分层耗散神经场：无穷多神经元群体活动流形的维数界
Layered Dissipative Neural Fields: Bounding the Dimension of the Population Activity Manifold with Infinitely Many Neurons
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- Technical University of Munich（慕尼黑工业大学）
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中文总结 AI 辅助
本文针对分层无穷神经元群体活动的高阶抛物型神经场模型，证明耗散诱导谱间隙并存在惯性流形，给出其维数上界，并证明普适逼近性及行波和平稳解存在条件。
中文摘要 AI 辅助
我们考虑一类神经场模型，该模型描述组织成有限多个堆叠层的无穷神经元群体的宏观活动，写成半线性高阶抛物型方程系统。对于这类受生物学启发的方程，我们证明高阶耗散（模拟间隙连接活动调节）诱导一个谱间隙，确保相关半群具有惯性流形，即一个有限维流形，吸引无穷维神经活动系统的所有轨道。我们的主要结果是该流形维数的上界，给出其关于耗散阶数和强度、泄漏率以及连接算子范数的显式标度。我们将其解释为朝着经验驱动的神经流形概念（即大型神经群体的集体活动位于低维流形上的想法）的数学建模迈出的一步。对于这类方程，我们进一步证明普适逼近性质，即关于输入的平稳模式表达能力和有限时间范围内依赖于输入的动态表达能力。最后，我们提供行波和非平凡平稳解存在的充分条件。
英文摘要
We consider a class of neural field models describing the macroscopic activity of an infinite population of neurons organized into finitely many stacked layers, written as a system of semi-linear higher-order parabolic equations. For this class of biologically inspired equations, we show that the higher-order dissipation, modeling gap junction activity regulation, induces a spectral gap ensuring that the associated semigroup possesses an inertial manifold, that is, a finite-dimensional manifold attracting all orbits of the infinite-dimensional system of neural activity. Our main result is an upper bound on the dimension of this manifold, giving its explicit scaling in the dissipation order and strength, the leaking rate, and the norm of the connection operator. We interpret this as a step toward the mathematical modeling of the empirically motivated concept of neural manifold, namely the idea that the collective activity of a large neural population lies on a low-dimensional manifold. For this class of equations, we further prove universal approximation properties, namely stationary pattern expressivity in terms of the input and input-dependent dynamical expressivity over finite time horizons. Finally, we provide sufficient conditions for the existence of traveling waves and nontrivial stationary solutions.
