Complex patterns of subcellular cardiac alternans

Cardiac alternans, in which the membrane potential and the intracellular calcium concentration exhibit alternating durations and peak amplitudes at consecutive beats, constitute a precursor to fatal cardiac arrhythmia such as sudden cardiac death. A crucial question therefore concerns the onset of cardiac alternans. Typically, alternans are only reported when they are fully developed. Here, we present a modelling approach to explore recently discovered microscopic alternans, which represent one of the earliest manifestations of cardiac alternans. In this case, the regular periodic dynamics of the local intracellular calcium concentration is already unstable, while the whole-cell behaviour suggests a healthy cell state. In particular, we use our model to investigate the impact of calcium diffusion in both the cytosol and the sarcoplasmic reticulum on the formation of microscopic calcium alternans. We find that for dominant cytosolic coupling, calcium alternans emerge via the traditional period doubling bifurcation. In contrast, dominant luminal coupling leads to a novel route to calcium alternans through a saddle-node bifurcation at the network level. Combining semi-analytical and computational approaches, we compute areas of stability in parameter space and find that as we cross from stable to unstable regions, the emergent patterns of the intracellular calcium concentration change abruptly in a fashion that is highly dependent upon position along the stability boundary. Our results demonstrate that microscopic calcium alternans may possess a much richer dynamical repertoire than previously thought and further strengthen the role of luminal calcium in shaping cardiac calcium dynamics

Using GAMMs to model trial-by-trial fluctuations in experimental data: more risks but hardly any benefit

Data from each subject in a repeated-measures experiment form a time series, which may include trial-by-trial fluctuations arising from human factors such as practice or fatigue. Concerns about the statistical implications of such effects have increased the popularity of Generalized Additive Mixed Models (GAMMs), a powerful technique for modeling wiggly patterns. We question these statistical concerns and investigate the costs and benefits of using GAMMs relative to linear mixed-effects models (LMEMs). In two sets of Monte Carlo simulations, LMEMs that ignored time-varying effects were no more prone to false positives than GAMMs. Although GAMMs generally boosted power for within-subject effects, they reduced power for between-subject effects, sometimes to a severe degree. Our results signal the importance of proper subject-level randomization as the main defense against statistical artifacts due to by-trial fluctuations

Calcium buffers and L-type calcium channels as modulators of cardiac subcellular alternans

In cardiac myocytes, calcium cycling links the dynamics of the membrane potential to the activation of the contractile filaments. Perturbations of the calcium signalling toolkit have been demonstrated to disrupt this connection and lead to numerous pathologies including cardiac alternans. This rhythm disturbance is characterised by alternations in the membrane potential and the intracellular calcium concentration, which in turn can lead to sudden cardiac death. In the present computational study, we make further inroads into understanding this severe condition by investigating the impact of calcium buffers and L-type calcium channels on the formation of subcellular calcium alternans when calcium diffusion in the cytosol is weak and the main route of transport in the myocyte is via the sarcoplasmic reticulum. Through numerical simulations of a two dimensional network of calcium release units, we show that increasing calcium entry is proarrhythmogenic and that this is modulated by the calcium-dependent inactivation of the L-type calcium channel. We also find that while calcium buffers can exert a stabilising force and abolish subcellular alternans, they can significantly shape the spatial patterning of subcellular calcium alternans. Taken together, our results demonstrate that subcellular calcium alternans can emerge via various routes and that calcium diffusion in the sarcoplasmic reticulum critically determines their spatial patterns

Analysis of networks where discontinuities and nonsmooth dynamics collide: understanding synchrony

Integrate-and-fire networks have proven remarkably useful in modelling the dynamics of real world phenomena ranging from earthquakes, to synchrony in neural networks, to cascading activity in social networks. The reset process means that such models are inherently discontinuous. Moreover, for jump interactions, which are a common choice for many physical systems, the models are also nonsmooth. For synchronous network states these processes can occur simultaneously, and care must be taken with the mathematical analysis of solution stability. This leads to an ordering problem, that has no counterpart in smoothly coupled limit cycle systems. Here we develop a set of network saltation matrices that can be used with an appropriate ordering to determine the instability of a synchronous network state. Moreover, we show that smoothed versions of jump interactions do not capture the behaviour of the nonsmooth model. Synchrony in the smoothed model with reset is analysed using a generalised master stability function (MSF), and the eigenspectra for smooth and nonsmooth interactions are compared. We find that the one determined by the MSF organises that found from the analysis of the nonsmooth model, though the latter has further eigenvalues that can destabilise the synchronous state

Calcium buffers and L-type calcium channels as modulators of cardiac subcellular alternans

In cardiac myocytes, calcium cycling links the dynamics of the membrane potential to the activation of the contractile filaments. Perturbations of the calcium signalling toolkit have been demonstrated to disrupt this connection and lead to numerous pathologies including cardiac alternans. This rhythm disturbance is characterised by alternations in the membrane potential and the intracellular calcium concentration, which in turn can lead to sudden cardiac death. In the present computational study, we make further inroads into understanding this severe condition by investigating the impact of calcium buffers and L-type calcium channels on the formation of subcellular calcium alternans when calcium diffusion in the cytosol is weak and the main route of transport in the myocyte is via the sarcoplasmic reticulum. Through numerical simulations of a two dimensional network of calcium release units, we show that increasing calcium entry is proarrhythmogenic and that this is modulated by the calcium-dependent inactivation of the L-type calcium channel. We also find that while calcium buffers can exert a stabilising force and abolish subcellular alternans, they can significantly shape the spatial patterning of subcellular calcium alternans. Taken together, our results demonstrate that subcellular calcium alternans can emerge via various routes and that calcium diffusion in the sarcoplasmic reticulum critically determines their spatial patterns

Synchrony in networks of coupled non-smooth dynamical systems: extending the master stability function

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period- doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs

Probabilistic encoding of stimulus strength in astrocyte global calcium signals

Astrocyte calcium signals can range in size from subcellular microdomains to waves that spread through the whole cell (and into connected cells). The differential roles of such local or global calcium signaling are under intense investigation, but the mechanisms by which local signals evolve into global signals in astrocytes are not well understood, nor are the computational rules by which physiological stimuli are transduced into a global signal. To investigate these questions, we transiently applied receptor agonists linked to calcium signaling to primary cultures of cerebellar astrocytes. Astrocytes repetitively tested with the same stimulus responded with global signals intermittently, indicating that each stimulus had a defined probability for triggering a response. The response probability varied between agonists, increased with agonist concentration, and could be positively and negatively modulated by crosstalk with other signaling pathways. To better understand the processes determining the evolution of a global signal, we recorded subcellular calcium “puffs” throughout the whole cell during stimulation. The key requirement for puffs to trigger a global calcium wave following receptor activation appeared to be the synchronous release of calcium from three or more sites, rather than an increasing calcium load accumulating in the cytosol due to increased puff size, amplitude, or frequency. These results suggest that the concentration of transient stimuli will be encoded into a probability of generating a global calcium response, determined by the likelihood of synchronous release from multiple subcellular sites

Neural Fields with Rebound Currents: Novel Routes to Patterning

The understanding of how spatio-temporal patterns of neural activity may arise in the cortex of the brain has advanced with the development and analysis of neural field models. To replicate this success for sub-cortical tissues, such as the thalamus, requires an extension to include relevant ionic currents that can further shape firing response. Here we advocate for one such approach that can accommodate slow currents. By way of illustration we focus on incorporating a T-type calcium current into the standard neural field framework. Direct numerical simulations are used to show that the resulting tissue model has many of the properties seen in more biophysically detailed model studies, and most importantly the generation of oscillations, waves, and patterns that arise from rebound firing. To explore the emergence of such solutions we focus on one-and two-dimensional spatial models and show that exact solutions describing homogeneous oscillations can be constructed n the limit that the firing rate nonlinearity is a Heaviside function. A linear stability analysis, using techniques from non-smooth dynamical systems, is used to determine the points at which bifurcations from synchrony can occur. Furthermore, we construct periodic travelling waves and investigate their stability with the use of an appropriate Evans function. The stable branches of the dispersion curve for periodic travelling waves are found to be in excellent agreement with simulations initiated from an unstable branch of the synchronous solution