Differential and Integral Equations in Neurobiology

研究人员研究的动力学某些类的积分微分方程模型中出现的皮质。 他研究了由于突触抑制、去极化阻滞和后超极化等多种效应，复发性活动如何终止。 他还考虑了循环连接如何导致双稳态，以及在什么条件下这足以产生局部空间态、波前和脉冲。 他使用平均和分叉理论的方法来推导和分析这些模型的简化动力学。 这些数学结果解释了在正常神经组织和人工操纵神经组织中的各种实验结果。这些方法被应用于Limax前脑叶，海龟嗅球和各种皮质切片制备。 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，并将其转换为运动动作。 它们通过产生尖峰信号进行交流，这些尖峰信号通过被称为递质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个部分转移到其他部分的。 他用模拟和数学推导出了如何做到这一点的方程式和一般原理。实验表明，在某些认知任务中，某些神经元群会同步放电。 他感兴趣的是获得不同神经元组之间精确计时所必需的条件。 为了研究这些，他开发了数学软件来研究复杂的生物和物理系统。 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的培训机会。

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.