Abstract: The ∞uctuations in the vertical displacement angle of a stick balanced at the flngertip exhibit on-ofi intermitency. However, even a skilled balancer cannot indeflnitely maintain a stick balanced at their flngertip. The survival function for stick balancing, P(tesc > t), is shown to have the form of a Weibul function, exp(i‚t) fl , where ‚ is a constant and fl > 1. The measured survival function can be reproduced by a stochastic delayed discrete map possessing only unstable solutions. These observations emphasize the importance of state{dependent, or paametric, noise in this balancing task.

Abstract: The remarkable success of deep brain stimulation (DBS) for treating neurological disorders has increased interest in the physiology and pathophysiology of the basal ganglia (BG) and has forced re-evaluation of present theories of brain function. Recent studies of BG neuronal physiology suggest that the BG form reentrant closed-loop networks with the cortex (Ctx) and thalamus (Th), generating non-linear oscillators. Further, these oscillators are dynamically coupled to form networks with unique local properties that have been described for coupled oscillators and larger-scale properties that have been described for scalefree networks. Here, I integrate results of neurophysiological studies with those of mathematical and computational studies of non-linear oscillators and scale-free networks and propose a new theory of BG-Th-Ctx function. I contrast the new theory with present theories, not only regarding BG-Th-Ctx function but also how function or behavior is represented in brain activity in general. A corollary to the new theory is that function is determined by the network and not by distinct anatomical structures. This notion contrasts with the current modulist explanation of normal function and disease pathophysiology.

Abstract: We investigate the skill of rhythmically bouncing a ball on a racket with a focus on the mathematical modeling of the stability of performance. As a first step we derive the deterministic ball bouncing map as a Poincar´e section of a sinusoidally driven bouncing ball. Subsequently, we show the ball bouncing map to have a passively stable regime. More precisely, for negative racket acceleration at impact, no control of racket amplitude or frequency is necessary for stable performance. Support for the model comes from a motor learning study, where a decrease in variability covaries with a change of mean acceleration at impact towards more negative values. For a more fine-grained test of the model we develop a stochastic version of it, by adding Gaussian white noise to the dynamics. We then test the model predictions for the correlation functions. We find that the observed correlation functions match the theoretical ones quite well, lending new support for the model. Lastly, we compare the observed recovery from a sudden change with which the ball leaves the racket with model predictions. We find a mismatch between data and model in the sense that the model is too “slow”. We take this failure of the ball bouncing model as an impetus to further develop the model. In the perturbation study, we observe a significant modulation of the racket period but not of the racket amplitude. Thus, racket period seems a candidate state variable that should be included the ball bouncing map.

Abstract: : Powerful nonlinear approaches for missile autopilot design have recently emerged in the literature, which have the potential to deliver improved missile performance. However, the lack of computational methods has made it difficult for the practicing engineers to exploit these techniques in routine applications. Another factor that has slowed their application is that the missile models are generally available in the form of simulations, rather than as compact set of differential-algebraic equations. This paper discusses five different approaches for computer-aided nonlinear control system design that ameliorate these difficulties. Since these design techniques are based on simulation models, they enable direct synthesis of nonlinear autopilots using missile models of arbitrary complexity. Airframe stabilization of a nonlinear, longitudinal missile model is used to illustrate the design techniques.

Abstract: Parkinson's disease is a complex disorder for which there is no known cure. Nor do we understand fully the origin of one of the disease's cardinal symptoms: tremor. A non-traditional approach to research in Parkinson's disease and in Parkinsonian tremor involves the application of mathematical models. The purpose of this article is to review briefly the contributions of mathematical models to the study of Parkinsonian tremor. There is little evidence that modelling attempts have built on previous ones but there has been a trend to move the focus of modelling from the periphery to the brain, and from abstracted to more physiologically detailed views. We hope that this review will encourage more mathematical modelling in the study of Parkinson's disease and Parkinsonian tremor.

Abstract: Methods for determining empirical linear transfer functions of experimentally obtained or model vestibular neurons from the modulation of their discharge firing rate by current input have been developed and shown to distinguish between regularly firing neurons with prominent after hyperpolarizations, AHP (Type A), and cells with more complex AHP's (Type B) An increase in magnitude of the modulated spike discharge rate with frequency has been observed to be greater in Type B compared to Type A neurons In order to better interpret this finding, simulations of spike frequency modulation were done with known Hodgkin-Huxley type neuronal models consisting of non-linear differential equations that show the essential behavior of vestibular neurons, namely previously published Type A and B neuronal models (AvRon and Vidal, 1999) An empirical spike rate transfer function was obtained from the analysis of the spike rate modulation at different stimulating sinusoidal frequencies, and was compared with the theoretical linear transfer function obtained from the exact linearized equations and type B neuronal models These linearized theoretical frequency domain functions reveal the underlying voltage dependent conductances by showing resonant behavior, increased impedance with activation of negative conductances and specific kinetic responses dependent on the time constants of the active conductances It is shown that the theoretical linear transfer functions can be approximated by empirical spike rate transfer functions, which indicates that the basic strategy of this analysis can be applied to data from real neurons The simulations demonstrate that such an approach is a valid experimental method that allows one to estimate membrane properties from frequency modulation of discharge rates measured extracellularly

Abstract: A realistic model of the human ventilatory controller during sleep is presented. The model consists of two chemoreflex feedback loops (central and peripheral) each with its own delay and gain. An initial stability analysis of the resultant model produces a thumb shape region of stability for the steady state solution. Changes in cerebral blood flow in the model had profound changes in the shape of the region of the stability. Increasing the gain of either the central or the peripheral loop yielded periodic solutions consistent with observed data from humans. The model provides a new tool for understanding human ventilation during sleep. The interaction of the two loops produces complex dynamics. Understanding these dynamics will assist in the treatment of sleep-disordered breathing.

Abstract: The monotone iterative technique is a fruitful method of immense value which played a crucial role in unifying a wide variety of nonlinear problems. This technique was developed for set differential equations in [11], in a very general set up which includes various known and unknown results. In this paper, we follow that approach and develop the monotone method for impulsive set differential equations in all its generality.

Abstract: The dynamical features of spike train generation in the presence of noise are investigated for three difierent models of neural rhythm generators: a single neuron model that simulates impulse pattern modulation for temperature encoding in mammalian cold receptors, a minimal neural network that describes transitions between beta and gamma rhythms in the brain and an electronic switching device that represents a simple breathing rhythm generator for a snail. We show that noise can explain a number of peculiarities in the observed spike trains, cause coherent switchings between difierent states, and induce new rhythms in small neural ensembles.

Abstract: In the framework of a simplified mathematical model, a technique for the control of the "path capture" and "steady descent" flight phases of the ALFLEX reentry vehicle (during its final approach and landing flight) is pre- sented. The technique consists of successive, prescribed and quick changes of the values of the aileron and elevator angles and lead of the state parameters of the vehicle along the stable manifolds of the hyperbolic equilibrium states, correspond- ing to the "path capture" and "steady descent" flight phases, to these equilibrium states. The obtained results are compared to those reported in the experimental flight data.

Abstract: The objective is to show, in a simplified mathematical model of the ALFLEX reentry vehicle, that it is possible to recover with a single maneuver a zero roll rate equilibrium state from a high roll rate state into which the vehicle has accidentally fallen during its final approach and landing phase. For this purpose, bifurcation analysis is undertaken along different equilibrium contours and it is shown that some unstable equilibrium states are very sensitive to the perturbations and to the change of the control surface angles. For the asymptotically stable zero roll rate equilibrium states the domains of attraction are estimated. Using the domain of attraction of a particular asymptotically stable zero roll rate equilibrium state, it is shown that "any" high roll rate state can be transferred to this equilibrium state by a single maneuver.

Abstract: The missile pitch-axis autopilot design is reconsidered using a classical result in Linear Parameter-Varying (LPV) control. We consider a nonlinear missile model with a highly nonlinear dependence on the scheduling variables on which the application of traditional finite-dimensional LPV methods as the LFT- or polytopic-based techniques is difficult or conservative. The main objective is to obtain gain-scheduled autopilots that guarantee closed-loop $L_2$-gain performances with little conservatism by providing a general LPV/gridding technique state) Lyapunov functions. An iterative and computationally feasible procedure is proposed to construct functional dependencies for the Lyapunov variables that are rich enough to ensure a suitably specified performance level over a wide range of flight conditions. Gain-scheduled autopilots are synthesized using the proposed approach and results are compared with different LPV synthesis strategies.

Abstract: Gain scheduled control is one very useful control technique for linear parameter-varying (LPV) and nonlinear systems. A disadvantage of gainscheduled control is that it is not easy to design a controller that guarantees the global stability of the closed-loop system over the entire operating range from the theoretical point of view. Another disadvantage is that the interpolation increases in complexity as number of scheduling parameters increases. As an improvement, this paper presents a gain-scheduling control technique, in which fuzzy logic is used to construct a model representing a quasi-LPV or a nonlinear missile and to perform a control law. The fuzzy inference system is generated using a multiobjective evolutionary algorithm to optimise the performance characteristics of the plant.

Abstract: This papers deals with the gain-scheduled control of the pitch channel of a missile with large variations of Mach and altitude. A reduced order $H_\infty$ loopshaping control technique allows designing discrete-time controllers of very small order for different flight points; the final control law consists in interpolating the parameters of the controllers so obtained.

Abstract: In this paper, we are concerned with the existence of traveling wave solutions to a system of reaction-diffusion equations. The latter describes the propagation of evolving fronts for a two-step chemical reaction. With the resulting system of two second order ODEs, three different limit conditions may be taken into account at positive infinity. For each case, we prove existence of bounded solutions on $(0,\infty)$ for any value of the wave speed; the analysis includes the case where one of the coefficients vanishes. Then we show that such solutions can be extended to the negative half-line to get unbounded solutions. Shooting type techniques, upper and lower solution method as well as fixed point arguments and Leray-Schauder topological degree are used.

Abstract: In this work, the convergence and stability analysis are investigated for stochastic parabolic partial differential equations of Ito-type. A very general comparison theorem is obtained in the context of vector Lyapunov-like functions and differential inequalities. Furthermore, this comparison theorem has been applied to derive sufficient conditions for various concepts of stability and convergence of the equilibrium state of the system. In addition, an example is given to illustrate the significance of the presented results.

Abstract: Drug therapies are often designed to reproduce the physiological uctuations in normal biological agents, including hormones. These regimens entail the need for a periodic, yet sustained administration, and thus require the generation of oscillatory variations in concentrations. We extend (in part, by the explicit use of time-delayed arguments) a system introduced recently along this design, in which the oscillations are generated by the delayed response of the permeability of a membrane at the boundary of a reaction chamber. We completely analyse the stability of the equilibrium, and present conditions under which Hopf bifurcations are present. We also provide further conditions for multiple mode instabilities to occur, by the occurence of double Hopf bifurcations.

Abstract: This paper presents a novel modeling technique of image edge detection based on the complexity of images. The method is composed of an edge level detector and an adaptive edge level analysis. Edge level detection is related to how much attention a person needs to use to detect an edge. The edge level stage is based on the analysis of gradient information of images that is considered as fuzzy information. The adaptive edge detector section defines which edges will be shown according to the complexity of the image. Complexity is considered as a subjective characteristic that is represented by a fuzzy interpretation of the percentage of edges in an image. The edges that appear in the final result are selected through an integration of edge levels using fuzzy rules. Results of the new algorithm indicate that the behavior of the method is adaptive according to the complexity of the image as stated in our theory and they turn to be better than those obtained with conventional methods as shown in the paper.

Abstract: We give an overview of the analysis of a new type of bursting ("ghost-bursting") seen in pyramidal cells of weakly electric fish. We start with the experimental observations and characterization of the bursting, describe a compartmental model of a pyramidal cell that undergoes ghostbursting and the development of another simplified yet realistic conductance-based model of this cell. This model then motivates a minimal leaky integrate-and-fire model that also has the qualitative features of ghostbursting.

Abstract: In this paper, we propose a novel attitude control strategy for a hydrofoil catamaran over its whole operating range based on Takagi-Sugeno (T-S) fuzzy controller via parallel distributed compensation (PDC) method, which aims to overcome some drawback of the conventional linear quadratic regulator (LQR). First, the T-S fuzzy system and PDC algorithm are introduced. The nonlinear mathematical model of the hydrofoil catamaran is then derived to verify the proposed control strategy. The advantage in interpolation of the T-S fuzzy model ensures that the feedback gain can be obtained smoothly, while the boat's speed is shifting over the operating envelope. At the same time, the merit of the LQR control is also well retained. By use of the Matlab's LMI toolbox, a common positive definite matrix P can be found so as to guarantee the global stability of the fuzzy control system of the hydrofoil catamaran. Finally, based on the simulation model of the boat HB200B-A1, the effectiveness of the nonlinear mathematical model and the proposed T-S fuzzy controller using PDC method is demonstrated.

Abstract: We consider a single-server closed queueing system with a finite number of permanent customers. This system can also be described in terms of servicing unreliable machines by a single repairman. In addition we assume that not every machine is vulnerable. Of the total quantity, there are some that are inactive and subject to cold standby. They are not on reserve though, and this situation alters randomly, i.e. at random epochs of time, each machine, which is intact, can change its status from being active (or vulnerable) to inactive (or invulnerable). Active machines, in turn, can fail, and in this case, they form a line of defective machines, which are processed one by one by a single repairman. If all machines are intact, the repairman is idle until a next breakdown. We will use a dual multi-channel open queueing model with a variable number of active channels to investigate the processes of intact and defective machines in the closed model and develop a duality principle that enables us to simplify the original system and arrive at a closed form solution. We also use semi-regenerative techniques and demonstrate applicability on a number of real world situations that include major epidemics (such as TB), human resources, and a calibration system.