对称微分方程神经元模型兼具信号传播与自激振荡双重功能。
From Propagator to Oscillator: The Dual Role of Symmetric Differential Equations in Neural Systems
- 用对称微分方程构建神经元，可实现稳定传播与持续振荡两种行为。
- 系统在参数调整下可在传播态与振荡态间切换，且振荡可被外部信号抑制。
- 该模型为类脑计算中的信息传递与节律生成提供统一理论框架。
在前期工作中，我们提出基于对称微分方程的新型神经元模型，并证明其作为高效信号传播器的潜力。本文在此基础上，通过系统探索参数空间并结合多种数学分析工具，理论上揭示了该系统的功能二元性：一类轨迹渐近稳定，表现为可靠信号传播；另一类为李雅普诺夫稳定，呈现持续自激振荡，充当信号发生器。为实现仿真中状态的有效监测与预测，我们引入一种新型中间态度量——行进能量。模拟结果表明，通过调节参数或修改连接结构可诱导两种功能模式间的转换；此外，外部信号可有效抑制振荡。这些发现与生物神经元在信息传递与节律生成中的双重角色形成强烈类比，为该模型在类脑工程中的广泛应用奠定了坚实的理论基础和明确的功能路线图。
原文摘要 · Abstract (English)
In our previous work, we proposed a novel neuron model based on symmetric differential equations and demonstrated its potential as an efficient signal propagator. Building upon that foundation, the present study delves deeper into the intrinsic dynamics and functional diversity of this model. By systematically exploring the parameter space and employing a range of mathematical analysis tools, we theoretically reveal the system 's core property of functional duality. Specifically, the model exhibits two distinct trajectory behaviors: one is asymptotically stable, corresponding to a reliable signal propagator; the other is Lyapunov stable, characterized by sustained self-excited oscillations, functioning as a signal generator. To enable effective monitoring and prediction of system states during simulations, we introduce a novel intermediate-state metric termed on-road energy. Simulation results confirm that transitions between the two functional modes can be induced through parameter adjustments or modifications to the connection structure. Moreover, we show that oscillations can be effectively suppressed by introducing external signals. These findings draw a compelling parallel to the dual roles of biological neurons in both information transmission and rhythm generation, thereby establishing a solid theoretical basis and a clear functional roadmap for the broader application of this model in neuromorphic engineering.
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