Differential and Integral Equations in Neurobiology

研究者研究了在大脑皮层模型中出现的某些类型的积分-微分方程组的动力学。他研究了由于突触抑制、去极化阻断和后超极化等几种影响，复发性活动是如何终止的。他还考虑了循环连接是如何导致双稳的，以及在什么条件下这种情况下才能产生局域的空间态、波前和脉冲。他使用平均化和分叉理论的方法，从这些模型中推导和分析简化的动力学。数学结果解释了在正常和药物操纵的神经组织中的各种实验发现。这些方法适用于Limax前脑叶、海尔蒙嗅球和各种皮质切片准备。神经元是大脑中的主要细胞，负责处理来自外部世界的输入并转换这种运动动作。它们通过产生尖峰信号进行交流，这些尖峰信号被称为发射器的化学物质传播到大脑的不同区域。研究者感兴趣的是这些电子和化学信息是如何从大脑的一部分传递到其他部分的。他使用模拟和数学来推导方程和如何做到这一点的一般原理。实验表明，在一些认知任务中，特定的神经元组同步放电。他感兴趣的是在不同的神经元组之间达到精确计时所必需的条件。为了研究这些，他开发了研究复杂生物和物理系统的数学软件。作为这个项目的一部分，他继续开发这个软件并使其广泛使用。该项目还为学生和博士后提供跨学科培训机会。

The investigator studies the dynamics of certain classes ofintegro-differential equations that arise in models of cortex. Heexamines how recurrent activity terminates due to any of severaleffects such as synaptic depression, depolarization block, andafter-hyperpolarization. He also considers how recurrentconnections lead to bistability and under what conditions this issufficient to produce localized spatial states, wave fronts, andpulses. He uses methods of averaging and bifurcation theory toderive and analyse simplified dynamics from these models. Themathematical results explain a variety of experimental findingsin both normal and pharmacologically manipulated neural tissue.These methods are applied to the Limax procerebral lobe, theturtle olfactory bulb, and various cortical slice preparations. Neurons, the main cells in the brain, are responsible forprocessing inputs from the outside world and converting this tomotor actions. They communicate by producing spikes that arepropagated to different regions of the brain by chemicals calledtransmitters. The nvestigator is interested in how thiselectrical and chemical information moves from one part of thebrain to others. He uses simulations and mathematics to deriveequations and general principles about how this is done.Experiments have shown that during some cognitive tasks, certaingroups of neurons fire synchronously. He is interested in theconditions that are necessary to attain precise timing betweendifferent groups of neurons. In order to study these, he hasdeveloped mathematical software for the study of complexbiological and physical systems. As part of this project, hecontinues to develop this software and to make it widelyavailable. The project also provides interdisciplinary trainingopportunities for students and postdocs.