What is "integrated spindle activity" in plain English?
Integrated spindle activity is just this paper's measure of the intensity of spindles. It isn't really a term of art, there is no defined thing "integrated spindle activity," it is just a description for their method of quantifying spindles, or in other words operationalizing their definition of spindle activity.

Spindles are brief bouts of (relatively) high-frequency oscillatory activity that occur occasionally during sleep - often they are called "sleep spindles."

Figure 1 of the paper you link shows their method for detecting and measuring these spindles.



Ferrarelli, F., Huber, R., Peterson, M. J., Massimini, M., Murphy, M., Riedner, B. A., ... &amp; Tononi, G. (2007). Reduced sleep spindle activity in schizophrenia patients. American Journal of Psychiatry, 164(3), 483-492.

In summary, the top trace is a raw EEG trace. Spindles are present in this trace, but the authors want to quantify spindle activity in some way. By filtering between 12-15Hz, the spindles become more obvious (middle): they are the high-amplitude things bracketed by vertical lines.

On the bottom, they have rectified and smoothed the signal to approximate the amplitude of the spindles (that is, they aren't interested in the area under the curve in the middle plot, which has negative and positive going components, they just want a measure of the whole event).

They are defining the integrated spindle activity as the integral (i.e., the area under the curve, just like in calculus) of bouts of activity that exceed some threshold, indicated by the area shaded blue.

In summary, to answer your question, there isn't really a better or simpler English way to say "integrated spindle activity" without losing information for the reader - they are indeed referring to mathematical integration. You could say something like "total spindle activity" but then it is much less clear how exactly you are quantifying spindle activity. Probably the best plain English description of integration is the 'area under the curve' of a function or measurement, though this isn't a perfect mathematical description.

As someone who is familiar with general methods for analyzing electrophysiological data, it was pretty clear to me what they meant by "integrated spindle activity" before opening the paper - this is a fairly common way to report the amplitude of oscillatory activity, because it captures both average amplitude and duration in a sensible way.