An Adaptive LIF is just one approximation of a biological learning rule. It is not a model for how biology actually works.
The most-studied biological learning rule is long-term potentiation in the hippocampus, though similar processes occur elsewhere including neocortex. "LTP" can come in shorter or longer forms, but the longest forms of LTP are potentially permanent. The processes for maintaining this permanence are referred to as maintenance and consolidation. Some of the longer lasting forms of LTP have been shown to last weeks to years (for example, see Abraham et al., 2002).
At that point, any 'adaptation' is just as well thought of as additional plasticity, because living neurons are constantly getting input and adjusting synaptic weights. Depression is just as important in biological circuits as is potentiation, and depressed circuits can atrophy completely, effectively being permanently lost.
All nervous connectivity is due to some sort of plasticity at some point in development, and some of those connections will persist throughout the lifetime of the organism. However, synapse weights are constantly being updated in a living system. It isn't really appropriate to think of any certain weight as being permanent, while also not appropriate to think of any arbitrary synapse as possessing an adaptation time constant.
Abraham, W. C., Logan, B., Greenwood, J. M., & Dragunow, M. (2002). Induction and experience-dependent consolidation of stable long-term potentiation lasting months in the hippocampus. Journal of Neuroscience, 22(21), 9626-9634.
Collingridge, G. L., Peineau, S., Howland, J. G., & Wang, Y. T. (2010). Long-term depression in the CNS. Nature reviews neuroscience, 11(7), 459.
Malenka, R. C., & Nicoll, R. A. (1999). Long-term potentiation--a decade of progress?. Science, 285(5435), 1870-1874.
Lynch, M. A. (2004). Long-term potentiation and memory. Physiological reviews, 84(1), 87-136.