Mathematical Sciences: "Analog Neural Networks and Delay Equations".

吉迪恩9622722模拟神经网络支持平稳、周期和混沌解决方案。到目前为止，定态解一直是研究人员最关注的问题。然而，近年来，非静态模式在大脑中的重要性变得明显起来。研究人员研究了允许周期轨道的网络，并解决了这类离散时间网络和连续时间网络之间的明显差异。在连续网络中，人们通常期望收敛到周期轨道，而在离散时间网络中，存在稳定的定常解。该项目研究了一类具有阶跃函数非线性的网络的混沌行为。这些可以被视为陡峭的乙状结肠非线性的极限情况，并且更容易分析处理。神经元突触反应的延迟是一个公认的事实，如果要解释它的影响，必须用延迟微分方程来表示适当的模型。研究人员对该方程的离散化进行了研究，证明了这种有限维近似刻画了时滞方程动力学的本质特征。人工神经网络是一个介于人工智能和神经生物学之间的领域。数学模型在这一领域发挥着重要作用。神经网络试图对大脑的一些基本功能(学习、识别训练模式)进行建模，而不试图理解神经元如何在化学或生理水平上工作的复杂细节。今天，神经网络被应用于科学和工程的许多领域。在给定轻微扰动的数据的情况下，该网络可以被训练成识别特定的模式。这种模式必须是固定的，即在时间上是固定的。另一方面，大脑中几乎没有任何固定的过程，而且大多数过程都是周期性的。研究人员研究了允许周期性模式的网络，以了解不同模型之间的关系，并确定哪种模型最适合应用。如果一个人能够训练网络识别复杂的模式(在空间和时间上)，这将对机器人和人工智能产生直接影响。

Gedeon 9622722 Analog neural networks admit stationary, periodic and chaotic solutions. So far the stationary solutions have received most attention of researchers. However, in recent years the importance of non-stationary patterns in brain became clear. The investigator studies networks which admit periodic orbits and addresses the apparent discrepancy between discrete and continuous time networks of this type. While in continuous networks one expects generally a convergence to a periodic orbit, in discrete time networks there are stable stationary solutions. The project investigates chaotic behaviour of a class of networks with step-function nonlinearities. These can be viewed as a limiting case of a steep sigmoid nonlinearities and are easier to handle analytically. The delay in synaptic response of a neuron is a recognized fact; if one wants to account for its affect the appropriate model must be expressed in terms of delay differential equation. The investigator studies the discretization of this equation and shows that this finite-dimensional approximation captures the essential features of the dynamics of the delay equation. Artificial neural networks is an area on a crossroads between artificial inteligence and neurobiology. Mathematical models play an important role in this area. Neural networks try to model some basic functions of brain (learning, recognizing trained patterns) without trying to understand the intricate details of how the neurons work on a chemical or physiological level. Neural networks are used today in many areas of apllied science and engineering. The network can be trained to recognize a certain pattern, given slightly perturbed data. This pattern must be stationary, i.e. fixed in time. On the other hand, there is hardly any process in the brain which is stationary, and most processes are periodic. The investigator studies networks that admit periodic patterns, to understand the relationship between differen t models and see which one is the most suitable for applications. If one were able to train the networks to recognize complicated patterns (in space and time), that would have an immediate impact on robotics and artificial inteligence.