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Multistability in neural networks with delayed feedback: Theory and applications.

机译：具有延迟反馈的神经网络中的多重稳定性：理论与应用。

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摘要

In this dissertation, we study the coexistence of multiple stable patterns (multistability) in recurrent neural networks with delayed feedback. The coexistence of multiple stable patterns such as equilibria and periodic orbits is the basis for associative content-addressable memory storage and retrieval in neural networks where each equilibrium is identified with static memory, while stable periodic orbits are associated with temporally patterned spike trains. Periodic patterns exhibited in recurrent neural networks are also associated with a variety of rhythms displayed in the nervous system. These rhythms have been linked to important behavioral and cognitive states in the nervous system, including attention, working memory, associative memory, object recognition, sensory motor integration and perception processing.;Our investigation addresses the questions of how we can efficiently increase the network's capacity for memory storage and retrieval, and what kind of mechanisms enable neural networks to generate a large number of coexisting stable oscillatory patterns. These questions are addressed in this dissertation from two aspects: (1) the incorporation of some important biological features of neurons such as the firing process and the absolute refractory period, (2) the introduction of a non-monotonic activation function. Our results show that the interaction of time lag, the recurrent feedback, biological features of individual neurons, and the non-monotonicity of synaptic update functions leads to a large number of stable periodic solutions with predictable patterns of oscillations, via interesting pattern transition or through a mode interaction of pitchfork, saddle-node and Hopf bifurcations.;First, we investigate the impact of the effective duration of a delayed feedback on multistability in a recurrent inhibitory loop when biological realities of firing and absolute refractory period are incorporated into an integrate-and-fire neuron model. Our analysis shows that the interaction of the delay, the inhibitory feedback and the absolute refractory period can generate four basic types of oscillations which give the basic building blocks of possible periodic patterns. We then show how these basic oscillations can be pinned together to form four types of periodic patterns, such as self-inhibitory patterns and nearest-neighbor-inhibitory patterns. The coexistence of these different types of periodic patterns leads to the occurrence of multistability in the recurrent inhibitory loop. Moreover, interesting pattern transitions occur as the time delay passes through certain critical values. These pattern transitions play a similar role to the standard bifurcation theory in terms of the birth and continuation of multiple periodic patterns. We also link the identified periodic patterns to certain neural rhythms, and use the average time of convergence to a periodic pattern to determine what kind of periodic patterns have the potential to be used for neural information transmission and cognition processing in the nervous system.;Second, we examine the effect of non-monotonic activation functions on the network's capacity for memory storage and retrieval. We first show how supercritical pitch-fork bifurcations and saddle-node bifurcations lead to the coexistence of multiple stable equilibria in the instantaneous updating network. We then study the effect of time delay on the local stability of these equilibria and show that four equilibria lose their stability at a certain critical value of time delay, and a Hopf bifurcation of four periodic solutions occurs, leading to multiple coexisting periodic orbits. We apply center manifold theory and normal form theory to determine the direction of this Hopf bifurcation and the stability of bifurcated periodic orbits. Numerical simulations show very interesting global patterns of periodic solutions as the time delay is varied. In particular, we observe that these four periodic solutions are glued together along the stable and unstable manifolds of saddle points to develop a butterfly structure through a complicated process of gluing bifurcations of periodic solutions with increasing frequencies crossing the stable manifolds of the saddle points.
机译：本文研究了具有延迟反馈的递归神经网络中多个稳定模式（多重稳定性）的共存。多个稳定模式（例如平衡和周期性轨道）的共存是在神经网络中关联内容可寻址存储器的存储和检索的基础，其中每个平衡都由静态记忆来标识，而稳定周期性轨道与时间模式化的尖峰序列相关。循环神经网络中表现出的周期性模式也与神经系统中显示的各种节律有关。这些节律与神经系统中的重要行为和认知状态有关，包括注意力，工作记忆，联想记忆，物体识别，感觉运动整合和知觉处理。;我们的研究解决了如何有效提高网络容量的问题存储器的存储和检索，以及什么样的机制使神经网络能够生成大量并存的稳定振荡模式。本文从两个方面解决了这些问题：（1）结合了神经元的一些重要生物学特征，例如放电过程和绝对不应期，（2）引入了非单调激活功能。我们的结果表明，时滞，递归反馈，单个神经元的生物学特征以及突触更新功能的非单调性之间的相互作用，通过有趣的模式转换或通过导致可预测的振荡模式，产生了大量稳定的周期解。干草叉，鞍形节点和霍夫夫分叉的一种模式相互作用。首先，我们研究了将放电的生物现实和绝对不应期合并到一个积分中后，延迟反馈的有效持续时间对循环抑制回路中多重稳定性的影响-和发射神经元模型。我们的分析表明，延迟，抑制性反馈和绝对不应期的相互作用可以产生四种基本类型的振荡，这些振荡给出了可能的周期性模式的基本构造块。然后，我们展示了如何将这些基本振荡固定在一起以形成四种类型的周期性模式，例如自抑制模式和最近邻抑制模式。这些不同类型的周期模式的共存导致循环抑制回路中出现多重稳定性。此外，随着时间延迟通过某些临界值，会发生有趣的模式转换。在多个周期性模式的产生和延续方面，这些模式过渡与标准分叉理论具有相似的作用。我们还将识别出的周期性模式与某些神经节律联系起来，并使用收敛的平均时间与周期性模式确定哪种周期性模式有潜力用于神经系统中的神经信息传递和认知处理。 ，我们研究了非单调激活函数对网络内存存储和检索能力的影响。我们首先显示超临界音叉分叉和鞍节点分叉如何导致瞬时更新网络中多个稳定平衡的共存。然后，我们研究了时间延​​迟对这些平衡的局部稳定性的影响，并表明四个平衡在一定的时间延迟临界值下失去了稳定性，并且出现了四个周期解的Hopf分叉，从而导致了多个同时存在的周期轨道。我们应用中心流形理论和正规形式理论来确定此Hopf分叉的方向以及分叉的周期性轨道的稳定性。数值模拟显示了随着时间延迟的变化，周期解的整体模式非常有趣。尤其是，我们观察到这四个周期解沿着鞍点的稳定和不稳定歧管胶合在一起，通过一个复杂的过程来胶合蝶形结构，该过程将周期解的分叉粘合，其频率越过鞍点的稳定歧管。

著录项

  • 作者

    Ma, Jianfu.;

  • 作者单位

    York University (Canada).;

  • 授予单位 York University (Canada).;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2008
  • 页码 239 p.
  • 总页数 239
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

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