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Many phenomena can be modeled as network dynamics with punctuateinteractions. However, most relevant dynamics do not allow for computationaltractability. To circumvent this difficulty, the Poisson Hypothesis regimereplaces interaction times between nodes by independent Poisson processes,allowing for tractability in several cases, such as intensity-based models fromcomputational neuroscience. This hypothesis is usually only conjectured, ornumerically validated. In this work, we introduce a class of processes incontinuous time called continuous fragmentation-interaction-aggregationprocesses, by analogy with previously introduced processes in discrete time.The state of each node, described by the stochastic intensity of an associatedpoint process, aggregates arrivals from its neighbors and is fragmented upondeparture. We consider the replica-mean-field version of such a process, whichis a physical system consisting of randomly interacting copies of the networkof interest. Generalizing results proved in discrete time and in the particularcase of excitatory intensity-based neural dynamics, we prove that the PoissonHypothesis holds at the limit of an infinite number of replicas.

Replica-mean-field limit of continuous-time fragmentation-interaction-aggregation processes