Calculating the Half Life of Awareness

John Creighto

I'm not sure if memory decays exponential or not but because linear systems are easy to work with then a linear decay model with a non linear output would be a simple way to describe the underlying dynamics.

That is we can describe the state of learning an idea in terms of short term, and long term reinforcement which is modeled by a linear differential equation.

Let

[tex]S[/tex] = short term reinforcement
[tex]L[/tex] = long term reinforcement
[tex]\lambda_S[/tex] be the forgetting rate for an idea reinforced in the short term
[tex]\lambda_L[/tex] be the forgetting rate for an idea reinforced in the long term
[tex]\beta_L[/tex] be the amount one review of the idea reinforces it in the long term
[tex]\beta_S[/tex] be the amount one review of the idea reinforces it in the short term
[tex]F[/tex] be the frequency at which the idea is encountered

Then one might propose the following simple model for idea/memory reinforcement

[tex]\dot{S} = -\lambda_S S + \beta_S F[/tex]
[tex]\dot{L} = -\lambda_L L + \beta_L F[/tex]

It is convenient to model the system without parameters which express coupling between S and L because most linear systems can be decoupled into independent modes though diagonialization of the state space matrix. Therefore coupling parameter would make the parameter set redundant.

Now how might we estimate our parameters? Well, first we need an output model to relate our reinforcement model to the probability of remembering something. For instance we could define the probabiliyt of remembering something as follows:

[tex]P_r=1-exp(-(S+L))[/tex]

This equation is invertible. So we can calculate [tex]S+L[/tex] for a given [tex]P_r[/tex] without knowing any of the underlying paramaters of the above differential equation. One could try various reinforcements schemes and test the probability of remembering something with this scheme. This gives a value of S+L for each reinforcement scheme. Once S+L is estimated for several reinforcement schemes then the parameters that describe the underlying linear dynamics can be estimated.

As a final we may wish to also include some input dynamics. For instance [tex]F[/tex] for long term memory maybe be roughly the number of times the idea is encountered within a year. While, [tex]F[/tex] for short term memory maybe the the number of times the idea is encountered in a day. Many possibilities are available and we could represent these input dynamics as a linear or nonlinear filter.

floppyflaps

I'm an undergrad in sociology and I've always been curious in mathematical models of human behaviour, but there's serious limitations to them.

Similar to other sciences, it's difficult to categorize exactly what information from what field(s) need to be used. To me, it seems like the best approach to this memory problem would be from a neuroscience perspective. Neuroscience is kind of an esoteric field to be quoting in your paper though. Not to mention such approximations would be completely nature, without any respect to nurture.

I'm sorry I can't help more, but such an endeavour would be tantamount to deriving an equation for all of human behaviour. An equation where you can input various variables, and have an output for the behaviour of what a person will do next. Specific theories, like game theory do a pretty good job at modelling the behaviour of what people should do in a conflict scenario, but it morphs into what people should do, rather than what they actually end up doing.

I'm sorry for not being much of help, so I'm going to give you a related concept that you can research. It's called narcotizing dysfunction, and it's relatively new research (maybe a decade old) into why people stop caring about things once they're given more information about the situation.