Many people experience mild stress in modern society which raises the need for an improved understanding of psychophysiological responses to stressors. Heart rate variability (HRV) may be associated with a flexible network of intricate neural structures which are dynamically organized to cope with diverse challenges. HRV was obtained in thirty-three healthy participants performing a cognitive task both with and without added stressors. Markers of neural autonomic control and neurovisceral complexity (entropy) were computed from HRV time series. Based on individual anxiety responses to the experimental stressors, two subgroups were identified: anxiety responders and non-responders. While both vagal and entropy markers rose during the cognitive task alone in both subgroups, only entropy decreased when stressors were added and exclusively in anxiety responders. We conclude that entropy may be a promising marker of cognitive tasks and acute mild stress. It brings out a new central question: why is entropy the only marker affected by mild stress? Based on the neurovisceral integration model, we hypothesized that neurophysiological complexity may be altered by mild stress, which is reflected in entropy of the cardiac output signal. The putative role of the amygdala during mild stress, in modulating the complexity of a coordinated neural network linking brain to heart, is discussed.