Transfer entropy in continuous time, with applications to jump and neural spiking processes

摘要

Transfer entropy has been used to quantify the directed flow of information between source and target variablesin many complex systems. While transfer entropy was originally formulated in discrete time, in this paper weprovide a framework for considering transfer entropy in continuous time systems, based on Radon-Nikodymderivatives between measures of complete path realizations. To describe the information dynamics of individualpath realizations, we introduce the pathwise transfer entropy, the expectation of which is the transfer entropyaccumulated over a finite time interval. We demonstrate that this formalism permits an instantaneous transferentropy rate. These properties are analogous to the behavior of physical quantities defined along paths such aswork and heat.We use this approach to produce an explicit form for the transfer entropy for pure jump processes,and highlight the simplified form in the specific case of point processes (frequently used in neuroscience to modelneural spike trains). Finally, we present two synthetic spiking neuron model examples to exhibit the pertinentfeatures of our formalism, namely, that the information flow for point processes consists of discontinuous jumpcontributions (at spikes in the target) interrupting a continuously varying contribution (relating to waiting timesbetween target spikes). Numerical schemes based on our formalism promise significant benefits over existingstrategies based on discrete time formalisms.