神经科学与心理学2018arxiv.org
大脑中的信息是以连续还是离散形式表征？
Is Information in the Brain Represented in Continuous or Discrete Form?
进阶 · 约 60 分钟读完前置知识：神经科学基础、信息论基础、计算神经科学模型
导读
这篇论文探讨大脑中的信息是以连续还是离散形式表示的问题。作者论证由于神经信号中存在噪声，神经元之间无法通过连续表示可靠地传递信息，从而支持离散表示的观点。
HN 关注点：这触及神经科学与信息论的核心争论——大脑是'模拟'还是'数字'系统，HN 读者对计算与生物学基础问题历来兴趣浓厚。
HN 评论精选
这几条评论里：质疑 3 补充 1
质疑噪声限制分辨率不等于必须放弃连续模型，类比计算机上用连续数学编程
Rushton (1961) 早就得出结论：由于噪声的存在，从手指到脊髓的典型有髓神经纤维无法采用连续表示。尽管有这些开创性工作，基于连续表示的计算模型仍主导当今神经科学文献，例如连续吸引子网络。当然，从基础信息论可知任何含噪系统本质上都是离散的。但取决于噪声大小，我不明白为什么不能用连续变量来描述模型。类比普通数值程序：我们常用实值向量空间推导数学，但实现时变量最终都是离散的。不用离散对象思考的原因是数学一旦离散就会变得极其复杂，而且计算基底在大多数情况下足够精细。类似地，即使噪声把大脑信号的有效分辨率限制在约 5 比特（论文中的数字），我认为也不意味着应该放弃用连续空间描述计算神经科学模型——至少在模型用实测噪声量验证过的情况下如此。
aurelian15查看英文原文
Rushton (1961) concluded that the neural signaling of a typical human myelinated nerve fiber spanning, say, between a finger and the spinal cord cannot employ a continuous representation due to the presence of noise. Despite these seminal works, computational models based on continuous representation dominate present day neuroscience literature – for example, continuous attractor networks (Eliasmith, 2005; Wang, 2009). Sure, we know from basic information theory that any noisy system is inherently discrete. However, depending on the magnitude of the noise, I don't see why we should not describe models in terms of continuous variables. As an analogy, consider the average numerical computer program. We often derive the underlying math in terms of real-valued vector spaces. However, when implementing the program on a computer system, all variables are ultimately discrete. A reason for not thinking about our problems in terms of discrete objects is that math just tends to get incredibly complicated as soon as objects are discrete; and that the computational substrate is fine-grained enough (in most cases). Similarly, if noise limits the effective resolution of individual signals or representations in the brain to 5 bits (number taken from the paper), that (in my opinion) still does not mean that we should stop describing computational neuroscience models in continuous spaces -- at least not, if they are validated with the empirically measured amount of noise; which is exactly what Eliasmith's lab (cited above) does (full disclosure: I'm one of his students). Furthermore, as soon as you code information in populations of neurons, the noise on individual connections becomes less important; one could argue that whenever the brain needs precise computations, it dedicates more neural resources to that problem; then, the noise will "average out".
补充这是控制论时代奠基性争论的延续，McCulloch、冯·诺依曼等人早已交锋
让我沮丧的是工程界对自己历史的记忆如此之少，甚至没有把这个问题置于本领域奠基性问题之一的语境中。大脑是离散（'数字'）还是连续（'模拟'）运作的问题，是 1940 年代 Macy 会议的三大主要辩论之一。McCulloch 和 Pitts 基本上把职业生涯押在大脑是全数字网络上；而冯·诺依曼更谨慎，认为大脑的数字功能建立在化学的模拟基础之上，信息究竟以数字还是模拟方式编码并不清楚。Julian Bigelow 则指出数学家和物理学家倾向于忽略神经元的生物结构，将其等同于其数字操作。这场辩论直指模拟与数字之别的核心，在我看来它不仅是生理学问题，更是哲学问题。推荐 Ronald Kline 的《The Cybernetic Moment》。
nanna查看英文原文
It frustrates me that the engineering community has so little recollection of its own history that it doesn't even contextualise /this/ question as one of the founding problematics of its own specialism, and our cybernetic age in general. The question of whether the brain functions discreetly ('digitally') or continuously ('analogue') was one of the three main debates at the Macy Meetings in the 1940s. [0] For example Warren McCulloch and Walter Pitts effectively staked their careers on the brain being a fully digital network, as a computer. But John Von Neumann was more cautious, arguing that the digital function of the brain rested on a chemical, analogue foundation, and that it was unclear whether messages were coded in a digital or analogue way. Julian Bigelow argued that mathematicians and physicists preferred to ignore the biological structure of neurons and identify them with their digital operation. This debate strikes at the heart of the debate between the difference between analogue and digital, and in my opinion it's a philosophical, not just physiological, problem. [0] For anyone interested in this history I'd highly recommend Ronald R. Kline's work, especially The Cybernetic Moment (p 46-47)
质疑$1.99 论证隐含了数字概念硬编码于物理信号的可疑假设
论文中有个有趣的说法：连续表示'无法令人满意地解释大量心理物理学数据'，例如 $1.99 和 $2.00 之间 1 美分的差异会导致购买决策的突变，这种阶跃变化无法用连续表示建模。这里隐含的假设似乎是：我们的数字概念在物理信号层面硬编码进大脑，即想到数字 2 时大脑某处会产生数量为 2 的物理信号。否则这个实验怎么能说明信号在大脑中的物理编码方式？据我所知，人们普遍认为数字概念处于更抽象的符号操作层面（尤其涉及分数时）。这个悖论（看起来像连锁悖论的变体：一堆沙最少要多少粒？）如果考虑人们倾向于忽略金额中的零头，就可以解决。声明：我还没读引用的论文。
mannykannot查看英文原文
An interesting statement here: 'One answer, as outlined by VanRullen and Koch (2003), is that continuous representation “cannot satisfactorily account for a large body of psychophysical data”. For example, 1 cent does not typically have much value to most people. However, a person may decide to buy a product if priced at $1.99 – yet, refuse to buy the same product if priced 1 cent higher at $2.00. Such an abrupt (or step) change in the brain’s purchasing decision cannot be modeled using a continuous representation despite extensive attempts to do so (Basu, 1997).' The tacit assumption made here seems to be that our concept of number is hard-coded into our brains at the physical signal level, and so that when we think of the number 2, some part of our brain generates a physical signal that has the quantity of 2 (absent that assumption, why would one think that this experiment tells us anything about how signals are physically encoded in the brain?) AFAIK, it is generally thought that our concept of numbers is at a more abstract, symbol-manipulation level (especially when, as in this example, fractions are involved.) It seems to me that this paradox (which looks like a variant of the sorites paradox, e.g. what is the minimum size of a heap of sand?) is resolved if we consider peoples' tendency to disregard the pennies in a monetary amount. Caveat / mea culpa: I have not read the papers referenced in the quote.
质疑论文忽略了放电频率作为连续变量及统计平均效应
论文称'信息无法用连续表示在神经元之间可靠传递'，但这似乎没有考虑放电频率（频率本身是连续变量），也没有考虑不同平均幅度的信号（即使每个信号因噪声而不一致）在多次迭代中产生统计显著效应的可能性。
lend000查看英文原文
> we show that information cannot be communicated reliably between neurons using a continuous representation It seems like this doesn't take into account spiking frequencies (where the frequency is a continuous variable), nor the potential for signals with different average magnitudes (even if each signal is inconsistent due to the presence of noise) to have a statistically significant effect over many iterations.
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