Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator

This is a pre-print. The definitive version: COOMBES, S. and OSBALDESTIN, A.H., 2000. Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. Physical Review E, 62(3), pp.4057-4066 Part B.The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation theory shows the existence of an attracting invariant manifold which is used to construct the Fenichel normal form for the system. This facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so-called extended isochronal map. The isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response: mode-locked, quasi-periodic and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system

Diffusion barriers for ADP in the cardiac cell.

The regulation of mitochondrial respiration in the intact heart may differ from that of isolated mitochondria if intracellular diffusion is restricted. Here we consider which factors may hinder diffusion in vivo and, based on computational analysis, design a reverse engineering approach to estimate the role of diffusional resistance in mitochondrial regulation from an experiment on the intact heart. Computational analysis of respiration measurements on skinned heart fibers shows that the outer mitochondrial membrane does not hinder diffusion enough to cause ADP gradients of tens of micromolars. A diffusion model further shows that the mesoscale structure of the myofibrillar space also does not hinder diffusion appreciably. However, ADP gradients are suggested by the measured activation time of oxidative phosphorylation and may be caused by diffusion restriction of other intracellulax structures or the in vivo microstructure of networks of physically interacting proteins. Based on computational modeling we propose an experiment on the intact heart that allows to estimate the effective diffusion restriction between ATP producing and consuming sites in the cardiac cell

Role of the creatine/phosphocreatine system in the regulation of mitochondrial respiration.

International audienceThe mechanism of metabolic regulation of mitochondrial respiration in cardiac muscle cells was studied experimentally in the permeabilized heart fibres of mice and by computer modelling in silico. The experiments showed that the rate of mitochondrial respiration could be controlled by local production of ADP by mitochondrial creatine kinase in the intermembrane space of mitochondria. The spatially inhomogenous reaction-diffusion model of compartmentalized energy transfer was used to analyse which metabolite level in cytoplasm may be important for regulation of respiration. At low and moderate workloads, up to VO2 equal to 70 micromol min-1 g-1 dry weight, the only factor to which respiration responded was inorganic phosphate. At the values of VO2 higher than 70 micromol min-1 g-1 dry weight, the respiration rate responded mostly to changes in creatine, phosphocreatine and then time-averaged (over the contractile cycle) ADP concentrations in the cytoplasm. These results are taken to show that under conditions of moderate workloads, creatine kinase activity at given physiological creatine and phosphocreatine concentrations (apparent maximal activity achievable under these conditions) is in excess to oxidative phosphorylation rate, which is controlled by Pi concentration changes starting from very low values of the latter. At higher workloads mi-CK should be upregulated by increasing creatine and decreasing phosphocreatine concentrations, and only at very high workloads the ADP diffusion flux should be increased to upregulate oxidative phosphorylation. Thus, on the basis of the study in silico of compartmentalized energy transfer by phophocreatine/creatine system, the authors conclude that there exist multiple parallel regulatory factors controlling the rate of oxygen consumption in dependence of the workload. If creatine kinase is inhibited (and there is no myokinase activity), respiration requires high diffusive flux of ADP back into mitochondria, which is the sole regulator of respiration. This needs, however, increased ADP concentrations in the cytoplasm, which in turn result in inhibition of contraction

The mitochondrial out membrane is not a major diffusion barrier for ADP in mouse heart skinned fibre bundles.

The response of mitochondrial oxygen consumption to ADP in saponin-skinned cardiac fibre bundles has an apparent K m an order of magnitude higher than that in isolated mitochondria. Here we report that incubating skinned cardiac fibre bundles from wild-type mice or double-knockout mice lacking both cytosolic and mitochondrial creatine kinase (CK) with CK and creatine or with yeast hexokinase and glucose as extramitochondrial ADP-producing systems decreases the apparent K m of the bundles for ADP severalfold. We conclude that the affinity of mitochondria for ADP in mouse heart is of the same order of magnitude as that of isolated mitochondria, while the high apparent K m of the bundles is caused by diffusion gradients outside the mitochondria