Co-evolutionary dynamics for two adaptively coupled Theta neurons

Felix Augustsson, Erik Andreas Martens
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Abstract

Natural and technological networks exhibit dynamics that can lead to complex cooperative behaviors, such as synchronization in coupled oscillators and rhythmic activity in neuronal networks. Understanding these collective dynamics is crucial for deciphering a range of phenomena from brain activity to power grid stability. Recent interest in co-evolutionary networks has highlighted the intricate interplay between dynamics on and of the network with mixed time scales. Here, we explore the collective behavior of excitable oscillators in a simple networks of two Theta neurons with adaptive coupling without self-interaction. Through a combination of bifurcation analysis and numerical simulations, we seek to understand how the level of adaptivity in the coupling strength, $a$, influences the dynamics. We first investigate the dynamics possible in the non-adaptive limit; our bifurcation analysis reveals stability regions of quiescence and spiking behaviors, where the spiking frequencies mode-lock in a variety of configurations. Second, as we increase the adaptivity $a$, we observe a widening of the associated Arnol'd tongues, which may overlap and give room for multi-stable configurations. For larger adaptivity, the mode-locked regions may further undergo a period-doubling cascade into chaos. Our findings contribute to the mathematical theory of adaptive networks and offer insights into the potential mechanisms underlying neuronal communication and synchronization.
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两个自适应耦合 Theta 神经元的协同进化动力学
自然和技术网络表现出的动态可导致复杂的合作行为,例如耦合振荡器中的同步和神经元网络中的节律活动。了解这些集体动力学对于解读从大脑活动到电网稳定性等一系列现象至关重要。近来,人们对协同进化网络的兴趣凸显了网络上和网络中不同时间尺度的动态之间错综复杂的相互作用。在这里,我们探索了由两个 Theta 神经元组成的简单网络中可兴奋振荡器的集体行为,这些神经元具有自适应耦合,但没有自我交互作用。通过结合分岔分析和数值模拟,我们试图了解耦合强度 $a$ 的适应性水平如何影响动力学。我们首先研究了非适应性极限下的动力学可能性;我们的分岔分析揭示了静止和尖峰行为的稳定区域,在这些区域中,尖峰频率在各种配置下模式锁定。其次,随着适应性$a$的增加,我们观察到相关的阿诺舌(Arnol'd tongues)不断扩大,可能会出现重叠,并为多种稳定配置提供了空间。我们的发现有助于自适应网络的数学理论,并为神经元通信和同步的潜在机制提供了见解。
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