Computational Models for Diffusion of Second Messengers in Visual Transduction

The process of phototransduction, whereby light is converted into an electrical response in retinal rod and cone photoreceptors, involves, as a crucial step, the diffusion of cytoplasmic signaling molecules, termed second messengers. A barrier to mathematical and computational modeling is the complex geometry of the rod outer segment which contains about 1000 thin discs. Most current investigations on the subject assume a well-stirred bulk aqueous environment thereby avoiding such geometrical complexity. We present theoretical and computational spatio-temporal models for phototransduction in vertebrate rod photoreceptors, which are pointwise in nature and thus take into account the complex geometry of the rod outer segment. We consider both the full model and two forms (strong and weak) of a homogenized limit model which involves simplified geometry. These, spatially resolved, models reduce to simpler (longitudinal and lumped) models proposed by physiologists. We establish well-possedness of the model problems using upper and lower solutions and their associated monotone iterations. Computational models of the mathematical problems have been developed, based on Finite Volume discretization of the partial differential equations and boundary conditions, and implemented in Fortran. Convergence of the finite difference system to the solution of the continuous problem for a similar problem is shown in [41]. Due to the intricate geometry of the cytosol, the full model involves very intensive computations. This is achieved via parallelization for distributed memory clusters of processors. The homogenized limit problem is also tested and the computational model based on its weak form is found to produce qualitatively similar results as from the full model. We also tested a spatialy adaptive mesh for the homogenized problem and numerical experiments convinced us that we could get the same qualitative solutions with much less computational effort. Numerical experiments are presented, simulating the single photon response for salamander, with certain activation parameters chosen to produce the experimental 0:8% peak suppression of dark current, kindly communicated by Fred Rieke [45]. The model exhibits highly localized response about the activation site, with longitudinal spread of about 172 discs out of 800 discs. The radial profiles of cGMP become progressively steeper during activation and recede back during recovery. Thus radial diffusion is not negligible

Multiphoton Response of Retinal Rod Photoreceptors

Phototransduction is the process by which light is converted into an electrical response in retinal photoreceptors. Rod photoreceptors contain a stack of (about 1000) disc membranes packed with photopigment rhodopsin molecules, which absorb the photons. We present computational experiments which show the profound effect on the response of the distances (how many discs apart) photons happen to be absorbed at. This photon-distribution effect alone can account for much of the observed variability in response

Solitary waves, periodic and elliptic solutions to the Benjamin, Bona & Mahony (BBM) equation modified by viscosity

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of \eqref{BBMv} experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.Comment: 14 pages, 11 figure

Numerical Simulations of Snake Dissipative Solitons in Complex Cubic-Quintic Ginzburg-Landau Equation

Numerical simulations of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal five entirely novel classes of pulse or solitary waves solutions, viz. pulsating, creeping, snaking, erupting, and chaotical solitons. Here, we develop a theoretical framework for analyzing the full spatio-temporal structure of one class of dissipative solution (snaking soliton) of the CCQGLE using the variational approximation technique and the dynamical systems theory. The qualitative behavior of the snaking soliton is investigated using the numerical simulations of (a) the full nonlinear complex partial differential equation and (b) a system of three ordinary differential equations resulting from the variational approximation