Stability and Oscillations in Mathematical Models of Neural Systems

大脑是由称为神经元的细胞组成的，它们通过电信号相互交流。众所周知，由单个神经元产生的信号是周期性的脉冲串，其特性取决于单个神经元的特性。神经元组形成网络，执行大脑的功能，如存储和检索记忆。每个网络的电活动通常也是周期性的，其性质可以根据网络执行的功能而改变。该提案包括对几种不同神经网络的数学模型的研究，其中一些对应于特定的大脑区域，另一些则包含许多大脑网络的共同属性。该研究将使用数学来理解网络如何将单个神经元中的周期性活动转化为网络的集体电活动。一个特别的焦点将是神经元之间通信的时间延迟如何影响产生的活动。本研究中的一个项目将有助于我们对人体平衡控制的理解。第二个项目将有助于我们理解正常的大脑节律是如何在大脑的海马区产生的。这反过来可能有助于理解破坏性节律，如癫痫，是如何发生的。

The brain is made up of cells called neurons which communicate with each other through electrical signals. It is well known that the signal produced by a single neuron is a periodic train of pulses, with properties that depend on the characteristics of the individual neuron. Groups of neurons form networks which perform the functions of the brain such as storing and retrieving memories. The electrical activity of each network is generally also periodic, with properties that may change depending on the function the network performs. This proposal comprises a study of mathematical models of several different neural networks, some corresponding to specific brain regions and others incorporating properties common to many brain networks. The study will use mathematics to understand how networks transform the periodic activity in a single neuron into the collective electrical activity of the network. One particular focus will be on how the time delay in the communication between neurons affects the activity that is produced. One project in this study will contribute to our understanding of human balance control. A second project will contribute to our understanding of how normal brain rhythms are produced in the hippocampus region of the brain. This in turn may help understand how destructive rhythms, such as epilepsy, occur.