Hermeneutic Approach to the Brain: Process versus Device?

Hermeneutics, the art of the interpretation, is applied to brain. The relevance of : ?ncepts of hermeneutic processes and hermeneutic devices to brain theory are explained

The Construction of Arbitrary Stable Dynamics in Non-Linear Neural Networks

In this paper, two methods for constructing systems of ordinary differential equations realizing any fixed finite set of equilibria in any fixed finite dimension are introduced; no spurious equilibria are possible for either method. By using the first method, one can construct a system with the fewest number of equilibria, given a fixed set of attractors. Using a strict Lyapunov function for each of these differential equations, a large class of systems with the same set of equilibria is constructed. A method of fitting these nonlinear systems to trajectories is proposed. In addition, a general method which will produce an arbitrary number of periodic orbits of shapes of arbitrary complexity is also discussed. A more general second method is given to construct a differential equation which converges to a fixed given finite set of equilibria. This technique is much more general in that it allows this set of equilibria to have any of a large class of indices which are consistent with the Morse Inequalities. It is clear that this class is not universal, because there is a large class of additional vector fields with convergent dynamics which cannot be constructed by the above method. The easiest way to see this is to enumerate the set of Morse indices which can be obtained by the above method and compare this class with the class of Morse indices of arbitrary differential equations with convergent dynamics. The former set of indices are a proper subclass of the latter, therefore, the above construction cannot be universal. In general, it is a difficult open problem to construct a specific example of a differential equation with a given fixed set of equilibria, permissible Morse indices, and permissible connections between stable and unstable manifolds. A strict Lyapunov function is given for this second case as well. This strict Lyapunov function as above enables construction of a large class of examples consistent with these more complicated dynamics and indices. The determination of all the basins of attraction in the general case for these systems is also difficult and open.Air Force Office of Scientific Research (F49620-86-C-0037

Broglie and Young, visionaries who shed light in the polar topology that grounds our reality: a hypothesis

Una observación matemática que relaciona los patrones fractales y la operación de convolución en el contexto del procesamiento de imágenes digitales interrumpió una investigación que nos lleva a plantear la hipótesis de que el concepto de onda de materia (o dualidad onda-partícula) se encuentra en la dicotomía entre el par débil y un topología fuerte en el ámbito del marco de atractores singulares continuos en ninguna parte diferenciables y el concepto de fotón-solitón de Vigier. Tal inferencia parece ser más evidente en la interpretación de Broglie-Bohm de la mecánica cuántica en el cruce de características locales x globales. De esto se deduce también que la relación de los fenómenos naturales (sin escala) con la ecuación de Schröder generalizada (en pie de iteración) está todavía por explorar por completo. La inflación cosmológica y los fenómenos cuánticos ordinarios (desde la perspectiva del colapso posterior de la función de onda) ambos posiblemente se enfrentarían a las soluciones de esa ecuación de Schröder generalizada ignorada que también se remontaría al marco del atractor singular-continuo en ninguna parte diferenciable. Tales conocimientos teóricos ofrecen una acomodación matemática prometedora que se ajusta a las observaciones recientes utilizando los conjuntos de datos PLK + BAO + SN + H (astronomía), lo que sugiere un cambio de signo espontáneo de la constante cosmológica que hipotetizamos que se remonta a una discontinuidad de salto entre 2 conjuntos de niveles que tiene sus raíces en el concepto de conjugación topológica. Tal discontinuidad señalaría la transición entre la llamada fase de radiación y las cosmologías oscuras fantasma. La ruta de razonamiento más prometedora ideada hasta ahora en un programa de investigación de este tipo parece ocurrir en el locus que abarca las conexiones entre los fractales y la medida de Lebesgue-Cantor (convolución de Bernoulli). Tales conceptos parecen acomodar una especie de sistema aislado no forzado cuya dinámica sería impulsada por la iteración y regida por la condición inicial en el ámbito de la teoría de la perturbación (casi integrable). Tal locus parece ser preferencial para una prospección sobre la unificación de la física clásica, la mecánica cuántica ordinaria y la teoría de números.A mathematical observation relating fractal patterns and the convolution operation in the context of digital image processing disrupted an investigation that drives us to hypothesize that the concept of the wave of matter (or duality wave-particle) stands to the dichotomy between the pair weak and a strong topology in the realm of Singular-continuous nowhere-differentiable attractors framework and the Photon-soliton concept by Vigier. Such inference appears to be more evident in the Broglie-Bohm interpretation of quantum mechanics in the local x global features crossover. Follows of this also that the relation of the natural phenomena (scale-free) with the generalized Schröder equation (standing to iteration) is still to be totally explored. The cosmological inflation and ordinary quantum phenomena (by the perspective of after wavefunction collapse) both of them would possibly stand to the solutions of such disregarded generalized Schröder equation that would also trace back to the Singular-continuous nowhere-differentiable attractor framework. Such theoretical insights offer a promising mathematical accommodation that fits recent observations using the PLK+BAO+SN+H datasets (astronomy) suggesting a spontaneous sign change of the cosmological constant that we hypothesize tracks back to a jump discontinuity between 2 Leve-Sets that would have its roots on the concept of topological conjugacy. Such discontinuity would point to the transition between the called radiation phase and ghost dark cosmologies. The most promising route of reasoning devised so far in such a research program seems to occur in the locus that encompasses the connections between fractals and Lebesgue-Cantor measure (Bernoulli convolution). Such concepts seem to accommodate a sort of insulated unforced system which dynamics would be driven by iteration and ruled by initial condition in the realm of perturbation theory (near-integrable). Such locus seems to be preferential for one prospecting about the unification of classical physics, ordinary quantum mechanics and number theory

Differential analysis of nonlinear systems: Revisiting the pendulum example

Differential analysis aims at inferring global properties of nonlinear behaviors from the local analysis of the linearized dynamics. The paper motivates and illustrates the use of differential analysis on the nonlinear pendulum model, an archetype example of nonlinear behavior. Special emphasis is put on recent work by the authors in this area, which includes a differential Lyapunov framework for contraction analysis and the concept of differential positivity

Tensegrity and Recurrent Neural Networks: Towards an Ecological Model of Postural Coordination

Tensegrity systems have been proposed as both the medium of haptic perception and the functional architecture of motor coordination in animals. However, a full working model integrating those two aspects with some form of neural implementation is still lacking. A basic two-dimensional cross-tensegrity plant is designed and its mechanics simulated. The plant is coupled to a Recurrent Neural Network (RNN). The model’s task is to maintain postural balance against gravity despite the intrinsically unstable configuration of the plant. The RNN takes only proprioceptive input about the springs’ lengths and rate of length change and outputs minimum lengths for each spring which modulates their interaction with the plant’s inertial kinetics. Four artificial agents are evolved to coordinate the patterns of spring contractions in order to maintain dynamic equilibrium. A first study assesses quiet standing performance and reveals coordinative patterns between the tensegrity rods akin to humans’ strategy of anti-phase hip-ankle relative phase. The agents show a mixture of periodic and aperiodic trajectories of their Center of Mass. Moreover, the agents seem to tune to the anticipatory “time-to-balance” quantity in order to maintain their movements within a region of reversibility. A second study perturbs the systems with mechanical platform shifts and sensorimotor degradation. The agents’ response to the mechanical perturbation is robust. Dimensionality analysis of the RNNs’ unit activations reveals a pattern of degree of freedom recruitment after perturbation. In the degradation sub-study, different levels of noise are added to the RNN inputs and different levels of weakening gain are applied to the forces generated by the springs to mimic haptic degradation and muscular weakening in elderly humans. As expected, the systems perform less well, falling earlier than without the insults. However, the same systems re-evolved again under the degraded conditions see significant functional recovery. Overall, the dissertation supports the plausibility of RNN cum tensegrity models of haptics-guided postural coordination in humans

Deep Kalman Filters Can Filter

Deep Kalman filters (DKFs) are a class of neural network models that generate Gaussian probability measures from sequential data. Though DKFs are inspired by the Kalman filter, they lack concrete theoretical ties to the stochastic filtering problem, thus limiting their applicability to areas where traditional model-based filters have been used, e.g.\ model calibration for bond and option prices in mathematical finance. We address this issue in the mathematical foundations of deep learning by exhibiting a class of continuous-time DKFs which can approximately implement the conditional law of a broad class of non-Markovian and conditionally Gaussian signal processes given noisy continuous-times measurements. Our approximation results hold uniformly over sufficiently regular compact subsets of paths, where the approximation error is quantified by the worst-case 2-Wasserstein distance computed uniformly over the given compact set of paths

Fundamental Structure of General Stochastic Dynamical Systems: High-Dimension Case

No one has proved that mathematically general stochastic dynamical systems have a special structure. Thus, we introduce a structure of a general stochastic dynamical system. According to scientific understanding, we assert that its deterministic part can be decomposed into three significant parts: the gradient of the potential function, friction matrix and Lorenz matrix. Our previous work proved this structure for the low-dimension case. In this paper, we prove this structure for the high-dimension case. Hence, this structure of general stochastic dynamical systems is fundamental