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 $z_ige2$. 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.
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