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) (泄漏整合放电)neuron.
激光表现出兴奋，其行为类似于放电神经元的泄漏整合。
This model is both fast and scalable（延展）, operating up to a billion timesfaster 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).
该模型快速可扩展，比生物等效物快10亿倍，可在协议下实现，垂直腔表面发射激光器（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 ，具有嵌入式饱和吸收体的激光器减少了速率方程控制LIF神经元的行为。
We simulate（仿真） the laser using realistic（实际） rateequations governing a VCSEL cavity, and show behavior representative（代表） of cortical（皮质） spiking （尖峰）algorithms（算法） simulated in small circuitsof excitable lasers.
实际速率方程控制VCSEL模拟激光，并显示代表小激光线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.
将该技术与超快神经学习算法相匹配，将开辟一个新的处理领域。
I. INTRODUCTION(导言)
1. Inan 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·诺依曼架构固有的局限性，最近的一些计算项目利用设备的基础物理学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，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) set ofcomputational principles, including hybrid（混血） analog-digital(模拟-数字) signalrepresentations（表示）, collocation of memory and processing, unsupervised（无监督） learning, and distributed representations of information.
这些新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，存储器和处理器的组合，无监督学习和信息分布式表示。
Fig. 1. Spiking neural networks encode information as events in time ratherthan 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.
图一与比特相比，尖峰神经网络将信息编码不是比特。 由于尖峰发生的时间是模拟的，其范围是数字的，因此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 eventsor spikes in time , a hybrid signal with both analog anddigital properties（特性） as illustrated（插图） in Fig. 1.
在细胞水平上，大脑将信息编码成具有模拟和数字特征的混合信号，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 includingthe implementation（实现） of robust chaotic communication (鲁棒混沌通信)and power efficient channel coding .
该编码方案相当于光学中模拟脉冲位置的调制（PPM），它已用于各种应用，包括实现鲁棒混沌通信和功率有效信道编码。
Spike processing hasevolved（发展） in biological (nervous systems) and engineered (neuromorphic(神经形态) analog VLSI(超大集成电路Very Large Scale Integration）) systems as a means to exploit（利用） the efficiencyof analog signals while overcoming the problem ofnoise accumulation（累积） inherent （固有）in analog computation .
生物(神经系统)和工程(神经形态模拟)已经进行了尖峰处理VLSI）系统的发展是利用模拟信号效率克服模拟计算中固有的噪声积累问题的手段。
Varioustechnologies 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)programand Neurogrid as part of Stanford’sBrains in Silicon program.
各种技术模拟了电子学中的尖峰神经网络，包括自适应可塑性扩展电子神经系统计划的一部分IBM作为斯坦福大学硅脑计划的一部分，神经突触的核心和部分Neurogrid。
Although these architectureshave garnered success in various applications, they aim to targetbiological time scales rather than exceed（超越） them.
尽管这些架构在各种应用中都取得了成功，但它们的目标是生物时间尺度而不是超过它们。
Microelectronicneural networks that are both fast and highly interconnected are subject to(受制于) a fundamental bandwidth fan-in（fan扇） tradeoff.
基本带宽风扇进入权衡会影响快速高度互连的微电子神经网络。
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