Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations

A partial algebra construction of Gr\"atzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that every finite distributive lattice is representable, seen as a special case of the Finite Lattice Representation Problem. The construction of this proof brings together Birkhoff's representation theorem for finite distributive lattices, an emphasis on boolean lattices when representing finite lattices, and a perspective based on inequalities of partially ordered sets. It may be possible to generalize the techniques used in this approach. Other than the aforementioned representation theorem only elementary tools are used for the two theorems of this note. In particular there is no reliance on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice to be representable (see William J. DeMeo), such as the closure method

Burst-Timing-Dependent Plasticity of NMDA Receptor-Mediated Transmission in Midbrain Dopamine Neurons

SummaryBursts of spikes triggered by sensory stimuli in midbrain dopamine neurons evoke phasic release of dopamine in target brain areas, driving reward-based reinforcement learning and goal-directed behavior. NMDA-type glutamate receptors (NMDARs) play a critical role in the generation of these bursts. Here we report LTP of NMDAR-mediated excitatory transmission onto dopamine neurons in the substantia nigra. Induction of LTP requires burst-evoked Ca2+ signals amplified by preceding metabotropic neurotransmitter inputs in addition to the activation of NMDARs themselves. PKA activity gates LTP induction by regulating the magnitude of Ca2+ signal amplification. This form of plasticity is associative, input specific, reversible, and depends on the relative timing of synaptic input and postsynaptic bursting in a manner analogous to the timing rule for cue-reward learning paradigms in behaving animals. NMDAR plasticity might thus represent a potential neural substrate for conditioned dopamine neuron burst responses to environmental stimuli acquired during reward-based learning