Canonical Models for Mathematical Neuroscience

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，而是考虑所有模型的家族。他们使用不变流形、分岔和奇异摄动理论来证明，在一定的自然条件下，存在一个连续的、不可逆的变量变化，它能将家族中的每一个成员转化为正则模型。他们在对丘脑、海马体和新皮层的研究中，推导出了可兴奋、振荡和爆发神经元弱连接网络的规范模型。描述生物系统的精确方程很少为人所知。因此，对于同一个生物系统，不同的人可以使用不同的模型，但得到的结果却相互矛盾。解决这个问题的一个合理方法是同时研究同一生物系统的所有可能模型。研究人员在系统科学中发展和使用最先进的方法来表明，如果一个生物系统接近一个临界状态，那么它的所有模型都与一个更简单的模型有关，称为规范模型。这种方法的一个优点是，规范模型的合理性只取决于临界状态的合理性，而不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复杂的大脑结构，如丘脑、海马体和新皮层，建立简单而精确的模型，这些结构在一个关键的状态下运行，比如节律性活动的开始。这些模型被用来研究行为动物的注意力、记忆力和导航能力。此外，这些研究可能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.