Yes, there is a threshold of taste.
According to this article
The taste system encodes information about the quantity as well as the identity of stimuli. In general, the higher the stimulus concentration, the greater the perceived intensity of taste. Threshold concentrations for most ingested tastants are quite high, however. For example, the threshold concentration for citric acid is about 2 mM; for salt (NaCl), 10 mM; and for sucrose, 20 mM. Since the body requires substantial concentrations of salts and carbohydrates, taste cells may respond only to relatively high concentrations of these essential substances to promote an adequate intake. Clearly, it is advantageous for the taste system to detect potentially dangerous substances (e.g., bitter-tasting plant compounds) at much lower concentrations. Thus, the threshold concentration for quinine is 0.008 mM, and for strychnine 0.0001 mM. As in olfaction, gustatory sensitivity declines with age. Adults tend to add more salt and spices to food than children. The decreased sensitivity to salt can be problematic for older people with electrolyte and/or fluid balance problems.
In fact, all of these tastes can be detected over the full surface the tongue. However, different regions of the tongue do have different thresholds. Because the tip of the tongue is most responsive to sweet-tasting compounds, and because these compounds produce pleasurable sensations, information from this region activates feeding behaviors such as mouth movements, salivary secretion, insulin release, and swallowing. In contrast, responses to bitter compounds are indeed greatest on the back of the tongue. Activation of this region by bitter-tasting substances elicits protrusion of the tongue and other protective reactions that prevent ingestion
And in most of the individuals, this threshold is same for both right and lefts parts of the tongue.
Now as stated by the first article, taste threshold depends on the age of the individual.
More interestingly, it also depends on the individual's state of mind, i.e. whether depressed or anxious.
Circumstances in which serotonin (5-HT) and noradrenaline (NA) are altered, such as in anxiety or depression, are associated with taste disturbances, indicating the importance of these transmitters in the determination of taste thresholds in health and disease.
Measurement of taste function in healthy humans before and after a 5-HT reuptake inhibitor, NA reuptake inhibitor, or placebo showed that enhancing 5-HT significantly reduced the sucrose taste threshold by 27% and the quinine taste threshold by 53%. In contrast, enhancing NA significantly reduced bitter taste threshold by 39% and sour threshold by 22%.
Hence threshold value varies from substance to substance as well as from individual to individual.
Here is a good article giving threshold values of different chemicals in various food items.
Hope all your questions are solved :)