Wave-front solution behaviour for continuous neural networks with lateral inhibition

^** Email: rsittipong{at}yahoo.com^*** Corresponding author. Email: jbell{at}math.umbc.edu^**** Email: scylb{at}mahidol.ac.th In this paper we discuss the shape of travelling wave-frontsolutions to a model for a single continuous layer of nervecells originally introduced by Amari (1977, Dynamics of patternformation in lateral inhibition-type neural fields. Biol. Cybern.,27, 77–87). The neural field is homogeneous and isotropic,and the connection function is one of lateral inhibition type,meaning that nearby connecting cells have an excitatory influence,while more spatially distant cells impose an inhibitory influence.We give results on the shape of the wave-front solutions, whichare non-monotone and exhibit different shapes depending on thesize of a threshold parameter. For a layer of excitatory cellsindirectly inhibited by a second layer of cells, we derive resultson the qualitative behaviour of wave-fronts for changes in parametersrepresenting the inhibitory firing threshold and the time-scaleof the inhibition process. This study shows how intrinsic celland network parameter can interact to shape global responseproperties.     

A mathematical model of prolactin secretion: Effects of dopamine and thyrotropin-releasing hormone

Prolactin (PRL) is secreted in a pulsatile manner by lactotroph cells in the anterior pituitary gland and displays a circadian rhythm as well as increases in response to stress, sexual intercourse, breast stimulation, and suckling. We propose a mathematical model of prolactin secretion which is mainly controlled by the inhibiting effect of dopamine (DA) and the stimulating effect of thyrotopin releasing hormone (TRH). By applying the singular perturbation technique, the conditions are derived under which our model exhibits a periodic solution corresponding to the normal secretory pattern of PRL which has been observed as a series of daily pulses, occurring every 2–3 hours. Numerical investigations also show that chaotic time series is admitted by our model which resembles irregular patterns observed in PRL concentration profiles of patients with microprolactinoma and macroprolactinoma. Explaining the conditions that delineate varying dynamic behavior in this nonlinear system in terms of the removal rates of the three state variables, the removal rate d₁ of TRH seems to play the most important role in identifying different physiological conditions.