平均场神经网络爆破的时变特征
Characterization of blowups via time change in a mean-field neural network
摘要：
理想化的具有类似脉冲交互作用的积分-激发神经元网络在平均场极限下服从McKean-Vlasov扩散方程。这些方程很容易崩溃：对于足够强的相互作用耦合，相互作用的平均场速率在有限时间内发散，同时有有限部分神经元发出尖峰，从而标志着宏观同步事件。解析地描述这些爆破奇点是理解平均场神经模型中尖峰同步的出现和持续的关键。然而，这样的决议是阻碍了经典考虑动力学的平均场相互作用的首次通过性质。在这里，我们介绍了一个延迟泊松变化的经典积分和消防动力学的爆破分析定义在平均场极限。虽然基本上是非线性的，我们表明，这种延迟泊松动力学可以转化为一个非相互作用的线性动力学通过一个确定性的时变。我们指定这个时变的解决方案的非线性，延迟积分方程通过更新分析的第一次穿越问题。该公式还表明，同时发放神经元的分数可以通过一个自洽的，概率守恒原理的时变线性动力学确定。我们利用所提出的框架，在一个配套文件中，分析表明存在的奇异平均场动力学与持续的同步足够大的相互作用耦合。
英文摘要：
Idealized networks of integrate-and-fire neurons with impulse-like interactions obey McKean-Vlasov diffusion equations in the mean-field limit. These equations are prone to blowups: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously, thereby marking a macroscopic synchronous event. Characterizing these blowup singularities analytically is the key to understanding the emergence and persistence of spiking synchrony in mean-field neural models. However, such a resolution is hindered by the first-passage nature of the mean-field interaction in classically considered dynamics. Here, we introduce a delayed Poissonian variation of the classical integrate-and-fire dynamics for which blowups are analytically well defined in the mean-field limit. Albeit fundamentally nonlinear, we show that this delayed Poissonian dynamics can be transformed into a noninteracting linear dynamics via a deterministic time change. We specify this time change as the solution of a nonlinear, delayed integral equation via renewal analysis of first-passage problems. This formulation also reveals that the fraction of simultaneously spiking neurons can be determined via a self-consistent, probability-conservation principle about the time-changed linear dynamics. We utilize the proposed framework in a companion paper to show analytically the existence of singular mean-field dynamics with sustained synchrony for large enough interaction coupling.
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