神经回路中的“多即不同”：生物神经网络典型循环基序中有效理论的代数涌现

中文摘要 AI 辅助

典型神经回路基序通常从功能上描述：除法归一化（divisive normalization）通过池化信号缩放群体活动，胜者通吃（winner-take-all, WTA）竞争通过循环兴奋和共享抑制选择一个模式。我们将这些基序及其组合代数地表示为有限变换系统，分析其输入条件更新生成的转换幺半群，区分生成器中已存在的结构与仅通过组合出现的结构，在联合状态空间上区分从一个因子继承的结构与存在于联合构型上的结构。单个非周期更新可生成非非周期幺半群。在WTA中，每个冻结驱动生成器都会坍缩为不动点，但短输入序列会产生依赖胜者的抑制门控的局部循环：具有可逆作用的全局耗散动力学。最强结果出现在WTA到DN（除法归一化）的组合中，此时组合幺半群包含一个真正的复合局部循环，其中归一化状态和胜者的门控状态共同变化，尽管每个原始生成器都是非周期的。完整的环绕分析（holonomy analysis）证实这是Krohn-Rhodes级联的群分量而非偶然循环，并发现联合构型上的大部分带像集（image sets）具有群结构，而未耦合乘积则无此类结构。详尽的接口扫描表明，复合循环是耦合的属性而非所选映射的属性。若基序是神经计算的构建块，将它们组合就是一种编程：选择原语和接口使生成的代数具有预期的功能范围。转换幺半群就是该功能范围——原语呈现给任何后续构造的内容。循环回路是组合变换系统，其代数约束了它们可被编程实现的计算。

英文摘要

Canonical neural circuit motifs are usually described functionally: divisive normalization rescales population activity by a pooled signal, and winner-take-all competition selects one pattern through recurrent excitation and shared inhibition. We represent them, and their compositions, algebraically as finite transformation systems and analyze the transition monoids generated by their input-conditioned updates, distinguishing structure already present in a generator from structure that appears only through composition, and, on a joint state space, structure inherited from one factor from structure that lives on a joint configuration. Individually aperiodic updates can generate non-aperiodic monoids. In the WTA, every frozen-drive generator collapses to fixed points, yet short input sequences create local cycles of winner-dependent inhibitory gating: globally dissipative dynamics with a reversible action. The strongest result arises in WTA-to-DN composition. The composed monoid then contains a genuinely composite local cycle in which normalization state and the winner's gating state change together, although every primitive generator is aperiodic. Holonomy analysis certifies this as a group component of the Krohn-Rhodes cascade rather than an incidental cycle, and finds most group-carrying image sets on joint configurations, whereas the uncoupled product has none. An exhaustive interface sweep shows that the composite cycle is a property of the coupling rather than of a chosen map. If motifs are building blocks of neural computation, composing them is a form of programming: one chooses primitives and interfaces so that the generated algebra has the intended repertoire. The transition monoid is that repertoire - what a primitive presents to any later construction. Recurrent circuits are compositional transformation systems; their algebra constrains what they can be programmed to compute.

发表机构

• McGovern Institute for Brain Research, Massachusetts Institute of Technology（麻省理工学院麦戈文脑研究所）
• The NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI)（美国国家科学基金会人工智能与基础相互作用AI研究所）

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