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Behavioral systems, understanding it as an emergent system comprising the environment and organism subsystems, includes the spatial dynamics as a basic dimension in natural settings. Nevertheless, under the standard approaches in the experimental analysis of behavior that are based on the single response paradigm and the temporal distribution of these discrete responses, the continuous analysis of spatial behavioral-dynamics has been a scarcely studied field. The technological advancements in computer vision have opened new methodological perspectives for the continuous sensing of spatial behavior. Derived from them, recent studies suggest that there are multiple features embedded in the spatial dynamics of behavior, such as entropy, and that they are affected by programmed stimuli (e.g., schedules of reinforcement), at least, as much as features related to discrete responses. Despite the progress, the characterization of behavioral systems is still segmented, and integrated data analysis and representations between discrete responses and continuous spatial behavior, are exiguous. Machine Learning advancements, such as t-SNE, variable ranking, among others, provide invaluable tools to crystallize an integrated approach for the analysis and representation of multidimensional behavioral-data. Under this rational, the present work: 1) proposes a multidisciplinary approach for the integrative and multilevel analysis of behavioral systems; 2) shows behavioral aspects usually ignored under the standard approaches in the experimental analysis of behavior; and 3) provides sensitive behavioral measures, based on spatial dynamics, and useful data representations for the study of behavioral systems. In order to exemplify and evaluate our approach, the spatial-dynamics of behavior embedded in phenomena relevant to the behavioral science, namely water-seeking behavior and motivational operations, is examined, showing aspects of behavioral systems hidden until now. Keywords: Behavioral systems; spatial-behavioral dynamics; time-based schedules; water-seeking behavior; motivational operations; Machine Learning; t-SNE; entropy

Understanding behavioral systems as emergent systems comprising the environment and organism subsystems, include spatial dynamics as a primary dimension in natural settings. Nevertheless, under the standard approaches, the experimental analysis of behavior is based on the single response paradigm and the temporal distribution of discrete responses. Thus, the continuous analysis of spatial behavioral dynamics is a scarcely studied field. The technological advancements in computer vision have opened new methodological perspectives for the continuous sensing of spatial behavior. With the application of such advancements, recent studies suggest that there are multiple features embedded in the spatial dynamics of behavior, such as entropy, and that they are affected by programmed stimuli (e.g., schedules of reinforcement) at least as much as features related to discrete responses. Despite the progress, the characterization of behavioral systems is still segmented, and integrated data analysis and representations between discrete responses and continuous spatial behavior are exiguous in the experimental analysis of behavior. Machine learning advancements, such as t-distributed stochastic neighbor embedding and variable ranking, provide invaluable tools to crystallize an integrated approach for analyzing and representing multidimensional behavioral data. Under this rationale, the present work (1) proposes a multidisciplinary approach for the integrative and multilevel analysis of behavioral systems, (2) provides sensitive behavioral measures based on spatial dynamics and helpful data representations to study behavioral systems, and (3) reveals behavioral aspects usually ignored under the standard approaches in the experimental analysis of behavior. To exemplify and evaluate our approach, the spatial dynamics embedded in phenomena relevant to behavioral science, namely, water-seeking behavior and motivational operations, are examined, showing aspects of behavioral systems hidden until now.

This paper presents an approximation scheme for the Kantorovich-Rubinstein mass transshipment (KR) problem on compact spaces. A sequence of finite-dimensional linear programs, minimal cost network flow problems with bounds, are introduced and it is proven that the limit of the sequence of the optimal values of these problems is the optimal value of the KR problem. Numerical results are presented approximating the Kantorovich metric between distributions on [0,1].

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