  We introduce and study a non-conserving sandpile model, the autonomously
adapting sandpile (AAS) model, for which a site topples whenever it has two or
more grains, distributing three or two grains randomly on its neighboring
sites, respectively with probability $p$ and $(1-p)$. The toppling process is
independent of the actual number of grains $z_i$ of the toppling site, as long
as $z_i\ge2$. For a periodic lattice the model evolves into an inactive state
for small $p$, with the number of active sites becoming stationary for larger
values of $p$. In one and two dimensions we find that the absorbing phase
transition occurs for $p_c\!\approx\!0.717$ and $p_c\!\approx\!0.275$.
  The symmetry of bipartite lattices allows states in which all active sites
are located alternatingly on one of the two sublattices, A and B, respectively
for even and odd times. We show that the AB-sublattice symmetry is
spontaneously broken for the AAS model, an observation that holds also for the
Manna model. One finds that a metastable AB-symmetry conserving state is
transiently observable and that it has the potential to influence the width of
the scaling regime, in particular in two dimensions.
  The AAS model mimics the behavior of integrate-and-fire neurons which
propagate activity independently of the input received, as long as the
threshold is crossed. Abstracting from regular lattices, one can identify sites
with neurons and consider quenched networks of neurons connected to a fixed
number $G$ of other neurons, with $G$ being drawn from a suitable distribution.
The neuronal activity is then propagated to $G$ other neurons. The AAS model is
hence well suited for theoretical studies of nearly critical brain dynamics. We
also point out that the waiting-time distribution allows an avalanche-free
experimental access to criticality.
