ABSTRACT
The Epilogue offers commentary on salient features of Chapters 1, 2, 4, and 5:
Chapters 1 and 5
For our model, its impaired memory and its working definition of the bifurcation value b 1 are consistent with the controversial school of thought that
the epileptogenesis of mesial temporal lobe epilepsy in part involves the the dysfunction of Group I metabotropic glutamate receptors Types 1 and 5; and
the treatment with (experimental) anticonvulsant agents, which are competitive antagonists to the Group I metabotropic glutamate receptors, might be beneficial to the individual with mesial temporal lobe epilepsy.
Chapter 2
We claim that our model is appropriate for application to normal and abnormal neuronal processes for the following reasons:
The human brain functions recursively in many respects. Our model, as a difference equation, is a recursive relation.
There, of course, is a bound on the number of neurons in the human brain. Also, immediately after being stimulated, neurons are refractory to further stimulation for a short period of time. As a consequence of these two features, if most (resp. few) neurons in a region of the brain are stimulated in one instance, then few (resp. most) neurons remain to be stimulated in the next instance. Due to the reciprocal nature of the model as a max-type difference equation, if the density of activated neurons in the past, xn and xn− 1, is large (resp. small), then the density of activated neurons in the present, xn+1, is small (resp. large).
Because of hippocampal sclerosis in the individual with mesial temporal lobe epilepsy, memory of the individual is impaired. Because the model as a max-type difference equation has the smallest delay possible and is only second-order, its output, xn+1, is dependent on the fewest past states, xn and xn− 1, possible.
Chapter 4
While biological implications of our model as related to mesial temporal lobe epilepsy have been our primary focus, we have had the good fortune to discover what we believe to be a purely mathematical novelty, namely, rippled almost periodic behavior. Also, while a nonpositive Lyapunov exponent has pretty much ruled out that solutions with rippled almost periodic behavior are chaotic orbits, such solutions exhibit complex behavior that is hard to predict.
