From nonlinear dynamics to proof of conceptfor Heteroclinic Computing

异宿动力学自然地出现在耦合的非线性动力系统中，特别是在由相耦合和脉冲耦合振子组成的网络中。作为一种动力学现象，它在数学上得到了很好的理解。异宿网络是通过异宿连接连接在一起的鞍态的集合，形成了决定一系列系统中大部分集体动力学的骨架。异宿网络附近的动力学已被提出，以提供生物和生物启发系统中的计算机制，并提供高效、通用的计算特征。异宿网络还实现了独立于特定实现的新的计算框架，例如相位耦合或脉冲耦合振荡器的网络或竞争的Lotka-Volterra类系统。编码的核心计算原理被合理地理解。具体地说，作用于鞍座附近状态的驱动信号向量的方向确定轨迹离开该鞍座的方向，从而确定该轨迹接近的下一个鞍座。在较长的时间尺度上，驱动信号由此确定鞍座的序列，反之，鞍座序列揭示关于驱动信号的特定属性的信息，即，已经对该输入信号执行了计算。这里，具有相同偏秩序的分量的信号彼此关联并且产生相同的计算结果，即，在此意义上计算是稳健的。最近的工作还展示了噪声如何影响相位耦合和脉冲耦合振荡器系统中的可靠开关，从而为计算可靠性提供了线索，并首次深入了解了异宿计算系统在噪声现实世界条件下的实际行为。然而，到目前为止，异宿计算还没有在任何真实的设备上得到演示，如何确切地实现它仍然是一个悬而未决的问题。在拟议的项目中，我们计划解决剩下的三个关键问题。一个是关于高效译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子系统中实现固有的、自组织的存储器，一个是关于识别潜在的衬底、体系结构和实现技术，以在硬件上为异质计算机提供概念证明。我们将结合有关耦合振子网络的已知性质，特别是脉冲耦合振子理论和一般的耦合动力系统理论，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将对异宿计算范例的非线性动力学的理论方面产生新的见解，而且还将向一种新形式的健壮的模拟计算机迈出关键的一步，从非线性动力学的数学水平和计算思想的概念水平向技术水平的备选方案建立一座桥梁。

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.