Spectral problem of block-rectangular hierarchical matrices

The spectral problem for matrices with a block-hierarchical structure is often considered in context of the theory of complex systems. In the present article, a new class of matrices with a block-rectangular non-symmetric hierarchical structure is introduced and the corresponding spectral problem is investigated. Using these results we study a model of error generation in information sequence where such block-rectangular hierarchical matrices appear in a natural way.Comment: 18 pages, 1 figur

A reinforcement learning theory for homeostatic regulation

Reinforcement learning models address animal’s behavioral adaptation to its changing “external” environment, and are based on the assumption that Pavlovian, habitual and goal-directed responses seek to maximize reward acquisition. Negative-feedback models of homeostatic regulation, on the other hand, are concerned with behavioral adaptation in response to the “internal” state of the animal, and assume that animals’ behavioral objective is to minimize deviations of some key physiological variables from their hypothetical setpoints. Building upon the drive-reduction theory of reward, we propose a new analytical framework that integrates learning and regulatory systems, such that the two seemingly unrelated objectives of reward maximization and physiological-stability prove to be identi- cal. The proposed theory shows behavioral adaptation to both internal and external states in a disciplined way. We further show that the proposed framework allows for a unified explanation of some behavioral pattern like motivational sensitivity of different associative learning mechanism, anticipatory responses, interaction among competing motivational systems, and risk aversion

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.Comment: 37 pages, 11 figure

Spectral Statistics of "Cellular" Billiards

For a bounded planar domain $\Omega^0$ whose boundary contains a number of flat pieces $\Gamma_i$ we consider a family of non-symmetric billiards $\Omega$ constructed by patching several copies of $\Omega^0$ along $\Gamma_i$'s. It is demonstrated that the length spectrum of the periodic orbits in $\Omega$ is degenerate with the multiplicities determined by a matrix group $G$. We study the energy spectrum of the corresponding quantum billiard problem in $\Omega$ and show that it can be split in a number of uncorrelated subspectra corresponding to a set of irreducible representations $\alpha$ of $G$. Assuming that the classical dynamics in $\Omega^0$ are chaotic, we derive a semiclassical trace formula for each spectral component and show that their energy level statistics are the same as in standard Random Matrix ensembles. Depending on whether ${\alpha}$ is real, pseudo-real or complex, the spectrum has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types of statistics, respectively.Comment: 18 pages, 4 figure

A bright future for financial agent-based models

The history of research in finance and economics has been widely impacted by the field of Agent-based Computational Economics (ACE). While at the same time being popular among natural science researchers for its proximity to the successful methods of physics and chemistry for example, the field of ACE has also received critics by a part of the social science community for its lack of empiricism. Yet recent trends have shifted the weights of these general arguments and potentially given ACE a whole new range of realism. At the base of these trends are found two present-day major scientific breakthroughs: the steady shift of psychology towards a hard science due to the advances of neuropsychology, and the progress of artificial intelligence and more specifically machine learning due to increasing computational power and big data. These two have also found common fields of study in the form of computational neuroscience, and human-computer interaction, among others. We outline here the main lines of a computational research study of collective economic behavior via Agent-Based Models (ABM) or Multi-Agent System (MAS), where each agent would be endowed with specific cognitive and behavioral biases known to the field of neuroeconomics, and at the same time autonomously implement rational quantitative financial strategies updated by machine learning. We postulate that such ABMs would offer a whole new range of realism

Cocaine Addiction as a Homeostatic Reinforcement Learning Disorder

Drug addiction implicates both reward learning and homeostatic regulation mechanisms of the brain. This has stimulated 2 partially successful theoretical perspectives on addiction. Many important aspects of addiction, however, remain to be explained within a single, unified framework that integrates the 2 mechanisms. Building upon a recently developed homeostatic reinforcement learning theory, the authors focus on a key transition stage of addiction that is well modeled in animals, escalation of drug use, and propose a computational theory of cocaine addiction where cocaine reinforces behavior due to its rapid homeostatic corrective effect, whereas its chronic use induces slow and long-lasting changes in homeostatic setpoint. Simulations show that our new theory accounts for key behavioral and neurobiological features of addiction, most notably, escalation of cocaine use, drug-primed craving and relapse, individual differences underlying dose-response curves, and dopamine D2-receptor downregulation in addicts. The theory also generates unique predictions about cocaine self-administration behavior in rats that are confirmed by new experimental results. Viewing addiction as a homeostatic reinforcement learning disorder coherently explains many behavioral and neurobiological aspects of the transition to cocaine addiction, and suggests a new perspective toward understanding addiction

Homeostatic reinforcement learning for integrating reward collection and physiological stability

Efficient regulation of internal homeostasis and defending it against perturbations requires adaptive behavioral strategies. However, the computational principles mediating the interaction between homeostatic and associative learning processes remain undefined. Here we use a definition of primary rewards, as outcomes fulfilling physiological needs, to build a normative theory showing how learning motivated behaviors may be modulated by internal states. Within this framework, we mathematically prove that seeking rewards is equivalent to the fundamental objective of physiological stability, defining the notion of physiological rationality of behavior. We further suggest a formal basis for temporal discounting of rewards by showing that discounting motivates animals to follow the shortest path in the space of physiological variables toward the desired setpoint. We also explain how animals learn to act predictively to preclude prospective homeostatic challenges, and several other behavioral patterns. Finally, we suggest a computational role for interaction between hypothalamus and the brain reward system

The Simplest Piston Problem II: Inelastic Collisions

We study the dynamics of three particles in a finite interval, in which two light particles are separated by a heavy ``piston'', with elastic collisions between particles but inelastic collisions between the light particles and the interval ends. A symmetry breaking occurs in which the piston migrates near one end of the interval and performs small-amplitude periodic oscillations on a logarithmic time scale. The properties of this dissipative limit cycle can be understood simply in terms of an effective restitution coefficient picture. Many dynamical features of the three-particle system closely resemble those of the many-body inelastic piston problem.Comment: 8 pages, 7 figures, 2-column revtex4 forma