时移神经元网络模型的弛豫和自持振荡
Relaxation and self-sustained oscillations in the time elapsed neuron network model
摘要：
时间流逝模型描述了一个均匀的神经元组件的放电活动，这要归功于自上次放电以来所经过的时间的分布。它给出了一个数学描述的概率密度的神经元结构的时间。在较早的工作中，广义相对熵方法的基础上，它被证明是高度或弱连接网络的模型表现出松弛的稳定状态和适度连接的网络，它是获得的自我维持的周期解的外观的数值证据。在这里，我们更进一步，并使用特定形式的模型，我们量化的制度，松弛到一个稳定的状态发生在网络连接。为了介绍我们的方法，我们首先考虑的情况下，神经元是不连接的，我们给出了一个新的声明，表明总的异步发射的神经元出现渐近。在第二步中，我们考虑连接的情况，并给出一个低连接条件，仍然导致异步触发。我们的低连通性条件在某种程度上是尖锐的，因为我们可以给出一个例子，当这个条件不满足时，同步节律活动发生。事实上，我们能够建立几个明确的家庭的周期解。我们的构造是完全非线性的，并且网络中神经活动的非线性化不遵循分叉分析。它依赖于一个代数非线性边界条件，发生在model.These解析结果进行了比较与数值模拟在更广泛的假设下，并被证明是强大的。
英文摘要：
The time elapsed model describes the firing activity of an homogeneous assembly of neurons thanks to the distribution of times elapsed since the last discharge. It gives a mathematical description of the probability density of neurons structured by this time. In an earlier work, based on generalized relative entropy methods, it is proved that for highly or weakly connected networks the model exhibits relaxation to the steady state and for moderately connected networks it is obtained numerical evidence of appearance of self-sustained periodic solutions. Here, we go further and, using the particular form of the model, we quantify the regime where relaxation to a stationary state occurs in terms of the network connectivity. To introduce our methodology, we first consider the case where the neurons are not connected and we give a new statement showing that total asynchronous firing of neurons appears asymptotically. In a second step, we consider the case with connections and give a low connectivity condition that still leads to asynchronous firing. Our low connectivity condition is somehow sharp because we can give an example, when this condition is not fulfilled, where synchronous rhythmic activity occurs. Indeed, we are able to build several explicit families of periodic solutions. Our construction is fully nonlinear and the resynchronization of the neural activity in the network does not follow from bifurcation analysis. It relies on an algebraically nonlinear boundary condition that occurs in the model.These analytic results are compared with numerical simulations under broader hypotheses and shown to be robust.
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