Emergent effect is crucial to understanding the properties of complex systems that do not appear in their basic units, but there has been a lack of theories to measure and understand its mechanisms. In this letter, we consider emergence as a kind of structural nonlinearity, discuss a framework based on homological algebra that encodes emergence as the mathematical structure of cohomologies, and then apply it to network models to develop a computational measure of emergence. This framework ties the potential for emergent effects of a system to its network topology and local structures, paving the way to predict and understand the cause of emergent effects. We show in our numerical experiment that our measure of emergence correlates with the existing information-theoretic measure of emergence.

We describe the construction and theoretical analysis of a framework derived from canonical neurophysiological principles that model the competing dynamics of incident signals into nodes along directed edges in a network. The framework describes the dynamics between the offset in the latencies of propagating signals, which reflect the geometry of the edges and conduction velocities, and the internal refractory dynamics and processing times of the downstream node receiving the signals. This framework naturally extends to the construction of a perceptron model that takes into account such dynamic geometric considerations. We first describe the model in detail, culminating with the model of a geometric dynamic perceptron. We then derive upper and lower bounds for a notion of optimal efficient signaling between vertex pairs based on the structure of the framework. Efficient signaling in the context of the framework we develop here means that there needs to be a temporal match between the arrival time of the signals relative to how quickly nodes can internally process signals. These bounds reflect numerical constraints on the compensation of the timing of signaling events of upstream nodes attempting to activate downstream nodes they connect into that preserve this notion of efficiency. When a mismatch between signal arrival times and the internal states of activated nodes occurs, it can cause a breakdown in the signaling dynamics of the network. In contrast to essentially all of the current state of the art in machine learning, this work provides a theoretical foundation for machine learning and intelligence architectures based on the timing of node activations and their abilities to respond rather than necessary changes in synaptic weights. At the same time, the theoretical ideas we developed are guiding the discovery of experimentally testable new structure-function principles in the biological brain.

We introduce a framework for simulating signal propagation in geometric networks (networks that can be mapped to geometric graphs in some space) and developing algorithms that estimate (i.e., map) the state and functional topology of complex dynamic geometric networks. Within the framework, we define the key features typically present in such networks and of particular relevance to biological cellular neural networks: dynamics, signaling, observation, and control. The framework is particularly well suited for estimating functional connectivity in cellular neural networks from experimentally observable data and has been implemented using graphics processing unit high-performance computing. Computationally, the framework can simulate cellular network signaling close to or faster than real time. We further propose a standard test set of networks to measure performance and compare different mapping algorithms.