论文标题
相互作用神经元系统的亚竞争力
Metastability for systems of interacting neurons
论文作者
论文摘要
我们研究了相互作用神经元及其亚稳态特性的随机系统。该系统由$ n $神经元组成，每个神经元都随机峰值，具体取决于其膜潜力。在尖峰时期，神经元电位重置为$ 0 $，所有其他神经元都将获得额外的$ h/n $潜力。在连续的尖峰时间之间，每个神经元在指数速度下会失去潜力。我们在超临界状态中研究了该系统，也就是说，在非常轻微的条件下，在峰值率函数的非常温和的条件下，$ h。$ $ h。$ h。$ h。$ h。在杜阿尔特和ost \ cite \ cite {do}中已显示，有限量系统的唯一不变分布是微不足道的度量$δ__{\ bf 0} $ question forsution forsution forsiuntion。在最小的条件下，关于峰值率功能在$ 0 $附近的行为的情况下，我们证明，灭绝时间以$ n $为单位的迟到时间到达，并讨论了根据非线性均值限制过程的平衡$δ_{\ bf 0} $的稳定性，该稳定性取决于动力学的参数。然后，我们将我们的研究指定为饱和峰值率的情况，并表明，在适当的条件下，在模型的参数上，1）非线性均值场限制承认，在非线性平衡速率范围内的非线性率范围内，与法律相互分配的非线速率的平均有限系统的平均有限系统的跨度尖峰速率恢复了均衡的时间，以实现法律的分配。换句话说，该系统表现出亚稳态的行为。
We study a stochastic system of interacting neurons and its metastable properties. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential. In between successive spike times, each neuron looses potential at exponential speed. We study this system in the supercritical regime, that is, for sufficiently high values of the synaptic weight $h.$ Under very mild conditions on the spiking rate function, is has been shown in Duarte and Ost \cite{do} that the only invariant distribution of the finite system is the trivial measure $ δ_{\bf 0}$ corresponding to extinction of the process. Under minimal conditions on the behavior of the spiking rate function in the vicinity of $0$, we prove that the extinction time arrives at exponentially late times in $ N$, and discuss the stability of the equilibrium $δ_{\bf 0}$ for the non-linear mean-field limit process depending on the parameters of the dynamics. We then specify our study to the case of saturating spiking rates and show that, under suitable conditions on the parameters of the model, 1) the non-linear mean-field limit admits a unique and globally attracting equilibrium and 2) the rescaled exit times for the mean spiking rate of a finite system from a neighbourhood of the non-linear equilibrium rate converge in law to an exponential distribution, as the system size diverges. In other words, the system exhibits a metastable behavior.
