CRCNS Research Project: Multiply and Conquer: Replica-Mean-Field Limit for Neural Networks

人工智能现在可以在语音或物体识别、语言翻译和自主导航等任务上与人类的表现相媲美。然而，与由脆弱的硬连线电路支持的人工计算相比，生物计算似乎在嘈杂、无序的神经网络中稳健地出现。理解有意义的计算是如何从看似随机的神经成分相互作用中产生的，仍然是一个挑战。为了解决这个问题，人们可以希望挖掘生物网络的设计原理。不幸的是，这种任务被神经回路的复杂性所阻碍。破译神经计算只能通过生物物理相关理论的简化透镜来实现。迄今为止，神经计算已经在理想化模型中进行了理论研究，其中无限数量的神经元通过微小的相互作用进行通信。这种方法忽略了神经计算是由有限数量的细胞通过有限数量的突触相互作用来完成的。这种方法排除了理解神经回路如何可靠地处理信息，尽管神经可变性依赖于这些有限的数字。为了弥补这一点，pi将开发一个新的理论框架，允许在具有有限大小结构的神经电路中分析神经计算。pi将利用通信网络理论的思想来理解生物物理相关的神经网络模型如何通过嘈杂、无序的电路可靠地处理信息。这种方法将为根据稳定性对不同的大脑操作方案进行分类提供基础，并将有助于设计在健康状态下稳定神经系统的策略。与“分而治之”的方法（将系统等同于其各部分的总和）不同，pi将通过“乘法和征服”的方法来破译神经网络的活动。该方法考虑由具有相同基本神经结构的无限多个副本组成的极限网络。关键是，这些所谓的复制平均场网络实际上是神经网络的简化、易于处理的版本，保留了我们感兴趣的有限网络结构的重要特征。神经元群体和突触相互作用的有限大小是神经活动的核心决定因素，负责非零相关的尖峰活动和亚稳神经状态之间的有限转换速率。考虑这些有限大小的现象是发展复制方法的核心动机。预期的结果是对有限大小的神经电路计算的约束的机械理解，特别是在它们的可靠性，速度和成本方面。这将涉及表征以下方面的有限结构依赖性：(i)峰值相关的机制，这是神经编码的基本决定因素；（ii）亚稳态神经状态之间的转换速率，这被认为控制着信息的处理和门控。在这两种情况下，方法将基于简化函数方程的解、离散事件模拟和神经数据集之间的比较。最终目标将是分析生物物理上的详细模型，以产生一个合适的框架，该框架具有足够的限制性，可以制定和验证实验预测。该奖项由MPS数学科学部和CISE信息与智能系统（IIS）通过CRCNA和BRAIN项目共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Artificial intelligence can now rival human performance in tasks such as speech or object recognition, language translation, and autonomous navigation. However, by contrast with artificial computations supported by fragile, hardwired circuits, biological computations appear to robustly emerge in noisy, disordered neural networks. Understanding how meaningful computations emerge from the seemingly random interactions of neural constituents remains a challenge. To solve this, one can hope to mine biological networks for their design principles. Unfortunately, such a task is hindered by the sheer complexity of neural circuits. Deciphering neural computations will only be achieved through the simplifying lens of a biophysically relevant theory. To date, neural computations have been studied theoretically in idealized models whereby an infinite number of neurons communicate via vanishingly small interactions. Such an approach neglects that neural computations are carried out by a finite number of cells interacting via a finite number of synapses. This approach precludes understanding how neural circuits reliably process information in spite of neural variability, which depends on these finite numbers. To remedy this point, the PIs will develop a novel theoretical framework allowing for the analysis of neural computations in neural circuits with finite-size structure. The PIs will leverage ideas from the theory of communication networks to understand how biophysically relevant neural network models can reliably process information via noisy, disordered circuits. This approach will provide the basis for categorizing distinct brain operating regimen based on their stability and will help designing strategies to stabilize neural systems in their healthy regime.In contrast to “divide and conquer” approaches, which equate a system with the mere sum of its parts, the PIs will decipher the activity of neural networks via a “multiply and conquer” approach. This approach considers limit networks made of infinitely many replicas with the same basic neural structure. The key point is that these so-called replica-mean-field networks are in fact simplified, tractable versions of neural networks that retain important features of the finite network structure of interest. The finite size of neuronal populations and synaptic interactions is a core determinant of neural activity, being responsible for non-zero correlation in the spiking activity and for finite transition rates between metastable neural states. Accounting for these finite-size phenomena is the core motivation for the development of the replica approach. The expected outcome is a mechanistic understanding of the constraints bearing on computations in finite-size neural circuits, especially in terms of their reliability, speed, and cost. This will involve characterizing the finite-structure dependence of: (i) the regime of spiking correlations, which is a fundamental determinant of the neural code and (ii) the transition rates between metastable neural states, which are thought to control the processing and gating of information. In both cases, the methodology will be based on the comparison between solutions to reduced functional equations, discrete-event simulations, and neural data sets. The ultimate goal will be to analyze biophysically detailed models in order to produce a fitting framework that is restrictive enough to formulate and validate experimental predictions.This award is being co-funded by the MPS Division of Mathematical Sciences and the CISE Information and Intelligent Systems (IIS) through the CRCNA and BRAIN Programs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.