H. P. Ekobena Fouda

Modulated pressure waves in large elastic tubes.

Modulational instability is the direct way for the emergence of wave patterns and localized structures in nonlinear systems. We show in this work that it can be explored in the framework of blood flow models. The whole modified Navier-Stokes equations are reduced to a difference-differential amplitude equation. The modulational instability criterion is therefore derived from the latter, and unstable patterns occurrence is discussed on the basis of the nonlinear parameter model of the vessel. It is found that the critical amplitude is an increasing function of α, whereas the region of instability expands. The subsequent modulated pressure waves are obtained through numerical simulations, in agreement with our analytical expectations. Different classes of modulated pressure waves are obtained, and their close relationship with Mayer waves is discussed.

Nonlinear charge transport in the helicoidal DNA molecule

Charge transport in the twist-opening model of DNA is explored via the modulational instability of a plane wave. The dynamics of charge is shown to be governed, in the adiabatic approximation, by a modified discrete nonlinear Schrödinger equation with next-nearest neighbor interactions. The linear stability analysis is performed on the latter and manifestations of the modulational instability are discussed according to the value of the parameter α, which measures hopping interaction correction. In so doing, increasing α leads to a reduction of the instability domain and, therefore, increases our chances of choosing appropriate values of parameters that could give rise to pattern formation in the twist-opening model. Our analytical predictions are verified numerically, where the generic equations for the radial and torsional dynamics are directly integrated. The impact of charge migration on the above degrees of freedom is discussed for different values of α. Soliton-like and localized structures are observed and thus confirm our analytical predictions. We also find that polaronic structures, as known in DNA charge transport, are generated through modulational instability, and hence reinforces the robustness of polaron in the model we study.

Intramolecular vibrations and noise effects on pattern formation in a molecular helix.

Modulational instability in a biexciton molecular chain is addressed. We show that the model can be reduced to a set of three coupled equations: two nonlinear Schrödinger equations and a Boussinesq equation. The linear stability analysis of continuous wave solutions of the coupled systems is performed and the growth rate of instability is found numerically. Simulations of the full discrete systems reveal some behaviors of modulational instability, since wave patterns are observed for the excitons and the phonon spectrum. We also take the effect of thermal fluctuations into account and we numerically study both the stability and the instability of the plane waves under 300 K. The plane wave is found to be stable under modulation, but displays a gradual increase of the wave amplitudes. Under modulation, the same behaviors are observed and wave patterns are found to resist thermal fluctuations, which is in agreement with earlier research on localized structure stability under thermal noise.

Wave instability of intercellular Ca2+ oscillations

Modulational instability is exclusively addressed in a minimal model for calcium oscillations in cells. The cells are considered to be coupled through paracrine signaling. The endoplasmic recticulum and cytosolic equations are reduced to a single differential-difference amplitude equation. The linear stability analysis of a plane wave is performed on the latter and the paracrine coupling parameter is shown to deeply influence the instability features. Our analytical expectations are confirmed by numerical simulations, as instability regions give rise to unstable wave patterns. We also discuss the possibility of perfect intercellular communication via the activation of modulational instability.

Wave patterns in α-helix proteins with interspine coupling

Modulational instability is a direct way by which localized structures emerge in nonlinear systems. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in α-helix proteins with three spines. Through numerical simulations, trains of pulses are found and confirm our analytical predictions. The presence of higher-order interactions between adjacent spines tends to suppress the formed localized structures for erratic ones to emerge. These erratic structures are highly localized and rather reinforce the idea that the energy to be used in metabolic processes is rather confined to specific regions for its efficiency.

Discrete impulses in ephaptically coupled nerve fibers

We exclusively analyze the condition for modulated waves to emerge in two ephaptically coupled nerve fibers. Through the multiple scale expansion, it is shown that a set of coupled cable-like Hodgkin-Huxley equations can be reduced to a single differential-difference nonlinear equation. The standard approach of linear stability analysis of a plane wave is used to predict regions of parameters where nonlinear structures can be observed. Instability features are shown to be importantly controlled not only by the ephaptic coupling parameter, but also by the discreteness parameter. Numerical simulations, to verify our analytical predictions, are performed, and we explore the longtime dynamics of slightly perturbed plane waves in the coupled nerve fibers. On initially exciting only one fiber, quasi-perfect interneuronal communication is discussed along with the possibility of recruiting damaged or non-myelinated nerve fibers, by myelinated ones, into conduction.

Wave propagation of coupled modes in the DNA double helix

The remarkable dynamics of waves propagating along the DNA molecule is described by the coupled nonlinear Schr?dinger equations. We consider both the single and the coupled nonlinear excitation modes and, under numerical simulations of the Peyrard?Bishop model, with the use of realistic values of parameters, their biological implications are studied. Furthermore, the characteristics of the coupled mode solution are discussed and we show that such a solution can describe the local opening observed within the transcription and the replication phenomena.