Particle bursts detected on the earth's surface during thunderstorms by various particle detectors originated from the relativistic runaway electron avalanches (RREAs) initiated by free electrons accelerated in the strong atmospheric electric fields. Two oppositely directed dipoles in the thundercloud accelerate electrons in the direction of the earth's surface, and to the open space. The particle bursts observed by orbiting gamma ray observatories are called terrestrial gamma ray flashes (TGFs, with energies of several MeV, only sometimes reaching tens of MeV); ones registered by particle detectors located on the ground – are called thunderstorm ground enhancements (TGEs, with energies, usually reaching 40-50 MeV). Balloons and aircraft in the troposphere register gamma ray glows (with energies of several MeV). Recently, high-energy atmospheric physics includes also, so-called, downward TGFs (DTGFs), intense particle bursts with a duration of a few milliseconds.Well-known extensive air showers (EASs) originate from the interactions of galactic protons and fully-stripped nuclei with the atmosphere atoms. EAS particles have very dense cores around the shower axes. However, high-energy particles in the EAS cores comprise a very thin disc of (a few tens of ns), and a particle detector traversed by an EAS core will not register a particle burst, but only one very large pulse. Only neutron monitor, by collecting delayed thermal neutrons from EAS core particle interactions with soil, can register particle bursts. We discuss the relation between short particle bursts available from the largest particle arrays with EAS phenomena. We demonstrate that the neutron monitors can extend the EAS “lifetime” up to a few milliseconds, a time comparable with DTGFs duration. The possibility to use the network of neutron monitors for high-energy cosmic ray research is also deliberated.Plain Language Summary: Short and extended particle bursts are registered in space, the troposphere, and the earth's surface. Coordinated monitoring of the particle fluxes, near-surface electric fields, and lightning flashes makes it possible to formulate a hypothesis on the origin of intense bursts and their relation to extensive air showers and atmospheric discharges. Analysis of the observational data and possible origination scenarios of particle bursts allows us to conclude that the bursts can be explained by the electron acceleration in the thunderous atmosphere and by gigantic showers developed in the terrestrial atmosphere by high-energy protons and fully-stripped nuclei accelerated in Galaxy.