We identify a new regime of time dependent helium burning for high accretion rate neutron stars and suggest that this burning is the origin of the low-level luminosity variations (on timescales of 10-10(4) s, designated the ''very low-frequency noise'' (VLFN) by van der Klis and collaborators) always detected in the brightest accreting X-ray sources. Only two nuclear burning regimes were previously recognized. At accretion rates in excess of the Eddington limit [M greater than or similar to (1-3) x 10(-8) M. yr(-1)], the accreted matter fuses steadily. At very low M, the star's entire surface is rapidly (less than or similar to 10 s) burned by a fast propagating convective burning front at regular intervals, giving quasi-periodic Type I X-ray bursts. We show that for the observationally interesting range of 5 x 10(-10) M. yr(-1) less than or similar to M less than or similar to 10(-8) M. yr(-1), parts of the stellar surface burn slowly. At these accretion rates, a local thermonuclear instability starts a fire which propagates horizontally at nu similar to 300 cm s(-1). The fire propagates around the flammable surface in roughly the same time it takes to accrete enough fuel for the next instability (similar to 10(3)-10(4) s), so that only a few fires are burning at once, giving rise to large luminosity flares. Nuclear burning is always time dependent for sub-Eddington local accretion rates: a local patch undergoes a recurrent cycle, accumulating fuel for hours until it becomes thermally unstable or is ''ignited'' by a nearby burning region. The global pattern of burning and the resulting luminosity are thus very dependent on how fast nuclear fires spread around the star. It is thought that in the presence of convection a high combustion speed (nu greater than or similar to 10(6) cm s(-1)) leads to rapid ignition (less than or similar to 10 s) of all connected parts of the surface which have enough fuel for burning to convectively propagate. This is the prevalent burning mode at very low M's, where the colder envelope must accumulate more fuel before self-igniting. Most of the star is then convectively combustible at any one time, so that the recurrent thermal instability leads to quasi-periodic Type I X-ray bursting. However, as M increases, the amount of fuel accumulated between instabilities is less and the slower, nonconvective combustion that we have studied in detail becomes more prevalent. This breaks the burst periodicity, reduces the amount of fuel available for Type I X-ray bursts, and enhances the power in the VLFN. Our collation of EXOSAT observations of the brightest X-ray sources supports this theoretical inference. The nuclear burning luminosity is not uniform over the stellar surface and so may provide a handle on measuring, or constraining, the spin periods of these neutron stars. We thus provide ''pulse-profiles'' from the slow mode of combustion, which dramatically change on timescales similar to R/nu similar to hours and may ''de-cohere'' the spin signal on shorter timescales comparable to a few radial thermal times (similar to minutes). We also predict that the weaker X-ray bursts are produced by the combustion of a small fraction of the star, making detection of rotational modulations during bursts a distinct possibility. The low flux of these weak bursts may preclude this analysis for rapidly (P-s similar to ms) rotating neutron stars until the deployment of the X-Ray Timing Explorer and the USA Experiment.
