Neural networks and physical systems with emergent collective computational abilities
打开互动全文版（逐段中英对照 + 图/公式 + 论文问答）→（联想记忆 / 并行处理 / 分类 / 内容可寻址内存 / 故障软降设备） 由 John J. Hopfield 投稿，1982 年 1 月 15 日 摘要 对生物体或计算机建造有用的计算性质可以作为集体
(associative memory / parallel processing / categorization / content-addressable memory / fail-soft devices) Contributed by John J. Hopfield, January 15, 1982 ABSTRACT Computational properties of use to biological organisms or to the construction of computers can emerge as collective
（联想记忆 / 并行处理 / 分类 / 内容可寻址内存 / 故障软降设备）
(associative memory / parallel processing / categorization / content-addressable memory / fail-soft devices)
由 John J. Hopfield 投稿，1982 年 1 月 15 日
Contributed by John J. Hopfield, January 15, 1982
摘要 对生物体或计算机建造有用的计算性质可以作为集体
ABSTRACT Computational properties of use to biological organisms or to the construction of computers can emerge as collective
具有大量简单组分的系统的性质
Properties of systems having a large number of simple
等价组件（或神经元）。con- 的物理意义
equivalent components (or neurons). The physical meaning of con-
内容可寻址记忆通过系统状态的适当相空间流来描述。基于神经生物学的某些方面但易于适应集成电路的此类系统模型被给出。该模型的集体特性产生了一个内容可寻址记忆，它正确地产生整个
Content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire
tent-addressable memory is described by an appropriate phase
space flow of the state of a system. A model of such a system is
given, based on aspects of neurobiology but readily adapted to in-
tegrated circuits. The collective properties of this model produce
来自任何足够大的子部分的记忆。该算法用于
Memory from any subpart of sufficient size. The algorithm for the
系统状态的时间演化基于异步
time evolution of the state of the system is based on asynchronous
并行处理。额外的涌现集体属性包括一些泛化、熟悉度识别、分类、纠错和时间序列保持的能力。
parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention.
集体属性对以下细节仅微弱敏感
The collective properties are only weakly sensitive to details of the
建模或单个设备的失效。
modeling or the failure of individual devices.
鉴于神经元的动态电化学特性及其相互连接（突触），我们很容易理解使用少量神经元来获得基础有用的生物功能的方案
Given the dynamical electrochemical properties of neurons and their interconnections (synapses), we readily understand schemes that use a few neurons to obtain elementary useful biological
行为(1-3)。我们对这类简单电路的理解在
behavior (1-3). Our understanding of such simple circuits in
电子学中使我们能够规划更大更复杂的电路，
electronics allows us to plan larger and more complex circuits
这些电路对于大型计算机至关重要。由于进化并不具有
which are essential to large computers. Because evolution has
这样的计划，因此我们有理由质疑是否这种能力
no such plan, it becomes relevant to ask whether the ability of
大量神经元集合用于执行“计算”任务。
Large collections of neurons to perform "computational" tasks.
可能部分是由于拥有...的自发集体结果。
May in part be a spontaneous collective consequence of having
大量相互作用的简单神经元。
A large number of interacting simple neurons.
在由大量简单元素组成的物理系统中。
In physical systems made from a large number of simple elements.
大量基本组件之间的相互作用。
Interactions among large numbers of elementary components.
组件产生集体现象，例如稳定的磁性
Components yield collective phenomena such as the stable magnetic
磁系统中的取向和畴，或者涡旋
orientations and domains in a magnetic system or the vortex
流体流动中的模式。类似的集体现象是否在
patterns in fluid flow. Do analogous collective phenomena in
一个由简单相互作用的神经元组成的系统中具有有用的“计算
a system of simple interacting neurons have useful "computational
”关联？例如，记忆的稳定性是否
" correlates? For example, are the stability of memories,
泛化类别的构建，或时间-
the construction of categories of generalization, or time-
序列记忆也是涌现性质和集体
sequential memory also emergent properties and collective in
起源？本文审视了这个古老而基础问题的新的建模
origin? This paper examines a new modeling of this old and fun-
基础问题(4-8)并展示了重要的计算-
fundamental question (4-8) and shows that important computa-
所有建模都基于细节，而神经科学的细节
All modeling is based on details, and the details of neuro-
解剖和神经功能都是众多且不完全
anatomy and neural function are both myriad and incompletely
已知[9]。在许多物理系统中，涌现（
known (9). In many physical systems, the nature of the emer-
出）的集体特性的本质对模型中插入的细节不敏感
gent collective properties is insensitive to the details inserted
（例如，碰撞对于产生声波是必要的
in the model (e.g., collisions are essential to generate sound
，但任何合理的原子间力定律都会产生近似（
waves, but any reasonable interatomic force law will yield ap-
适当的碰撞）。本着同样的精神，我将寻找对模型细节变化鲁棒的集体性质。该模型可以容易地通过集成电路硬件实现。
appropriate collisions). In the same spirit, I will seek collective properties that are robust against change in the model details. The model could be readily implemented by integrated circuit
硬件。结论表明设计了一个去局域化的
hardware. The conclusions suggest the design of a delo-
内容可寻址存储器或分类器，使用大
calized content-addressable memory or categorizer using ex-
量异步并行处理。一般内容可寻址存储器的物理
tensive asynchronous parallel processing. The general content-addressable memory of a physical
假设存储在存储器中的一个条目是“H. A. Kramers &”
Suppose that an item stored in memory is "H. A. Kramers &"
G. H. Wannier Phys. Rev. 60, 252 (1941)。一个通用的内容-
G. H. Wannier Phys. Rev. 60, 252 (1941). A general content-
可寻址存储器能够检索整个
addressable memory would be capable of retrieving this entire
记忆项基于足够的局部信息。这个
memory item on the basis of sufficient partial information. The
输入“& Wannier, (1941)”可能就足够了。一个理想的存储器
input "& Wannier, (1941)" might suffice. An ideal memory
能够处理错误，甚至能从输入“Vannier, (1941)”中检索到该参考文献。在计算机中，只有相对简单的
could deal with errors and retrieve this reference even from the input "Vannier, (1941)". In computers, only relatively simple
已经以硬件形式实现了内容可寻址存储器的形式（10, 11）。
forms of content-addressable memory have been made in hardware (10, 11).
类似在信息访问中纠错的复杂想法通常以软件的形式引入（10）。
Sophisticated ideas like error correction in accessing information are usually introduced as software (10).
存在一类物理系统，其自发行为可用作一种通用（且纠错）的
There are classes of physical systems whose spontaneous behavior can be used as a form of general (and error-correcting)
forms of content-addressable memory have been made in hard-
ware (10, 11). Sophisticated ideas like error correction in ac-
内容可寻址存储器。考虑时间演化
content-addressable memory. Consider the time evolution of
一个可以由一组广义坐标描述的物理系统
a physical system that can be described by a set of general coordinates
状态空间中的一个点表示瞬时
A point in state space then represents the instantaneous
系统的状态。这个状态空间可以是
condition of the system. This state space may be either
连续或离散的（例如在 N 个 Ising 自旋的情况下）。
continuous or discrete (as in the case of N Ising spins).
系统的运动方程描述了状态空间中的流。各种类型的流模式都是可能的，但用于记忆的系统特别包括那些从这些点周围区域内的任何位置流向局部稳定点的系统。在具有两个极小值的势阱中移动的摩擦阻尼粒子体现了这种动力学。
The equations of motion of the system describe a flow in state space. Various classes of flow patterns are possible, but the systems of use for memory particularly include those that flow toward locally stable points from anywhere within regions around those points. A particle with frictional damping moving in a potential well with two minima exemplifies such a dynamics.
The equations of motion ofthe system describe a flow in state
space. Various classes offlow patterns are possible, but the sys-
tems of use for memory particularly include those that flow to-
ward locally stable points from anywhere within regions around those points. A particle with frictional damping moving in a
如果流动不是完全确定性的，描述
If the flow is not completely deterministic, the description
就更复杂。在上述双阱问题中，如果摩擦力由温度表征，则它也必须
is more complicated. In the two-well problems above, if the frictional force is characterized by a temperature, it must also
产生随机驱动力。极限点变成小的
produce a random driving force. The limit points become small
极限区域，稳定性不再是绝对的。但
limiting regions, and the stability becomes not absolute. But
只要随机效应很小，局部的本质
as long as the stochastic effects are small, the essence of local
考虑一个由多个坐标描述的物理系统
Consider a physical system described by many coordinates
X₁, ..., X_N，即状态向量 X 的分量。设该系统
X_1, ..., X_N, the components of a state vector X. Let the system
具有局部稳定的极限点 X_a, X_b, …。那么，如果系统
have locally stable limit points X_a, X_b, \ldots. Then, if the system
从足够接近某个 X_a 的地方开始，比如 X = X_a + A，它将
is started sufficiently near any X_a, as at X = X_a + A, it will
以向量\(X_a\)、\(X_b\)、\(\dots\)等形式存储在系统中。起始
stored in the system as the vectors \(X_a\), \(X_b\), \(\dots\). The starting
点\(X = X_a + A\)表示对条目\(X_a\)的部分知识，
point \(X = X_a + A\) represents a partial knowledge of the item
然后系统生成完整信息\(X_a\)。
\(X_a\), and the system then generates the total information \(X_a\).
任何相空间动力学受大量局部稳定态支配的物理系统，这些稳定态
Any physical system whose dynamics in phase space is dominated by a substantial number of locally stable states to which
吸引着该系统，因此可被视为一个通用内容-
it is attracted can therefore be regarded as a general content-
可寻址记忆。物理系统将是一个潜在
Addressable memory. The physical system will be a potentially
有用的记忆，如果除此之外，任何规定的状态集都可以很容易地成为系统的稳定状态。
useful memory if, in addition, any prescribed set of states can readily be made the stable states of the system.
处理设备将被称为神经元。每个神经元 i
The processing devices will be called neurons. Each neuron i
具有两个状态，类似于 McCulloch 和 Pitts（12）的模型：\(V_i = 0\)
has two states like those of McCulloch and Pitts (12): \(V_i = 0\)
本文的出版费用部分由版面费支付
The publication costs of this article were defrayed in part by page charge
(\"未激发\") 且 \(V_i = 1\) (\"以最大速率激发\")。当神经元
(\"not firing\") and \(V_i = 1\) (\"firing at maximum rate\"). When neuron
神经元 i 有一个从神经元 j 到它的连接，连接强度定义为 \(T_{ij}\)。（非连接神经元具有 \(T_{ij}\）
neuron i has a connection made to it from neuron j, the strength of connection is defined as \(T_{ij}\). (Nonconnected neurons have \(T_{ij}\)
The publication costs ofthis article were defrayed in part by page charge
V_i 的 N 个值，因此它由一个长度为 N 的二进制词表示。
the N values of V_i, so it is represented by a binary word of length N.
状态随时间根据以下算法变化。对于每个神经元 i，有一个固定的阈值 U_i。每个
The state changes in time according to the following algorithm. For each neuron i there is a fixed threshold U_i. Each
神经元 i 随机地在时间上调整其状态，但具有平均尝试率 W_i，设置
neuron i readjusts its state randomly in time but with a mean attempt rate W_i, setting
因此，每个神经元随机且异步地评估其是否高于或低于阈值，并相应地进行调整。（除非另有说明，我们选择 \(U_i = 0\)。）
Thus, each neuron randomly and asynchronously evaluates whether it is above or below threshold and readjusts accordingly. (Unless otherwise stated, we choose \(U_i = 0\).)
尽管该模型与感知器（Perceptron）(13,14)有表面上的相似之处，但本质差异导致了新的结果。首先，感知器主要建模为
Although this model has superficial similarities to the Perceptron (13, 14), the essential differences are responsible for the new results. First, Perceptrons were modeled chiefly with
神经连接呈“前向”方向 \(A \rightarrow B \rightarrow C \rightarrow D\)。
neural connections in a "forward" direction \(A \rightarrow B \rightarrow C \rightarrow D\).
对具有强后向耦合的网络的分析被证明是棘手的。我们所有有趣的结果都源于
The analysis of networks with strong backward coupling proved intractable. All our interesting results arise
Thus, each neuron randomly and asynchronously evaluates whether it is above or below threshold and readjusts accord-ingly. (Unless otherwise stated, we choose Ui = 0.)
作为强后向耦合的结果。其次，感知机
as consequences of the strong back-coupling. Second, Perceptron
研究通常使用随机神经元网络直接处理真实物理世界，并未提出对于发现更抽象涌现计算特性至关重要的
studies usually made a random net of neurons deal directly with a real physical world and did not ask the questions essential
问题。最后，感知机建模要求神经元像传统数字计算机一样同步运行。没有证据表明
to finding the more abstract emergent computational properties
存在这种全局同步性，而且考虑到神经信号的延迟
Finally, Perceptron modeling required synchronous neurons like a conventional digital computer. There is no evidence
，这种同步性不太可能实现。
for such global synchrony and, given the delays of nerve signal.
传播中，将无法使用全局同步
propagation, there would be no way to use global synchrony
有效。主要存在的计算特性
effectively. Chiefly computational properties which can exist
尽管异步性在生物学中具有有趣的启示。
in spite of asynchrony have interesting implications in biology.
假设我们希望存储状态集 \(V_8, s = 1 \ldots n\)。我们
Suppose we wish to store the set of states \(V_8, s = 1 \ldots n\). We
方程 3 中括号项的均值为 0，除非 \(s = s'\),
The mean value of the bracketed term in Eq. 3 is 0 unless \(s = s'\),
此时均值为 \(N/2\)。这种伪正交性
for which the mean is \(N/2\). This pseudoorthogonality
当 \(VW' = 1\) 时为正，当 \(V' = 0\) 时为负。除
and is positive if \(VW' = 1\) and negative if \(V' = 0\). Except for the
了来自 \(s = s'\) 项的噪声外，存储状态将始终
noise coming from the \(s = s'\) terms, the stored state would always
在我们的处理算法下保持稳定。
be stable under our processing algorithm.
这样的矩阵 T_{ij} 已被用于线性联想理论中，
Such matrices T_{ij} have been used in theories of linear asso-
联想网络[15-19]从成对输入刺激产生输出模式。
associative nets (15-19) to produce an output pattern from a paired
输入刺激 S1 → O1。第二个关联 S2 → O2 可以同时存储在同一网络中。
input stimulus, \(S_1 \rightarrow O_1\). A second association \(S_2 \rightarrow O_2\) can be
但是，混淆的
simultaneously stored in the same network. But the confusing
刺激 0.6 S1 + 0.4 S2 将产生通常无意义的结果。
stimulus \(0.6 S_1 + 0.4 S_2\) will produce a generally meaningless
混合输出 0.6 O1 + 0.4 O2。相反，我们的模型将利用
Mixed output 0.6 O1 + 0.4 O2. Our model, in contrast, will use
其强非线性来做出选择、生成类别，并
its strong nonlinearity to make choices, produce categories, and
重新生成信息，并且以高概率生成
regenerate information and, with high probability, will generate
输出 O1 来自这样一个混乱的混合刺激。
the output O1 from such a confusing mixed stimulus.
线性关联网络必须以复杂的方式连接
A linear associative net must be connected in a complex way
借助一个外部非线性逻辑处理器来产生真正的计算（20, 21）。复杂的电路易于设计，但更多
with an external nonlinear logic processor in order to yield true computation (20, 21). Complex circuitry is easy to plan but more
难以从进化角度讨论。相比之下，我们的模型从简单的
difficult to discuss in evolutionary terms. In contrast, our model obtains its emergent computational properties from simple
许多细胞而非电路的特性中获取其涌现计算特性。
properties of many cells rather than circuitry.
模型的生物学解释
The biological interpretation of the model
大多数神经元能够产生一连串的动作电位
Most neurons are capable of generating a train of action potentials
动作电位——电化学活动的传播脉冲——当
Action potentials - propagating pulses of electrochemical activity - when the
其膜上的平均电位被保持在远高于
average potential across their membrane is held well above its
正常静息值时。动作电位产生的平均速率
normal resting value. The mean rate at which action potentials
是平均膜电位的平滑函数，
are generated is a smooth function of the mean membrane potential,
其一般形式如图 1 所示。
having the general form shown in Fig. 1.
发送到其他神经元的生物信息通常位于
The biological information sent to other neurons often lies
在发放率的短时平均中（22）。当如此时，
in a short-time average of the firing rate (22). When this is so,
可以忽略单个动作电位的细节并
can neglect the details of individual action potentials and
将图 1 视为平滑的输入-输出关系。[并行
regard Fig. 1 as a smooth input-output relationship. [Parallel
携带相同信息的通路将增强
pathways carrying the same information would enhance the
系统提取短期平均发放率的能力
ability of the system to extract a short-term average firing rate
对涌现集体效应和自发计算的研究
A study of emergent collective effects and spontaneous computation
必须聚焦于输入-输出关系的非线性
must necessarily focus on the nonlinearity of the input-output relationship
计算的本质是非线性逻辑运算
The essence of computation is nonlinear logical operations
粒子相互作用产生
The particle interactions that produce
粒子动力学中的真实集体效应来自力的非线性依赖位置。而线性联想网络则强调了线性中心区域
true collective effects in particle dynamics come from a nonlinear
（图 1 中 14-19 区域），我们将用点划线阶跃替代输入-输出关系。那些操作主要为线性的神经元仅仅提供通信路径
dependence of forces on positions of the particles. Whereas linear associative networks have emphasized the linear central
（图 1 中 14-19 区域），我们将用点划线阶跃替代输入-输出关系。那些操作主要为线性的神经元仅仅提供通信路径
region (14-19) of Fig. 1, we will replace the input-output relationship
（图 1 中 14-19 区域），我们将用点划线阶跃替代输入-输出关系。那些操作主要为线性的神经元仅仅提供通信路径
by the dot-dash step. Those neurons whose operation
（图 1 中 14-19 区域），我们将用点划线阶跃替代输入-输出关系。那些操作主要为线性的神经元仅仅提供通信路径
is dominantly linear merely provide a pathway of communication
非线性神经元之间的相互作用。因此，我们考虑一个网络
interaction between nonlinear neurons. Thus, we consider a network
由“开或关”神经元组成，允许部分互连
of "on or off" neurons, granting that some of the interconnections
互连可以通过在线性范围内工作的神经元来实现。
Interconnections may be by way of neurons operating in the linear regime.
突触传递的延迟（部分随机特性）
Delays in synaptic transmission (of partially stochastic character
以及沿轴突和树突传递冲动的延迟，在神经元的输入与输出之间产生延迟
character) and in the transmission of impulses along axons and dendrites produce a delay between the input of a neuron and the
有效输出的生成。所有这些延迟都由单个参数——随机平均处理建模
generation of an effective output. All such delays have been modeled by a single parameter, the stochastic mean processing
特定神经元的输入来自电流泄漏
The input to a particular neuron arises from the current leaks
来自该神经元的突触，影响细胞平均（电位）
of the synapses to that neuron, which influence the cell mean
电位。突触由到达的动作电位激活。
potential. The synapses are activated by arriving action potentials
细胞 i 的输入信号可以认为是
The input signal to a cell i can be taken to be
其中 T_{ij} 表示突触的有效性。图 1 因此
where T_{ij} represents the effectiveness of a synapse. Fig. 1 thus
Little、Shaw 和 Roney（8, 25, 26）提出了基于“开/关”神经元和同步处理的神经网络集体功能的思想。然而，在他们的模型中，
Little, Shaw, and Roney (8, 25, 26) have developed ideas on the collective functioning of neural nets based on "on/off" neurons and synchronous processing. However, in their model the
动作电位尖峰的相对时序是核心，并且是回
relative timing of action potential spikes was central and re-
导致回响的动作电位序列。我们的模型及其
resulted in reverberating action potential trains. Our model and
模型仅在形式上有限相似，尽管可能在更深层次上存在联系。
theirs have limited formal similarity, although there may be connections at a deeper level.
大多数神经学习网络的建模基于
Most modeling of neural learning networks has been based
关于由 Hebb (27)和 Eccles (28)描述的一般类型的突触。
on synapses of a general type described by Hebb (27) and Eccles
（28）。关键要素是通过相关性修改 T_{ij}，
(28). The essential ingredient is the modification of T_{ij} by cor-
其中平均值是对过去历史进行的某种适当计算。
where the average is some appropriate calculation over past
还允许随时间衰减以及[V_i(t)]_avg 或[V_j(t)]_avg 的影响。
history. Decay in time and effects of [V_i(t)]_{avg} or [V_j(t)]_{avg} are also
具有这种突触的模型网络（16，20，21）可以
allowed. Model networks with such synapses (16, 20, 21) can
构建方程 2 的联想记忆\( T_{ij} \)。因此我们首先
construct the associative \( T_{ij} \) of Eq. 2. We will therefore initially
假设这样的\( T_{ij} \)是通过先前的经验（或遗传）产生的。赫布性质不必存在于
assume that such a \( T_{ij} \) has been produced by previous experience (or inheritance). The Hebbian property need not reside
单个突触中；产生这种净效应的小细胞群就足够了。我们描述的细胞网络执行一个抽象的计算。
in single synapses; small groups of cells which produce such a net effect would suffice. The network of cells we describe performs an abstract calculation.
construct the associative T., of Eq. 2. We will therefore initially
assume that such a Ty1 has been produced by previous experi-
计算及应用方面，输入应适当编码。
Calculation and, for applications, the inputs should be appropriately
在视觉处理中，例如特征提取应事先完成。当前建模可能与如何根据代表集合的输入记忆或分类一个实体或格式塔有关。
coded. In visual processing, for example, feature extraction should previously have been done. The present modeling might then be related to how an entity or Gestalt is remembered
或基于表示集合的输入进行分类。
or categorized on the basis of inputs representing a collection
对模型集体行为的研究表明，该模型具有稳定极限点。考虑特殊情况 T。
Studies of the collective behaviors of the model. The model has stable limit points. Consider the special case T
因此，改变 Vi 的算法使得 E 成为一个单调递减函数。
Thus, the algorithm for altering Vi causes E to be a monotonically
严格递减函数。状态变化将持续直到达到一个最小（局部）E。
strictly decreasing function. State changes will continue until a least (local) E is reached.
这种情况与 Ising 模型同构。\(T_{ij}\) 扮演交换耦合的角色，并且每个位点也有一个外部局域场。当 \(T_{ij}\) 是对称的
This case is isomorphic with an Ising model. \(T_{ij}\) provides the role of the exchange coupling, and there is also an external local field at each site. When \(T_{ij}\) is symmetric
但具有随机性（自旋玻璃），已知有
but has a random character (the spin glass) there are known to
cally decreasing function. State changes will continue until a
least (local) E is reached. This case is isomorphic with an Ising
在 \(N = 30\) 的系统上进行了蒙特卡洛计算。
Monte Carlo calculations were made on systems of \(N = 30\).
以及 \(N = 100\)，以检验去除 \(T_{ij} = T_{ji}\) 限制的效果。
and \(N = 100\), to examine the effect of removing the \(T_{ij} = T_{ji}\) restriction.
每个 \(T_{ij}\) 元素被选为 -1 到 1 之间的随机数。
Each element of \(T_{ij}\) was chosen as a random number between -1 and 1.
典型皮层区域（30, 31）以及无脊椎动物简单神经节（32）的神经结构
The neural architecture of typical cortical regions (30, 31) and also of simple ganglia of invertebrates (32)
表明具有强烈相互抑制的 \(100–10,000\) 个细胞的重要性。
suggests the importance of \(100–10,000\) cells with intense mutual inhibition.
基本处理中的实际互连，因此我们的规模
actual interconnections in elementary processing, so our scale of
动力学算法从随机选择的初始配置启动
The dynamics algorithm was initiated from randomly chosen
初始起始配置。对于 N=30，系统从未
initial starting configurations. For N = 30 the system never
表现出遍历性游走遍状态空间。在大约
displayed an ergodic wandering through state space. Within a
4/W 的时间后，它稳定到极限行为，这些共
time of about 4/W it settled into limiting behaviors, the com-
最常见的情况是一个稳定状态。当对 50 次试验进行检验时
Most common being a stable state. When 50 trials were examined for
某个特定的此类随机矩阵，所有试验都会导致两个
A particular such random matrix, all would result in one of two
或三个终止状态。少数稳定状态从而收集了来自
Or three end states. A few stable states thus collect the flow from
大部分初始状态空间的流。一个简单循环也发生了
Most of the initial state space. A simple cycle also occurred
偶尔 - 例如，A → B → A → B。
occasionally - for example, \(A \rightarrow B \rightarrow A \rightarrow B\).
观察到的第三种行为是在状态空间的小区域内的混沌游走。两个二进制状态 A 和 B 之间的汉明距离定义为
The third behavior seen was chaotic wandering in a small region of state space. The Hamming distance between two binary
状态 A 和 B 定义为对应位上数字不同的位置数量，其中
states A and B is defined as the number of places in which
数字不同。混沌游走发生在
the digits are different. The chaotic wandering occurred within
离某个特定状态的一个短汉明距离内。对状态出现的概率π进行了统计
a short Hamming distance of one particular state. Statistics were
在时间上对状态出现的概率π进行了统计（原文可能不完整）
done on the probability pi of the occurrence of a state in a time
围绕这个最小值游荡，以及一个熵度量
of wandering around this minimum, and an entropic measure
对于 N=30，M=25 的值被发现。相空间中的流
A value of M = 25 was found for N = 30. The flow in phase space
由该模型算法产生的具有必要的性质
produced by this model algorithm has the properties necessary
对于物理的内容可寻址存储器，无论 T 是否
for a physical content-addressable memory whether or not T
N=100 的模拟要慢得多，并且不是定量的
Simulations with N = 100 were much slower and not quantitative
定量地进行了研究。他们展示了与 N=的定性相似性。
Quantitatively pursued. They showed qualitative similarity to N =
30. 为什么当\(T_{ij}\)时稳定的极限点或区域会持续存在？
30. Why should stable limit points or regions persist when \(T_{ij}\)?
拆分为两项，其中一项始终为负。第二项
split into two terms, one of which is always negative. The second
在 Tij 对称时相同，若 Tij 和 Tji 为随机选择，则其是均值为零的随机项
is identical if Tij is symmetric and is stochastic with mean 0
如果随机选择 Tij 和 Tji。对于 Tij ≠ Tji 的算法，
if Tij and Tji are randomly chosen. The algorithm for Tij ≠ Tji,
因此，E 的变化方式类似于对称 Tij 情况下 E 随时间变化的方式，但算法
therefore changes E in a fashion similar to the way E would
对应于不对称情况的变化。
change in time for a symmetric Tij but with an algorithm corresponding to the asymmetry.
大约 0.15 N 个状态可以同时被记住，在
About 0.15 N states can be simultaneously remembered before
回忆错误严重之前。根据方程 2 对记忆存储的计算机建模针对 N=30 和 N=100 进行。
Before error in recall is severe. Computer modeling of memory storage according to Eq. 2 was carried out for N = 30 and N = 100.
随机选择了 n 个记忆状态，并计算了相应的
n random memory states were chosen and the corresponding
相应的 \(T_{ij}\) 被生成。如果神经系统预处理了
corresponding \(T_{ij}\) was generated. If a nervous system preprocessed
信号以便有效存储，预处理的信息
signals for efficient storage, the preprocessed information
会显得随机（例如，DNA 的编码序列具有
would appear random (e.g., the coding sequences of DNA have
随机性）。因此，随机记忆向量模拟
a random character). The random memory vectors thus simulate
高效编码的真实信息，并代表我们的
efficiently encoded real information, as well as representing our
无知。系统在每一指定的名义
ignorance. The system was started at each assigned nominal
记忆状态下启动，并让系统演化至稳定。
memory state, and the state was allowed to evolve until stationary.
典型结果如图 2 所示。统计量是平均值
Typical results are shown in Fig. 2. The statistics are averages
涵盖给定矩阵中的状态以及不同矩阵。
over both the states in a given matrix and different matrices.
当 n=5 时，分配的记忆状态几乎总是稳定的
With n = 5, the assigned memory states are almost always stable
（且可精确回忆）。当 n=15 时，大约一半的名义上
(and exactly recallable). For n = 15, about half of the nominally
被记忆的状态演变为稳定状态，误差少于 5 个。
remembered states evolved to stable states with less than 5 errors.
错误，但其余部分演化到了与起始状态截然不同的状态。
Errors, but the rest evolved to states quite different from the start.
这些结果可以从对效应的分析来理解。
These results can be understood from an analysis of the effect
噪声项。在公式 (3) 中，H' 是神经元上的“有效场”
of the noise terms. In Eq. (3), H' is the "effective field" on neuron
当系统状态为 s'（标称记忆状态之一）时，
i when the state of the system is s', one of the nominal memory states.
状态的期望值，公式 (4)，为 ±N/2，如
states. The expectation value of this sum, Eq. (4), is \pm N/2 as
适当。式(2)中 s·s'求和没有贡献
appropriate. The s · s' summation in Eq. 2 contributes no
平均值，但均方根噪声为\sqrt{(n-1)N/2}\。对于 nN 很大的情况，
mean, but has a rms noise of \sqrt{(n-1)N/2}\. For nN large,
该噪声近似服从高斯分布，且某个特定记忆的单个特定位出错的概率
this noise is approximately Gaussian and the probability of an
某个特定记忆的单个特定位的错误将是
error in a single particular bit of a particular memory will be
对于 n=10, N=100, P=0.0091 的情况，该概率
For the case n = 10, N = 100, P = 0.0091, the probability that
在其 100 位中没有错误的记忆的概率应约为 exp(−0.40)。
had no errors in its 100 bits should be about exp(−0.40).
在图 2 的模拟中，实验值为 0.6。
In the simulation of Fig. 2, the experimental number was 0.6.
在固定 P 下 n 随 N 的理论缩放通过在 N=30 到 N=100 之间的模拟得到了证明。
The theoretical scaling of n with N at fixed P was demonstrated in the simulations going between N = 30 and N = 100.
一半记忆被良好保留的实验结果
The experimental results of half the memories being well retained
在 n=0.15N 时保留良好而其余保留较差，预计会
at n = 0.15 N and the rest badly retained is expected to
错误出现的概率分布在
probability distribution of the occurrence of errors in
对所有大的 N 成立。给定级别上的信息存储
be true for all large N. The information storage at a given level
准确度可以通过一个明智的选择提高 2 倍
of accuracy can be increased by a factor of 2 by a judicious choice
单个神经元阈值的。这一选择等价于
of individual neuron thresholds. This choice is equivalent to
使用变量 V_i = ±1，T_{ij} = \frac{1}{2} \sum_{k} J_{ik} J_{kj}，以及一个阈值水平
using variables V_i = ±1, T_{ij} = \frac{1}{2} \sum_{k} J_{ik} J_{kj}, and a threshold level
给定某个任意起始状态，最终状态是什么？
Given some arbitrary starting state, what is the resulting final state?
（或统计意义上的状态）？为了研究这一点，我们从随机
(or statistically, states)? To study this, evolutions from random
选择的初始状态被制表，其中 N = 30，n =
randomly chosen initial states were tabulated for N = 30 and n
5。从算法的（非本质）对称性出发，如果
5. From the (inessential) symmetry of the algorithm, if
(101110) 是一个指定的稳定状态，(010001) 也是稳定的。
(101110) is an assigned stable state, (010001) is also stable.
因此，这些矩阵共有 10 个名义稳定状态。大约
Therefore, the matrices had 10 nominal stable states. Approximately
85%的试验以指定记忆结束，而 10%
85% of the trials ended in assigned memories, and 10%
以无明显含义的稳定状态结束。另有模糊的 5%
Ended in stable states of no obvious meaning. An ambiguous 5%
落在了非常接近指定记忆的稳定状态上。这些 10 个状态被找到的可能性范围有 20 倍差异。
Landed in stable states very near assigned memories. There was a range of a factor of 20 of the likelihood of finding these 10
该算法导致记忆接近初始状态。对于
The algorithm leads to memories near the starting state. For
N=30，n=5，通过随机修改已知记忆生成了部分随机的初始状态。概率
N = 30, n = 5, partially random starting states were generated by random modification of known memories. The probability
最终状态是最接近初始状态的那个被研究
that the final state was that closest to the initial state was studied
作为初始状态与（某个状态）之间距离的函数
as a function of the distance between the initial state and the
达到了超过 90%的时间。超出那个距离后，
reached more than 90% of the time. Beyond that distance, the
概率平稳下降，降至 0.2 的水平（2 倍
probability fell off smoothly, dropping to a level of 0.2 (2 times
这些是名义上分配的记忆状态，每个都主导着
which are the nominally assigned memories, each of which dominates
周围的一大片区域。流动并非完全
a substantial region around it. The flow is not entirely
确定性的，系统对模糊的起始状态作出响应
deterministic, and the system responds to an ambiguous starting
通过统计选择在记忆状态之间进行选择
state by a statistical choice between the memory states it
如果希望将这样的系统用于基于硅的内容可寻址存储器，则应使用该算法并进行修改，以保持已知信息位，同时让其他位自由变化。
Were it desired to use such a system in an Si-based content-addressable memory, the algorithm should be used and modified to hold the known bits of information while letting the others vary.
通过使用“截断”的 T_{ij}（将 Eq. 3 中的 T_{ij}替换为±1，即 T_{ij}的代数符号）对该模型进行了研究。其目的是
The model was studied by using a "clipped" T_{ij}, replacing T_{ij} in Eq. 3 by ±1, the algebraic sign of T_{ij}. The purposes were to
检验线性突触假设的必要性（通过构建
examine the necessity of a linear synapse supposition (by making
一个高度非线性的突触）并检验存储效率。
a highly nonlinear one) and to examine the efficiency of storage.
Were it desired to use such a system in an Si-based content-
只能在这个对称矩阵中存储 \(N(N/2)\) 比特的信息。
Only \(N(N/2)\) bits of information can possibly be stored in this
对称矩阵。实验表明，对于 \(N=100, n=9\)，错误水平
symmetric matrix. Experimentally, for \(N = 100, n = 9\), the level
与 \(n=\) 时的普通算法相似。
of errors was similar to that for the ordinary algorithm at \(n =\)
\(12\). 信噪比可以解析地评估为
\(12\). The signal-to-noise ratio can be evaluated analytically for
这种裁剪算法，并且与普通算法相比缩小了因子 \( (2/\pi)^{1/2} \)。
this clipped algorithm and is reduced by a factor of \( (2/\pi)^{1/2} \) com-
与未裁剪的情况相比。对于固定的错误概率，
Compared with the unclipped case. For a fixed error probability, the
记忆的数量必须减少 \(2/\pi\)。
number of memories must be reduced by \(2/\pi\).
使用第 4 种算法和裁剪后的 T_{ij}，分析和
With the 4th algorithm and the clipped T_{ij}, both analysis and
建模表明，对于 N 存储的最大信息
modeling showed that the maximal information stored for N
N = 100 出现在大约 n = 13 处。存在一些错误，并且
= 100 occurred at about n = 13. Some errors were present, and
存储的香农信息约对应 N(N/
the Shannon information stored corresponded to about N(N/
新记忆可以连续添加到 T_i 中。添加
New memories can be continually added to T_i. The addition
超出容量的新记忆会使系统过载并
of new memories beyond the capacity overloads the system and
使所有记忆状态不可检索，除非有某种机制
makes all memory states irretrievable unless there is a provision
用于遗忘旧记忆（16, 27, 28）。
for forgetting old memories (16, 27, 28).
Tij 可能尺寸的饱和本身就会导致遗忘。
The saturation of the possible size of Tij will itself cause forgetting.
设 Tij 的可能值为 0, ±1, ±2, ±3, 等等。
Let the possible values of Tij be 0, ±1, ±2, ±3, and so on.
+1 的增量将被忽略，而下一个 -1 的增量会将 T_{ij} 减至 2。
increment of +1 would be ignored and a next increment of -1 would reduce T_{ij} to 2.
-1 会将 T_{ij} 减至 2。当 T_y 如此构造时，只有
-1 would reduce T_{ij} to 2. When T_y is so constructed, only the
最近的记忆状态得以保留，噪声水平略有增加。
recent memory states are retained, with a slightly increased noise level.
噪声水平。来自遥远过去的记忆不再稳定。
Noise level. Memories from the distant past are no longer stable.
过去状态被记住的时效取决于
How far into the past states are remembered depends on the
数字化 T 的深度，而 0、±3 是一个合适的层次
digitizing depth of T, and 0, ±3 is an appropriate level for
N = 100。其他方案可用于防止太多 mem-
N = 100. Other schemes can be used to keep too many mem-
ories 同时写入，但这一特定方案
ories from being simultaneously written, but this particular one
因其不需要精细平衡且是一种
is attractive because it requires no delicate balances and is a
真实神经元无需同时形成 i -- j 和 j 的突触
Real neurons need not make synapses both of i -- j and j
i. 特定突触被限制为单一输出符号。我们
i. Particular synapses are restricted to one sign of output. We
因此探究了 T_{ij} = T_{ji} 是否重要。进行了
therefore asked whether T_{ij} = T_{ji} is important. Simulations were
仅保留一个 ij 连接的仿真：若 T_{ij}=0，则 T_{ji}=0。
carried out with only one ij connection: if T_{ij}=0, T_{ji}=0. The
出错概率增加，但该算法仍
probability of making errors increased, but the algorithm con-
继续生成稳定的极小值。一个高斯噪声描述
tinued to generate stable minima. A Gaussian noise description
对错误率的分析表明，给定条件下的信噪比
Analysis of the error rate shows that the signal-to-noise ratio for a given
n 和 N 应减少因子\(1/F^2\)，并且模拟
n and N should be decreased by the factor \(1/F^2\), and the simulations
结果与这一因子一致。同样的分析
were consistent with such a factor. This same analysis
表明系统通常以“软故障”的方式失败，伴随着
shows that the system generally fails in a "soft" fashion, with
信噪比和错误率随着更多（数据）的加入而缓慢增加
signal-to-noise ratio and error rate increasing slowly as more
记忆之间过于接近会导致混淆并倾向于合并。对于 N=100，一对随机记忆应相隔 50±5 汉明单位。研究了 N=100、n=8 的情况，其中七个随机记忆与第八个记忆（与七个记忆之一）的汉明距离仅为 30、20 或 10。在距离为 30 时，两个相似记忆
Memories too close to each other are confused and tend to merge. For N = 100, a pair of random memories should be separated by 50 ± 5 Hamming units. The case N = 100, n = 8, was studied with seven random memories and the eighth made up a Hamming distance of only 30, 20, or 10 from one of the other seven memories. At a distance of 30, both similar memories
Memories too close to each other are confused and tend to
merge. For N = 100, a pair ofrandom memories should be sep-
arated by 50 ± 5 Hamming units. The case N = 100, n = 8,
was studied with seven random memories and the eighth made
记忆状态通常是稳定的。在距离 20 处，最小值是
Memories were usually stable. At a distance of 20, the minima were
通常明显但发生了偏移。在距离 10 处，最小值
usually distinct but displaced. At a distance of 10, the minima
该算法根据相似性对初始状态进行分类，
The algorithm categorizes initial states according to the similarity,
到记忆状态。当阈值为 0 时，系统变成
to memory states. With a threshold of 0, the system becomes
状态 00000...始终稳定。对于阈值为 0 的情况，这
The state 00000 ... is always stable. For a threshold of 0, this
稳定状态的能量远高于存储的记忆
stable state is much higher in energy than the stored memory
状态且很少发生。在算法中添加一个统一阈值
states and very seldom occurs. Adding a uniform threshold in
相当于相比于 0000 状态提高了存储记忆的有效能量，并且 0000 也成为一个可能的稳定状态。然后 0000 状态被生成
the algorithm is equivalent to raising the effective energy of the stored memories compared to the 0000 state, and 0000 also becomes a likely stable state. The 0000 state is then generated
由任何与指定记忆不够相似的初始状态产生，并代表正识别
by any initial state that does not resemble adequately closely one of the assigned memories and represents positive recognition
stable state is much higher in energy than the stored memory
定义起始状态并不熟悉。
definition that the starting state is not familiar.
当记忆严重过载时，可以通过其他方式识别熟悉度。我们研究了情形 \(N\
Familiarity can be recognized by other means when the memory is drastically overloaded. We examined the case \(N\
\(N = 100, n = 500\)，其中记忆过载因子为
\(N = 100, n = 500\), in which there is a memory overload of a factor
25。所有分配的记忆状态都不稳定。初
of 25. None of the memory states assigned were stable. The ini-
始状态的初始处理速率定义为在时间 \(1/2W\) 内发生的神经元状态调整次数。Fa-
Initial rate of processing of a starting state is defined as the number
of neuron state readjustments that occur in a time \(1/2W\). Fa-
在这种过载水平下，熟悉和不熟悉状态大多数时候可根据初始处理速率区分，不熟悉状态的处理速率更快。这种熟悉度只能通过一类抽象处理组平均属性的神经元或设备从系统中读取。
Familiar and unfamiliar states were distinguishable most of the time at this level of overload on the basis of the initial processing rate, which was faster for unfamiliar states. This kind of familiarity can only be read out of the system by a class of neurons or devices abstracting average properties of the processing group.
对于迄今考虑的情况，当 \(i \neq j\) 时，\(T_{ij}\) 的期望值为 0。一组记忆可以通过平均相关性存储，且 \(T_{ij} = C_{ij} \neq 0\)，因为记忆中存在一致的内部相关性。
For the cases so far considered, the expectation value of \(T_{ij}\) was 0 for \(i \neq j\). A set of memories can be stored with average correlations, and \(T_{ij} = C_{ij} \neq 0\) because there is a consistent internal correlation in the memories.
miliar and unfamiliar states were distinguishable most of the
time at this level ofoverload on the basis ofthe initial processing rate, which was faster for unfamiliar states. This kind of famil-iarity can only be read out of the system by a class of neurons
只使用 k 个神经元而非 N 个，尝试重构
Using only k of the neurons rather than N, an attempt to reconstruct
它将在所有 N 个神经元上生成一个稳定点。得到的\(X_{k+1}, \dots, X_N\)的值将主要由……决定
it will generate a stable point for all N neurons. The values of \(X_{k+1}, \dots, X_N\) that result will be determined primarily
并且 X 根据其他记忆的平均相关性完成。
and X is completed according to the mean correlations of the other memories.
这种容量的最有效实现方式是存储大量弱存储的相关矩阵。
The most effective implementation of this capacity stores a large number of correlated matrices weakly stored.
using only k of the neurons rather than N, an attempt to re-
一个非对称的 T 可能导致一个最小值
A nonsymmetric T can lead to the possibility that a minimum
将只是亚稳态并且会随时间被另一个
will be only metastable and will be replaced in time by another
最小值所取代。额外的非对称项，这些项可以通过
minimum. Additional nonsymmetric terms which could be eas
容易地由 Hebb 突触的微小修改生成。
ily generated by a minor modification of Hebb synapses
被添加到 T 中。当 A 被明智地调整时，系统
were added to T. When A was judiciously adjusted, the system
会在\(V_s\)附近停留一段时间，然后离开并前往一个点
would spend a while near \(V_s\) and then leave and go to a point
靠近\(V_{s+1}\)。但长度超过四个状态的序列被证明是不
near \(V_{s+1}\). But sequences longer than four states proved im-
可能的，甚至这些序列也不能忠实再现
possible to generate, and even these were not faithfully
在模型网络中，每个“神经元”都具有基本属性，
In the model network each "neuron" has elementary properties,
并且网络几乎没有结构。尽管如此，集体性的
and the network has little structure. Nonetheless, collective
计算性质自发涌现。记忆是
computational properties spontaneously arose. Memories are
作为稳定的实体或格式塔被保留，并且可以从任何合理大小的子部分正确回忆。歧义在统计基础上得到解决。具有一定的泛化能力
retained as stable entities or Gestalts and can be correctly recalled from any reasonably sized subpart. Ambiguities are resolved on a statistical basis. Some capacity for generalization is
存在，并且记忆的时间顺序也可以被编码。这些特性源于处理算法产生的相空间中流的性质，该流并不
present, and time ordering of memories can also be encoded. These properties follow from the nature of the flow in phase space produced by the processing algorithm, which does not
似乎强烈依赖于模型的精确细节
appear to be strongly dependent on precise details of the model
这种鲁棒性表明，即使添加更多神经生物学细节，类似效果也会出现。
This robustness suggests that similar effects will obtain even when more neurobiological details are added.
高等动物大脑区域的结构在很大程度上必须由大量具有明确功能的简单局部电路组成。简单电路与高等动物的复杂计算特性之间的桥梁
Much of the architecture of regions of the brains of higher animals must be made from a proliferation of simple local circuits with well-defined functions. The bridge between simple circuits and the complex computational properties of higher
eling. This robustness suggests that similar effects will obtain
even when more neurobiological details are added.
Much of the architecture of regions of the brains of higher
神经系统可能是新的自发性涌现
Nervous systems may be the spontaneous emergence of new
计算能力来自于大规模集体行为
computational capabilities from the collective behavior of large
通过使用集成电路实现类似模型
Implementation of a similar model by using integrated circuits
将导致芯片对元件
would lead to chips which are much less sensitive to element
故障和软故障的敏感度低于正常电路。这种芯片
failure and soft-failure than are normal circuits. Such chips
会浪费门电路，但可以做得大许多倍。
would be wasteful of gates but could be made many times larger
在给定良率下比标准设计大。它们的异步并行
than standard designs at a given yield. Their asynchronous parallel
处理能力将提供快速解决方案给某些
processing capability would provide rapid solutions to some
特殊类别的计算问题。
special classes of computational problems.
加州理工学院的部分工作得到了资助。
The work at California Institute of Technology was supported in part
由国家科学基金会拨款 DMR-8107494 资助。这是贡献
by National Science Foundation Grant DMR-8107494. This is contribution
来自化学与化学部的第 6580 号
no. 6580 from the Division of Chemistry and Chemical
Willows, A. O. D., Dorsett, D. A. 和 Hoyle, G. (1973) J. Neuro
Willows, A. O. D., Dorsett, D. A. & Hoyle, G. (1973) J. Neuro
(1980) 收录于《信息处理在神经》
(1980) in Information Processing in the Nervous
编：Pinsker, H. M. 和 Willis, W. D. (Raven, 纽约),
eds. Pinsker, H. M. & Willis, W. D. (Raven, New York),
Knight, B. W. (1975) 生命科学数学讲座 5, 111-144.
Knight, B. W. (1975) Lect. Math. Life Sci. 5, 111-144.
Smith, D. R. & Davidson, C. H. (1962) 计算机协会杂志
Smith, D. R. & Davidson, C. H. (1962) J. Assoc. Comput. Mach.
(斯坦福大学出版社, 加利福尼亚州斯坦福), 第 23-24 页.
(Stanford Univ. Press, Stanford, CA), pp. 23-24.
Kohonen, T. (1980) Content Addressable Memories (Springer,
Kohonen, T. (1980) Content Addressable Memories (Springer,
McCulloch, W. S. 和 Pitts, W. (1943) Bull. Math. Biophys. 5,
McCulloch, W. S. & Pitts, W. (1943) Bull. Math. Biophys. 5,
Minsky, M. & Papert, S. (1969) 感知器：导论
Minsky, M. & Papert, S. (1969) Perceptrons: An Introduction to
计算几何 (MIT 出版社, 剑桥, MA).
Computational Geometry (MIT Press, Cambridge, MA).
Rosenblatt, F. (1962) 感知器原理 (Spartan 出版社, 华盛-
Rosenblatt, F. (1962) Principles of Perceptrons (Spartan, Wash-
Cooper, L. N. (1973) 在诺贝尔研讨会论文集中
Cooper, L. N. (1973) in Proceedings of the Nobel Symposium on
物理系统的集体性质, 编辑 Lundqvist, B. &
Collective Properties of Physical Systems, eds. Lundqvist, B. &
Lundqvist, S. (学术出版社，纽约)，第 252-264 页。
Lundqvist, S. (Academic, New York), 252-264.
Cooper, L. N., Liberman, F. & Oja, E. (1979) 《生物控制论》第 33 期，
Cooper, L. N., Liberman, F. & Oja, E. (1979) Biol Cybern. 33,
J. C. (1968) 《伦敦皇家学会会刊 B 辑》第 171 期，
J. C. (1968) Proc. Roy. Soc. London Ser. B 171,
J. C. (1968) 《自然》(伦敦)第 217 卷，第 104-105 页。
J. C. (1968) Nature (London) 217, 104-105.
Kohonen, T. (1977) 《联想记忆——一种系统理论》
Kohonen, T. (1977) Associative Memory-A System-Theoretic
Willwacher, G. (1976) 生物控制论, 24, 181-198.
Willwacher, G. (1976) Biol Cybern. 24, 181-198.
Anderson, J. A. (1977) 心理学评论, 84, 413-451.
Anderson, J. A. (1977) Psych. Rev. 84, 413-451.
Perkel, D. H. & Bullock, T. H. (1969) 神经科学研究研讨会.
Perkel, D. H. & Bullock, T. H. (1969) Neurosci. Res. Symp.
J., Scheibel, A. B. & Shaw, G. L. (1979) 大脑研究.
J., Scheibel, A. B. & Shaw, G. L. (1979) Brain Res.
Little, W. A. & Shaw, G. L. (1978) 数学生物科学, 39, 281-289.
Little, W. A. & Shaw, G. L. (1978) Math. Biosci. 39, 281-289.
Shaw, G. L. & Roney, K. J. (1979) Phys. Rev. Lett. 74, 146-150.
Shaw, G. L. & Roney, K. J. (1979) Phys. Rev. Lett. 74, 146-150.
Hebb, D. O. (1949) 《行为的组织》(Wiley, 纽约).
Hebb, D. O. (1949) The Organization of Behavior (Wiley, New York).
Eccles, J. C. (1953) 《心灵的神经生理学基础》(Clarendon Press, 牛津).
Eccles, J. C. (1953) The Neurophysiological Basis of Mind (Clarendon Press, Oxford).
Kirkpatrick, S. & Sherrington, D. (1978) Phys. Rev. B 17, 4384-4403.
Kirkpatrick, S. & Sherrington, D. (1978) Phys. Rev. B 17, 4384-4403.
Mountcastle, V. B. (1978) 载于《有意识的大脑》, 编 Edelman, G. M. & Mountcastle, V. B. (MIT Press, 剑桥, 马萨诸塞州).
Mountcastle, V. B. (1978) in The Mindful Brain, eds. Edelman, G. M. & Mountcastle, V. B. (MIT Press, Cambridge, MA).
Mountcastle, V. B. (MIT 出版社，马萨诸塞州剑桥)，页码。
Mountcastle, V. B. (MIT Press, Cambridge, MA), pp.
Goldman, P. S. & Nauta, W. J. H. (1977) Brain Res. 122,
Goldman, P. S. & Nauta, W. J. H. (1977) Brain Res. 122,
Kandel, E. R. (1979) Sci. Am. 241, 61-70.
Kandel, E. R. (1979) Sci. Am. 241, 61-70.