Nonlinear models of development, amplification and compression in the mammalian cochlea.

This paper reviews current understanding and presents new results on some of the nonlinear processes that underlie the function of the mammalian cochlea. These processes occur within mechano-sensory hair cells that form part of the organ of Corti. After a general overview of cochlear physiology, mathematical modelling results are presented in three parts. First, the dynamic interplay between ion channels within the sensory inner hair cells is used to explain some new electrophysiological recordings from early development. Next, the state of the art is reviewed in modelling the electro-motility present within the outer hair cells (OHCs), including the current debate concerning the role of cell body motility versus active hair bundle dynamics. A simplified model is introduced that combines both effects in order to explain observed amplification and compression in experiments. Finally, new modelling evidence is presented that structural longitudinal coupling between OHCs may be necessary in order to capture all features of the observed mechanical responses.

Encoding and decoding mechanisms of pulsatile hormone secretion.

Ultradian pulsatile hormone secretion underlies the activity of most neuroendocrine systems, including the hypothalamic-pituitary adrenal (HPA) and gonadal (HPG) axes, and this pulsatile mode of signalling permits the encoding of information through both amplitude and frequency modulation. In the HPA axis, glucocorticoid pulse amplitude increases in anticipation of waking, and, in the HPG axis, changing gonadotrophin-releasing hormone pulse frequency is the primary means by which the body alters its reproductive status during development (i.e. puberty). The prevalence of hormone pulsatility raises two crucial questions: how are ultradian pulses encoded (or generated) by these systems, and how are these pulses decoded (or interpreted) at their target sites? We have looked at mechanisms within the HPA axis responsible for encoding the pulsatile mode of glucocorticoid signalling that we observe in vivo. We review evidence regarding the 'hypothalamic pulse generator' hypothesis, and describe an alternative model for pulse generation, which involves steroid feedback-dependent endogenous rhythmic activity throughout the HPA axis. We consider the decoding of hormone pulsatility by taking the HPG axis as a model system and focussing on molecular mechanisms of frequency decoding by pituitary gonadotrophs.

Dynamics of plateau bursting depending on the location of its equilibrium.

We present a mathematical analysis of the dynamics that underlies plateau bursting in models of endocrine cells under variation of the location of the (unstable) equilibrium around which these bursting patterns are organised. We focus primarily on the less well-studied case of pseudo-plateau bursting, but also consider the square-wave case. The behaviour of such models is explained using the theory for systems with multiple time scales and it is well known that the underlying so-called fast subsystem organises their dynamics. However, such results are valid only in a sufficiently small neighbourhood of the singular limit that defines the fast subsystem. Hence, the slow variable (intracellular calcium concentration) must be very slow, which is actually not the case for pseudo-plateau bursting. Furthermore, the theoretical predictions are also only valid for parameter values such that the equilibrium is close to a homoclinic bifurcation occuring in the fast subsystem. In the present study, we use numerical explorations to discuss what happens outside this theoretically known neighbourhood of parameter space. In particular, we consider what happens as the equilibrium moves outside a small neighbourhood of the homoclinic bifurcation that occurs in the fast subsystem, and relatively fast speeds are allowed for the slow variable which is controlled by a relatively large value of a parameter ε. The results obtained complement our earlier work [Tsaneva-Atanasova et al. (2010) J Theor Biol264, 1133-1146], which focussed on how the bursting patterns vary with the rate of change ε of the slow variable: we fix ε and move the equilibrium over the full range of the bursting regime. Our findings show that the transitions between different bursting patterns are rather similar for square-wave and pseudo-plateau bursting, provided that the value of ε for the pseudo-plateau-bursting model is chosen so that it is much larger than for the square-wave bursting model. Furthermore, the two families of tonic spiking and plateau bursting, which are generally viewed as two separately generated families, are actually connected into a single family in the two-parameter plane through branches of unstable periodic orbits.

A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations.

In some cell types, oscillations in the concentration of free intracellular calcium ([Ca2+]) are accompanied by oscillations in the concentration of inositol 1,4,5-trisphosphate ([IP3]). However, in most cell types it is still an open question as to whether oscillations in [IP3] are necessary for Ca2+ oscillations in vivo, or whether they merely follow passively. Using a wide range of models, we show that the response to an artificially applied pulse of IP3 can be used to distinguish between these two cases. Hence, we show that muscarinic receptor-mediated, long-period Ca2+ oscillations in pancreatic acinar cells depend on [IP3] oscillations, whereas short-period Ca2+ oscillations in airway smooth muscle do not.

Calcium oscillations in a triplet of pancreatic acinar cells.

We use a mathematical model of calcium dynamics in pancreatic acinar cells to investigate calcium oscillations in a ring of three coupled cells. A connected group of cells is modeled in two different ways: 1), as coupled point oscillators, each oscillator being described by a spatially homogeneous model; and 2), as spatially distributed cells coupled along their common boundaries by gap-junctional diffusion of inositol trisphosphate and/or calcium. We show that, although the point-oscillator model gives a reasonably accurate general picture, the behavior of the spatially distributed cells cannot always be predicted from the simpler analysis; spatially distributed diffusion and cell geometry both play important roles in determining behavior. In particular, oscillations in which two cells are in synchrony, with the third phase-locked but not synchronous, appears to be more dominant in the spatially distributed model than in the point-oscillator model. In both types of model, intercellular coupling leads to a variety of synchronous, phase-locked, or asynchronous behaviors. For some parameter values there are multiple, simultaneous stable types of oscillation. We predict 1), that intercellular calcium diffusion is necessary and sufficient to coordinate the responses in neighboring cells; 2), that the function of intercellular inositol trisphosphate diffusion is to smooth out any concentration differences between the cells, thus making it easier for the diffusion of calcium to synchronize the oscillations; 3), that groups of coupled cells will tend to respond in a clumped manner, with groups of synchronized cells, rather than with regular phase-locked periodic intercellular waves; and 4), that enzyme secretion is maximized by the presence of a pacemaker cell in each cluster which drives the other cells at a frequency greater than their intrinsic frequency.

Control of calcium oscillations by membrane fluxes.

It is known that Ca(2+) influx plays an important role in the modulation of inositol trisphosphate-generated Ca(2+) oscillations, but controversy over the mechanisms underlying these effects exists. In addition, the effects of blocking membrane transport or reducing Ca(2+) entry vary from one cell type to another; in some cell types oscillations persist in the absence of Ca(2+) entry (although their frequency is affected), whereas in other cell types oscillations depend on Ca(2+) entry. We present theoretical and experimental evidence that membrane transport can control oscillations by controlling the total amount of Ca(2+) in the cell (the Ca(2+) load). Our model predicts that the cell can be balanced at a point where small changes in the Ca(2+) load can move the cell into or out of oscillatory regions, resulting in the appearance or disappearance of oscillations. Our theoretical predictions are verified by experimental results from HEK293 cells. We predict that the role of Ca(2+) influx during an oscillation is to replenish the Ca(2+) load of the cell. Despite this prediction, even during the peak of an oscillation the cell or the endoplasmic reticulum may not be measurably depleted of Ca(2+).

A model of calcium waves in pancreatic and parotid acinar cells.

We construct a mathematical model of Ca(2+) wave propagation in pancreatic and parotid acinar cells. Ca(2+) release is via inositol trisphosphate receptors and ryanodine receptors that are distributed heterogeneously through the cell. The apical and basal regions are separated by a region containing the mitochondria. In response to a whole-cell, homogeneous application of inositol trisphosphate (IP(3)), the model predicts that 1), at lower concentrations of IP(3), the intracellular waves in pancreatic cells begin in the apical region and are actively propagated across the basal region by Ca(2+) release through ryanodine receptors; 2), at higher [IP(3)], the waves in pancreatic and parotid cells are not true waves but rather apparent waves, formed as the result of sequential activation of inositol trisphosphate receptors in the apical and basal regions; 3), the differences in wave propagation in pancreatic and parotid cells can be explained in part by differences in inositol trisphosphate receptor density; 4), in pancreatic cells, increased Ca(2+) uptake by the mitochondria is capable of restricting Ca(2+) responses to the apical region, but that this happens only for a relatively narrow range of [IP(3)]; and 5), at higher [IP(3)], the apical and basal regions of the cell act as coupled Ca(2+) oscillators, with the basal region partially entrained to the apical region.