A Logical Calculus of the Ideas Immanent in Nervous Activity

Source of this text: the 1990 reprint in Bulletin of Mathematical Biology, Vol. 52, No. 1/2, pp. 99–115 (Pergamon Press plc, Society for Mathematical Biology). The paper was first published in Bulletin of Mathematical Biophysics, Vol. 5, pp. 115–133 (1943). Symbols and apparent misprints are reproduced as printed in the 1990 reprint.

本对照所据原文：1990 年《Bulletin of Mathematical Biology》第 52 卷第 1/2 期第 99–115 页的重印版（Pergamon Press plc，数学生物学学会）。论文最初发表于 1943 年《Bulletin of Mathematical Biophysics》第 5 卷第 115–133 页。记号以及疑似排印错误一律按 1990 年重印版原样保留。

A Logical Calculus of the Ideas Immanent in Nervous Activity*

神经活动中内在思想的逻辑演算*

Warren S. McCulloch and Walter Pitts

University of Illinois, College of Medicine, Department of Psychiatry at the Illinois Neuropsychiatric Institute, University of Chicago, Chicago, U.S.A.

* Reprinted from the Bulletin of Mathematical Biophysics, Vol. 5, pp. 115–133 (1943).

沃伦·S·麦卡洛克（Warren S. McCulloch）、沃尔特·皮茨（Walter Pitts）

伊利诺伊大学医学院伊利诺伊神经精神病学研究所精神病学系；芝加哥大学，美国芝加哥

* 重印自《Bulletin of Mathematical Biophysics》第 5 卷第 115–133 页（1943）。

Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.

由于神经活动具有“全或无”的特性，神经事件以及它们之间的关系可以用命题逻辑来处理。我们发现，每一个网络的行为都可以用这些术语来描述，对含有环路的网络则需要再加上更复杂的逻辑手段；并且对于任何满足一定条件的逻辑表达式，都能找到一个按其所描述的方式运作的网络。我们证明，在若干可能的神经生理学假设之间作出的许多特定选择是等价的，其含义是：对于在一种假设下运作的每个网络，都存在另一个在另一种假设下运作、并给出相同结果的网络，尽管所用时间也许不同。文中还讨论了这一演算的多种应用。

1. Introduction

1. 引言

Theoretical neurophysiology rests on certain cardinal assumptions. The nervous system is a net of neurons, each having a soma and an axon. Their adjunctions, or synapses, are always between the axon of one neuron and the soma of another. At any instant a neuron has some threshold, which excitation must exceed to initiate an impulse. This, except for the fact and the time of its occurence, is determined by the neuron, not by the excitation. From the point of excitation the impulse is propagated to all parts of the neuron. The velocity along the axon varies directly with its diameter, from $<1\ \mathrm{ms^{-1}}$ in thin axons, which are usually short, to $>150\ \mathrm{ms^{-1}}$ in thick axons, which are usually long. The time for axonal conduction is consequently of little importance in determining the time of arrival of impulses at points unequally remote from the same source. Excitation across synapses occurs predominantly from axonal terminations to somata. It is still a moot point whether this depends upon irreciprocity of individual synapses or merely upon prevalent anatomical configurations. To suppose the latter requires no hypothesis ad hoc and explains known exceptions, but any assumption as to cause is compatible with the calculus to come. No case is known in which excitation through a single synapse has elicited a nervous impulse in any neuron, whereas any neuron may be excited by impulses arriving at a sufficient number of neighboring synapses within the period of latent addition, which lasts $<0.25$ ms. Observed temporal summation of impulses at greater intervals is impossible for single neurons and empirically depends upon structural properties of the net. Between the arrival of impulses upon a neuron and its own propagated impulse there is a synaptic delay of $>0.5$ ms. During the first part of the nervous impulse the neuron is absolutely refractory to any stimulation. Thereafter its excitability returns rapidly, in some cases reaching a value above normal from which it sinks again to a subnormal value, whence it returns slowly to normal. Frequent activity augments this subnormality. Such specificity as is possessed by nervous impulses depends solely upon their time and place and not on any other specificity of nervous energies. Of late only inhibition has been seriously adduced to contravene this thesis. Inhibition is the termination or prevention of the activity of one group of neurons by concurrent or antecedent activity of a second group. Until recently this could be explained on the supposition that previous activity of neurons of the second group might so raise the thresholds of internuncial neurons that they could no longer be excited by neurons of the first group, whereas the impulses of the first group must sum with the impulses of these internuncials to excite the now inhibited neurons. Today, some inhibitions have been shown to consume $<1$ ms. This excludes internuncials and requires synapses through which impulses inhibit that neuron which is being stimulated by impulses through other synapses. As yet experiment has not shown whether the refractoriness is relative or absolute. We will assume the latter and demonstrate that the difference is immaterial to our argument. Either variety of refractoriness can be accounted for in either of two ways. The “inhibitory synapse” may be of such a kind as to produce a substance which raises the threshold of the neuron, or it may be so placed that the local disturbance produced by its excitation opposes the alteration induced by the otherwise excitatory synapses. Inasmuch as position is already known to have such effects in the cases of electrical stimulation, the first hypothesis is to be excluded unless and until it be subtantiated, for the second involves no new hypothesis. We have, then, two explanations of inhibition based on the same general premises, differing only in the assumed nervous nets and, consequently, in the time required for inhibition. Hereafter we shall refer to such nervous nets as equivalent in the extended sense. Since we are concerned with properties of nets which are invarient under equivalence, we may make the physical assumptions which are most convenient for the calculus.

理论神经生理学建立在若干基本假设之上。神经系统是由神经元构成的网络，每个神经元都有一个胞体和一条轴突。它们的连接处，即突触，总是位于一个神经元的轴突与另一个神经元的胞体之间。在任一时刻，神经元都有某个阈值，兴奋必须超过它才能引发冲动。除了冲动是否发生及其发生时间之外，冲动由神经元本身决定，而不由兴奋决定。冲动从兴奋点传播到神经元的所有部分。沿轴突的传导速度与轴突直径成正比，从细轴突（通常较短）中的 $<1\ \mathrm{ms^{-1}}$ 到粗轴突（通常较长）中的 $>150\ \mathrm{ms^{-1}}$ 不等。因此，在决定冲动到达与同一来源距离不等的各点的时间时，轴突传导所需的时间无关紧要。跨突触的兴奋主要是从轴突末梢传向胞体。这究竟取决于单个突触的不可逆性，还是仅仅取决于普遍的解剖构型，目前仍有争议。假设是后者不需要任何特设假说，并能解释已知的例外；不过，关于原因的任何假设都与下文的演算相容。目前尚无已知案例表明，经由单个突触的兴奋曾在任何神经元中引发神经冲动；而任何神经元都可能被在潜伏总和期（持续 $<0.25$ 毫秒）内到达足够数量相邻突触的冲动所兴奋。观察到的、间隔更长的冲动的时间总和，对单个神经元来说是不可能的，在经验上取决于网络的结构特性。从冲动到达一个神经元到该神经元自身发出传播性冲动之间，存在 $>0.5$ 毫秒的突触延迟。在神经冲动的最初阶段，神经元对任何刺激都处于绝对不应期。此后其兴奋性迅速恢复，在某些情况下会升到高于正常的水平，随后又降到低于正常的水平，再从那里缓慢恢复正常。频繁的活动会加重这种低于正常的状态。神经冲动所具有的特异性仅取决于其时间和位置，而不取决于神经能量的任何其他特异性。近来，只有抑制被认真地援引来反驳这一论点。抑制是指一组神经元的活动因第二组神经元同时或先前的活动而终止或被阻止。直到不久前，这还可以这样解释：第二组神经元先前的活动可能把中间神经元的阈值提高到不再能被第一组神经元兴奋的程度，而第一组的冲动必须与这些中间神经元的冲动相加，才能兴奋此时受到抑制的神经元。如今已经证明，某些抑制所耗时间 $<1$ 毫秒。这就排除了中间神经元的可能，而要求存在这样的突触：冲动经由它们去抑制正被经由其他突触的冲动所刺激的那个神经元。实验尚未表明不应性是相对的还是绝对的。我们将假设是后者，并证明这一区别对我们的论证无关紧要。两种不应性中的任何一种都可以用两种方式中的任一种来解释。“抑制性突触”可能属于这样一类：它产生某种提高神经元阈值的物质；也可能其位置使得其兴奋所产生的局部扰动与其他兴奋性突触所引起的改变相抵。由于在电刺激的情形中已知位置具有这种效应，在第一种假说得到证实之前应当排除它，因为第二种假说不涉及任何新的假设。于是，我们对抑制有两种基于相同一般前提的解释，它们只在所假定的神经网络上不同，因而在抑制所需的时间上不同。今后我们将称这样的神经网络在扩展意义上等价。由于我们关心的是网络在等价变换下不变的性质，我们可以采用对演算最方便的物理假设。

Many years ago one of us, by considerations impertinent to this argument, was led to conceive of the response of any neuron as factually equivalnt to a proposition which proposed its adequate stimulus. He therefore attempted to record the behavior of complicated nets in the notation of the symbolic logic of propositions. The “all-or-none” law of nervous activity is sufficient to insure that the activity of any neuron may be represented as a proposition. Physiological relations existing among nervous activities correspond, of course, to relations among the propositions; and the utility of the representation depends upon the identity of these relations with those of the logic of propositions. To each reaction of any neuron there is a corresponding assertion of a simple proposition. This, in turn, implies either some other simple proposition or the disjunction of the conjunction, with or without negation, of similar propositions, according to the configuration of the synapses upon and the threshold of the neuron in question. Two difficulties appeared. The first concerns facilitation and extinction, in which antecedent activity temporarily alters responsiveness to subsequent stimulation of one and the same part of the net. The second concerns learning, in which activities concurrent at some previous time have altered the net permanently, so that a stimulus which would previously have been inadequate is now adequate. But for nets undergoing both alterations, we can substitute equivalent fictitious nets composed of neurons whose connections and thresholds are unaltered. But one point must be made clear: neither of us conceives the formal equivalence to be a factual explanation. Per contra!—we regard facilitation and extinction as dependent upon continuous changes in threshold related to electrical and chemical variables, such as after-potentials and ionic concentrations; and learning as an enduring change which can survive sleep, anaesthesia, convulsions and coma. The importance of the formal equivalence lies in this: that the alterations actually underlying facilitation, extinction and learning in no way affect the conclusions which follow from the formal treatment of the activity of nervous nets, and the relations of the corresponding propositions remain those of the logic of propositions.

多年前，我们中的一位出于与本论证无关的考虑，设想任何神经元的反应在事实上都等价于一个命题，该命题陈述的是它的充分刺激。因此他尝试用命题符号逻辑的记法来记录复杂网络的行为。神经活动的“全或无”定律足以保证任何神经元的活动都可以表示为一个命题。神经活动之间存在的生理关系当然对应于命题之间的关系；而这种表示是否有用，取决于这些关系与命题逻辑中的关系是否相同。任何神经元的每一次反应，都对应着对一个简单命题的断言。根据该神经元上突触的构型及其阈值，这个命题又蕴涵另一个简单命题，或者蕴涵若干类似命题（带或不带否定）的合取的析取。这里出现了两个困难。第一个涉及易化和消退：先前的活动会暂时改变网络同一部分对后续刺激的反应性。第二个涉及学习：某个先前时刻同时发生的活动永久地改变了网络，使得原本不充分的刺激如今变得充分。不过，对于经历这两种改变的网络，我们可以代之以由连接和阈值都不变的神经元构成的、等价的虚构网络。但有一点必须说清楚：我们两人都不认为这种形式等价就是事实上的解释。恰恰相反！我们认为易化和消退取决于与电学和化学变量（如后电位和离子浓度）相关的阈值连续变化；而学习是一种能够经受睡眠、麻醉、惊厥和昏迷而保存下来的持久变化。形式等价的重要性在于：真正构成易化、消退和学习基础的那些改变，丝毫不影响对神经网络活动作形式处理所得出的结论，相应命题之间的关系仍然是命题逻辑的关系。

The nervous system contains many circular paths, whose activity so regenerates the excitation of any participant neuron that reference to time past becomes indefinite, although it still implies that afferent activity has realized one of a certain class of configurations over time. Precise specification of these implications by means of recursive functions, and determination of those that can be embodied in the activity of nervous nets, completes the theory.

神经系统包含许多环形通路，其活动不断再生其中任一神经元的兴奋，以致对过去时间的指涉变得不确定；尽管如此，它仍然蕴涵传入活动在一段时间内实现了某一类构型中的一种。借助递归函数精确刻画这些蕴涵，并确定其中哪些能够体现在神经网络的活动中，便完成了这一理论。

2. The Theory: Nets Without Circles

2. 理论：无环网络

We shall make the following physical assumptions for our calculus.

(1) The activity of the neuron is an “all-or-none” process.

(2) A certain fixed number of synapses must be excited within the period of latent addition in order to excite a neuron at any time, and this number is independent of previous activity and position on the neuron.

(3) The only significant delay within the nervous sytem is synaptic delay.

(4) The activity of any inhibitory synapse absolutely prevents excitation of the neuron at that time.

(5) The structure of the net does not change with time.

我们为这一演算作如下物理假设。

(1) 神经元的活动是一个“全或无”过程。

(2) 在任何时刻，要兴奋一个神经元，必须在潜伏总和期内兴奋某个固定数目的突触，这个数目与先前的活动以及突触在神经元上的位置无关。

(3) 神经系统内唯一显著的延迟是突触延迟。

(4) 任何抑制性突触的活动都会绝对地阻止该神经元在那一时刻被兴奋。

(5) 网络的结构不随时间改变。

To present the theory, the most appropriate symbolism is that of Language II of Carnap (1938), augmented with various notations drawn from Russell and Whitehead (1927), including the Principia conventions for dots. Typographical necessity, however, will compel us to use the upright ‘$E$’ for the existential operator instead of the inverted, and an arrow (“$\rightarrow$”) for implication instead of the horseshoe. We shall also use the Carnap syntactical notations, but print them in boldface rather than German type; and we shall introduce a functor $S$, whose value for a property $P$ is the property which holds of a number when $P$ holds of its predecessor; it is defined by “$S(P)(t) . \equiv . P(Kx) . t = x'$”; the brackets around its argument will often be omitted, in which case this is understood to be the nearest predicate-expression [$\boldsymbol{Pr}$] on the right. Moreover, we shall write $S^2\boldsymbol{Pr}$ for $S(S(\boldsymbol{Pr}))$, etc.

表述这一理论最合适的符号体系是卡尔纳普（Carnap，1938）的语言 II，并补充了取自罗素与怀特海（Russell and Whitehead，1927）的多种记法，包括《数学原理》（Principia）中关于圆点的约定。不过出于排版上的需要，我们将用正体的‘$E$’而不是倒写的 E 表示存在量词，用箭头（“$\rightarrow$”）而不是马蹄形符号表示蕴涵。我们也将使用卡尔纳普的句法记号，但用粗体而非德文花体印出；我们还将引入一个函子 $S$：对于性质 $P$，它的值是这样一个性质，即当 $P$ 对某数的前驱成立时，该性质对这个数成立；其定义为“$S(P)(t) . \equiv . P(Kx) . t = x'$”。其自变量两侧的括号常常省略，此时理解为指右侧最近的谓词表达式 [$\boldsymbol{Pr}$]。此外，我们把 $S(S(\boldsymbol{Pr}))$ 写作 $S^2\boldsymbol{Pr}$，以此类推。

The neurons of a given net $\mathcal{N}$ may be assigned designations “$c_1$”, “$c_2$”, …, “$c_n$”. This done, we shall denote the property of a number, that a neuron $c_i$ fires at a time which is that number of synaptic delays from the origin of time, by “$N$” with the numeral $i$ as subscript, so that $N_i(t)$ asserts that $c_i$ fires at the time $t$. $N_i$ is called the action of $c_i$. We shall sometimes regard the subscripted numeral of “$N$” as if it belonged to the object-language, and were in a place for a functoral argument, so that it might be replaced by a number-variable [$\boldsymbol{z}$] and quantified; this enables us to abbreviate long but finite disjunctions and conjunctions by the use of an operator. We shall employ this locution quite generally for sequences of $\boldsymbol{Pr}$; it may be secured formally by an obvious disjunctive definition. The predicates “$N_1$”, “$N_2$”, …, comprise the syntactical class “$\boldsymbol{N}$”.

给定网络 $\mathcal{N}$ 中的神经元可以分别指定名称“$c_1$”“$c_2$”……“$c_n$”。这样做之后，我们用带下标数字 $i$ 的“$N$”来表示数的这样一个性质：神经元 $c_i$ 在距时间原点该数目个突触延迟的时刻放电；于是 $N_i(t)$ 断言 $c_i$ 在时刻 $t$ 放电。$N_i$ 称为 $c_i$ 的动作（action）。我们有时会把“$N$”的下标数字看作仿佛属于对象语言、并处在函子自变量的位置上，从而可以用数变元 [$\boldsymbol{z}$] 替换并加以量化；这使我们能够用一个算子来缩写冗长但有限的析取和合取。我们将相当普遍地对 $\boldsymbol{Pr}$ 的序列使用这种说法；它可以通过一个显而易见的析取定义在形式上得到保证。谓词“$N_1$”“$N_2$”……构成句法类“$\boldsymbol{N}$”。

Let us define the peripheral afferents of $\mathcal{N}$ as the neurons of $\mathcal{N}$ with no axons synapsing upon them. Let $N_1, \ldots, N_p$ denote the actions of such neurons and $N_{p+1}, N_{p+2}, \ldots, N_n$ those of the rest. Then a solution of $\mathcal{N}$ will be a class of sentences of the form $\boldsymbol{S}_i$: $N_{p+1}(z_1) . \equiv . \boldsymbol{Pr}_i(N_1, N_2, \ldots, N_p, z_1)$, where $\boldsymbol{Pr}_i$ contains no free variable save $z_1$ and no descriptive symbols save the $N$ in the argument [$\boldsymbol{Arg}$], and possibly some constant sentences [$\boldsymbol{sa}$]; and such that each $\boldsymbol{S}_i$ is true of $\mathcal{N}$. Conversely, given a $\boldsymbol{Pr}_1({}^1p_1^1, {}^1p_2^1, \ldots, {}^1p_p^1, z_1, s)$, containing no free variable save those in its $\boldsymbol{Arg}$, we shall say that it is realizable in the narrow sense if there exists a net $\mathcal{N}$ and a series of $N_i$ in it such that $N_1(z_1) . \equiv . \boldsymbol{PR}_1(N_1, N_2, \ldots, z_1, \boldsymbol{sa}_1)$ is true of it, where $\boldsymbol{sa}_1$ has the form $N(0)$. We shall call it realizable in the extended sense, or simply realizable, if for some $n$ $S^n(\boldsymbol{Pr}_1)(\boldsymbol{p}_1, \ldots, \boldsymbol{p}_p, z_1, s)$ is realizable in the above sense. $c_{pi}$ is here the realizing neuron. We shall say of two laws of nervous excitation which are such that every $\boldsymbol{S}$ which is realizable in either sense upon one supposition is also realizable, perhaps by a different net, upon the other, that they are equivalent assumptions, in that sense.

我们把 $\mathcal{N}$ 中没有任何轴突与之形成突触的神经元定义为 $\mathcal{N}$ 的外周传入元（peripheral afferents）。令 $N_1, \ldots, N_p$ 表示这类神经元的动作，$N_{p+1}, N_{p+2}, \ldots, N_n$ 表示其余神经元的动作。于是 $\mathcal{N}$ 的一个解是这样一类语句 $\boldsymbol{S}_i$：$N_{p+1}(z_1) . \equiv . \boldsymbol{Pr}_i(N_1, N_2, \ldots, N_p, z_1)$，其中 $\boldsymbol{Pr}_i$ 除 $z_1$ 外不含自由变元，除自变量 [$\boldsymbol{Arg}$] 中的 $N$ 以及可能的某些常语句 [$\boldsymbol{sa}$] 外不含描述性符号；并且每个 $\boldsymbol{S}_i$ 对 $\mathcal{N}$ 都为真。反过来，给定一个除其 $\boldsymbol{Arg}$ 中的变元外不含自由变元的 $\boldsymbol{Pr}_1({}^1p_1^1, {}^1p_2^1, \ldots, {}^1p_p^1, z_1, s)$，如果存在一个网络 $\mathcal{N}$ 以及其中的一列 $N_i$，使得 $N_1(z_1) . \equiv . \boldsymbol{PR}_1(N_1, N_2, \ldots, z_1, \boldsymbol{sa}_1)$ 对它为真，其中 $\boldsymbol{sa}_1$ 具有 $N(0)$ 的形式，我们就说它是狭义可实现的。如果对某个 $n$，$S^n(\boldsymbol{Pr}_1)(\boldsymbol{p}_1, \ldots, \boldsymbol{p}_p, z_1, s)$ 在上述意义下可实现，我们就称它是广义可实现的，或简称可实现的。这里 $c_{pi}$ 是实现神经元。对于两条神经兴奋定律，如果在一种假定下以任一意义可实现的每个 $\boldsymbol{S}$，在另一种假定下也可实现（也许通过不同的网络），我们就说它们在该意义上是等价的假定。

The following theorems about realizability all refer to the extended sense. In some cases, sharper theorems about narrow realizability can be obtained; but in addition to greater complication in statement this were of little practical value, since our present neurophysiological knowledge determines the law of excitation only to extended equivalence, and the more precise theorems differ according to which possible assumption we make. Our less precise theorems, however, are invariant under equivalence, and are still sufficient for all purposes in which the exact time for impulses to pass through the whole net is not crucial.

下面关于可实现性的定理都指广义而言。在某些情形下，可以得到关于狭义可实现性的更精确的定理；但除了表述更加复杂之外，这也没有多大实际价值，因为我们目前的神经生理学知识只能把兴奋定律确定到广义等价的程度，而更精确的定理会因我们采用哪一种可能的假设而不同。我们这些不那么精确的定理在等价变换下不变，并且对于冲动穿过整个网络的确切时间并不关键的一切用途来说已经足够。

Our central problems may now be stated exactly: first, to find an effective method of obtaining a set of computable $\boldsymbol{S}$ constituting a solution of any given net; and second, to characterize the class of realizable $\boldsymbol{S}$ in an effective fashion. Materially stated, the problems are to calculate the behavior of any net, and to find a net which will behave in a specified way, when such a net exists.

现在可以确切地陈述我们的核心问题：第一，找到一种能行的方法，求得构成任意给定网络之解的一组可计算的 $\boldsymbol{S}$；第二，以能行的方式刻画可实现的 $\boldsymbol{S}$ 所构成的类。用实质的话说，问题就是计算任意网络的行为，以及在存在这样一个网络时，找到一个按指定方式运作的网络。

A net will be called cyclic if it contains a circle, i.e. if there exists a chain $c_i, c_{i+1}, \ldots$ of neurons on it, each member of the chain synapsing upon the next, with the same beginning and end. If a set of its neurons $c_1, c_2, \ldots, c_p$ is such that its removal from $\mathcal{N}$ leaves it without circles, and no smaller class of neurons has this property, the set is called a cyclic set, and its cardinality is the order of $\mathcal{N}$. In an important sense, as we shall see, the order of a net is an index of the complexity of its behaviour. In particular, nets of zero order have especially simple properties; we shall discuss them first.

如果一个网络含有环路，即其中存在一条神经元链 $c_i, c_{i+1}, \ldots$，链中每个成员都与下一个成员形成突触，且首尾相同，就称该网络是有环的（cyclic）。如果网络中的一组神经元 $c_1, c_2, \ldots, c_p$ 从 $\mathcal{N}$ 中移除后网络不再有环，并且没有更小的神经元类具有这一性质，就称这组神经元为环集（cyclic set），其基数称为 $\mathcal{N}$ 的阶（order）。我们将会看到，在一个重要的意义上，网络的阶是其行为复杂程度的指标。特别是，零阶网络具有格外简单的性质；我们先讨论它们。

Let us define a temporal propositional expression (a TPE), designating a temporal propositional function (TPF), by the following recursion.

(1) A ${}^1\boldsymbol{p}^1[z_1]$ is a TPE, where $\boldsymbol{p}_1$ is a predicate-variable.

(2) If $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$ are TPE containing the same free individual variable, so are $S\boldsymbol{S}_1$, $\boldsymbol{S}_1 \vee \boldsymbol{S}_2$, $\boldsymbol{S}_1 . \boldsymbol{S}_2$ and $\boldsymbol{S}_i . \sim . \boldsymbol{S}_2$.

(3) Nothing else is a TPE.

我们用如下递归来定义时间命题表达式（TPE），它指称一个时间命题函数（TPF）。

(1) ${}^1\boldsymbol{p}^1[z_1]$ 是一个 TPE，其中 $\boldsymbol{p}_1$ 是一个谓词变元。

(2) 如果 $\boldsymbol{S}_1$ 和 $\boldsymbol{S}_2$ 是含有同一个自由个体变元的 TPE，那么 $S\boldsymbol{S}_1$、$\boldsymbol{S}_1 \vee \boldsymbol{S}_2$、$\boldsymbol{S}_1 . \boldsymbol{S}_2$ 和 $\boldsymbol{S}_i . \sim . \boldsymbol{S}_2$ 也都是 TPE。

(3) 除此之外没有别的 TPE。

Theorem 1. Every net of order 0 can be solved in terms of temporal propositional expressions.

Let $c_i$ be any neuron of $\mathcal{N}$ with a threshold $\beta_i > 0$, and let $c_{i1}, c_{i2}, \ldots, c_{ip}$ have respectively $n_{i1}, n_{i2}, \ldots, n_{ip}$ excitatory synapses upon it. Let $c_{j1}, c_{j2}, \ldots, c_{jq}$ have inhibitory synapses upon it. Let $\kappa_i$ be the set of the subclasses of $\{n_{i1}, n_{i2}, \ldots, n_{ip}\}$ such that the sum of their members exceeds $\theta_i$. We shall then be able to write, in accordance with the assumptions mentioned above:

$$N_i(z_1) . \equiv . S\left\{\prod_{m=1}^{q} \sim N_{jm}(z_1) . \sum_{\alpha\varepsilon\kappa_i} \prod_{s\varepsilon\alpha} N_{is}(z_1)\right\}, \tag{1}$$

where the “$\sum$” and “$\Pi$” are syntactical symbols for disjunctions and conjunctions which are finite in each case. Since an expression of this form can be written for each $c_i$ which is not a peripheral afferent, we can, by substituting the corresponding expression in (1) for each $N_{jm}$ or $N_{is}$ whose neuron is not a peripheral afferent, and repeating the process on the result, ultimately come to an expression for $N_i$ in terms solely of peripherally afferent $N$, since $\mathcal{N}$ is without circles. Moreover, this expression will be a TPE, since obviously (1) is; and it follows immediately from the definition that the result of substituting a TPE for a constituent $\boldsymbol{p}(z)$ in a TPE is also one.

定理 1. 每个 0 阶网络都可以用时间命题表达式求解。

设 $c_i$ 是 $\mathcal{N}$ 中阈值为 $\beta_i > 0$ 的任一神经元，并设 $c_{i1}, c_{i2}, \ldots, c_{ip}$ 分别在它上面有 $n_{i1}, n_{i2}, \ldots, n_{ip}$ 个兴奋性突触，$c_{j1}, c_{j2}, \ldots, c_{jq}$ 在它上面有抑制性突触。令 $\kappa_i$ 为 $\{n_{i1}, n_{i2}, \ldots, n_{ip}\}$ 中其成员之和超过 $\theta_i$ 的那些子类所构成的集合。（译注：此处原文阈值先记作 $\beta_i$、后记作 $\theta_i$，按原样保留。）于是根据上述假设，我们可以写出：

$$N_i(z_1) . \equiv . S\left\{\prod_{m=1}^{q} \sim N_{jm}(z_1) . \sum_{\alpha\varepsilon\kappa_i} \prod_{s\varepsilon\alpha} N_{is}(z_1)\right\}, \tag{1}$$

其中“$\sum$”和“$\Pi$”是表示析取与合取的句法符号，在每种情形下都是有限的。由于对每个不是外周传入元的 $c_i$ 都能写出这种形式的表达式，因此对于 (1) 中每个其神经元不是外周传入元的 $N_{jm}$ 或 $N_{is}$，我们可以代入相应的表达式，并对结果重复这一过程；由于 $\mathcal{N}$ 无环，最终会得到一个仅用外周传入的 $N$ 表示 $N_i$ 的表达式。而且，这个表达式是一个 TPE，因为 (1) 显然是；由定义立即可知，在一个 TPE 中用 TPE 替换其成分 $\boldsymbol{p}(z)$，所得结果仍是 TPE。

Theorem 2. Every TPE is realizable by a net of order zero.

The functor $S$ obviously commutes with disjunction, conjunction, and negation. It is obvious that the result of substituting any $\boldsymbol{S}_i$, realizable in the narrow sense (i.n.s.), for the $\boldsymbol{p}(z)$ in a realizable expression $\boldsymbol{S}_1$ is itself realizable i.n.s.; one constructs the realizing net by replacing the peripheral afferents in the net for $\boldsymbol{S}_1$ by the realizing neurons in the nets for the $\boldsymbol{S}_i$. The one neuron net realizes $\boldsymbol{p}_1(z_1)$ i.n.s., and Fig. 1a shows a net that realizes $S\boldsymbol{p}_1(z_1)$ and hence $S\boldsymbol{S}_2$, i.n.s., if $\boldsymbol{S}_2$ can be realized i.n.s. Now if $\boldsymbol{S}_2$ and $\boldsymbol{S}_3$ are realizable then $S^m\boldsymbol{S}_2$ and $S^n\boldsymbol{S}_3$ are realizable i.n.s., for suitable $m$ and $n$. Hence so are $S^{m+2}\boldsymbol{S}_2$ and $S^{m+n}\boldsymbol{S}_3$. Now the nets of Figs 1b–d respectively realize $S(\boldsymbol{p}_1(z_1) \vee \boldsymbol{p}_2(z_1))$, $S(\boldsymbol{p}_1(z_1) . \boldsymbol{p}_2(z_1))$, and $zs(\boldsymbol{p}_1(z_1 . \sim \boldsymbol{p}_2(z_1))$ i.n.s. Hence $S^{m+n+1}(\boldsymbol{S}_1 \vee \boldsymbol{S}_2)$, $S^{m+n+1}(\boldsymbol{S}_1 . \boldsymbol{S}_2)$, and $S^{m+n+1}(\boldsymbol{S}_1 . \sim \boldsymbol{S}_2)$ are realizable i.n.s. Therefore $\boldsymbol{S}_1 \vee \boldsymbol{S}_2 \boldsymbol{S}_1 . \boldsymbol{S}_2 \boldsymbol{S}_1 . \sim \boldsymbol{S}_2$ are realizable if $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$ are. By complete induction, all TPE are realizable. In this way all nets may be regarded as built out of the fundamental elements of Figs 1a–d, precisely as the temporal propositional expressions are generated out of the operations of precession, disjunction, conjunction, and conjoined negation. In particular, corresponding to any description of state, or distribution of the values true and false for the actions of all the neurons of a net save that which makes them all false, a single neuron is constructible whose firing is a necessary and sufficient condition for the validity of that description. Moreover, there is always an indefinite number of topologically different nets realizing any TPE.

定理 2. 每个 TPE 都可以由一个零阶网络实现。

函子 $S$ 显然与析取、合取和否定可交换。显然，在一个可实现的表达式 $\boldsymbol{S}_1$ 中，用任何狭义可实现（i.n.s.）的 $\boldsymbol{S}_i$ 替换 $\boldsymbol{p}(z)$，结果本身也是狭义可实现的；构造实现网络的方法是，把 $\boldsymbol{S}_1$ 的网络中的外周传入元替换为各 $\boldsymbol{S}_i$ 的网络中的实现神经元。单神经元网络狭义地实现 $\boldsymbol{p}_1(z_1)$；图 1a 给出一个网络，它狭义地实现 $S\boldsymbol{p}_1(z_1)$，从而在 $\boldsymbol{S}_2$ 狭义可实现时也狭义地实现 $S\boldsymbol{S}_2$。如果 $\boldsymbol{S}_2$ 和 $\boldsymbol{S}_3$ 可实现，那么对适当的 $m$ 和 $n$，$S^m\boldsymbol{S}_2$ 和 $S^n\boldsymbol{S}_3$ 是狭义可实现的。因此 $S^{m+2}\boldsymbol{S}_2$ 和 $S^{m+n}\boldsymbol{S}_3$ 也是。图 1b–d 中的网络分别狭义地实现 $S(\boldsymbol{p}_1(z_1) \vee \boldsymbol{p}_2(z_1))$、$S(\boldsymbol{p}_1(z_1) . \boldsymbol{p}_2(z_1))$ 和 $zs(\boldsymbol{p}_1(z_1 . \sim \boldsymbol{p}_2(z_1))$。因此 $S^{m+n+1}(\boldsymbol{S}_1 \vee \boldsymbol{S}_2)$、$S^{m+n+1}(\boldsymbol{S}_1 . \boldsymbol{S}_2)$ 和 $S^{m+n+1}(\boldsymbol{S}_1 . \sim \boldsymbol{S}_2)$ 都是狭义可实现的。所以，只要 $\boldsymbol{S}_1$ 和 $\boldsymbol{S}_2$ 可实现，$\boldsymbol{S}_1 \vee \boldsymbol{S}_2 \boldsymbol{S}_1 . \boldsymbol{S}_2 \boldsymbol{S}_1 . \sim \boldsymbol{S}_2$ 就可实现。（译注：本段 $S^{m+2}$、$zs(\ldots)$ 及末一串表达式缺少分隔符，均按 1990 年重印版原样保留，未作改动。）由完全归纳法，一切 TPE 都可实现。这样，一切网络都可以看作由图 1a–d 中的基本元件搭建而成，恰如时间命题表达式是由前移、析取、合取和合取否定这些运算生成的一样。特别是，对于任何状态描述，即对网络中所有神经元的动作所作的真、假取值分配（全部为假的那一种除外），都可以构造出一个单个神经元，其放电是该描述成立的充分必要条件。此外，实现任一 TPE 的拓扑上不同的网络总有无限多个。

Theorem 3. Let there be a complex sentence $\mathbf{S}_1$ built up in any manner out of elementary sentences of the form $\mathbf{p}(\mathbf{z}_1 - \mathbf{zz})$ where $\mathbf{zz}$ is any numeral, by any of the propositional connections: negation, disjunction, conjunction, implication, and equivalence. Then $\mathbf{S}_1$ is a TPE and only if it is false when its constituent $\mathbf{p}(\mathbf{z}_1 - \mathbf{zz})$ are all assumed false—i.e. replaced by false sentences—or that the last line in its truth-table contains an ‘F’—or there is no term in its Hilbert disjunctive normal form composed exclusively of negated terms.

定理 3. 设复合语句 $\mathbf{S}_1$ 以任意方式、借助任何命题联结词（否定、析取、合取、蕴涵和等值），由形如 $\mathbf{p}(\mathbf{z}_1 - \mathbf{zz})$ 的基本语句构成，其中 $\mathbf{zz}$ 是任意数字。那么， $\mathbf{S}_1$ 是 TPE 当且仅当：当其成分 $\mathbf{p}(\mathbf{z}_1 - \mathbf{zz})$ 全部被假定为假（即替换为假语句）时它为假；或者说，它的真值表最后一行含有‘F’；或者说，它的希尔伯特析取范式中没有完全由否定项构成的项。（译注：原文“is a TPE and only if”按原样保留，意为“当且仅当”。）

These latter three conditions are of course equivalent (Hilbert and Ackermann, 1938). We see by induction that the first of them is necessary, since $\boldsymbol{p}(z_1 - zz)$ becomes false when it is replaced by a false sentence, and $\boldsymbol{S}_1 \vee \boldsymbol{S}_2$, $S^1 . \boldsymbol{S}_1$ and $\boldsymbol{S}_1 . \sim \boldsymbol{S}_2$ are all false if both their constituents are. We see that the last condition is sufficient by remarking that a disjunction is a TPE when its constituents are, and that any term:

$$\boldsymbol{S}_1 . \boldsymbol{S}_2 \ldots \boldsymbol{S}_m . \sim \boldsymbol{S}_{m+1} . \sim \ldots \sim \boldsymbol{S}_n,$$

can be written as:

$$(\boldsymbol{S}_1 . \boldsymbol{S}_2 \ldots \boldsymbol{S}_m) . \sim (\boldsymbol{S}_{m+1} \vee \boldsymbol{S}_{m+2} \vee \ldots \vee \boldsymbol{S}_m),$$

which is clearly a TPE.

后面这三个条件当然是等价的（Hilbert and Ackermann, 1938）。用归纳法可见第一个条件是必要的：因为 $\boldsymbol{p}(z_1 - zz)$ 被替换为假语句时变为假，而 $\boldsymbol{S}_1 \vee \boldsymbol{S}_2$、$S^1 . \boldsymbol{S}_1$ 和 $\boldsymbol{S}_1 . \sim \boldsymbol{S}_2$ 在两个成分都为假时都为假。注意到当成分都是 TPE 时析取也是 TPE，并且任何项

$$\boldsymbol{S}_1 . \boldsymbol{S}_2 \ldots \boldsymbol{S}_m . \sim \boldsymbol{S}_{m+1} . \sim \ldots \sim \boldsymbol{S}_n,$$

都可以写成

$$(\boldsymbol{S}_1 . \boldsymbol{S}_2 \ldots \boldsymbol{S}_m) . \sim (\boldsymbol{S}_{m+1} \vee \boldsymbol{S}_{m+2} \vee \ldots \vee \boldsymbol{S}_m),$$

而这显然是一个 TPE，由此可见最后一个条件是充分的。（译注：第二个式子末项原文印作 $\boldsymbol{S}_m$，按原样保留。）

Figure 1. The neuron $c_i$ is always marked with the numeral $i$ upon the body of the cell, and the corresponding action is denoted by “$N$” with $i$ as subscript, as in the text:

(a) $N_2(t) . \equiv . N_1(t-1)$;

(b) $N_3(t) . \equiv . N_1(t-1) \vee N_2(t-1)$;

(c) $N_3(t) . \equiv . N_1(t-1) . N_2(t-1)$;

(d) $N_3(t) . \equiv N_1(t-1) . \sim N_2(t-1)$;

(e) $N_3(t) :\equiv: N_1(t-1) . \vee . N_2(t-3) . \sim N_2(t-2)$; $N_4(t) . \equiv . N_2(t-2) . N_2(t-1)$;

(f) $N_4(t) :\equiv: \sim N_1(t-1) . N_2(t-1) \vee N_3(t-1) . \vee . N_1(t-1) . N_2(t-1) . N_3(t-1)$
$N_4(t) :\equiv: \sim N_1(t-2) . N_2(t-2) \vee N_3(t-2) . \vee . N_1(t-2) . N_2(t-2) . N_3(t-2)$;

(g) $N_3(t) . \equiv . N_2(t-2) . \sim N_1(t-3)$;

(h) $N_2(t) . \equiv . N_1(t-1) . N_1(t-2)$;

(i) $N_3(t) :\equiv: N_2(t-1) . \vee . N_1(t-1) . (Ex)t-1 . N_1(x) . N_2(x)$.

[Figure 1 (a)–(i): network diagrams; see the original PDF.]

图 1. 神经元 $c_i$ 总是在细胞体上标以数字 $i$，相应的动作如正文中那样记为带下标 $i$ 的“$N$”：

(a) $N_2(t) . \equiv . N_1(t-1)$；

(b) $N_3(t) . \equiv . N_1(t-1) \vee N_2(t-1)$；

(c) $N_3(t) . \equiv . N_1(t-1) . N_2(t-1)$；

(d) $N_3(t) . \equiv N_1(t-1) . \sim N_2(t-1)$；

(e) $N_3(t) :\equiv: N_1(t-1) . \vee . N_2(t-3) . \sim N_2(t-2)$；$N_4(t) . \equiv . N_2(t-2) . N_2(t-1)$；

(f) $N_4(t) :\equiv: \sim N_1(t-1) . N_2(t-1) \vee N_3(t-1) . \vee . N_1(t-1) . N_2(t-1) . N_3(t-1)$
$N_4(t) :\equiv: \sim N_1(t-2) . N_2(t-2) \vee N_3(t-2) . \vee . N_1(t-2) . N_2(t-2) . N_3(t-2)$；

(g) $N_3(t) . \equiv . N_2(t-2) . \sim N_1(t-3)$；

(h) $N_2(t) . \equiv . N_1(t-1) . N_1(t-2)$；

(i) $N_3(t) :\equiv: N_2(t-1) . \vee . N_1(t-1) . (Ex)t-1 . N_1(x) . N_2(x)$。

［图 1 (a)–(i)：网络示意图，见原文 PDF。］

The method of the last theorems does in fact provide a very convenient and workable procedure for constructing nervous nets to order, for those cases where there is no reference to events indefinitely far in the past in the specification of the conditions. By way of example, we may consider the case of heat produced by a transient cooling.

前面几条定理的方法，对于条件的规定中不涉及无限久远过去事件的那些情形，确实提供了一种按要求构造神经网络的非常方便、可行的程序。作为例子，我们可以考虑短暂冷却产生热感的情形。

If a cold object is held to the skin for a moment and removed, a sensation of heat will be felt; if it is applied for a longer time, the sensation will be only of cold, with no preliminary warmth, however transient. It is known that one cutaneous receptor is affected by heat, and another by cold. If we let $N_1$ and $N_2$ be the actions of the respective receptors and $N_3$ and $N_4$ of neurons whose activity implies a sensation of heat and cold, our requirements may be written as:

$$N_3(t) :\equiv: N_1(t-1) . \vee . N_2(t-3) . \sim N_2(t-2),$$
$$N_4(t) . \equiv . N_2(t-2) . N_2(t-1),$$

where we suppose for simplicity that the required persistence in the sensation of cold is say two synaptic delays, compared with one for that of heat. These conditions clearly fall under Theorem 3. A net may consequently be constructed to realize them, by the method of Theorem 2. We begin by writing them in a fashion which exhibits them as built out of their constituents by the operations realized in Figs 1a–d, i.e. in the form:

$$N_3(t) . \equiv . S(N_1(t) \vee S[(SN_2(t)) . \sim N_2(t)]\}$$
$$N_4(t) . \equiv . S\{[SN_2(t)] . N_2(t)\}.$$

如果把一个冷的物体在皮肤上放一会儿再拿开，会感到热；如果放得更久，就只会感到冷，而不会先有哪怕短暂的温热感。已知有一种皮肤感受器受热影响，另一种受冷影响。令 $N_1$ 和 $N_2$ 分别为这两种感受器的动作，$N_3$ 和 $N_4$ 分别为其活动意味着热感和冷感的神经元的动作，那么我们的要求可以写成：

$$N_3(t) :\equiv: N_1(t-1) . \vee . N_2(t-3) . \sim N_2(t-2),$$
$$N_4(t) . \equiv . N_2(t-2) . N_2(t-1),$$

这里为简单起见，假定冷感所需的持续时间是两个突触延迟，而热感是一个。这些条件显然属于定理 3 的范围，因此可以用定理 2 的方法构造一个实现它们的网络。我们先把它们写成能显示其如何由各成分经图 1a–d 所实现的运算构成的形式，即：

$$N_3(t) . \equiv . S(N_1(t) \vee S[(SN_2(t)) . \sim N_2(t)]\}$$
$$N_4(t) . \equiv . S\{[SN_2(t)] . N_2(t)\}.$$

（译注：第一式括号不配对，按原样保留。）

First we construct a net for the function enclosed in the greatest number of brackets and proceed outward; in this case we run a net of the form shown in Fig. 1a from $c_2$ to some neuron $c_n$, say, so that:

$$N_a(t) . \equiv . SN_2(t).$$

Next introduce two nets of the forms 1c and 1d, both running from $c_a$ and $c_2$, and ending respectively at $c_1$ and say $c_b$. Then:

$$N_4(t) . \equiv . S[N_a(t) . N_2(t)] . \equiv . S[SN_2(t)) . N_2(t)].,$$
$$N_b(t) . \equiv . S[N_a(t) . \sim N_2(t)] . \equiv . S[(SN_2(t)) . \sim N_2(t)].$$

Finally, run a net of the form 1b from $c_1$ and $c_b$ to $c_3$, and derive:

$$N_3(t) . \equiv . S[N_1(t) \vee N_b(t)]$$
$$. \equiv . S(N_1(t) \vee S[SN_2(t)] . \sim N_2(t)\}.$$

These expressions for $N_3(t)$ and $N_4(t)$ are the ones desired; and the realizing net in toto is shown in Fig. 1e.

我们先为被括号嵌套最深的函数构造网络，再逐层向外；在本例中，我们从 $c_2$ 引出一个图 1a 形式的网络到某个神经元，比如 $c_n$，使得：

$$N_a(t) . \equiv . SN_2(t).$$

接着引入两个分别为图 1c 和 1d 形式的网络，都从 $c_a$ 和 $c_2$ 出发，分别终止于 $c_1$ 和比如 $c_b$。于是：

$$N_4(t) . \equiv . S[N_a(t) . N_2(t)] . \equiv . S[SN_2(t)) . N_2(t)].,$$
$$N_b(t) . \equiv . S[N_a(t) . \sim N_2(t)] . \equiv . S[(SN_2(t)) . \sim N_2(t)].$$

最后，从 $c_1$ 和 $c_b$ 引出一个图 1b 形式的网络到 $c_3$，得到：

$$N_3(t) . \equiv . S[N_1(t) \vee N_b(t)]$$
$$. \equiv . S(N_1(t) \vee S[SN_2(t)] . \sim N_2(t)\}.$$

这两个关于 $N_3(t)$ 和 $N_4(t)$ 的表达式正是所求；实现它们的整个网络见图 1e。（译注：原文先写 $c_n$ 后写 $N_a$、$c_a$，又写“终止于 $c_1$”，均按 1990 年重印版原样保留。）

This illusion makes very clear the dependence of the correspondence between perception and the “external world” upon the specific structural properties of the intervening nervous net. The same illusion, of course, could also have been produced under various other assumptions about the behavior of the cutaneous receptors, with corresponding different nets.

这种错觉非常清楚地表明，知觉与“外部世界”之间的对应取决于居间神经网络的特定结构性质。当然，在关于皮肤感受器行为的各种其他假设下，配以相应不同的网络，也能产生同样的错觉。

We shall now consider some theorems of equivalence, i.e. theorems which demonstrate the essential identity, save for time, of various alternative laws of nervous excitation. Let us first discuss the case of relative inhibition. By this we mean the supposition that the firing of an inhibitory synapse does not absolutely prevent the firing of the neuron, but merely raises its threshold, so that a greater number of excitatory synapses must fire concurrently to fire it than would otherwise be needed. We may suppose, losing no generality, that the increase in threshold is unity for the firing of each such synapse; we then have Theorem 4.

现在我们来考虑一些等价定理，即证明各种可供替代的神经兴奋定律除时间外本质上相同的定理。先讨论相对抑制的情形。所谓相对抑制，是指这样的假设：抑制性突触的放电并不绝对阻止神经元放电，而只是提高其阈值，使得要让它放电，必须有比平常更多的兴奋性突触同时放电。不失一般性，我们可以假定每个这样的突触放电使阈值增加 1；于是得到定理 4。

Theorem 4. Relative and absolute inhibition are equivalent in the extended sense.

We may write out a law of nervous excitation after the fashion of (1), but employing the assumption of relative inhibition instead; inspection then shows that this expression is a TPE. An example of the replacement of relative inhibition by absolute is given by Fig. 1f. The reverse replacement is even easier; we give the inhibitory axons afferent to $c_i$ any sufficiently large number of inhibitory synapses apiece.

定理 4. 相对抑制与绝对抑制在广义上等价。

我们可以仿照 (1) 的方式写出一条神经兴奋定律，但改用相对抑制的假设；检视可知该表达式是一个 TPE。用绝对抑制替代相对抑制的一个例子见图 1f。反向替代更为容易：只需让传入 $c_i$ 的每条抑制性轴突各自具有任意足够多的抑制性突触即可。

Second, we consider the case of extinction. We may write this in the form of a variation in the threshold $\theta_i$; after the neuron $c_i$ has fired; to the nearest integer—and only to this approximation is the variation in threshold significant in natural forms of excitation—this may be written as a sequence $\theta_i + b_j$ for $j$ synaptic delays after firing, where $b_j = 0$ for $j$ large enough, say $j = M$ or greater. We may then state Theorem 5.

Theorem 5. Extinction is equivalent to absolute inhibition.

For, assuming relative inhibition to hold for the moment, we need merely run $M$ circuits $\mathscr{T}_1, \mathscr{T}_2, \ldots \mathscr{T}_M$ containing respectively $1, 2, \ldots, M$ neurons, such that the firing of each link in any is sufficient to fire the next, from the neuron $c_i$ back to it, where the end of the circuit $\mathscr{T}_j$ has just $b_j$ inhibitory synapses upon $c_i$. It is evident that this will produce the desired results. The reverse substitution may be accomplished by the diagram of Fig. 1g. From the transitivity of replacement, we infer the theorem. To this group of theorems also belongs the following well-known theorem.

其次，我们考虑消退的情形。我们可以把它写成神经元 $c_i$ 放电后阈值 $\theta_i$ 的变化；取到最接近的整数（在自然的兴奋形式中，阈值变化也只在这种近似下才有意义），它可以写成放电后第 $j$ 个突触延迟时的数列 $\theta_i + b_j$，其中当 $j$ 足够大，比如 $j = M$ 或更大时，$b_j = 0$。于是可以陈述定理 5。

定理 5. 消退等价于绝对抑制。

因为暂且假定相对抑制成立，我们只需从神经元 $c_i$ 出发再回到它，引出 $M$ 个分别含有 $1, 2, \ldots, M$ 个神经元的回路 $\mathscr{T}_1, \mathscr{T}_2, \ldots \mathscr{T}_M$，使得任一回路中每一环节的放电都足以使下一环节放电，并且回路 $\mathscr{T}_j$ 的末端在 $c_i$ 上恰有 $b_j$ 个抑制性突触。显然这将产生所需的结果。反向替代可以用图 1g 的线路图完成。由替代的传递性，我们推出该定理。下面这条众所周知的定理也属于这一组定理。

Theorem 6. Facilitation and temporal summation may be replaced by spatial summation.

This is obvious: one need merely introduce a suitable sequence of delaying chains, of increasing numbers of synapses, between the exciting cell and the neuron whereon temporal summation is desired to hold. The assumption of spatial summation will then give the required results (see e.g. Fig. 1h). This procedure had application in showing that the observed temporal summation in gross nets does not imply such a mechanism in the interaction of individual neurons.

定理 6. 易化和时间总和可以用空间总和来替代。

这是显然的：只需在兴奋细胞与希望其上发生时间总和的神经元之间，引入一系列适当的、突触数目逐渐增加的延迟链。空间总和的假设就会给出所需的结果（例如见图 1h）。这一程序曾用于表明：在宏观网络中观察到的时间总和，并不意味着单个神经元之间的相互作用中存在这样的机制。

The phenomena of learning, which are of a character persisting over most physiological changes in nervous activity, seem to require the possibility of permanent alterations in the structure of nets. The simplest such alteration is the formation of new synapses or equivalent local depressions of threshold. We suppose that some axonal terminations cannot at first excite the succeeding neuron; but if at any time the neuron fires, and the axonal terminations are simultaneously excited, they become synapses of the ordinary kind, henceforth capable of exciting the neuron. The loss of an inhibitory synapse gives an entirely equivalent result. We shall then have

Theorem 7. Alterable synapses can be replaced by circles.

This is accomplished by the method of Fig. 1i. It is also to be remarked that a neuron which becomes and remains spontaneously active can likewise be replaced by a circle, which is set into activity by a peripheral afferent when the activity commences, and inhibited by one when it ceases.

学习现象的特点是能够在神经活动的大多数生理变化中持续存在，这似乎要求网络结构可以发生永久性的改变。最简单的这类改变是形成新的突触，或与之等价的局部阈值降低。我们假定某些轴突末梢起初不能兴奋其后的神经元；但如果在任何时刻该神经元放电，同时这些轴突末梢也受到兴奋，它们就变成普通的突触，此后能够兴奋该神经元。失去一个抑制性突触会产生完全等价的结果。于是我们得到

定理 7. 可改变的突触可以用环路来替代。

这可以用图 1i 的方法实现。还应指出，一个变得自发活动并持续活动的神经元，同样可以用一个环路来替代：该环路在活动开始时由一个外周传入元启动，在活动停止时由另一个外周传入元加以抑制。

3. The Theory: Nets with Circles

3. 理论：有环网络

The treatment of nets which do not satisfy our previous assumption of freedom from circles is very much more difficult than that case. This is largely a consequence of the possibility that activity may be set up in a circuit and continue reverberating around it for an indefinite period of time, so that the realizable $\boldsymbol{Pr}$ may involve reference to past events of an indefinite degree of remoteness. Consider such a net $\mathcal{N}$, say of order $p$, and let $c_1, c_2, \ldots, c_p$ be a cyclic set of neurons of $\mathcal{N}$. It is first of all clear from the definition that every $N_3$ of $\mathcal{N}$ can be expressed as a TPE, of $N_1, N_2, \ldots, N_p$ and the absolute afferents; the solution of $\mathcal{N}$ involves then only the determination of expressions for the cyclic set. This done, we shall derive a set of expressions [$\boldsymbol{A}$]:

$$N_i(z_1) . \equiv . \boldsymbol{Pr}_i[S^{n_{i1}}N_1(z_1), S^{n_{i2}}N_2(z_1), \ldots, S^{n_{ip}}N_p(z_1)], \tag{2}$$

where $\boldsymbol{Pr}_i$ also involves peripheral afferents. Now if $n$ is the least common multiple of the $n_{ij}$, we shall, by substituting their equivalents according to (2) in (3) for the $N_j$, and repeating this process often enough on the result, obtain $\boldsymbol{S}$ of the form:

$$N_i(z_1) . \equiv . \boldsymbol{Pr}_1[S^nN_1(z_1), S^nN_2(z_1), \ldots, S^nN_p(z_1)]. \tag{3}$$

处理不满足我们先前无环假设的网络，要比无环情形困难得多。这在很大程度上是因为，活动可能在一个回路中建立起来，并在其中无限期地往复回荡，因而可实现的 $\boldsymbol{Pr}$ 可能涉及任意久远的过去事件。考虑这样一个网络 $\mathcal{N}$，设其阶为 $p$，并令 $c_1, c_2, \ldots, c_p$ 为 $\mathcal{N}$ 的一个环集。首先，由定义可知，$\mathcal{N}$ 的每个 $N_3$ 都可以表示为 $N_1, N_2, \ldots, N_p$ 与绝对传入元的一个 TPE；于是求 $\mathcal{N}$ 的解只需确定环集的表达式。这样做之后，我们将推出一组表达式 [$\boldsymbol{A}$]：

$$N_i(z_1) . \equiv . \boldsymbol{Pr}_i[S^{n_{i1}}N_1(z_1), S^{n_{i2}}N_2(z_1), \ldots, S^{n_{ip}}N_p(z_1)], \tag{2}$$

其中 $\boldsymbol{Pr}_i$ 也涉及外周传入元。现在，若 $n$ 是各 $n_{ij}$ 的最小公倍数，那么按照 (2) 在 (3) 中用其等价式替换各 $N_j$，并对结果充分多次重复这一过程，就得到如下形式的 $\boldsymbol{S}$：

$$N_i(z_1) . \equiv . \boldsymbol{Pr}_1[S^nN_1(z_1), S^nN_2(z_1), \ldots, S^nN_p(z_1)]. \tag{3}$$

（译注：“每个 $N_3$”“在 (3) 中”以及 (3) 式中的 $\boldsymbol{Pr}_1$ 均按原样保留；按上下文，前者应泛指环集以外神经元的动作。）

These expressions may be written in the Hilbert disjunctive normal form as:

$$N_i(z_1) . \equiv . \sum_{\substack{\alpha\varepsilon\kappa \\ \beta_\alpha\varepsilon\kappa}} \boldsymbol{S}_\alpha \prod_{j\varepsilon\kappa} S^nN_j(z_1) \prod_{j\varepsilon\beta_\alpha} \sim S^nN_j(z_1), \text{ for suitable } \kappa, \tag{4}$$

where $\boldsymbol{S}_\alpha$ is a TPE of the absolute afferents of $\mathcal{N}$. There exist some $2^p$ different sentences formed out of the $pN_i$ by conjoining to the conjunction of some set of them the conjunction of the negations of the rest. Denumerating these by $X_1(z_1), X_2(z_1), \ldots, X_{2^p}(z_1)$, we may, by use of the expressions (4), arrive at an equipollent set of equations of the form:

$$X_i(z_1) . \equiv . \sum_{j=1}^{2^p} \boldsymbol{Pr}_{ij}(z_1) . S^nX_j(z_1). \tag{5}$$

这些表达式可以写成希尔伯特析取范式：

$$N_i(z_1) . \equiv . \sum_{\substack{\alpha\varepsilon\kappa \\ \beta_\alpha\varepsilon\kappa}} \boldsymbol{S}_\alpha \prod_{j\varepsilon\kappa} S^nN_j(z_1) \prod_{j\varepsilon\beta_\alpha} \sim S^nN_j(z_1), \text{ 对适当的 } \kappa, \tag{4}$$

其中 $\boldsymbol{S}_\alpha$ 是 $\mathcal{N}$ 的绝对传入元的一个 TPE。由 $p$ 个 $N_i$ 可以组成 $2^p$ 个不同的语句：取其中某一组的合取，再合取其余各个的否定。把它们编号为 $X_1(z_1), X_2(z_1), \ldots, X_{2^p}(z_1)$，利用表达式 (4)，我们可以得到一组等势的方程，其形式为：

$$X_i(z_1) . \equiv . \sum_{j=1}^{2^p} \boldsymbol{Pr}_{ij}(z_1) . S^nX_j(z_1). \tag{5}$$

（译注：(4) 式第一个连乘号的下标原文印作 $j\varepsilon\kappa$，按原样保留。）

Now we import the subscripted numerals $i, j$ into the object-language, i.e. define $\boldsymbol{Pr}_1$ and $\boldsymbol{Pr}_2$ such that $\boldsymbol{Pr}_1(zz_1, z_1) . \equiv . X_i(z_1)$ and $\boldsymbol{Pr}_2(zz_1, zz_2, z_1) . \equiv . \boldsymbol{Pr}_{ij}(z_1)$ are provable whenever $zz_1$ and $zz_2$ denote $i$ and $j$ respectively. Then we may rewrite (5) as:

$$\begin{aligned}&(z_1)zz_p : \boldsymbol{Pr}_1(z_1, z_3) \\ &\quad . \equiv . (Ez_2)zz_p . \boldsymbol{Pr}_2(z_1, z_2, z_3 - zz_n) . \boldsymbol{Pr}_1(z_2, z_3 - zz_n),\end{aligned} \tag{6}$$

where $zz_n$ denotes $n$ and $zz_p$ denotes $2^p$. By repeated substitution we arrive at an expression:

$$\begin{aligned}&(z_1)zz_p : \boldsymbol{Pr}_1(z_1, zz_nzz_2) . \equiv . (Ez_2)zz_p(Ez_3)zz_p \ldots (Ez_m)zz_p . \\ &\quad \boldsymbol{Pr}_2(z_1, z_2, zz_n(zz_2 - 1)) . \boldsymbol{Pr}_2(z_2, z_3, zz_n(zz_3 - 1)) \ldots\end{aligned} \tag{7}$$

$\boldsymbol{Pr}_2(z_{n-1}, z_n, 0)$, for any numeral $zz_2$ which denotes $s$. This is easily shown by induction to be equipollent to:

$$\begin{aligned}&(z_1)zz_p :. \boldsymbol{Pr}_1(z_1, zz_nzz_2) :\equiv: (Ef)(z_2)zz_2 - 1f(z_2zz_n) \\ &\quad \leqq zz_p . f(zz_nzz_2) = z_1 . \boldsymbol{Pr}_2(f(zz_n(z_2+1)), \\ &\quad f(zz_nz_2)) . \boldsymbol{Pr}_1(f(0), 0),\end{aligned} \tag{8}$$

and since this is the case for all $zz_2$, it is also true that:

$$\begin{aligned}&(z_4)(z_1)zz_p : \boldsymbol{Pr}_1(z_1, z_4) . \equiv . (Ef)(z_2)(z_4 - 1) . f(z_2) \\ &\quad \leqq zz_p . f(z_4) = z_1 f(z_4) = z_1 . \boldsymbol{Pr}_2[f(z_2+1), f(z_2), z_2] . \\ &\quad \boldsymbol{Pr}_1[f(\mathrm{res}(z_4, zz_n)), \mathrm{res}(z_4, zz_n)],\end{aligned} \tag{9}$$

where $zz_n$ denotes $n$, $\mathrm{res}(r, s)$ is the residue of $r$ mod $s$ and $zz_p$ denotes $2^p$.