Writing in the past about evaluating bi-valence of stimuli for determining behavioral response did leave a sense of something missing in our argument of using only the static sense of the stimuli in a summation. What about if the stimuli are changing, how one accounts for the rate of change of the stimuli should be almost as important as the static strength of the stimuli. Hence, we have to find some basis of analyzing rate of change of stimuli that is negative, one that causes us to run away, or rate of change of the positive stimuli which encourages us to go for it and of course more importantly, when both stimuli are changing. It is when both are changing, we need to find a way to account for the static strength of the stimuli as modified by its rate of change in our evaluation of the resulting bi-valence of the stimuli.
To make this adjustment, we shall meander a bit and look at a simple control problem in a physical material world. It is one that is a fairly common in industry and requires implementing a control system, the control algorithm is fairly easy to understand. So, no need for any outstanding mathematics to worry about. The control problem is to control the level of some liquid in a tank, where the inflow of the liquid to the tank can be controlled while the outflow of the liquid from the tank is the uncontrolled variable and therefore tends to be random. There is a level measuring instrument and a controller whose output controls the inflow.
We do have a required value to keep the level in the tank, and so when the level rises there will be a negative error between the measured level and the required level on which the controller will act and hopefully decrease the inflow to bring the level back to its required value. The reverse will take place if the outflow increases and level falls, which would give us a positive error and the controller output should increase to increase the inflow to compensate. But note, if the error was zero, then the output from the controller too would be zero and the inflow would stop. That of course is not a good thing and therefore we add to the controller an integrator, which integrates (continuously adds) the error to add to the output such that the error can go to zero, a highly desirable condition. When the error is zero, the integrator also stops, quite clever.
But now we have introduced some dynamics into the system which will tend to make the level fluctuate. The output of the controller does not change as long as the sign of the error signal does not change, though we would like it to change in anticipation of the level of the tank coasting to the required level without overshooting the target. The fluctuations, as one can imagine, take place because the level overshoots, the sign of the error changes, the controller than acts in the opposite sense to what it was doing previously and so on and so forth. In the worst case the fluctuations can continue, a not very desirable state. There are ways to compensate, but it makes the algorithm complex and tuning it for all conditions quite impossible.
An option is to create a control law such that it flips the sign of the error signal so that the controller output changes direction before the level reaches the required level. A simple control law is the sum of the fractional value of the error and its rate of change. Hence, when the error signal is small and the rate of change is large, the output of the control law would change sign and it is possible to use this change of sign to flip the controller to act opposite to what it was doing before. Just what we wanted.
Now to get to our problem of evaluating bi-valence stimuli. We could simply use the output of the control law for each stimuli, to evaluate the sum of the stimuli and the resultant sign from the evaluation would tell us what action should have been taken. Hence, we have brought into our equation the rate of change of the stimuli, where even if one of the stimuli is small in magnitude but changing rapidly, its effect could override the larger but more stationary stimuli, something that we all should have experienced and be aware off.
Hence it is fairly well known that one of the worse punishments to be awarded is one of isolation where the unchanging environment causes the worse of mental anguish. A lot of selling is based on creating an environment of rushed decision making by the con artists and we all are liable to loose. We are built that way, it is simply a cleverness on the part of the con artist to find out what to change fast enough that you would react illogically.
Next comes, or should come the question is it possible to consider the evaluation when there are many stimuli. There seems to be no reason why not, we are mostly concerned by the sign of the resulting evaluation, the magnitude simply tells us how much effort to put in our reaction. Do we or can we come to the conclusion that we are logical and rational beings, one would hardly think so, and one hopes that there are others who read this and extend this thinking to actually create mathematical models to prove that we react like any other biological system who have been considered to have no free will or consciousness or whatever is the latest that defines humans as special.