Differential and Integral Equations in Neurobiology

研究者研究在大脑皮层模型中出现的某些类型的神经微分方程的动力学。他研究了反复活动是如何由于突触抑制、去极化阻滞和后超极化等影响而终止的。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状态、波前和脉冲。他使用平均和分岔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理学处理的神经组织中的各种实验发现。这些方法应用于极限前脑叶、海龟嗅球和各种皮层切片制备。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化为运动动作。它们通过产生脉冲来交流，这种脉冲通过一种叫做“传递器”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转移到另一部分的。他利用模拟和数学推导出方程和一般原理来说明如何做到这一点。实验表明，在一些认知任务中，某些神经元群会同步放电。他感兴趣的是在不同神经元组之间获得精确定时所必需的条件。为了研究这些问题，他开发了用于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可用。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.