Integral transform solution of random coupled parabolic partial differential models

[EN] Random coupled parabolic partial differential models are solved numerically using random cosine Fourier transform together with non-Gaussian random numerical integration that captures the highly oscillatory behaviour of the involved integrands. Sufficient condition of spectral type imposed on the random matrices of the system is given so that the approximated stochastic process solution and its statistical moments are numerically convergent. Numerical experiments illustrate the results.Spanish Ministerio de Economia, Industria y Competitividad (MINECO); Agencia Estatal de Investigacion (AEI); Fondo Europeo de Desarrollo Regional (FEDER UE), Grant/Award Number: MTM2017-89664-PCasabán Bartual, MC.; Company Rossi, R.; Egorova, VN.; Jódar Sánchez, LA. (2020). Integral transform solution of random coupled parabolic partial differential models. Mathematical Methods in the Applied Sciences. 43(14):8223-8236. https://doi.org/10.1002/mma.6492S822382364314Bäck, J., Nobile, F., Tamellini, L., & Tempone, R. (2010). Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison. Spectral and High Order Methods for Partial Differential Equations, 43-62. doi:10.1007/978-3-642-15337-2_3Bachmayr, M., Cohen, A., & Migliorati, G. (2016). Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. ESAIM: Mathematical Modelling and Numerical Analysis, 51(1), 321-339. doi:10.1051/m2an/2016045Ernst, O. G., Sprungk, B., & Tamellini, L. (2018). Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs). SIAM Journal on Numerical Analysis, 56(2), 877-905. doi:10.1137/17m1123079Sheng, D., & Axelsson, K. (1995). Uncoupling of coupled flows in soil—a finite element method. International Journal for Numerical and Analytical Methods in Geomechanics, 19(8), 537-553. doi:10.1002/nag.1610190804Mitchell, J. K. (1991). Conduction phenomena: from theory to geotechnical practice. Géotechnique, 41(3), 299-340. doi:10.1680/geot.1991.41.3.299Das, P. K. (1991). Optical Signal Processing. doi:10.1007/978-3-642-74962-9Ashkenazy, Y. (2017). Energy transfer of surface wind-induced currents to the deep ocean via resonance with the Coriolis force. Journal of Marine Systems, 167, 93-104. doi:10.1016/j.jmarsys.2016.11.019Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500-544. doi:10.1113/jphysiol.1952.sp004764Galiano, G. (2012). On a cross-diffusion population model deduced from mutation and splitting of a single species. Computers & Mathematics with Applications, 64(6), 1927-1936. doi:10.1016/j.camwa.2012.03.045Casabán, M. C., Company, R., & Jódar, L. (2019). Numerical solutions of random mean square Fisher‐KPP models with advection. Mathematical Methods in the Applied Sciences, 43(14), 8015-8031. doi:10.1002/mma.5942Casabán, M. C., Company, R., & Jódar, L. (2019). Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness. Mathematics, 7(9), 853. doi:10.3390/math7090853Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131-140. doi:10.1016/j.cam.2006.11.021Iserles, A. (2004). On the numerical quadrature of highly-oscillating integrals I: Fourier transforms. IMA Journal of Numerical Analysis, 24(3), 365-391. doi:10.1093/imanum/24.3.365Ma, J., & Liu, H. (2018). On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels. Symmetry, 10(7), 239. doi:10.3390/sym10070239Jódar, L., & Goberna, D. (1996). Exact and analytic numerical solution of coupled diffusion problems in a semi-infinite medium. Computers & Mathematics with Applications, 31(9), 17-24. doi:10.1016/0898-1221(96)00038-7Jódar, L., & Goberna, D. (1998). A matrix D’Alembert formula for coupled wave initial value problems. Computers & Mathematics with Applications, 35(9), 1-15. doi:10.1016/s0898-1221(98)00052-2Ostrowski, A. M. (1959). A QUANTITATIVE FORMULATION OF SYLVESTER’S LAW OF INERTIA. Proceedings of the National Academy of Sciences, 45(5), 740-744. doi:10.1073/pnas.45.5.740Ashkenazy, Y., Gildor, H., & Bel, G. (2015). The effect of stochastic wind on the infinite depth Ekman layer model. EPL (Europhysics Letters), 111(3), 39001. doi:10.1209/0295-5075/111/3900

Sensitivity analysis of circadian entrainment in the space of phase response curves

Sensitivity analysis is a classical and fundamental tool to evaluate the role of a given parameter in a given system characteristic. Because the phase response curve is a fundamental input--output characteristic of oscillators, we developed a sensitivity analysis for oscillator models in the space of phase response curves. The proposed tool can be applied to high-dimensional oscillator models without facing the curse of dimensionality obstacle associated with numerical exploration of the parameter space. Application of this tool to a state-of-the-art model of circadian rhythms suggests that it can be useful and instrumental to biological investigations.Comment: 22 pages, 8 figures. Correction of a mistake in Definition 2.1. arXiv admin note: text overlap with arXiv:1206.414

Multiple dynamical time-scales in networks with hierarchically nested modular organization

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intra-modular and slow inter-modular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.Comment: 10 pages, 4 figure

Regional differences in APD restitution can initiate wavebreak and re-entry in cardiac tissue: A computational study

Background Regional differences in action potential duration (APD) restitution in the heart favour arrhythmias, but the mechanism is not well understood. Methods We simulated a 150 × 150 mm 2D sheet of cardiac ventricular tissue using a simplified computational model. We investigated wavebreak and re-entry initiated by an S1S2S3 stimulus protocol in tissue sheets with two regions, each with different APD restitution. The two regions had a different APD at short diastolic interval (DI), but similar APD at long DI. Simulations were performed twice; once with both regions having steep (slope > 1), and once with both regions having flat (slope < 1) APD restitution. Results Wavebreak and re-entry were readily initiated using the S1S2S3 protocol in tissue sheets with two regions having different APD restitution properties. Initiation occurred irrespective of whether the APD restitution slopes were steep or flat. With steep APD restitution, the range of S2S3 intervals resulting in wavebreak increased from 1 ms with S1S2 of 250 ms, to 75 ms (S1S2 180 ms). With flat APD restitution, the range of S2S3 intervals resulting in wavebreak increased from 1 ms (S1S2 250 ms), to 21 ms (S1S2 340 ms) and then 11 ms (S1S2 400 ms). Conclusion Regional differences in APD restitution are an arrhythmogenic substrate that can be concealed at normal heart rates. A premature stimulus produces regional differences in repolarisation, and a further premature stimulus can then result in wavebreak and initiate re-entry. This mechanism for initiating re-entry is independent of the steepness of the APD restitution curve

Nonlinearity of Mechanochemical Motions in Motor Proteins

The assumption of linear response of protein molecules to thermal noise or structural perturbations, such as ligand binding or detachment, is broadly used in the studies of protein dynamics. Conformational motions in proteins are traditionally analyzed in terms of normal modes and experimental data on thermal fluctuations in such macromolecules is also usually interpreted in terms of the excitation of normal modes. We have chosen two important protein motors - myosin V and kinesin KIF1A - and performed numerical investigations of their conformational relaxation properties within the coarse-grained elastic network approximation. We have found that the linearity assumption is deficient for ligand-induced conformational motions and can even be violated for characteristic thermal fluctuations. The deficiency is particularly pronounced in KIF1A where the normal mode description fails completely in describing functional mechanochemical motions. These results indicate that important assumptions of the theory of protein dynamics may need to be reconsidered. Neither a single normal mode, nor a superposition of such modes yield an approximation of strongly nonlinear dynamics.Comment: 10 pages, 6 figure

A mathematical model for electrical stimulation of a monolayer of cardiac cells

BACKGROUND: The goal of our study is to examine the effect of stimulating a two-dimensional sheet of myocardial cells. We assume that the stimulating electrode is located in a bath perfusing the tissue. METHODS: An equation governing the transmembrane potential, based on the continuity equation and Ohm's law, is solved numerically using a finite difference technique. RESULTS: The sheet is depolarized under the stimulating electrode and is hyperpolarized on each side of the electrode along the fiber axis. CONCLUSIONS: The results are similar to those obtained previously by Sepulveda et al. (Biophys J, 55: 987–999, 1989) for stimulation of a two-dimensional sheet of tissue with no perfusing bath present

Global parameter search reveals design principles of the mammalian circadian clock

Background: Virtually all living organisms have evolved a circadian (~24 hour) clock that controls physiological and behavioural processes with exquisite precision throughout the day/night cycle. The suprachiasmatic nucleus (SCN), which generates these ~24 h rhythms in mammals, consists of several thousand neurons. Each neuron contains a gene-regulatory network generating molecular oscillations, and the individual neuron oscillations are synchronised by intercellular coupling, presumably via neurotransmitters. Although this basic mechanism is currently accepted and has been recapitulated in mathematical models, several fundamental questions about the design principles of the SCN remain little understood. For example, a remarkable property of the SCN is that the phase of the SCN rhythm resets rapidly after a 'jet lag' type experiment, i.e. when the light/ dark (LD) cycle is abruptly advanced or delayed by several hours. Results: Here, we describe an extensive parameter optimization of a previously constructed simplified model of the SCN in order to further understand its design principles. By examining the top 50 solutions from the parameter optimization, we show that the neurotransmitters' role in generating the molecular circadian rhythms is extremely important. In addition, we show that when a neurotransmitter drives the rhythm of a system of coupled damped oscillators, it exhibits very robust synchronization and is much more easily entrained to light/dark cycles. We were also able to recreate in our simulations the fast rhythm resetting seen after a 'jet lag' type experiment. Conclusion: Our work shows that a careful exploration of parameter space for even an extremely simplified model of the mammalian clock can reveal unexpected behaviours and non-trivial predictions. Our results suggest that the neurotransmitter feedback loop plays a crucial role in the robustness and phase resetting properties of the mammalian clock, even at the single neuron level

High Amplitude Phase Resetting in Rev-Erbα/Per1 Double Mutant Mice

Over time, organisms developed various strategies to adapt to their environment. Circadian clocks are thought to have evolved to adjust to the predictable rhythms of the light-dark cycle caused by the rotation of the Earth around its own axis. The rhythms these clocks generate persist even in the absence of environmental cues with a period of about 24 hours. To tick in time, they continuously synchronize themselves to the prevailing photoperiod by appropriate phase shifts. In this study, we disrupted two molecular components of the mammalian circadian oscillator, Rev-Erbα and Period1 (Per1). We found that mice lacking these genes displayed robust circadian rhythms with significantly shorter periods under constant darkness conditions. Strikingly, they showed high amplitude resetting in response to a brief light pulse at the end of their subjective night phase, which is rare in mammals. Surprisingly, Cry1, a clock component not inducible by light in mammals, became slightly inducible in these mice. Taken together, Rev-Erbα and Per1 may be part of a mechanism preventing drastic phase shifts in mammals