化學與電子耦合的 Hindmarsh-Rose 神經細胞其產生的多狀態、層級同步現象

Abstract

本文研究化學與電子耦合的 Hindmarsh-Rose (HR) 神經細胞產生的多狀態同步(混沌共存的穩定和周期/穩態同步)和多層級同步複雜的神經網絡。我們明確的得到HR 神經元網路的完全同步狀態的全局穩定和局部穩定的同步區間。同時有化學和電子耦合才會有共存的穩定多狀態同步，包括混沌同步、週期/穩態同步。相比之下耦合振盪器系統或耦合映象格子模型只有單一狀態的同步現像。對於局部同步，我們得到即使沒有電子耦合，不管化學耦合的連結網络是如何稀疏，耦合神經元可以達到穩定穩態，我們預測最小的化學耦合強度和每個神經元接收的信號成正比。此外，我們提供化學耦合的連結網络要如何的濃密，化學突觸才會扮演增強同步化的角色來達到系統的同步。另外我們證明HR 耦合神經元是有界耗散，來得到全局同步的區域，這樣的結果可以應用於一般的單神經元模型和複雜的網絡。我們的方法很一般，例如，它可以立即應用到其他單神經元模型，如FitzHugh-Nagumo 模型。所需的工具和概念，包括坐標變換，矩陣 測量，單調動力學和時間平均估計。

The multi-state synchronization, the coexistence of stably chaotic and periodic/steadystate synchronization, and multi-stage synchronization of Hindmarsh-Rose(HR) neurons with both chemical and electrical synapses over the complex network are analytically studied. The synchronization regions for both global and local stability of the completely synchronous state in such networks of HR neurons are explicitly obtained. The coexistence of the stably multi-state synchronization, including chaotic synchronization and periodic/steady-state synchronization, is provided with the presence of both chemical and electrical synapses in the network. This is in contrast with Coupled Oscillator Systems or Coupled Map Lattices where only single-state synchronization is found. For local synchronization, we are able to show that even without electrical coupling, the coupled neurons may reach stably steady-state synchrony regardless of how sparsely the chemically coupling networks is coupled and that the minimum chemically coupling bound as predicted is inversely proportional to the number of signals each neuron receives. Moreover, we provide a measurement of how densely coupled the system should be so as to have the chemical synapses to play an enforcing role in achieving the synchrony of the system. We establish the bounded dissipation of the coupled HR neurons, followed by the attainment of its global synchronization region. Such a result can be applied to a general class of single neuron models and the complex networks. Our method employed here is quite general. For instance, it can be immediately applied to other single neuron models such as the FitzHugh-Nagumo model. The analytical tools and concepts needed include coordinate transformations, matrix measures, monotone dynamics and time averaging estimates.