Mathematical Medicine and Biology - Advance Access

Analysis of extrema of heartbeat time series in exercise test

Cammarota, C., Curione, M. — 2008-05-01

The heartbeat time series of the electrocardiogram recorded during exercise test clearly reflects the physiological control mechanism of the autonomic nervous system on heart rate. This series shows both decreasing and increasing trends and variability of the variance. We analyse the series of intervals between two consecutive extrema, i.e. the durations of accelerations or decelerations of heart rate. We compute the distribution of the length of these intervals and their mean in a model of stationary independent variables, where they are independent of the variables’ distribution. We use the mean length as discriminant statistics to compare stress and recovery phases. Data analysis performed over the heartbeat series of 14 healthy subjects shows significant difference between stress and recovery.

Single-cell-based models in biology and medicine

Anderson, A. R. A., Chaplain, M. A. J., Rejniak, K. A., Fozard, J. A. — 2008-04-28

Control of invasive hosts by generalist parasitoids

Magal, C., Cosner, C., Ruan, S., Casas, J — 2008-04-28

This article was motivated by the invasion of leaf-mining microlepidopteron attacking horse chestnut trees in Europe and the need for a biological control. Following Owen & Lewis (2001, Bull. Math. Biol., 63, 655–684), we consider predation of leafminers by a generalist parasitoid with a Holling Type II functional response. We first identified six equilibrium points and discussed their stabilities in the non-spatial model. The model always predicts persistence of the parasitoid. Depending on the parameter values, the model may predict that the host persists and goes extinct or there is something like an Allee effect where the outcome depends on the initial host density. Special cases were also studied for small carrying capacities leading to complex dynamical behaviours. Then, numerical simulations of the spatial reaction--diffusion model enabled us to identify the conditions for which the leafminer's advance can be stopped and reversed by parasitoids. Compared to the ordinary differential equation model, the incorporation of space, combined with the polyphagy of the parasitoid, leads to a decrease of the parameter domain of coexistence. This is in stark to several other models in which space promotes coexistence by enabling hosts to escape.

Neuronal currents and EEG-MEG fields

Dassios, G. — 2008-04-15

In a recent paper by the author, Fokas and Hadjiloizi proved that a neuronal current within a spherical homogeneous conductor can be split into two orthogonal components in such a way that one component provides the electroencephalography (EEG)-related fields and the other component provides the fields related to magnetoencephalography (MEG). Hence, in spherical geometry, the EEG and MEG measurements contain no overlapping information about the current. In the present work, we utilize a new integral representation for the magnetic potential, introduced recently by Fokas, Kariotou and the author, to prove that this elegant property is not true once the highly symmetric spherical environment is abandoned. It seems that any ambiguity concerning overlapping information coming from EEG and MEG measurements has its origin in the fact that in most clinical applications the spherical model is used although the actual data never come from a perfect sphere.

Calcium release site ultrastructure and the dynamics of puffs and sparks

DeRemigio, H., Groff, J. R., Smith, G. D. — 2008-03-21

When Markov chain models of intracellular Ca²⁺-regulated Ca²⁺ channels are coupled via a mathematical representation of a Ca²⁺ microdomain, simulated Ca²⁺ release sites may exhibit the phenomenon of ‘stochastic Ca²⁺ excitability’ reminiscent of Ca²⁺ puffs and sparks. Interestingly, some single-channel models that include Ca²⁺ inactivation are not particularly sensitive to channel density, so long as the requirement for inter-channel communication is satisfied, while other single-channel models that do not include Ca²⁺ inactivation open and close synchronously only when the channel density is in a prescribed range. This observation led us to hypothesize that single-channel models with Ca²⁺ inactivation would be less sensitive to the details of release site ultrastructure than models that lack a slow Ca²⁺ inactivation process. To determine if this was the case, we simulated Ca²⁺ release sites composed of instantaneously coupled Ca²⁺-regulated Ca²⁺ channels whose random spatial locations were chosen from a uniform distribution on a disc of specified radius and compared the resulting release site dynamics to simulations with channels arranged on hexagonal lattices. Analysis of puff/spark statistics confirmed our hypothesis that puffs and sparks are less sensitive to the spatial organization of release sites when the single-channel model includes a slow inactivation process. We also investigated the validity of several different mean-field reductions that do not explicitly account for the details of release site ultrastructure. The most successful approximation maintains a distinction between each channel's substantial influence on its own stochastic gating and the collective contribution of elevated [Ca²⁺] from neighbouring channels.

Variability in the secretion of corticotropin-releasing hormone, adrenocorticotropic hormone and cortisol and understandability of the hypothalamic-pituitary-adrenal axis dynamics--a mathematical study based on clinical evidence

Bairagi, N., Chatterjee, S., Chattopadhyay, J. — 2008-03-14

In this article, we have developed a simple mathematical model that captures the vital mechanisms of the hypothalamic-pituitary-adrenal (HPA) axis self-regulatory activities. For this, a system of three-component non-linear delay differential equations has been proposed and analysed to observe the ultradian and circadian variabilities of the hormone secretion of the HPA axis in normal subjects. Our analysis reveals that a feedback mechanism is sufficient to show the ultradian variability of the hormone secretion pattern but fails to show the circadian variability. A central nervous system-driven pulse generator coupled with the primary feedback mechanism can exhibit the ultradian as well as circadian variability in the hormone secretion of the HPA axis. The model can also predict different dynamics of the normal HPA axis following physiological changes (viz. adrenalectomy and hypophysectomy) and pathological changes (viz. infusion of different hormones).

Shortening of cardiac action potential duration near an insulating boundary

Cain, J. W., Schaeffer, D. G. — 2008-03-14

It is known, from both experiments and simulations, that cardiac action potentials are shortened near a non-conducting boundary. In the present paper, this effect is studied in a simple, two-current ionic model, with propagation restricted to a 1D fibre. An asymptotic approximation for the dependence of action potential duration on distance to the boundary is derived. This estimate agrees well with simulations.