We propose a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis (TB). Uniform asymptotic stability of the unique endemic equilibrium of the fractional-order TB model is proved, for any α ∈ (0, 1). Numerical simulations for the stability of the endemic equilibrium are provided.Uniform asymptotic stability of a fractional tuberculosis model Communicable diseases have always been an important part of human history [6]. Fractional-order differential system models for infectious disease dynamics have been introduced in recent years [1,14,17,28,30,44]. In [30], the authors propose a fractional-order model and show, through numerical simulations, that the fractional models fit better the first dengue epidemic recorded in the Cape Verde islands off the coast of west Africa, in 2009, when compared with the standard differential model. The authors of [17] show that a nonlinear fractional order epidemic model is well suited to provide numerical results that agree very well with real data of influenza A (H1N1) at the population level. In [28], a fractional model for malaria transmission is considered and numerical simulations are done for the variation of the values of the fractional derivative and of the parameter that models personal protection. In [44], fractionalorder derivatives are introduced into an HIV infection model and local asymptotic stability is proved. The authors of [14] introduce fractional-order derivatives into a model of HIV infection of CD4 + T-cells and analyze the local asymptotic stability of the equilibrium points. In [41], the uniform asymptotic stability is proved extending the Volterra-type Lyapunov functions to fractional-order epidemic systems. Fractional-order predator-prey models are investigated in [2]. In particular, existence and uniqueness of solutions are proved, and stability of equilibrium points studied. Numerical solutions of such models were obtained [2]. Here we are interested to investigate fractional calculus with respect to tuberculosis.

Many of our everyday tasks require the control of the serial order and the timing of component actions. Using the dynamic neural field (DNF) framework, we address the learning of representations that support the performance of precisely time action sequences. In continuation of previous modeling work and robotics implementations, we ask specifically the question how feedback about executed actions might be used by the learning system to fine tune a joint memory representation of the ordinal and the temporal structure which has been initially acquired by observation. The perceptual memory is represented by a self-stabilized, multi-bump activity pattern of neurons encoding instances of a sensory event (e.g., color, position or pitch) which guides sequence learning. The strength of the population representation of each event is a function of elapsed time since sequence onset. We propose and test in simulations a simple learning rule that detects a mismatch between the expected and realized timing of events and adapts the activation strengths in order to compensate for the movement time needed to achieve the desired effect. The simulation results show that the effector-specific memory representation can be robustly recalled. We discuss the impact of the fast, activationbased learning that the DNF framework provides for robotics applications.

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Many of our sequential activities require that behaviors must be both precisely timed and put in the proper order. This paper presents a neuro-computational model based on the theoretical framework of Dynamic Neural Fields that supports the rapid learning and flexible adaptation of coupled order-timing representations of sequential events. A key assumption is that elapsed time is encoded in the monotonic buildup of self-stabilized neural population activity representing event memory. A stable activation gradient over subpopulations carries the information of an entire sequence. With robotics applications in mind, we test the model in simulations of a learning by observation paradigm, in which the cognitive agent first memorizes the order and relative timing of observed events and, subsequently, recalls the information from memory taking potential speed constraints into account. Model robustness is tested by systematically varying sequence complexity along the temporal and the ordinal dimension. Furthermore, an adaptation rule is proposed that allows the agent to adjust in a single trial a learned timing pattern to a changing temporal context. The simulation results are discussed with respect to our goal to endow autonomous robots with the capacity to efficiently learn complex sequences with time constraints, supporting more natural human-robot interactions.

Abstract. We present a novel dynamic neural field model consisting of two coupled fields of Amari-type which supports the existence of localized activity patterns or "bumps" with a continuum of amplitudes. Bump solutions have been used in the past to model spatial working memory. We apply the model to explain input-specific persistent activity that increases monotonically with the time integral of the input (parametric working memory). In numerical simulations of a multi-item memory task, we show that the model robustly memorizes the strength and/or duration of inputs. Moreover, and important for adaptive behavior in dynamic environments, the memory strength can be changed at any time by new behaviorally relevant information. A direct comparison of model behaviors shows that the 2-field model does not suffer the problems of the classical Amari model when the inputs are presented sequentially as opposed to simultaneously.