Canonical Models for Mathematical Neuroscience

9805544研究人员引入了一种用于数学神经科学的新方法：他们不是研究特定大脑结构的单一模型，而是考虑其所有模型的家族。他们使用不变流形、分支和奇异摄动理论来证明，在某些自然条件下，存在将族中的每个成员转换为规范模型的连续的、不可逆的变量变化。在丘脑、海马体和新皮质的研究中，他们得出了由可兴奋、振荡和爆裂的神经元组成的弱连接网络的典型模型。精确的方程式很少被用来描述生物系统。因此，不同的人可以对同一生物系统使用不同的模型，但得到的结果是相互矛盾的。解决这一问题的合理方法是同时研究同一生物系统的所有可能模型。研究人员开发并使用系统科学中最先进的方法来证明，如果一个生物系统接近临界状态，那么它的所有模型都与一个更简单的模型有关，称为正则模型。这种方法的一个优点是，正则模型的似是而非仅仅取决于临界状态的似是而非，而不是取决于描述生物系统细节的方程的选择。这种方法使他们能够为丘脑、海马体和新皮质等复杂的大脑结构推导出简单但准确的模型，这些结构在关键区域附近运行，例如启动节律性活动。这些模型被用来研究行为动物的注意力、记忆和导航。此外，这些研究可能会揭示这些大脑结构的神经计算特性，并可能应用于神经工程和计算机设计。

Hoppensteadt 9805544 The investigators introduce a novel methodology for use in mathematical neuroscience: Rather than studying a single model of a particular brain structure, they consider the family of all its models. They use invariant manifold, bifurcation, and singular perturbation theories to prove that under certain natural conditions there is a continuous, noninvertible change of variables that converts every member of the family into a canonical model. They derive canonical models for weakly connected networks of excitable, oscillatory and bursting neurons, arising in studies of the thalamus, hippocampus and neocortex. Exact equations are rarely known to describe biological systems. As a result, various people can use various models for the same biological system, but obtain contradictory results. A reasonable way to resolve this is to study ALL possible models of the same biological system at once. The investigators develop and use state-of-the-art methodologies in system science to show that if a biological system is near a critical regime, then all its models are related to a simpler one, called a canonical model. An advantage of this approach is that the plausibility of the canonical model rests only on the plausibility of the critical regime, but not on the choice of equations describing details of the biological system. This approach enables them to derive simple but exact models for such complex brain structures as the thalamus, hippocampus and neocortex operating near a critical regime, such as the onset of rhythmic activity. These models are used to study attention, memorization and navigation in behaving animals. In addition, these studies may shed light on neurocomputational properties of these brain structures with possible applications to neuroengineering and computer design.