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In this paper we study the pure-slow and pure-fast dynamics of the disparity convergence of the eye movements second-order linear dynamic mathematical model under time varying stimuli. Performing simulation of the isolated pure-slow and pure-fast dynamics, it has been observed that the pure-fast component corresponding to the eye angular velocity displays abrupt and very fast changes in a very broad range of values. The result obtained is specific for the considered second-order mathematical model that does not include any saturation elements nor time-delay elements. The importance of presented results is in their mathematical simplicity and exactness. More complex mathematical models can be built starting with the presented pure-slow and pure-fast first-order models by appropriately adding saturation and time-delay elements independently to the identified isolated pure-slow and pure-fast first-order models., Competing Interests: The author(s) declare(s) that the contents of the article are in agreement with the ethics described in http://biblio.unibe.ch/portale/elibrary/BOP/jemr/ethics.html and that there is no conflict of interest regarding the publication of this paper.

In this paper we consider a mathematical model of HIV-virus dynamics and propose an efficient control strategy to keep the number of HIV virons under a pre-specified level and to reduce the total amount of medications that patients receive. The model considered is a nonlinear third-order model. The third-order model describes dynamics of three most dominant variables: number of healthy white blood cells (T-cells), number of infected T-cells, and number of virus particles. There are two control variables in this model corresponding to two categories of antiviral drugs: reverse transcriptase inhibitors (RTI) and protease inhibitors (PI). The proposed strategy is based on linearization of the nonlinear model at the equilibrium point (steady state). The corresponding controller has two components: the first one that keeps the system state variables at the desired equilibrium (set-point controller) and the second-one that reduces in an optimal way deviations of the system state variables from their desired equilibrium values. The second controller is based on minimization of the square of the error between the actual and desired (equilibrium) values for the linearized system (linear-quadratic optimal controller). The obtained control strategy recommends to HIV researchers and experimentalists that the constant dosages of drugs have to be administrated at all times (set point controller, open-loop controller) and that the variable dosages of drugs have to be administrated on a daily basis (closed-loop controller, feedback controller).

In this paper we show how to separate the slow and fast dynamics of the disparity convergence of the eye movements dynamic model. The dynamic equations obtained determine the modified slow dynamics that takes into account the impact of the fast dynamics and the modified fast dynamics that takes into account the impact of the slow dynamics. The slow fast decoupling is achieved by finding analytical solutions of the transformation equations used. The transformed slow and fast subsystems have very simple forms. Having separated the slow and fast dynamics completely, neural control problems for the slow and fast eye movements dynamics can be independently studied and better understood.