[Modern trends in mathematical modeling of the hemispheric channels functions].

The review deals with new trends in modeling the vestibular function of the hemispheric channels (HC) involved in the head angular movements. The trends were spurred by the growing significance of computers in both mathematical modeling and direct simulation of the HC structure and processes, and conditions of experiments. Literature analysis reveals the following trends in the CH modeling: 1) reconstruction, 2) micromodeling, 3) integral modeling, 4) simulation (imitation), and 5) alternative modeling. The article gives several examples of the modern trends in HC modeling.

The distribution of otolith polarization vectors in mammals: comparison between model predictions and single cell recordings.

The transformation of head-movements into neural signals represents a multi-stage process. It depends on orientation and movement of the head, the geometry and mechanics of the vestibular sensors, and the ensuing processing of the peripheral vestibular signals. While this process is well understood for the semicircular canals, where each canal transduces the angular velocity in the corresponding canal plane, the contributions of the individual otoliths, our linear acceleration sensors, are still under debate. This is in part due to the complex geometrical structure of the otoliths. To improve our understanding of the otoliths, we have developed a new technique to visualize otolith function: using measured 3D-shapes of human otoliths and the observed 2D patterns of hair cell orientation over the epithelia, morphological polarization vectors are predicted. To visualize the geometric distribution of these vectors, we have created distribution plots which indicate the density of hair cell polarization vectors for the different directions. In many respects, our results closely agree with earlier recordings of polarization vectors of vestibular afferents in squirrel monkeys: for example, hair cells on the saccule do not cover the sagittal plane equally, but show a strong concentration in the dorso-ventral directions. Some discrepancies exist in the density distribution of otolith, which could provide valuable information for future anatomical investigations of the otoliths.

Effect of difference of cupula and endolymph densities on the dynamics of semicircular canal.

The effect of different densities of a cupula and endolymph on the dynamics of the semicircular canals is considered within the framework of a simplified one-dimensional mathematical model where the canal is approximated by a torus. If the densities are equal, the model is represented by Steinhausen's phenomenological equation. The difference of densities results in the complex dynamics of the cupulo-endolymphatic system, and leads to a dependence on the orientation of both the gravity vector relative to the canal plane and the axis of rotation, as well as on the distance between the axis of rotation and the center of the semicircular canal. Our analysis focused on two cases of canal stimulation: rotation with a constant velocity and a time-dependent (harmonically oscillating) angular velocity. Two types of spatial orientation of the axis of rotation, the axis of canal symmetry, and the vector of gravity were considered: i) the gravity vector and axis of rotation lie in the canal plane, and ii) the axis of rotation and gravity vector are normal to the canal plane. The difference of the cupula and endolymph densities reveals new features of cupula dynamics, for instance--a shift of the cupula to a new position of equilibrium that depends on the gravity vector and the parameters of head rotation, and the onset of cupula oscillations with multiple frequencies that results in the distortion of cupula dynamics relative to harmonic stimulation. Factors that might influence the density difference effects and the conditions under which these effects occur are discussed.

The model of cupular dynamics in the case of the difference of the cupula and endolymph densities.

The function of semicircular canals (SC) is based on the precise equality of densities of the cupula and endolymph. Otherwise the information provided by SC would depend on the orientation of both the gravity vector relative to canal plane and the axis of rotation. It would also depend on the distance between the axis of rotation and the center of the SC. We believe that the equality of densities is approximate and expect that due to the high sensitivity of the SC, even the small differences of densities (approximately 10(-4) g/cm3) can influence the SC dynamics, and this influence depends on the conditions of canal stimulation. The work aims to examine this hypothesis and analyze the parameters of the SC and mechanical stimulation under which the effect of the difference of densities on the SC functioning could be observed.

[Modeling of the Otolith's structure and functions].

Physical and chemical processes of the transformation of mechanic stimuli into nerve signals in the otolith responsible for gravity and linear accelerations perception are of interest of equally biologists and experts in aerospace medicine. A severe experimental difficulty is the small size of the otolith and, therefore, mathematical and computer modeling has become a powerful tool for examination of the otolith organs. Most of the above processes are spatially distributed; consequently, spatial distribution models will provide the exact description of the existing biological structures and processes. The review is dedicated to these models and their applicability to demonstrate the variety of processes associated with the otolith.

Feedback hypothesis and the effects of altered gravity on formation and function of gravireceptors of mollusks and fish.

Popular hypothesis based on the idea of simple feedback mechanism that correlates gravity level and weight of test mass cannot explain the variety of the effects of altered gravity on development and function of gravireceptors. The reaction of organisms to the change of gravity depends on the gravisensitivity of the physical and chemical processes corresponding to specific phases of development and may have no relation to any feedback mechanisms of compensation of altered weight of the test mass. The present work analyzes the hypothesis of feedback and shows the ambiguity of possible effects of the altered gravity on formation and function of gravireceptors basing on the data from mollusks and fish.

Mass and mechanical sensitivity of otoliths.

It has been suggested that the changes of otolith mass during the otolith development in altered gravity conditions as well as the growth of otoliths in fishes in normal conditions are determined by the feedback between the otolith dynamics and the processes that regulate otolith growth. This hypothesis originates from the pendulum model of an otolith (de Vries, 1950), in which otolith mass is a parameters. The validity of this hypothesis is tested by comparing the pendulum model with a simplified spatially distributed model of an otolith. It was shown that when the otolith plate (otoconial layer) was spatially distributed and fixed to the macular surface, the mechanical sensitivity of the otolith does not depend on the total otolith mass and its longitudinal dimensions. It is determined by otolith thickness, Young's modulus, and the viscosity of the gel layer of the growing otolith. These parameters may change in order to secure otolith sensitivity under altered dynamic conditions (e.g., in microgravity). Possible hypotheses regarding the relationship between the otolith growth, otolith dynamics and animal growth are proposed and discussed here.

New model of hair cell bundle functioning in otoliths.

Transformation of the mechanical input in the chain: acceleration of otolithic membrane (OM)-displacement of the OM gel layer -deflection of hair cell bundle (HCB) -deformation of the system of tip-links- formation of temporal pattern of polarization was studied using simplified analytical models of these stages of conversion of mechanical stimulus into the HCB electrical response. The process of transformation of information in this chain was considered for two extreme cases of OM gel-HCB interaction: 1) the HCBs exactly follow the gel displacement; 2) stiff stereocilia and weak surrounding gel allow the relative motion of the bundle with regard to the gel. The analysis of a simplified model of cell polarization based on threshold triggering of the HCB tip-links allows to hypothesize that spatially nonhomogeneous HCB structure with a set of stereocilia of varying heights is designed to perceive spatially nonhomogeneous gel displacements caused by external acceleration. Thus, the HCB-OM gel interaction in the first case leads to formation of temporal pattern of depolarization that corresponds to the temporal pattern of gel displacement. In the second case the kinetics of depolarization reflects time dependence of gel displacement velocity.

[Sensitivity and parameters of the otolith as a spatially distributed system]. / Chuvstvitel'nost' i parametry otolita kak prostranstvenno-raspredelennoi sistemy.

Comparison of the de Vries pendulum and spatially distributed otolith models allowed physical interpretation of the pendulum model parameters and determination of the influence of otolith structure on its mechanical sensitivity when otolith dynamics is represented by a parallel displacement relative to the receptor surface. Proof was obtained that in this case otolith dynamics is independent of total mass, and mechanic sensitivity is not proportional to size of the organ. Possible explanations of the effects of changed dynamics conditions on otolith parameters in the fish are discussed.

Otoliths as biomechanical gravisensors.

This paper analyzes experimental data related to the reaction of otolith afferents in response to acceleration (Fernandez and Goldberg, 1976). It considers the assumptions that were the basis of the interpretation of the stimulus-response characteristics of afferents proposed by Fernandez and Goldberg. Comparing these experimental data with the results of modeling the otolith structures of vertebrates indicates that some peculiarities of the neural responses may be explained by the spatial dependence of the material parameters of the otolithic membrane across its thickness and within the volume of the membrane corresponding to the terminal field. The importance of the spatial dependence of the material parameters of the otolithic membrane for otolith functioning is discussed.

Theoretical considerations of plant gravisensing.

The mechanisms proposed to explain gravity sensing can be divided into two groups, "statolith" and "non-statolith" mechanisms. The traditional estimates of the plausibility of these mechanisms are based on the analysis of the signal-to-noise ratio. The existing data indicate that the problem of plant gravisensing may be related to the general problem of the detection of weak signals in mechanoreceptors. This paper reviews the known mechanisms of plant gravisensing as well as the latest nonlinear stochastic models of mechanoreception in which noise promotes detection and amplification of weak signals. These models based on nonlinear stochastic phenomena may be used to explain plant gravisensing, if the cell is considered a dynamic, spatially distributed system of active intracellular cytoskeletal networks and mechanosensitive proteins.

The effects of HGMFs on the plant gravisensing system.

High Gradient Magnetic Fields (HGMFs) offer new opportunities for studying the gravitropic system of plants. However, it is necessary to analyze the influence that HGMF can have on cellular processes and structures that may not be related to amyloplasts displacement. This paper considers possible HGMF effects on plants, which may accompany HGMF stimulation of amyloplasts and contribute to the mechanisms of the HGMF-induced curvature.

Finite element modeling of the 3D otolith structure.

A 3D finite element model (FEM) of the mammalian utricular otolith corresponding to spatial structure, shape and size of the otolith from the guinea pig was developed. The otolithic membrane (OM) was considered as consisting of gel and otoconial layers. The macular surface was approximated as a plane. The deformation of the OM under static loads such as gravity and the change of endolymphatic pressure was analyzed using the FEM for different mechanical parameters of the OM and for different gravity vector orientations. The analytical dependence of OM displacements caused by the acceleration parallel to the macular plane was obtained. By comparison of the results of calculations with the known experimental data Young's modulus of the gel layer was estimated to be of order of 10 N/m(2). It was shown that static loads result in 3D local otolith displacements inhomogeneously distributed along the macular surface and across otolith thickness. Their distribution depends on the geometrical and mechanical parameters of the otolith components. The influences of the finite size of the OM, the Young's modulus, Poisson's ratio and thickness of the gel layer on the local displacements distribution of the OM were analyzed. The results of simulation suggest that: a) the Young's modulus of the thin lowest part of the gel layer adjacent to the macular surface is much smaller than that of the rest of the OM; b) the structure of the border is designed to reduce the spatial inhomogeneity of the gel layer displacement; c) a change of the endolymphatic pressure may result in significant deformation of the OM.

Models of the dynamics of otolithic membrane and hair cell bundle mechanics.

Dynamic behavior of an otolithic membrane (OM) was studied analytically using simplified homogeneous viscoelastic (Kelvin-Voight body) model of the OM. The OM was represented by a thin plate attached to a macular plane. Viscoelastic properties of the OM determine the specific times (T(1) and T(2)) and frequency-dependent behavior of the local displacements of the membrane caused by the inertial time-dependent forces. Two kinds of an otolith stimulation were analyzed: step-function and harmonic accelerations of the membrane. Results of the modeling were compared with the known experimental data to estimate the Young's modulus E and viscosity mu of a gel layer: E is of order of 10 N/m(2), mu is of order of 1 poise in the range of frequency 0.2-2 Hz. It has allowed us to estimate the values of T(1) (10(-5)-10(-6) sec) and T(2) ( approximately 3 x 10(-2) sec). A relationship of the motion equation of the OM with well-known overdamped pendulum model of the otolith was discussed. The model of stereocilia tip-links deformation in the case, when the HCBs passively follow gel deformation, was proposed and analyzed. It was shown that for slender and long HCBs with the lengths comparable to a thickness of effective gel layer, a relative deformation of the tip-links of stereocilia caused by OM acceleration depends on time and the distance from the macular plane. The results of the modeling suggest that this type of the HCB may be responsible for analysis of fine temporal (frequency) structure of the OM acceleration.

Mathematical model of the evolution of statoconia.

A mathematical model of the evolution of statoconia in statocysts of freshwater snails based on the analysis of experimental data [Wiederhold et al., 1990; Pedrozo et al., 1996; Gao et al., 1997; Gao and Wiederhold, 1997; Wiederhold et al., 1999] is proposed. The growth of statoconia is considered as the process of solution crystallization. The model proposed assumes that two main processes determine the evolution of statoconia in developing snails: the generation of new statoconia and the linear growth of statoconia sizes. The analytical solution of the model and qualitative comparison of theoretical results with the experimental data show: (1) there are at least three periods of statoconia evolution; (2) the generation of new statoconia mainly determines the first period of evolution when the shell diameter of snails D < 4 mm; (3) when D > 6 mm the size distribution of statoconia is determined by the growth of their sizes with a constant rate; (4) on the interval deltaD = 4-6 mm the transformation of size distribution with selective dissolution of statoconia takes place. The model agrees well with the experimental data and makes it possible to estimate some parameters of the statoconia kinetics. Additional experiments, which are necessary for further development of the model, and quantitative estimates of the mechanisms of statoconia evolution are formulated.

Simulation and interpretation of experiments with otoliths.

The function of otolith as a gravisensor comprises a sequence of different physical and chemical processes, which experimental investigation is difficult or unfeasible. To understand the relationship between the input and output in otolith organ, certain assumptions concerning the mechanisms of transformation of information on the inner, intermediate stages have to be made. Their validity has to be tested by comparison of the properties of output characteristics, which follow from these assumptions, with experimental data. The goals of the present paper are: 1) to analyze some of the assumptions used in and related to the intermediate stages of transformation of mechanical stimulus in neural response; 2) to demonstrate by comparison with the results of modeling of otolith structure that some peculiarities of the experiment may be caused by spatial inhomogeneity of otolithic membrane (OM).

Model of statoconia accumulation in gravireceptors of mollusks.

The kinetics of formation and accumulation of statoconia are different for Aplysia californica and Biomphalaria glabrata. In Aplysia californica, the fast growth of statoconia number occurs after the critical size (approximately 45 micrometers) of statocyst is reached; then the increase of statoconia number is proceeding with the nonmonotonic rate during the life of an animal. In Biomphalaria the growth of statoconia number occurs only in the initial phase. Then long-term evolution of statoconia in the absence of their generation is the result of their growth in the cyst lumen. In the case of Aplysia californica it is not clear whether a temporal change of the statoconia size distribution (SSD) is caused by statoconia growth in the cyst lumen similar to that in Biomphalaria (Model 1) or statoconia growth takes place in supporting cells until their release into the cyst lumen occurs. (Model 2). This problem is of practical importance because the majority of experiments related to the development of molluscan gravireceptors in altered gravity dealt with an initial phase of statoconia evolution in Aplysia californica and Biomphalaria glabrata. The purpose of the present work is the application of mathematical modeling to the analysis of mechanisms of statoconia formation by supporting cells.

Computer simulation of the mechanical stimulation of the saccular membrane of bullfrog.

A three-dimensional computer simulation of the experiment (Benser M. E., Issa N.P., Hudspeth A.J., 1993. Hear. Res. 68, 243-252), devoted to the mechanical stimulation of the surface of saccular membrane of bullfrog with the otoconial mass removed was carried out by finite-element method. Comparison of the calculated distribution of the membrane displacements with the experimental data indicates that the gel layer of the saccular membrane is inhomogeneous. Its lower, thin (6-10 microm) sublayer bordering the macular surface has the smaller Young's modulus which is 20 times less than the modulus of the upper part of the gel membrane (2.5x10(2) N/m(2) and 6.6x10(3) N/m(2), respectively). Possible consequences of this result and modification of the experiment are being discussed. The estimates based on the results of simulation support the conclusion that hair bundle stiffness may dominate the elastic reactance to otolithic membrane shear (Benser et al., 1993).

[Modeling of the structure and mechanics of the otolith membrane]. / Modelirovanie struktury i mekhaniki otolitovoi membrany.

Behavior of otoliths of mammals against static (gravitation and changed pressure in the surrounding endolymph) and dynamic loads was surveyed using analytic and computerized (the finite difference method) models of the otolith membrane (OM). It was presumed that OM consists of gel-like and otoconial layers differing in mechanic and thickness. Comparison with available experimental data allowed to assess magnitudes of mechanic parameters of the gel-like layer responsible for OM interaction with the receptor hair cells (Yung's module for the layer is 1-10 N/m2 with the viscosity in the order of 1 poise), the characteristic times of otolith dynamics (T2 approximately 0.03 s, T1 approximately 10(-6)-10(-5) s), and the impact of changed endolymph pressure on the OM behavior. As was shown, inertial drift of mammalian OM is not so dependent on OM mass as on the relation of OM pressure on the molecular surface to Yung' module of the gel-like layer overlieing the macula.

[The imitation of the effects of otolith-ocular asymmetry by eccentric rotation]. / Modelirovanie éffektov otolit-okuliarnoi asimmetrii v usloviiakh ékstsentricheskogo vrashcheniia.

Today, investigation of the vestibuloocular reactions is the mainstream method of studying the vestibular asymmetry. Analysis of experimental data requires a model of otolith-ocular interaction. The proposed model is based on the literary data concerning measurements of ocular counter-rotation (OCR) and luminous line rotation (LLR) in experiments with eccentric rotation. The method utilizes a number of simplifications and suppositions, the basic of which are linearity of all phases of transformation of mechanic stimulus with the exception of the afferents' transmission function (proportionality of the nervous response to acceleration; the otolith-ocular response is proportional to the nervous response). It was demonstrated that the model qualitatively imitates the behavior of OCR and LLR in response to axifugal acceleration of the utricular otoliths and permits analysis of the role of various parameters of the otolith-ocular interaction. Comparison of calculated and experimental dependence of OCR and LLR on acceleration can help understanding of the otolith asymmetry.