形式为y（t）=δ（t-tf），其中tf是尖峰的时间点火，Vm（t）设定为Vreset。接下来是相对不应期，在此期间Vm（t）从Vreset恢复到静止电位VL，其中很难，但可能诱发尖峰的发射。因此，神经元的输出包括一个由尖峰组成的连续时间序列，其形式为尖峰发射时间ti的y（t）=Σiδ（t-ti）。

B. Excitable Laser Model 可激发的激光模型

Our starting point is a set of dimensionless（无量纲） equations（方程）describing SA lasers that can generalize to a variety of different systems, including passively Q-switched microchip lasers [28],distributed Bragg reflector（反射器） lasers [29], and VCSELs [30].Below,we will show that a series of approximations（近似）leads to behavior that is isomorphic（同构） with LIF neurons.

我们的出发点是一组描述SA激光器的无量纲方程，可以推广到各种不同的系统，包括被动Q开关微芯片激光器[28]，分布式布拉格反射器激光器[29]和VCSEL [30]。我们将表明一系列近似导致与LIF神经元同构的行为。

1There may also be a short, absolute refractory period τrefrac for which
Vm (tf + Δt) = Vreset if Δt ≤ τrefrac , and during which no spikes may be fired. Although this condition typically precedes（先于） the relative(相对的) refractory period,we have omitted(省略) this from the model since it does not significantly affect the underlying(潜在) dynamics.

1也可能存在短的绝对不应期τrefrac，如果Δt≤τrefrac则Vm（tf +Δt）= Vreset，并且在此期间不会发出尖峰。虽然这种情况通常在相对不应期之前，但我们已从模型中省略了这一点，因为它不会显着影响潜在的动态。

Fig. 4. A simple schematic of an SA laser. The device is composed（组成）of (i) a gain section, (ii) a saturable absorber, and (iii) mirrors for cavity feedback. In the LIF excitable model inputs selectively perturb（干扰） the gain optically or electrically.

图4. SA激光器的简单示意图。该装置包括（i）增益部分，（ii）可饱和吸收器，和（iii）腔反馈镜。在LIF可激励模型中，输入选择性地光学或电学地扰乱增益。

We begin with the Yamada model [31], which describe the behavior of lasers with independent gain and SA sections with an approximately constant intensity(强度) profile(轮廓) across the cavity as illustrated in Fig. 4. We assume that the SA has a very short relaxation time on the order of the cavity intensity, which can be implemented either through doping(掺杂) or special material properties.The dynamics now operate such that the gain is a slow variable, while the intensity and loss are both fast. This 3-D dynamical system can be described with the following equations:

我们从Yamada模型[31]开始，它描述了具有独立增益和如图4所示在腔体上具有近似恒定的强度分布SA截面的激光器的行为。我们假设SA在腔强度的量级上具有非常短的弛豫时间，这可以通过掺杂或特殊材料特性来实现。动态现在运行使得增益是缓慢变量，而强度和损失都很快。这个三维动力系统可以用以下公式描述：

where G(t) models the gain, Q(t) is the absorption, I(t) is the laser intensity, A is the bias(偏压) current of the gain, B is the level of absorption, a describes the differential absorption relative to the differential gain, γG is the relaxation rate of the gain, γQ
is the relaxation rate of the absorber, γI is the reverse(相反) photon lifetime, and f(G) represents the small contributions to the intensity made by spontaneous（自发） emission, (noise term) where is very small.2

其中G（t）模拟增益，Q（t）是吸收，I（t）是激光强度，A是增益的偏置电流，B是吸收水平，a描述了相对于差分增益的差分吸收，γG是增益的弛豫率，γQ是吸收体的弛豫率，γI是反向光子寿命，f（G）代表自发辐射强度的小贡献，（噪声项）很小.2

2 Nondimensionalization allows us to set γI to 1, but we include this variable in our description to explicitly compare time scales between variables G, Q,and I.

非尺寸化允许我们将γI设置为1，但我们在描述中包含此变量以明确比较变量G，Q和I之间的时间尺度。

We further assume that inputs to the system cause perturbations to the gain G(t) only. Pulses—from other excitable lasers,for example—will induce a change G as illustrated by the arrows in Fig. 5 and analog inputs will modulate G(t) continuously.
This can be achieved either injection via the optical pulses that selectively modulate the gain medium or through electrical current injection. We also make the additional assumption that the laser exhibits behavior similar to region 2 of the bifurcation（分枝）diagram（图）presented in [31], but with a fast absorber.

我们进一步假设系统的输入仅引起对增益G（t）的扰动。例如，来自其他可激发激光器的脉冲将引起变化G，如图5中的箭头所示，并且模拟输入将连续地调制G（t）。
这可以通过选择性地调制增益介质的光脉冲或通过电流注入来实现。我们还做出了额外的假设，即激光表现出的行为类似于[31]中提出的分叉图的区域2，但具有快速吸收器。

1) Before Pulse Formation: Since the loss Q(t) and the intensity I(t) are fast, they will quickly settle to their equilibrium（平衡） values. On slower time scales, our system behaves as:

1）在脉冲形成之前：由于损失Q（t）和强度I（t）很快，它们将很快稳定到它们的平衡值。在较慢的时间尺度上，我们的系统表现如下：

Fig. 5. Simulation results of an SA laser behaving as an LIF neuron. Arrows indicate excitatory pulses and inhibitory pulses that change the gain by some amount G. Enough excitatory input causes the system to enter fast dynamics in which a spike is generated, followed by the fast recover of the absorption Q(t) and the slow recover of the gain G(t). Variables were rescaled to fit within the desired range. Values used: A = 4.3;B = 3.52; a = 1.8; γG =.05; γL , γI .05.

图5. SA激光器作为LIF神经元的模拟结果。箭头表示通过量G改变增益的兴奋脉冲和抑制脉冲.足够的兴奋性输入使系统进入快速动态，其中产生尖峰，然后是吸收Q（t）的快速恢复和增益G（t）的缓慢恢复。重新调整变量以适合所需范围。使用的值：A = 4.3; B = 3.52; a = 1.8; γG= .05; γL，γI.05。

with θ(t) representing possible inputs, and the equilibrium values Qeq = B and Ieq = f(G)/γI [1 − G(t) + Q(t)]. Since is quite small, Ieq ≈ 0.With zero intensity in the cavity, theG(t) andQ(t) variables are dynamically decoupled. The result is that if inputs are incident on the gain, they will only perturb G(t) unless I(t) becomes sufficiently large to couple the dynamics together.

用θ（t）表示可能的输入，平衡值Qeq = B，Ieq = f（G）/γI[1-G（t）+ Q（t）]。因为非常小，Ieq≈0。在腔中具有零强度，G（t）和Q（t）变量是动态解耦的。结果是，如果输入入射在增益上，它们将仅扰动G（t），除非I（t）变得足够大以将动态耦合在一起。

If I(t) increases, the slow dynamics will break. Since I˙(t) ≈ γI [G(t) − Q(t) − 1] I(t), I(t) will reach instability when G(t) − Q(t) − 1 > 0. Given our perturbations to G(t),we can define a threshold condition:

如果I（t）增加，那么缓慢的动态就会破裂。由于I˙（t）≈γI[G（t） - Q（t） - 1] I（t），当G（t） - Q（t）-1> 0时，I（t）将达到不稳定性。鉴于我们的对G（t）的扰动，我们可以定义一个阈值条件：

above which fast dynamics will take effect. This occurs after the third excitatory pulse in Fig. 5.

快速动态将在其上生效。这发生在图5中的第三个兴奋脉冲之后。

Pulse Generation: Perturbations that cause G(t) >Gthresh will result in the release of a short pulse. Once I(t) is lifted above the invariant plane {I = 0}, I(t) will increase
exponentially（指数）. This results in the saturation of Q(t) and the depletion of the gain G(t). Once G(t) − Q(t) − 1 < 0, I(t) will hit its peak intensity Imax and Q(t) will reach its minimum Q ≈ 0, followed by a fast decay（衰变） of both I and Q on the order of 1/γI and 1/γQ in time, respectively. I(t) will eventually reach I ≈ 0 as it further depletes（耗尽） the gain to a final value Greset ,which—with a large enough intensity—is often close to the transparency（透明度） level, i.e., Greset ≈ 0.

脉冲产生：导致G（t）> Gthresh的扰动将导致短脉冲的释放。一旦I（t）被提升到不变平面{I = 0}之上，I（t）将呈指数增长。这导致Q（t）的饱和和增益G（t）的耗尽。一旦G（t） - Q（t） - 1 <0，I（t）将达到其峰值强度Imax，Q（t）将达到其最小Q≈0，然后I和Q的快速衰减时间顺序为1 /γI和1 /γQ。 I（t）最终将达到I≈0，因为它进一步耗尽增益到最终值Greset，其具有足够大的强度 - 通常接近透明度水平，即Greset≈0。

A given pulse derives（获得） its energy from excited carriers in the cavity. The total energy of the pulse is Epulse = Nhν, where N is the number of excited carriers that have been depleted（耗尽） and hν is the energy of a single photon at the lasing frequency. Because the gain is proportional（成比例的） to the inversion（相反的）population, N must be proportional to the amount that the gain G(t) has depleted during the formation of a pulse. Thus, if Gfire is the gain that causes the release of a pulse, we can expect that an output pulse will take the approximate form:

给定脉冲从腔中的激发载流子获得其能量。脉冲的总能量是Epulse =Nhν，其中N是已经耗尽的激发载流子的数量，并且hν是激光频率下单个光子的能量。因为增益与反转总体成比例，所以N必须与在脉冲形成期间增益G（t）耗尽的量成比例。因此，如果Gfire是导致脉冲释放的增益，我们可以预期输出脉冲将采用近似形式：

Fig. 6. Normalized, simulated transfer functions for a single pulse, operating the laser with a low equilibrium state (red curve) and a near-threshold equilibrium (blue curve). When a perturbation G brings G(t) above Gthresh (i.e.G = Gthresh − Geq ), the neuron fires a pulse with energy Epulse . Setting Geq close to Gthresh reduces the required perturbation G to initiate（发起） a pulse and thereby minimizes the impact it has on the resulting output pulse, leading to the flatter（平坦） region above threshold on the blue curve. A laser operating near threshold would minimize amplitude variations in the output.

图6.单脉冲的归一化模拟传递函数，以低平衡状态（红色曲线）和近阈值平衡（蓝色曲线）操作激光。当扰动G使G（t）高于Gthresh时（即G = Gthresh - Geq），神经元用能量Epulse发射脉冲。将Geq设置为接近Gthresh可减少所需的扰动G以启动脉冲，从而最小化其对所得输出脉冲的影响，从而导致蓝色曲线上的平坦区域高于阈值。在阈值附近工作的激光器将使输出中的幅度变化最小化。

where τf is the time atwhich a pulse is triggered to fire and δ(t) is a delta function. One of the properties of spike-encoded channels is that spike energies are encoded digitally. Spikes must have a constant amplitude every iteration（迭代）, a characteristic property of the all-or-nothing response shared by biological neurons.We can normalize our output pulses if we set our system to operate close to threshold Gthresh − Geq Gthresh . Since the threshold is effectively lowered, the size of input perturbations Gmust be scaled smaller. This impliesGfire ≈ Gthresh ,which
helps in suppressing variations in the output pulse amplitude by reducing the input perturbation to the system. This leads to a step-function like response, as illustrated in Fig. 6, which is the desired behavior.

其中τf是触发脉冲发射的时间，δ（t）是δ函数。尖峰编码通道的一个特性是尖峰能量以数字方式编码。尖峰必须在每次迭代时具有恒定的振幅，这是生物神经元共有的全有或全无响应的特征属性。
如果我们将系统设置为接近阈值Gthresh - Geq Gthresh，我们可以规范化输出脉冲。由于阈值被有效地降低，因此输入扰动G的大小必须缩小。这意味着Gfire≈Gthresh，它有助于通过减少对系统的输入扰动来抑制输出脉冲幅度的变化。这导致类似于阶跃函数的响应，如图6所示，这是期望的行为。

After a pulse is released, I(t) → 0 and Q(t) will quickly recover to Qeq . The fast dynamics will give way to slower dynamics, in which G(t) will slowly creep（慢慢移动） from Greset to Geq .The fast dynamics of Q(t) assure that the threshold Gthresh =1 + Q(t) recovers quickly after a pulse is generated, preventing partial pulse release during the recovery period. In addition, the laser will experience a relative refractory period in which it is difficult—but not impossible—to fire another pulse.

释放脉冲后，I（t）→0和Q（t）将快速恢复到Qeq。快速动态将让位于较慢的动态，其中G（t）将从Greset慢慢地蠕变到Geq。
Q（t）的快速动态确保在产生脉冲之后阈值Gthresh = 1 + Q（t）快速恢复，从而防止在恢复期间部分脉冲释放。此外，激光器将经历相对不应期，其中难以 - 但不是不可能 - 发射另一个脉冲。

3) LIF Analogy: If we assume the fast dynamics are nearly instantaneous（瞬间）, then we can compress the behavior of our system into the following set of equations and conditions:

3）LIF类比：如果我们假设快速动力学几乎是瞬时的，那么我们可以将系统的行为压缩成下面的一组方程和条件：

where θ(t) represent input perturbations. This behavior can be seen qualitatively in Fig. 5. The conditional statements account for the fast dynamics of the system that occur on times scales of order 1/γI , and other various assumptions—including the
fast Q(t) variable and operation close to threshold—assure that Gthresh,Greset and the pulse amplitude Epulse remain constant.If we compare this to the LIF model, or equation (1):

其中θ（t）代表输入扰动。在图5中可以定性地看到这种行为。条件语句解释了在1 /γI阶的时间尺度上发生的系统的快速动态，以及其他各种假设 - 包括快速Q（t）变量和接近于阈值 - 确保Gthresh，Greset和脉冲幅度Epulse保持不变。
如果我们将其与LIF模型或等式（1）进行比较：

The analogy between the equations becomes clear. Setting the variables γG = 1/RmCm,A = VL, θ(t) = Iapp(t)/RmCm,and G(t) = Vm(t) shows their algebraic equivalence. Thus, the gain of the laser G(t) can be thought of as a virtual membrane
voltage, the input current A as a virtual leakage voltage, etc.3 There is a key difference; however—both dynamical systems operate on vastly different time scales. Whereas biological neurons have time constants τm = CmRm on order of milliseconds,carrier lifetimes of laser gain sections are typically in the nanosecond range and can go down to picosecond.

方程之间的类比变得清晰。设置变量γG= 1 / RmCm，A = VL，θ（t）= Iapp（t）/ RmCm，G（t）= Vm（t）表示它们的代数等价。因此，激光器G（t）的增益可以被认为是虚拟膜电压，输入电流A被认为是虚拟泄漏电压等.3存在关键差异;然而，两个动力系统都在非常不同的时间尺度上运行。尽管生物神经元具有毫秒级的时间常数τm= CmRm，但激光增益部分的载流子寿命通常在纳秒范围内并且可以下降到皮秒。

III. EXCITABLE VCSELS

Although the excitable model is generalizable to a variety of different laser types, VCSELs are a particularly attractive candidate for our computational primitive as they occupy（占据） small footprints, can be fabricated（制造） in large arrays allowing for massive（大规模的） scalability, and use low powers [32]. An excitable, VCSEL with an intracavity（腔内） SA that operates using the same rate equation model described previously has already been experimentally realized [33]. In addition, the technology is amenable（适合） to a variety of different interconnect schemes: VCSELs can send signals upward and form 3-D interconnects [34], can emit downward into an interconnection layer via grating（光栅） couplers（耦合器） [35] or connect monolithically（单片） through intracavity（腔内）holographic（全息） gratings [36].

尽管可激励模型可以推广到各种不同的激光器类型，但VCSEL对于我们的计算原型来说是一个特别有吸引力的候选者，因为它们占据小的占地面积，可以在大型阵列中制造，允许形成可扩展性，并且使用低功率[32]。具有腔内SA的可激发的VCSEL使用前面描述的相同速率方程模型进行操作已经通过实验实现[33]。此外，该技术适用于各种不同的互连方案：VCSEL可向上发送信号并形成3-D互连[34]，可通过光栅耦合器向下发射到互连层[35]或通过腔内全息光栅单片连接[36]。

A schematic of our VCSEL structure, which includes an intracavity SA, is illustrated in Fig. 7. To simulate the device, we use a typical two-section rate equation model such as the one described in [30]:

我们的VCSEL结构示意图包括腔内SA，如图7所示。为了模拟器件，我们使用典型的两段速率方程模型，如[30]中描述的模型：

3 Our laser lacks an absolute refractory period variable τrefrac seen in some LIF models, but the absence of this condition does not significantly affect its qualitative behavior.

3 我们的激光缺乏在一些LIF模型中看到的绝对不应期变量τrefrac，但缺乏这种情况并不会显着影响其定性行为。

Fig. 7. A schematic（概要） diagram（图） of a VCSEL-SA embedded in a network. In this configuration, inputs λ1 , λ2 , . . . , λn modulate the gain selectively. Various
frequencies lie on different parts of the gain spectrum, leading to different excitatory and inhibitory responses. The weights and delays are applied by amplifiers and delay lines within the fiber network. If excited, a pulse at wavelength λ0 is emitted upward and is eventually incident on other excitable lasers.

图7.嵌入网络中的VCSEL-SA的示意图。在这种配置中，输入λ1，λ2，....。。，λn选择性地调制增益。各种频率位于增益谱的不同部分，导致不同的兴奋和抑制反应。权重和延迟由光纤网络内的放大器和延迟线施加。如果被激发，则波长λ0的脉冲向上发射并最终入射到其他可激发的激光器上。

where Nph(t) is the total number of photons in the cavity, na (t) is the number of carriers in the gain region, and ns (t) is the number of carriers in the absorber. Subscripts（下标） a and s identify the active and absorber regions, respectively（分别）. The remaining device parameters are summarized in Table I. We add an additional input term φ(t) to account for optical inputs selectively coupled（耦合） into the gain, an additional modulation term ie (t) to represent electrical modulation in the gain, and an SA current injection term Is/eVs to allow for an adjustable threshold. For small perturbations, φ(t) and ie (t) possess similar functionalities and represent equally valid（有效） ways of modulating our laser with analog inputs.

其中Nph（t）是腔中光子的总数，na（t）是增益区中的载流子数，ns（t）是吸收器中的载流子数。下标a和s分别标识活动区域和吸收区域。其余的器件参数总结在表I中。我们添加一个额外的输入项φ（t）以考虑选择性地耦合到增益中的光输入，附加调制项即（t）来表示增益中的电调制，以及SA电流注入项Is / eVs允许可调阈值。对于小扰动，φ（t）和ie（t）具有相似的功能，并且代表了用模拟输入调制激光器的同样有效的方法。

These equations are analogous to the dimensionless set of equations (2) provided that the following coordinate（坐标） transformations are made:

这些方程类似于无量纲方程组（2），条件是进行以下坐标变换：

where differentiation is now with respect（方面） to ˜t rather than t. The dimensionless parameters are now

现在差别在于t而不是t。无量纲参数现在

For the simulation, we set the input currents to Ia = 2 mA and Is = 0 mA for the gain and absorber regions, respectively.The output power is proportional（成比例的） to the photon number Nph inside the cavity via the following formula（式子）:

对于模拟，我们分别将增益和吸收区域的输入电流设置为Ia = 2 mA和Is = 0 mA。输出功率通过以下公式与腔内光子数Nph成比例：

in which ηc is the output power coupling coefficient, c the speed of light, and hc/λ is the energy of a single photon at wavelength λ. We assume the structure is grown on a typical GaAs-based substrate（基质） and emits at a wavelength of 850 nm.

其中ηc是输出功率耦合系数，c是光速，hc /λ是波长λ下单个光子的能量。我们假设该结构在典型的基于GaAs的衬底上生长并且发射波长为850nm。

Using the parameters described previously, we simulated the device with optical injection into the gain as shown in Fig. 8. Input perturbations that cause gain depletion（消耗） or enhancement—represented by positive and negative input pulses—modulate the carrier concentration（浓度） inside the gain section. Enough excitation eventually causes the laser to enter fast dynamics and fire a pulse. This behavior matches an LIF neuron model as described in Section II-B3.

使用前面描述的参数，我们模拟了光学注入增益的器件，如图8所示。
导致增益耗尽或增强的输入扰动 - 由正和负输入脉冲表示 - 调制增益部分内的载流子浓度。足够的激发最终导致激光进入快速动态并发射脉冲。此行为与第II-B3节中描述的LIF神经元模型匹配。

Our simulation effectively shows that an excitable LIF neuron is physically realizable in a VCSEL-SA cavity structure. The carrier lifetime of the gain is on the order of 1 ns, which as we have shown in Section II-B3 is analogous to the RmCm time constant of a biological neuron—typically on the order of 10 ms. Thus, our device already exhibits speeds that are 10 million times faster than a biological equivalent. Lifetimes could go as short as a picosecond, making the potential factor speed increase between biology and photonics up to a billion.

我们的模拟有效地表明，可激活的LIF神经元在VCSEL-SA腔结构中是物理可实现的。增益的载流子寿命大约为1ns，如我们在II-B3节中所示，其类似于生物神经元的RmCm时间常数 - 通常在10ms的量级。因此，我们的设备已经展示出比生物等效物快1000万倍的速度。寿命可能短至皮秒，使得生物学和光子学之间的潜在因子速度增加高达10亿。

IV. CORTICAL（皮层） SPIKE ALGORITHMS—SMALL-CIRCUIT DEMONSTRATIONS

线性尖峰算法 - 小电路演示

Since our laser behaves identically（相同） to an LIF model, we can create a wide variety of useful networks that can implement a diversity（多样） of cortical functions. This section describes implementation of biologically inspired circuits with the excitable laser computational primitive.We have constructed circuits with unique properties as a proof of concept of system creation and wireability. These examples form a basis for a small-scale validity（有效） of any theoretical or experimental demonstration of important processing tasks that underlie many spiking neural networks.Though rudimentary(初步), the circuits presented here are undamental(基本的) exemplars of three spike processing functions: multistable operation, synfire（同步） processing [41], and spatiotemporal pattern recognition [42]. Multistability forms the basis of attractor networks [43], synfire chains describe a mechanism with which neurons can form distributed representations of information to avoid noise degradation [44], and pattern recognition has been implicated（牵连） in playing a crucial component in working memory [45].

由于我们的激光器与LIF模型的行为相同，我们可以创建各种有用的网络，可以实现多种皮质功能。本节描述了具有可激励激光计算原语的生物启发电路的实现。我们构建了具有独特属性的电路，作为系统创建和可连线性概念的证明。这些例子构成了许多尖峰神经网络基础的重要处理任务的任何理论或实验演示的小规模有效性的基础。虽然是初步的，但这里介绍的电路是三种尖峰处理功能的基本范例：多稳态操作，同步处理[41]和时空模式识别[42]。
多重性构成了吸引子网络的基础[43]，同步链描述了一种机制，神经元可以利用该机制形成信息的分布式表示，以避免噪声降级[44]，模式识别与工作记忆中的关键组成部分有关[45]。

Fig. 8. Simulation of an excitable, LIF VCSEL-SA with realistic parameters.Inputs (top) selectively modulate the carrier concentration in the gain section (middle). Enough excitation leads to the saturation of the absorber to transparency (bottom) and the release of a pulse, followed by a relative refractory period while the pump current recovers the carrier concentration back to its equilibrium（平衡） value.

图8.具有实际参数的可激励LIF VCSEL-SA的仿真。输入（顶部）选择性地调制增益部分（中间）中的载流子浓度。足够的激发导致吸收器饱和到透明度（底部）和脉冲释放，随后是相对不应期，而泵电流将载流子浓度恢复到其平衡值。

We stipulate（规定） a mechanism for optical outputs of excitable lasers to selectively modulate the gain of others through both excitatory (gain enhancement) and inhibitory (gain depletion（消耗）) pulses as illustrated in Fig. 7. Selective coupling into the gain can be achieved by positioning the gain and saturable absorber regions to interact only with specific optical frequencies as experimentally（实验） demonstrated（证明） in [33]. Excitation and inhibition can be achieved via the gain section’s frequency dependent absorption spectrum—different frequencies can induce gain enhancement or depletion. This phenomenon has been experimentally demonstrated in semiconductor optical amplifiers (SOAs) [46] and could generalize to laser gain sections if the cavity modes are accounted for. Alternatives to these proposed solutions include photodetectors with short electrical connections and injection into an extended gain region in which excited carriers are swept（迅速并且顺利的移动sweep） into the cavity via carrier transport mechanisms.

我们规定了可激发激光器的光输出机制，通过激发（增益增强）和抑制（增益耗尽）脉冲选择性地调制其他激光的增益，如图7所示。通过定位增益可以实现对增益的选择性耦合和可饱和吸收体区域仅与特定的光学频率相互作用，如[33]中的实验所示。激发和抑制可以通过增益部分的频率相关吸收光谱来实现 - 不同的频率可以引起增益增强或耗尽。这种现象已经在半导体光放大器（SOA）[46]中进行了实验证明，并且如果考虑了腔模式，可以推广到激光增益部分。这些提出的解决方案的替代方案包括具有短电连接的光电探测器和注入扩展增益区域的光电探测器，其中激发的载流子通过载流子传输机制扫入腔体。

A network of excitable lasers connected via weights and delays—consistent with the model described in Section II-A—can be described as a delayed differential equation (DDE) of the form:

通过权重和延迟连接的可激发激光器网络 - 与第II-A节中描述的模型一致 - 可以描述为以下形式的延迟微分方程（DDE）：

where the vector x(t) contains all the state variable associated with the system. The output to our system is simply the output power Pout(t),4 while the input is a set of weighted and delayed outputs from the network σ(t) = k Wk Pout(t − τk ). We can
construct weight and delay matrices（矩阵） W,D such that the Wij element of W represents the strength of the connection between excitable lasers i, j, and the Dij element of D represents the delay between lasers i, j. If we recast（重铸） (8) in a vector form, we can formulate（制定） our system in (10) given（假设） that the input function vector φ(t) is

其中向量x（t）包含与系统关联的所有状态变量。我们系统的输出只是输出功率P out（t），4而输入是来自网络的一组加权和延迟输出σ（t）= k Wk Pout（t-τk）。我们可以构造权重和延迟矩阵W，D，使得W的Wij元素表示可激励激光器i，j之间的连接强度，并且D的Dij元件表示激光器i，j之间的延迟。如果我们以矢量形式重铸（8），我们可以在（10）中表示我们的系统，假设输入函数矢量φ（t）是

where we create a sparse（稀疏） matrix Ω containing information for both W and D, and a vector Θ(t) that contains all the past outputs from the system during unique delays U =[τ1, τ2, τ3, . . . , τn ]:

我们创建一个稀疏矩阵Ω，包含W和D的信息，以及一个向量Θ（t），它包含在唯一延迟期间系统的所有过去输出U = [τ1，τ2，τ3，...。。，τn]：

Wk describes a sparse matrix of weights associated with the delay in element k of the unique delay vector U. To simulate various system configurations, we used Runge–Kutta methods iteratively（迭代） within a standard DDE solver in MATLAB. This formulation（公式） allows the simulation of arbitrary（随意） networks of excitable lasers.

Wk描述了与唯一延迟向量U的元素k的延迟相关联的稀疏权重矩阵。为了模拟各种系统配置，我们在MATLAB中的标准DDE求解器内迭代地使用Runge-Kutta方法。该公式允许模拟可激发激光的任意网络。

Since weighing and delaying are both linear operations, they can be implemented optically with passive devices. A physical architecture of a tunable（可调） weight-delay network is illustrated in Fig. 9. Excitable lasers send pulses into an optical network,which may use amplifiers, filters, or switching technologies to sort and distribute the spikes en route to other excitable lasers.The combined inputs incident on a single laser embedded within the network are then weighted and delayed individually by tunable optical attenuators（衰减器） and delay lines before arrival.

由于称重和延迟都是线性操作，因此可以使用无源设备进行光学实现。
可调谐权重延迟网络的物理结构如图9所示。可激发激光器将脉冲发送到光网络，光网络可以使用放大器，滤波器或开关技术来分配和分配到其他可激发激光器的尖峰。
然后，入射在网络内的单个激光器上的组合输入在到达之前由可调谐光学衰减器和延迟线单独加权和延迟。

A tunable weight-delay input array as depicted（描绘） in Fig. 9—which can be thought of as the photonic equivalent（等效物） of a dendritic（树突） tree—has been experimentally realized in a photonic beam（光束） former [47]. With the appropriate integrated, tunable attenuators and delay lines—which can be implemented using ring（环）resonators（谐振器） structures [48], [49] or other technologies [50], [51]—this array could be compacted（压缩） into a small footprint, allowing for massive network integration. Further work will explore the scalability of this approach. Described next are several circuits that could potentially utilize（利用） this architecture to perform tasks specific to spike processing.

如图9所示的可调权重延迟输入阵列 - 可以被认为是树枝状树的光子等效物 - 已经在光子束形成器中实验性地实现[47]。使用适当的集成，可调衰减器和延迟线 - 可以使用环形谐振器结构[48]，[49]或其他技术[50]，[51]实现 - 这个阵列可以压缩成一个小的占地面积，允许大规模网络集成。进一步的工作将探索这种方法的可扩展性。接下来描述的是可能利用该架构来执行尖峰处理特定任务的若干电路。

4We absorb the attenuation（衰减） or amplification the pulse experiences en route to its destination along with the responsivity of the perturbation to the incident pulse into a single weight parameter Wij .

我们吸收脉冲在到达目的地途中经历的衰减或放大以及对入射脉冲的扰动对单个权重参数Wij的响应。

Fig. 9. A physical architecture of a photonic neural network with tunable weights and delays (two lasers displayed). Laser outputs are separated from inputs with use of a circulator and travel into an optical network that evenly（均匀） distributes spiking signals across the entire device landscape（环境）. Before signals arrive at their respective destinations, they interface with a front-end control unit that applies weights wij and delays τij to signals traveling from lasers i to j.

图9.具有可调权重和延迟的光子神经网络的物理结构（显示两个激光器）。激光输出通过使用循环器与输入分离，并进入光学网络，在整个设备环境中均匀分布尖峰信号。在信号到达它们各自的目的地之前，它们与前端控制单元连接，该前端控制单元将权重wij和延迟τij延迟到从激光器i到j的信号。

A. Multistable System

多稳态系统

Multistability represents a crucial property of dynamical systems and arises out of the formation of hysteric（过度的狂烈的） attractors. This phenomenon plays an important role in the formation of memory in processing systems. Here, we describe a network of two interconnected excitable lasers, each with two incoming connections and identical weights and delays, as illustrated in Fig. 10(a). The system is recursive（递归） rather than feedforward, possessing a network path that contains a closed loop. This allows the system to exhibit hysteresis（滞后）.

多稳态性代表了动力系统的一个重要特性，并且是由于过度的狂烈的吸引子的形成而产生的。这种现象在处理系统中的存储器形成中起着重要作用。在这里，我们描述了两个互连的可激发激光器的网络，每个激光器具有两个输入连接和相同的权重和延迟，如图10（a）所示。系统是递归的而不是前馈的，拥有包含闭环的网络路径。这允许系统表现出滞后现象。

Results for the two laser multistable system are shown in Fig. 10(b). The network is composed of two lasers, interconnected via optical connections with a delay of 1 ns.An excitatory pulse travels to the first unit at t = 5 ns, initiating the system to settle to a new attractor. The units fire pulses repetitively at fixed intervals（间隔） before being deactivated（停用） by a precisely（精确） timed inhibitory pulse at t = 24 ns. It is worth noting that the system is also capable of stabilizing to other states, including those with multiple pulses or different pulse intervals. It, therefore, acts as a kind of optical pattern buffer over longer time scales.Ultimately（最终）,this circuit represents a test of the network’s ability to handle recursive feedback. In addition, the stability of the system implies that a network is cascadable since a self-referent connection is isomorphic（同构） to an infinite（无限） chain of identical lasers with identical weights W between every node. Because this system successfully maintains the stability of self-pulsations, processing networks of excitable VCSELs are theoretically capable of cascadibility and information retention（保留） during computations.

两个激光多稳态系统的结果如图10（b）所示。该网络由两个激光器组成，通过光学连接互连，延迟为1 ns。激发脉冲在t = 5 ns时传播到第一个单元，启动系统稳定到新的吸引子。在t = 24 ns时被精确定时的抑制脉冲去停止之前，这些单元在固定的时间间隔内重复发射脉冲。值得注意的是，该系统还能够稳定到其他状态，包括具有多个脉冲或不同脉冲间隔的状态。因此，它在较长时间尺度上充当一种光学模式缓冲器。最后，该电路代表了对网络处理递归反馈能力的测试。此外，系统的稳定性意味着网络是可级联的，因为自参考连接与每个节点之间具有相同权重W的无限链相同激光同构。因为该系统成功地保持了自脉冲的稳定性，所以可激励VCSEL的处理网络在理论上能够在计算期间具有级联性和信息保持性。

Fig. 10. (a) Bistability schematic—In this configuration, two lasers are connected
symmetrically（对称地） to each other. (b) A simulation of a two laser system exhibiting bistability with connection delays of 1 ns. The input perturbations to unit 1 are plotted, followed by the output powers of units 1 and 2, which include scaled version of the carrier concentrations of their gain sections as the dotted（点） blue lines. Excitatory pulses are represented by positive perturbations while inhibitory pulses are represented by negative perturbations. An excitatory input excites the first unit, causing a pulse to be passed back and forth between the nodes. A precisely timed inhibitory pulse terminates（终止） the sequence.

图10.（a）双稳态示意图 - 在这种配置中，两个激光器彼此对称连接。（b）两个激光系统的模拟，表现出双稳态，连接延迟为1ns。绘制对单元1的输入扰动，然后绘制单元1和2的输出功率，其包括其增益部分的载流子浓度的缩放版本作为虚线蓝线。兴奋脉冲由正扰动表示，而抑制脉冲由负扰动表示。兴奋性输入激发第一个单元，导致脉冲在节点之间来回传递。精确定时的抑制脉冲终止序列。

B. Synfire Chain 同步链

Synfire chains have been proposed by Abeles [52] as a model of cortical function. A synfire chain is essentially a feedforward network of neurons with many layers (or pools). Each neuron in one pool feeds many excitatory connections to neurons in the
next pool, and each neuron in the receiving pool is excited by many neurons in the previous pool, so that a wave of activity can propagate（传播） from pool to pool in the chain. It has been postulated（假设）that such a wave corresponds to an elementary（初级） cognitive event[53].

Abeles [52]提出同步链作为皮质功能的模型。同步链本质上是具有许多层（或池）的神经元的前馈网络。一个池中的每个神经元为下一个池中的神经元提供许多兴奋性连接，并且接收池中的每个神经元都被前一个池中的许多神经元激发，因此活动波可以从链中的池传播到池中。据推测，这种波对应于基本的认知事件[53]。

Synfire chains can use population encoding to reduce jitter（抖动） accumulation when sending, receiving, or storing a spatiotemporal bit pattern of spikes [41]. Population encoding works by making multiple copies of a pulse and distributing it via several distinct （不同）channels. When these copies arrive and recombine onto the same subsequent（随后） processing unit, jitter and amplitude noise accumulated in statistically uncorrelated channels are averaged and therefore reduced. One of the key features of a hybrid analog-digital system such as an spiking neural networks (SNN) is that many analog nodes can process in a distributed and redundant（冗余） way to reduce noise accumulation. Recruiting（招募） a higher number of neurons to accomplish the same computation is an effective and simple way of reducing spike error rates.

Synfire链可以使用填充编码来减少发送，接收或存储尖峰的时空位模式时的抖动累积[41]。群体编码通过制作脉冲的多个副本并通过几个不同的通道分发来工作。当这些副本到达并重新组合到相同的后续处理单元上时，在统计上不相关的信道中累积的抖动和幅度噪声被平均并因此减小。混合模拟 - 数字系统（例如尖峰神经网络（SNN））的关键特征之一是许多模拟节点可以以分布式和冗余方式处理以减少噪声累积。招募更多数量的神经元来完成相同的计算是降低尖峰错误率的有效且简单的方法。

Fig. 11. (a) Synfire schematic—In this configuration, two groups of lasers are
connected symmetrically two each other. (b) Simulation of a four laser circuit
forming synfire chains with connection delays of 14 ns. The input perturbations
to units 1, 2 are plotted over time, followed by the output powers of units 1–4
with the scaled carrier concentrations of their gain sections as the dotted blue
lines. A characteristic spike pattern is repeatedly passed back and forth between
the left and right set of nodes.

图11.（a）Synfire原理图 - 在这种配置中，两组激光器彼此对称地连接。（b）模拟形成具有14ns连接延迟的同步链的四个激光器电路。对单元1,2的输入扰动随时间绘制，接着是单元1-4的输出功率，其增益部分的缩放载流子浓度为蓝色虚线。特征尖峰模式在左右节点集之间反复传递。

Fig. 11(a) shows a demonstration of a simple four-laser synfire chain with feedback connections. The chain is simply a two unit expansion of each node in the multistability circuit from Fig. 10(a). Like the multistability circuit, recursion allows the synfire chain to possess hysteric properties; however, the use of two lasers for each logical node provides processing redundancy and increases reliability. Once the spike pattern is input into the system as excitatory inputs injected simultaneously （同时）into the first two lasers, it is continuously passed back and forth between each set of two nodes. The spatiotemporal bit pattern persists after several iterations（迭代） and is thereby stored in the network as depicted（描绘） in Fig. 11(b).

图11（a）示出了具有反馈连接的简单四激光同心链的演示。该链简单地是图10（a）所示的多稳态电路中每个节点的两个单元扩展。与多稳态电路一样，递归允许同步链具有歇斯底里特性;但是，为每个逻辑节点使用两个激光器可提供处理冗余并提高可靠性。一旦将尖峰模式作为同时注入前两个激光器的激发输入输入系统，它就会在每组两个节点之间连续地来回传递。时空比特模式在几次迭代后持续存在，从而存储在网络中，如图11（b）所示。

C. Spatiotemporal Pattern Recognition Circuit时空模式识别电路

The concept of polychrony, proposed by Izhikevich [42] is defined as an event relationship that is precisely time-locked to firing patterns but not necessarily synchronous（同步）. Polychronization presents a minimal spiking network that consists of cortical spiking neurons with axonal delays and synaptic time dependent
plasticity (STDP), an important learning rule for spike-encoded neurons. As a result of the interplay（相互作用） between the delays and STDP, spiking neurons spontaneously（自发） self-organize into groups and generate patterns of stereotypical（定型） polychronous activity.

由Izhikevich [42]提出的多重概念被定义为一种事件关系，它精确地时间锁定到激发模式但不一定是同步的。多重同步呈现最小的尖峰网络，其由具有轴突延迟和突触时间依赖性可塑性（STDP）的皮质尖峰神经元组成，这是针对尖峰编码神经元的重要学习规则。由于延迟和STDP之间的相互作用，尖峰神经元自发地自组织成组并产生刻板的多重活动模式。

One of the key properties of polychronization is the ability to perform delay logic to perform spatiotemporal pattern recognition.As shown in Fig. 12(a), we construct a simple three unit pattern recognition circuit of excitable lasers with carefully tuned delay lines, where each subsequent neuron in the chain requires stronger perturbations to fire. The resulting simulation is shown in Fig. 12(b). Three excitatory inputs separated sequentially（顺序） by Δt1 = 5 ns and Δt2 = 10 ns are incident（入射） on all three units. The third is configured only to fire if it receives an input pulse and pulses from the other two simultaneously（同时）. The system, therefore, only reacts to a specific spatiotemporal bit pattern.

多重同步的关键特性之一是能够执行延迟逻辑以执行时空模式识别。如图12（a）所示，我们构建了一个简单的三单元模式识别电路，其中可激励激光器具有仔细调谐的延迟线，其中链中的每个后续神经元需要更强的激发扰动。得到的模拟如图12（b）所示。依次分开Δt1= 5ns和Δt2= 10ns的三个兴奋输入入射在所有三个单元上。第三个配置为仅在接收到输入脉冲并同时接收来自其他两个脉冲的脉冲时才触发。因此，系统仅对特定的时空位模式作出反应。

Although this circuit is simple, the ability to perform temporal logic implies that excitable, neuromorphic systems are capable of categorization and decision making. Two existing applications utilize temporal logic, including light detection and ranging
sensitivity （灵敏度）that is analogous to an owl’s（猫头鹰） echolocation system and the escape response of a crayfish（小龙虾） [54], [55]. Combined with learning algorithms such as STDP which has recently been demonstrated in optics [56], networks could potentially perform more complex tasks such as spike-pattern cluster（聚类） analysis.

尽管该电路很简单，但执行时间逻辑的能力意味着可激励的神经形态系统能够进行分类和决策。两个现有的应用程序利用时间逻辑，包括光检测和测距灵敏度，类似于猫头鹰的回声定位系统和小龙虾的逃逸反应[54]，[55]。结合最近在光学[56]中演示的STDP等学习算法，网络可能会执行更复杂的任务，如尖峰模式聚类分析。

Fig. 12. (a) Schematic of a three-laser circuit that can recognize specific
spatiotemporal bit patterns. (b) A simulation of a spatiotemporal recognition
circuit with t1 = 5 ns and t2 = 10 ns. The input perturbation to unit 1 is plotted, along with the output powers of units 1–3 with the scaled carrier concentrations of their gain sections as the dotted blue lines. The third neuron fires during the triplet spike pattern with time delays t1 and t2 between spikes.

图12.（a）可以识别特定时空位模式的三激光电路的示意图。（b）时空识别电路的模拟，其中t1 = 5ns且t2 = 10ns。绘制对单元1的输入扰动以及单元1-3的输出功率，其增益部分的缩放载流子浓度为虚线蓝线。第三个神经元在三重尖峰模式期间发射，尖峰之间具有时间延迟t1和t2

V. DISCUSSION
A. Comparing Technological Platforms比较技术平台

Cortically-inspired microelectronic architectures have traditionally targeted biological time scales. Several proposals [3],[57] suggest using a crossbar（交叉） array to network neurons together,essentially（实质上） a dense（稠密） mesh（网孔） of wires overlaying the CMOS (processor) substrate（基质）. This is to achieve a massive fan-in and fan-out per connection, which is typical in neural networks but less critical in conventional（常规） processors. Several popular approaches aim to achieve clock rates comparable to biological time scales,but transmitting high-bandwidth spikes at the speeds of current processors (gigahertz)—which tend to have high bandwidth requirements—could overrun（泛滥） the system with electromagnetic（电磁） interference（干扰） (EMI). Signals would quickly attenuate（衰减）, disperse,or couple together unfavorably（不利）, especially on a crossbar array,which has a large area of closely packed signal wires. In contrast,light can support the high frequency components of spikes and large fan-in and fan-out per connection with almost no crosstalk through techniques such as wavelength division （区分）multiplexing.Achieving both high speeds and large interconnection densities（密度） simultaneously（同时） is comparably impossible in electronics.

皮质启发的微电子架构传统上以生物时间尺度为目标。一些提议[3]，[57]建议使用交叉开关阵列将神经元联网在一起，基本上是一层覆盖CMOS（处理器）基板的密集网格线。这是为了实现每个连接的大量扇入和扇出，这在神经网络中是典型的，但在传统处理器中不那么重要。几种流行的方法旨在实现与生物时间尺度相当的时钟速率，但是以当前处理器（千兆赫兹）的速度传输高带宽尖峰 - 这往往具有高带宽要求 - 可能会因电磁干扰（EMI）而超出系统。信号会很快地衰减，分散或耦合在一起，尤其是在交叉阵列上，该阵列具有大面积紧密堆积的信号线。相比之下，光可以支持尖峰的高频分量以及每个连接的大扇入和扇出，并且通过诸如波分复用的技术几乎没有串扰。同时实现高速和大互连密度在电子学中是相对不可能的。

The ability to make a technology that complements（补充） the physical constraints （限制） that guide it, rather than abstracting them away entirely, represents an important step in streamlining（精简） efficiency and performance. Optics is a perfect fit for high bandwidth spike information and could represent a highly efficient processing scheme that ties closely to its underlying physics.

能够制造一种技术来补充引导它的物理约束，而不是完全抽象它们，这是简化效率和性能的重要一步。光学器件非常适合高带宽尖峰信息，可以代表一种高效的处理方案，与其基础物理学密切相关。

B. Improvements Over Previous Models 对先前模型的改进

Past photonic neurons have demonstrated important features of biological neurons but did not integrate enough properties together to make effective processors. One of the first implementations of a photonic spiking neuron [15] achieved noise
suppression（抑制）and thresholding through a nonlinear optical loop mirror, generated pulses synchronously via a mode-locked laser,and utilized an SOA for integration. This model demonstrated temporal integration and spike thresholding, but too much excitation could lead to the release of multiple pulses, degrading spike-encoded information. In addition, spikes were not asynchronous（异步）,making the output of the system digital.

过去的光子神经元已经证明了生物神经元的重要特征，但没有将足够的特性整合在一起以形成有效的处理器。光子尖峰神经元的第一个实现之一[15]通过非线性光学环路镜实现了噪声抑制和阈值处理，通过锁模激光器同步产生脉冲，并利用SOA进行集成。该模型展示了时间积分和尖峰阈值，但过多的激发可能导致多个脉冲的释放，从而降低尖峰编码信息。此外，尖峰不是异步的，使系统的输出成为数字。

The fully functioning photonic neuron demonstrated by Kravstsov [16] integrated both excitation and inhibition, but also suffered from the problems mentioned previously. A newer asynchronous model was proposed in [17] that generated spikes
based on incoming spikes. Although the system could emit a pulse at analog times, because the system did not generate its own spikes through internal mechanisms, the spikes could eventually degrade into noise as nonspike inputs led to nonspike
outputs.

由Kravstsov [16]证明的完全功能的光子神经元整合了激发和抑制，但也遇到了前面提到的问题。在[17]中提出了一种新的异步模型，它根据传入的尖峰产生尖峰。虽然系统可以在模拟时间发出脉冲，因为系统没有通过内部机制产生自己的尖峰，但是当非吸收输入导致非吸收输出时，尖峰最终会降级为噪声。

The excitable models are a step in the right direction, given the similarities between lasers and biological phenomena and the stability of feedback systems. However, both of the recent proposals [22], [24] have a large, base intensity level that underlies
spike outputs. Spikes resemble（类似） variations（变化） of the output intensity rather than pulses. A base-level intensity can potentially be debilitating（衰弱）: by adding constant optical power into the inputs, one can change the equilibrium（平衡） levels of internal state variables. Programming the synaptic weights wij would simultaneously modulate the internal dynamics of each laser in addition to the strength between connections, causing forward propagating（传播） system dependences. In addition, a constant base level intensity could increase amplified spontaneous（自发） emission noise within the network, decreasing its robustness.

鉴于激光和生物现象之间的相似性以及反馈系统的稳定性，可激发模型是朝着正确方向迈出的一步。然而，最近的两个提议[22]，[24]都具有较大的基于尖峰输出强度水平。尖峰类似于输出强度的变化而不是脉冲。基准强度可能会使人衰弱：通过向输入中添加恒定的光功率，可以改变内部状态变量的均衡水平。对突触权重进行编程wij将同时调制每个激光器的内部动态以及连接之间的强度，从而导致向前传播的系统依赖性。此外，恒定的基准电平强度可能增加网络内放大的自发发射噪声，降低其稳健性。

Although biological neurons also have a continuously varying state variable (voltage) during an an action potential, the actual response is thresholded by voltage-gated reversal potentials that only induce neurotransmitter release between cells during a
spike. It is unclear if the semispiking signals emitted by these other lasers could be processed effectively by subsequent units in a network. These lasers would probably require an optical thresholder such as [58] or a nonlinear OEO (optical–electrical–
optical) connection in addition to the units already proposed to work effectively.

尽管生物神经元在动作电位期间也具有连续变化的状态变量（电压），但实际响应通过电压门控反转电位来阈值化，该电压门控反转电位仅在尖峰期间诱导细胞之间的神经递质释放。目前还不清楚这些其他激光器发出的半导体信号是否可以被网络中的后续单元有效地处理。除了已经提出的有效工作的单元之外，这些激光器可能还需要光学阈值器，如[58]或非线性OEO（光电光学）连接。

The neuron model described here avoids many previous issues by combining an excitable approach with some of the ideas of the feedforward model. The integration of a laser gain section—which is dynamically analogous to an SOA—and a saturable absorber, which is essentially a feedback version of the thresholder
used in the fiber model, leads to many desirable properties and a close analogy with biology, including the formation of highly stereotyped and well-defined optical spikes.

这里描述的神经元模型通过将可兴奋的方法与前馈模型的一些想法相结合，避免了许多先前的问题。激光增益部分- 动态类似于SOA - 和可饱和吸收器的集成，它基本上是光纤模型中使用的阈值器的反馈版本，导致许多理想的属性和与生物学的近似类比，包括形成高度刻板和明确定义的光学尖峰。

VI. CONCLUSION

In summary, we have proposed and simulated a novel optical signal processing device that is capable of implementing cortical-inspired algorithms. We have shown that, unlike previous models, our device can effectively perform cognitive algorithms
at ultrafast time scales. This model is demonstrably analogous to an LIF neuron, and networks of such devices are stable, robust to noise, and can recognize patterns.

总之，我们已经提出并模拟了一种新颖的光学信号处理设备，其能够实现皮质启发的算法。我们已经证明，与以前的模型不同，我们的设备可以在超快时间尺度上有效地执行认知算法。该模型明显类似于LIF神经元，并且这种装置的网络稳定，对噪声稳健，并且可以识别图案。

Spike processing algorithms are well understood in a number of important biological sensory processing systems and are finding growing use in signal processing applications [27]. The combination of these physiological principles with engineering
not only helps in studying biological neural circuits [59],but the pairing of computational technology with underlying physics could achieve new domains of application and study.

尖峰处理算法在许多重要的生物传感处理系统中得到了很好的理解，并且正在信号处理应用中得到越来越多的应用[27]。这些生理学原理与工程学的结合不仅有助于研究生物神经回路[59]，而且计算技术与基础物理学的配对可以实现新的应用和研究领域。

Ultrafast optical STDP, one of the most important algorithms for spike-based learning, has recently been demonstrated experimentally [56]. All the components used in this experiment—including the Mach–Zehnder configuration, the EAM, and the SOA—can be fabricated with a small footprint in planar（平面） photonics.Integrating this together with LIF excitable neurons on a single chip could lead to systems that emulate（仿真） a well established paradigm（范例） for adaptive computing on a scalable platform.These compact, adaptive, and unconventional processing systems would operate on unprecedented（史无前例） time scales. Large networks could potentially be constructed as liquid state machines for optical reservoir（池） computing [60] to aid in the study of biology or open up new ultrafast environments such as the RF spectrum for neuromorphic experimentation.

超快光学STDP是最重要的基于尖峰学习的算法之一，最近已通过实验证明[56]。本实验中使用的所有组件 - 包括Mach-Zehnder配置，EAM和SOA--可以在平面光子学中以小占地面积制造。
将LIF可激活神经元与单个芯片集成在一起可以使系统模拟在可扩展平台上进行自适应计算的完善范例。
这些紧凑，自适应和非传统的处理系统将在前所未有的时间尺度上运行。大型网络可能被构建为用于光学储层计算的液态机[60]，以帮助研究生物学或开辟新的超快环境，例如用于神经形态实验的RF频谱。