Abstract-Optical mapping studies have suggested that intracellular Ca 2ϩ and T-wave alternans are linked through underlying alternations in Ca 2ϩ cycling-inducing oscillations in action potential duration through Ca 2ϩ -sensitive conductances. However, these studies cannot measure single-cell behavior; therefore, the Ca 2ϩ cycling heterogeneities within microscopic ventricular regions are unknown. The goal of this study was to measure cellular activity in intact myocardium during rapid pacing and arrhythmias. We used single-photon laser-scanning confocal microscopy to measure Ca 2ϩ signaling in individual myocytes of intact rat myocardium during rapid pacing and during pacing-induced ventricular arrhythmias. At low rates, all myocytes demonstrate Ca 2ϩ alternans that is synchronized but whose magnitude varies depending on recovery kinetics of Ca 2ϩ cycling for each individual myocyte. As rate increases, some cells reverse alternans phase, giving a dyssynchronous activation pattern, even in adjoining myocytes. Increased pacing rate also induces subcellular alternans where Ca 2ϩ alternates out of phase with different regions within the same cell. These forms of heterogeneous Ca 2ϩ signaling also occurred during pacing-induced ventricular tachycardia. Our results demonstrate highly nonuniform Ca 2ϩ signaling among and within individual myocytes in intact heart during rapid pacing and arrhythmias. Thus, certain pathophysiological conditions that alter Ca 2ϩ cycling kinetics, such as heart failure, might promote ventricular arrhythmias by exaggerating these cellular heterogeneities in Ca 2ϩ signaling. (Circ Res. 2006;99:e65-e73.) Key Words: calcium transients Ⅲ calcium alternans Ⅲ subcellular alternans Ⅲ arrhythmias O ne of the most important clues to the mechanisms responsible for repolarization alternans was derived from the fact that action potential duration (APD) alternans occurs at the cellular level in intact heart. 1-3 It is now widely accepted that T-wave alternans (TWA) on the surface ECG reflects tissue repolarization alternans at the level of the whole heart. In contrast to a purely electrophysiological explanation involving ion channel kinetics, 4,5 evidence suggests that APD and T-wave alternans are in fact associated with changes in intracellular Ca 2ϩ dynamics. 2,[5][6][7] The link between alternations in intracellular Ca 2ϩ dynamics and TWA has recently been summarized 2 as possibly arising from underlying alternans in Ca 2ϩ cycling. Intracellular Ca 2ϩ release enters into an alternating pattern based on the balance between the dynamics of Ca 2ϩ release, reuptake, and recovery rates that induce oscillations in APD as a result of Ca 2ϩ -sensitive conductances. Theoretically, a large contraction occurs as the result of a large release of Ca 2ϩ from stores in the sarcoplasmic reticulum (SR), which would in turn cause a large inward Na/Ca exchange current (I NCX ) and a long APD. Because the large SR Ca 2ϩ release would have the effect of temporary depletion of SR Ca 2ϩ content, the next beat would activ...
Abstract. We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in combination succeed in proving #P-hardness; and (3) algebraic symmetrization, which significantly lowers the symbolic complexity of the proof for computational complexity. With holographic reductions the classification theorem also applies to problems beyond the basic model.
We introduce an idea called anti-gadgets in complexity reductions. These combinatorial gadgets have the effect of erasing the presence of some other graph fragment, as if we had managed to include a negative copy of a graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z(G) on 3-regular directed graphs G, where each edge is given a complex-valued binary function f : {0, 1} 2 → C. We show thatis either computable in polynomial time or #P-hard, depending explicitly on f .To state the dichotomy theorem more explicitly, we show that the partition function Z(G) on 3-regular directed graphs G is computable in polynomial time when f belongs to one of four classes, which can be described as (1) degenerate, (2) generalized disequality, (3) generalized equality, and (4) affine after a holographic transformation. In all other cases it is #P-hard. Here class (4), after a holographic transformation, can also be described as an exponential quadratic polynomial of the form i Q(x,y) , where i = √ −1 and the cross term xy in the quadratic polynomial Q(x, y) has an even coefficient. If the input graph G is planar, then an additional class of functions becomes computable in polynomial time, and everything else remains #P-hard. This additional class is precisely those which can be computed by holographic algorithms with matchgates, making use of the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.There is a long history in the study of "Exactly Solved Models" in statistical physics. In the language of complexity theory, physicists' notion of an "Exactly Solvable" system corresponds to a system with a polynomial time computable partition function. A central question is to identify which "systems" can be solved "exactly" and which "systems" are "difficult". While in physics, there is no rigorous definition of being "difficult", complexity theory supplies the proper notion-#P-hardness.The main innovation in this paper is the idea of the anti-gadget. It is analogous to the pairing of a particle and its anti-particle in physics. Coupled with the idea of anti-gadgets, we also introduce a general way of proving #P-hardness by two types of gadgets called recursive gadgets and projector gadgets. We prove a Group Lemma which spells out a general condition for the technique to succeed. This Group Lemma states that as long as the group generated by the transition matrices of the constructed gadgets is infinite, then one can interpolate all unary functions-a key step in the proof of #P-hardness. Interpolation is carried out by forming a Vandermonde system and proving that it is of full rank. The anti-gadget concept makes the transition to group theory very natural and seamless.Not only is the idea of anti-gadgets useful in proving a new complexity dichotomy theorem in counting complexity, we also show that anti-gadgets provide a simple explanation for some miraculous cancellations that were observed in previous results. Furthermore, anti-gadgets can also guide the search for gadget sets m...
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