From nonlinear dynamics to proof of conceptfor Heteroclinic Computing

在耦合非线性动力系统中，特别是在相耦合和脉冲耦合振荡器网络中，异斜动力学是自然产生的。作为一种动力现象，它在数学上可以很好地理解。异斜网络是通过异斜连接连接起来的鞍态的集合，形成了决定一系列系统中许多集体动力学的骨架。异质网络动力学已被提出为生物和仿生系统提供计算机制，并提供高效、通用的计算特征。异斜网络还支持独立于特定实现的新计算框架，例如相位耦合或脉冲耦合振荡器网络或竞争性的lotka - voltera -类系统。编码的核心计算原理被合理地理解。特别是，作用于鞍座附近状态的驱动信号矢量的方向决定了轨迹离开该鞍座的方向，从而决定了轨迹所接近的下一个鞍座。在更长的时间尺度上，驱动信号因此决定了鞍座序列，反过来，鞍座序列揭示了驱动信号的特定属性的信息，即对该输入信号进行了计算。在这里，具有相同偏秩阶分量的信号相互关联并产生相同的计算结果，即在这个意义上计算是鲁棒的。最近的研究还展示了噪声如何影响相位耦合和脉冲耦合振荡器系统的可靠切换，从而为计算可靠性提供了线索，并首次提供了关于异斜计算系统在嘈杂的现实世界条件下的实际行为的见解。然而，到目前为止，异斜计算还没有在任何实际设备上得到证明，如何准确地实现它仍然是一个悬而未决的问题。在拟议的项目中，我们计划解决三个剩下的关键问题。一个是关于有效的解码及其与编码的相互作用，一个是关于如何在合适的耦合振荡器系统中实现内在的、自组织的存储器，还有一个是关于识别潜在的衬底、架构和实现技术，以提供硬件上的异斜计算机的概念证明。我们将结合有关耦合振荡器网络的已知性质，特别是脉冲耦合振荡器理论和耦合动力系统理论，以及神经网络理论的思想来解决这些问题。一项成功的研究不仅会对异斜计算范式的非线性动力学的理论方面产生新的见解，而且还会为一种新的鲁棒模拟计算机形式提供关键的步骤，在非线性动力学的数学层面和计算思想的概念层面之间架起一座桥梁，通往技术层面的选择。

Heteroclinic dynamics naturally emerges in coupled nonlinear dynamical systems, in particular in networks of phase-coupled and pulse-coupled oscillators. As a dynamical phenomenon it is reasonably well understood mathematically. Heteroclinic networks, a collection of saddle states linked via heteroclinic connections, form the skeleton determining much of the collective dynamics in a range of systems. Dynamics near heteroclinic networks has been proposed to provide computing mechanisms in biological and bio-inspired systems and to offer highly effective, universal computational features. Heteroclinic networks also enable a new computing framework that is independent of specific implementations, such as networks of phase-coupled or pulse-coupled oscillators or competitive Lotka-Volterra-like systems. The core computational principles of encoding are reasonably understood. In particular, the direction of the driving signal vector acting on a state near a saddle determines the direction a trajectory leaves that saddle and thus the next saddle approached by the trajectory. On longer time scales, the driving signal thereby determines the sequence of saddles and reversely, the sequence of saddles reveals information about specific properties of the driving signal, i.e. a computation has been performed on that input signal. Here, signals with components in the same partial rank order are associated with each other and yield the same computational result, i.e. the computation is robust in this sense. Recent works also demonstrated how noise affects reliable switching in both phase-coupled and pulse-coupled oscillator systems, thereby offering hints into computational reliability and providing first insights about how heteroclinic computing systems may actually behave under noisy, real world conditions. Heteroclinic computing, however, has so far not been demonstrated in any real device and it remains an open question how exactly to realize it.In the proposed project, we plan to address three remaining key questions. One about efficient decoding and its interplay with encoding, one about how to realize intrinsic, self-organized memory in suitable coupled oscillator systems, and one about identifying potential substrates, architectures and implementation techniques to provide a proof of concept for heteroclinic computers in hardware. We will combine known properties about coupled oscillator networks, especially the theory of pulse-coupled oscillators and generally the theory of coupled dynamical systems, with ideas from neural network theory to address these questions. A successful study would not only yield new insights into theoretical aspects of the nonlinear dynamics of the heteroclinic computing paradigm but also provide crucial steps towards a new form of robust analogue computers, creating a bridge from the mathematical level of nonlinear dynamics and the conceptual level of the computing idea towards options for the technological level.