34 research outputs found
Mathematical Foundations of Consciousness
We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical
primitives. The Anti-foundation Axiom plays a significant role in our
development, since among other of its features, its replacement for the Axiom
of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic
interpretations. These interpretations also depend on such allied notions for
sets as pictures, graphs, decorations, labelings and various mappings that we
use. A syntax and semantics of operators acting on sets is developed. Such
features enable construction of a theory of non-well-founded sets that we use
to frame mathematical foundations of consciousness. To do this we introduce a
supplementary axiomatic system that characterizes experience and consciousness
as primitives. The new axioms proceed through characterization of so- called
consciousness operators. The Russell operator plays a central role and is shown
to be one example of a consciousness operator. Neural networks supply striking
examples of non-well-founded graphs the decorations of which generate
associated sets, each with a Platonic aspect. Employing our foundations, we
show how the supervening of consciousness on its neural correlates in the brain
enables the framing of a theory of consciousness by applying appropriate
consciousness operators to the generated sets in question
Modeling the clonal heterogeneity of stem cells
Recent experimental studies suggest that tissue stem cell pools are composed of functionally diverse clones. Metapopulation models in ecology concentrate on collections of populations and their role in stabilizing coexistence and maintaining selected genetic or epigenetic variation. Such models are characterized by expansion and extinction of spatially distributed populations. We develop a mathematical framework derived from the multispecies metapopulation model of Tilman et al (1994) to study the dynamics of heterogeneous stem cell metapopulations. In addition to normal stem cells, the model can be applied to cancer cell populations and their response to treatment. In our model disturbances may lead to expansion or contraction of cells with distinct properties, reflecting proliferation, apoptosis, and clonal competition. We first present closed-form expressions for the basic model which defines clonal dynamics in the presence of exogenous global disturbances. We then extend the model to include disturbances which are periodic and which may affect clones differently. Within the model framework, we propose a method to devise an optimal strategy of treatments to regulate expansion, contraction, or mutual maintenance of cells with specific properties
Differential equations having rapidly changing solutions: Analytic methods for weakly nonlinear systems
FAST HYBRID SOLUTION OF ALGEBRAIC SYSTEMS
We propose and analyze the error and timing of solvers consisting of both analog and digital circuitry for sparse linear systems of equations. We obtain high speed, but low precision from the analog circuits. We combine this with low speed, but high precision from the digital circuits. The hybrid circuit should be faster than digital circuits alone. As a preconditioner to standard iterative solution methods, the hybrid circuit makes the cost of the preconditioning step negligible. We also apply the hybrid circuit to a standard multilevel algorithm