I haven't done any measurements; but, knowing that 1 kcal = 4184 Joules and that the basal metabolic rate (BMR) in an adult is 1.1 kcal/min, it means that you are looking at something that consumes 4184000 lesser energy.
As an educated guess, and if we are talking about movement, I believe that we are looking at something as peristalsis in smooth muscle or the like.
I would be interested in the findings you achieve!
Two more possibilities: axonal transport and pupillary dilator muscle.
Isothermal microcalorimetry (IMC) is used to measure real-time metabolic rates in cultures of mammalian cells, bacteria, fungi and even small multi-cell creatures. The magnitudes measured (W=J/s) and cumulative energy over time (J) depend on the size of the specimen. (A single mammalian cell in culture has a metabolic rate of ca. 30 pico watts. A single bacterium has a metabolic rate of ca. 3 pico watts.) IMC instruments have huge dynamic ranges and the same instrument can often measure flows ranging from milli J/s to micro J/s. For a complete description of IMC and its applications, with 60+ references, see
https://en.wikipedia.org/wiki/Isothermal_microcalorimetry
I found this paper about locust jumping (you can see mJ in the abstract): http://jeb.biologists.org/content/63/1/53.1.abstract
Also, energy consumed by a single wingbeat of a hummingbird (or similar-sized insects like hawkmoth) during hovering is of the order of 10 mJ. You can estimate this by the available data from e.g. Chai & Dudley (1995) Nature where they showed the mechanical power as 100-200 W/kg which is mW/g = mJ/(g s). The ruby-throated hummingbird weighs around 4 grams and hovers at around 50 Hz (=1/s), so 150*4/50 = 12 mJ/wingbeat, though this is very rough.
http://jeb.biologists.org/content/63/1/53.1.abstract
I have attached a paper which provides a grand overview of biological heat--from the molecular scale to the mammalian scale. It is a true classic.
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