论文标题
广义接触过程的缩放限制
Scaling limit of a generalized contact process
论文作者
论文摘要
我们为广义接触过程得出了宏观方程，该过程的灵感来自晶格$ \ mathbb {z}^d $上的神经元集成和火灾模型。每个晶格站点的州都可以在$ 0，\ ldots，k $中获得值。这些可以解释为神经元膜的潜力，而状态$ k $对应于射击阈值。在我们将在本文中使用的接触过程的术语中，状态$ k $对应于传染性的个人（所有其他州都不感染）。为了达到射击阈值，或者要变得传染性，该网站必须从$ 0 $逐渐发展到$ k $。它爬升的速率由状态$ k $的其他神经元确定，并通过KAC型电势（范围$γ^{ - 1} $）与其结合。流体动力方程是在极限$γ\ rightarrow 0 $中获得的。还考虑了微观模型的扩展，包括兴奋性和抑制性神经元类型以及其他生物物理机制。
We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice $\mathbb{Z}^d$. The states at each lattice site can take values in $0,\ldots,k$. These can be interpreted as neuronal membrane potential, with the state $k$ corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state $k$ corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from $0$ to $k$. The rate at which it climbs is determined by other neurons at state $k$, coupled to it through a Kac-type potential, of range $γ^{-1}$. The hydrodynamic equations are obtained in the limit $γ\rightarrow 0$. Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered.
