Pattern formation in a 2-population homogenized neuronal network model.

IF 2.3 4区 医学 Q1 Neuroscience
Karina Kolodina, John Wyller, Anna Oleynik, Mads Peter Sørensen
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引用次数: 1

Abstract

We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of [Formula: see text]-stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

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2种群均质神经网络模型的模式形成。
本文研究了具有周期微观结构的Hopfield型2种群均质神经场模型在一维空间上的模式形成。连接功能在突触足迹和空间尺度上都是周期性调节的。结果表明,非局域突触相互作用促进了有限带宽的不稳定性。稳定性方法依赖于[公式:见文本]稳定性矩阵的波数不变量序列,稳定性矩阵表示施加在齐次背景上的扰动的傅立叶变换线性化演化方程序列。不稳定结构的一般图像由一组分离良好的增益带组成。在浅发射速率下,用相应周期平均连通性函数代替连通性核的平移不变模型确定了不稳定性的非线性发展。在急剧燃烧速率下,模式形成过程敏感地依赖于连通性核的空间局部化:对于强局部化核,该过程由具有周期平均连通性核的平移不变模型决定,而在弱和中等局部化互补状态下,需要均质化模型作为分析的起点。当连通性核用指数衰减函数建模时,我们从数值上跟踪陡射率函数和浅射率函数的不稳定性发展到非线性状态。我们还用数值方法研究了从弱调制到强非均质情况下四种不同情况下的非均质参数的函数模式形成过程。对于弱调制区,我们观察到在陡峭发射速率区形成了稳定的空间振荡,而在发射速率函数的浅区则形成了时空振荡。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Mathematical Neuroscience
Journal of Mathematical Neuroscience Neuroscience-Neuroscience (miscellaneous)
自引率
0.00%
发文量
0
审稿时长
13 weeks
期刊介绍: The Journal of Mathematical Neuroscience (JMN) publishes research articles on the mathematical modeling and analysis of all areas of neuroscience, i.e., the study of the nervous system and its dysfunctions. The focus is on using mathematics as the primary tool for elucidating the fundamental mechanisms responsible for experimentally observed behaviours in neuroscience at all relevant scales, from the molecular world to that of cognition. The aim is to publish work that uses advanced mathematical techniques to illuminate these questions. It publishes full length original papers, rapid communications and review articles. Papers that combine theoretical results supported by convincing numerical experiments are especially encouraged. Papers that introduce and help develop those new pieces of mathematical theory which are likely to be relevant to future studies of the nervous system in general and the human brain in particular are also welcome.
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