Mechanisms of human static spatial orientation

S. B. Bortolami; S. Rocca; S. Daros; P. DiZio; J. R. Lackner

doi:10.1007/s00221-006-0387-9

Experimental Brain Research

August 2006, Volume 173, Issue 3, pp 374–388 | Cite as

Mechanisms of human static spatial orientation

Authors
Authors and affiliations

S. B. BortolamiEmail author
S. Rocca
S. Daros
P. DiZio
J. R. Lackner

Research Article

First Online: 21 April 2006

Received: 13 May 2005

Accepted: 13 January 2006

180 Downloads
17 Citations

Abstract

We have developed a tri-axial model of spatial orientation applicable to static 1g and non-1g environments. The model attempts to capture the mechanics of otolith organ transduction of static linear forces and the perceptual computations performed on these sensor signals to yield subjective orientation of the vertical direction relative to the head. Our model differs from other treatments that involve computing the gravitoinertial force (GIF) vector in three independent dimensions. The perceptual component of our model embodies the idea that the central nervous system processes utricular and saccular stimuli as if they were produced by a GIF vector equal to 1g, even when it differs in magnitude, because in the course of evolution living creatures have always experienced gravity as a constant. We determine just two independent angles of head orientation relative to the vertical that are GIF dependent, the third angle being derived from the first two and being GIF independent. Somatosensory stimulation is used to resolve our vestibular model’s ambiguity of the up–down directions. Our otolith mechanical model takes into account recently established non-linear behavior of the force–displacement relationship of the otoconia, and possible otoconial deflections that are not co-linear with the direction of the input force (cross-talk). The free parameters of our model relate entirely to the mechanical otolith model. They were determined by fitting the integrated mechanical/perceptual model to subjective indications of the vertical obtained during pitch and roll body tilts in 1g and 2g force backgrounds and during recumbent yaw tilts in 1g. The complete data set was fit with very little residual error. A novel prediction of the model is that background force magnitude either lower or higher than 1g will not affect subjective vertical judgments during recumbent yaw tilt. These predictions have been confirmed in recent parabolic flight experiments.

Keywords

Spatial orientation Otolith organs Somatosensation Subjective vertical Gravity Orientation model

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Notes

Acknowledgments

This research has been supported by AFSOR grant F49620110171 and NASA grant NAG9-1483.

Appendix

We model the otolith organs as if they were a single sensor located in the center of the head. The response of the combined sensor is the equal-weight sum or the difference of the signals coming from the right and left sides. The input to the sensor is the GIF (f′) with its components expressed as f′₁, f′₂, and f′₃ in otolith coordinates. The natural orientation of f′ is downward. The otolith organs individually are modeled as sets of three accelerometers capable of measuring the three components of the GIF (Fig. 11a). The response of the compound otolithic sensor $ \tilde{\mathbf{x}}'$ is defined as $(\tilde{x}^{\prime}_{1}, \tilde{x}^{\prime}_{2}, \tilde{x}^{\prime}_{3}),$ which is a function of f′. The tilde symbol, ∼, indicates internal representations of the physical variables which are presented without the tilde.

Fig. 11
a Reference frame and definition of variables for the right and left otolith organs and their respective arrangements; f′=(f′₁, f′₂, f′₃) represents the stimulus in otolith coordinates. The stimulus is the same on the left and right otoliths; however, the otoconial deflections are different for the left (x′_1L, x′_2L, x′_3L) and right (x′_1R, x′_2R, x′_3R) otoconia. b Schema of the cross-talks along the x′₂ and x′₃ axes produced by a stimulus (*thick arrow*) parallel to the i′₁ axis. Forces along the other cardinal otolith axes would produce analogous cross-displacements in the respective orthogonal axes

Sensor errors affect perceived orientation. Possible error sources include, but are not limited to, crosstalk and non-linearity of the otolith response. Displacement of the otoconia can be proportional to the applied force in relation to a stiffness k value:

$$ \tilde{x}^{\prime} = \frac{{f^{\prime}}}{k}$$

Linearity is rare in nature and generally does not extend over a broad range. Consequently, a more realistic way of addressing the mechanical characteristics of the otoliths is by introducing other terms in addition to the stiffness k:

$$ \tilde{x}^{\prime} = \frac{{f^{\prime}}}{k} + \frac{{(f^{\prime})^{n}}}{q}.$$

(11)

where 1/q is the coefficient of the nonlinear contribution (n>0). Non-linearity contributes to orientation errors in the following manner. Two GIFs having the same direction but different magnitudes would, because of non-linearity, generate non-proportional sets of sensed/estimated variables $(\tilde{x}^{\prime}_{1}, \tilde{x}^{\prime}_{2}, \tilde{x}^{\prime}_{3})$ and therefore different perceived directions.

Cross-talk among sensor axes, e.g.,

$$ \tilde{x}^{\prime}_{2} = \frac{{f^{\prime}_{2}}}{{k_{{22}}}} + \frac{{f^{\prime}_{3}}}{{k_{{23}}}}, $$

(12)

may produce sensor discharges in directions for which the GIF is not acting and consequently perception errors. An example of cross-talk is provided by Kondrachuk’s (2001) finite element modeling of the deformation of the gel layer and the otolithic membrane of the utricle of the guinea pig in response to a static force stimulus. He demonstrated that the direction of displacement of a generic point of the utricle is not necessarily parallel to the direction of the applied force and that a force parallel to one axis can also elicit a displacement along the other two axes.

We treat the mechanical response of the otoliths within their own reference frame (Fig. 1a). To transform the stimulus force f from the head reference frame (Figs. 1a, 2, 3) to the otolith frame we multiply it by the following rotation matrix:

$${\mathbf{f}}^{\prime} \triangleq {\mathbf{R}}_{y} (- 30)^{{\rm T}} \cdot {\mathbf{f}};\; {\mathbf{R}}_{y} (- 30)^{{\rm T}} = {\left[ {\begin{array}{*{20}c} {{\cos 30^\circ}} & {0} & {{\sin 30^\circ}} \\ {0} & {1} & {0} \\ {{- \sin 30^\circ}} & {0} & {{\cos 30^\circ}} \\ \end{array}} \right]}.$$

(13)

Taking relationships 11, 12, and 13 into account for all three axes, we obtain the following comprehensive representation of the compound otolith sensor:

$$\begin{aligned} \tilde{x}'_{1} &= a_{{11}} f^{\prime}_{a} + a_{{12}} f^{\prime}_{2} + a_{{13}} f^{\prime}_{3} + b_{{11}} (f^{\prime}_{1})^{n} + b_{{12}} (f^{\prime}_{2})^{n} + b_{{13}} (f^{\prime}_{3})^{n} \\ \tilde{x}^{\prime}_{2} &= a_{{21}} f^{\prime}_{1} + a_{{22}} f^{\prime}_{2} + a_{{23}} f^{\prime}_{3} + b_{{21}} (f^{\prime}_{1})^{n} + b_{{22}} (f^{\prime}_{2})^{n} + b_{{23}} (f^{\prime}_{3})^{n} \\ \tilde{x}^{\prime}_{3} &= a_{{31}} f^{\prime}_{1} + a_{{32}} f^{\prime}_{2} + a_{{33}} f^{\prime}_{3} + b_{{31}} (f^{\prime}_{1})^{n} + b_{{32}} (f^{\prime}_{2})^{n} + b_{{33}} (f^{\prime}_{3})^{n} \\ \end{aligned}, $$

which can be simplified as:

$$ \tilde{\mathbf{x}}^{\prime} = {\mathbf{Af}}^{\prime} + {\mathbf{B}}({\mathbf{f}}')^{n} $$

(14)

The coefficients a _ij and b _ij are the “compliances” ⁶ of the i axis associated with the f′_i force.

The paired otolith organs are oriented opposite to each other along the inter-aural axis and in the same direction along the sagittal plane (cf. also Spoendlin 1965; Wilson and Melvill Jones 1979; see Fig. 11a). This layout is appropriate if the combined signal along the inter-aural axis (i′₂) is the difference of the signals from the left and right otoliths (push–pull; cf. also Colenbrander 1964, Benson and Barnes 1970). Similarly, we can speculate that along the i′₁ and i′₃ axes the signals are the sums of signals of the two individual otoliths (pull–pull, cf. also Benson and Barnes 1970; Wilson and Melvill Jones 1979). Formally speaking, x′ = (x′_1L+x′_1R, x′_2L−x′_2R, x′_3L+x′_3R) represents the displacement of the combined otoconia. Hence, the inter-aural axis behaves symmetrically while the other two axes do not. In fact, along the inter-aural axis, at any given time, the stimulus on one utricle is just the opposite of that on the other (cf. Fig. 11a). Thus, whatever the response characteristics of a single utricle, the overall response of the pair, when combined, is symmetrical. Similar considerations apply to the saccule. Our experimental data for roll and recumbent yaw orientations also appear to be symmetrical, while the pitch data are not (Bortolami et al. 2006). This is consistent with the pull-pull signal configuration for the pitch axis, which is not necessarily bound to any symmetry.

Anatomical symmetries can help us identify potential patterns of cross-talk between mechanical axes. The sagittal plane is a plane of symmetry for the otolith organs. Figure 11b shows the pattern of displacements that would be produced by a force in the i′₁ direction. The components along the stimulated axis yield the main response, which represents the diagonal terms of the matrices A and B. The cross-talks, x′₂₁ and x′₃₁ relate to the coefficients (a ₂₁, b ₂₁) and (a ₃₁, b ₃₁) of the A and B matrices. The left and right contributions x′_21L and x′_21R are opposite to one another because of the symmetry of the two otoliths, but are equivalent in magnitude (at least in statistical terms, the otolith organs should have the same otoconial masses). Consequently, for the combined otoliths we obtain x′_21L+x′_21R=x′₂₁ ≅ 0. In turn, this implies a ₂₁=a ₂₁=0 and b ₂₁=b ₂₁=0. The same reasoning does not hold true for the x′₃₁ component since x′_31L has the same sign as x′_31R which yields a ₃₁ ≠ 0 and b ₃₁ ≠ 0.

Figure 11a presents the pattern for the inter-aural axis. Along this axis the cross-talks are opposite in sign, which yields a ₁₂=b ₁₂=a ₃₂=b ₃₂=0. Figure 11b shows the case for stimulation along the i′₃ axis. Here, a ₂₃ and b ₂₃ will be equal to zero, while a ₁₃ and b ₁₃ will be different from zero. This analysis indicates that all off-diagonal coefficients of the matrices A and B besides the two pairs (a ₁₃, b ₁₃) and (a ₃₁, b ₃₁) must be statistically zero. Our formulation would therefore predict that during roll body reorientation there might also be a slight perception of pitch tilt (a ₁₃, b ₁₃). We have not yet experimentally looked for the presence of such effects but shall in the future. In conclusion, the structure of the model’s matrices is as follows:

$${\mathbf{A}} = {\left[ {\begin{array}{*{20}c} {{a_{{11}}}} & {0} & {{a_{{13}}}} \\ {0} & {{a_{{22}}}} & {0} \\ {{a_{{31}}}} & {0} & {{a_{{33}}}} \\ \end{array}} \right]} \quad {\mathbf{B}} = {\left[ {\begin{array}{*{20}c} {{b_{{11}}}} & {0} & {{b_{{13}}}} \\ {0} & {{b_{{22}}}} & {0} \\ {{b_{{31}}}} & {0} & {{b_{{33}}}} \\ \end{array}} \right]}$$

(15)

The values of the coefficients of A and B must be compatible with the fact that each axis cannot move opposite to the applied force. This can be formulated as follows (cf. also Fig. 6) and requires that the response should be monotonic within physiological ranges.

$${\sum\limits_{j = 1}^3 {a_{{ij}} f^{\prime}_{j} + b_{{ij}} (f^{\prime}_{j})^{n}}} > 0; \quad i = 1\ldots3$$

A least-squares algorithm (MATLAB^TM 7.0: lsqnonlin) was used to determine the coefficients of A and B and the exponent n that best fit all of the experimental data for roll, pitch and recumbent yaw shown in Fig. 4. The model estimates the perceived vertical and not an Euler angle sequence of the head orientation. From the components of $\hat{\mathbf{g}},$ we derived roll, pitch and recumbent yaw projection angles. The derived angles $(\hat{\varphi}_{R}, \hat{\varphi}_{P}, \hat{\varphi}_{Y})$ were compared against the experimental data to calculate the goodness of the prediction/fit. The least-squares algorithm then adjusted the matrices A and B and the exponent n until the fit did not improve with additional iterations.

We identified the coefficients of the model by both imposing (9 parameters) and not imposing (11 parameters) a ₁₃=a ₃₁ and b ₁₃=b ₃₁. Both led to comparable fits (Fig. 4). Further reduction of the number of parameters resulted in a worse fit as shown in Fig. 10. The identified characterization of the input–output model of the otolith (nine parameters) relative to the plots displayed in Figs. 5 and 6 is as follows:

$${\left\{{\begin{array}{*{20}c} {{\tilde{x}^{\prime}_{1}}} \\ {{\tilde{x}^{\prime}_{2}}} \\ {{\tilde{x}^{\prime}_{3}}} \\ \end{array}} \right\}} = {\left[ {\begin{array}{*{20}c} 1.7527 & {0} & - 1.7268 \\ {0} & 2.5561 & {0} \\ - 1.7268 & {0} & 1.7693 \\ \end{array}} \right]}{\left\{{\begin{array}{*{20}c} {{f^{\prime}_{1}}} \\ {{f^{\prime}_{2}}} \\ {{f^{\prime}_{3}}} \\ \end{array}} \right\}} + {\left[ {\begin{array}{*{20}c} 0.6489 & {0} & 1.7450 \\ {0} & - 0.5545 & {0} \\ 1.7450 & {0} & 0.5927 \\ \end{array}} \right]}{\left\{{\begin{array}{*{20}c} {{f^{\prime}_{1}}} \\ {{f^{\prime}_{2}}} \\ {{f^{\prime}_{3}}} \\ \end{array}} \right\}}^{1.5217}$$

(16)

The terms f′_i have dimensions of accelerations while the matrix coefficients, “compliances”, have dimensions [s²]. Finally, for clarity we report the flow chart of the actual implementation of the model.

$$ \tilde{\mathbf{x}} = {\mathbf{R}}_{y} (30)^{{\rm T}} \cdot \tilde{\mathbf{x}}^{\prime}$$

$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{{RP}} = atan2 {\left( {{\left| {\frac{f} {g}} \right|}\frac{{{\sqrt { \tilde{x}^{2}_{1} + \tilde{x}^{2}_{2} } }}} {{ - \tilde{x}_{3} }}} \right)}$ ⁷

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{Y} = atan2{\left({\frac{{\tilde{x}_{2}}}{{\tilde{x}_{1}}}} \right)}$$

These lead to the following estimates of the gravity vector with respect to the head:

$$\begin{aligned} \hat{g}_{1} &= \sin (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{{RP}}) \cdot \cos(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{Y})\\ \hat{g}_{2} &= \sin (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{{RP}}) \cdot \sin(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{Y})\\ \hat{g}_{3} &= - \cos (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi} _{{RP}})\\ \end{aligned}$$

This is the model output. However, to compare with the experimental data for the purpose of parameter identification we used the following projection angles, which are not an Euler sequence.

$$\begin{aligned} \hat{\varphi}_{R} &= atan2 {\left({\frac{- \hat{g}_{2}}{- \hat{g}_{3}}} \right)}\\ \hat{\varphi}_{P} &= atan2{\left({\frac{\hat{g}_{1}}{-\hat{g}_{3}}} \right)}\\ \hat{\varphi}_{Y} &= atan2{\left({\frac{\hat{g}_{2}}{-\hat{g}_{1}}} \right)}\\ \end{aligned}$$

The above representation is singular for $\hat{\mathbf{g}} \equiv (1,0,0),$ $\hat{\mathbf{g}} \equiv (0,1,0),$ and $\hat{\mathbf{g}} \equiv (0,0,1),$ for which the estimations of $\hat{\varphi}_{R},$ $\hat{\varphi}_{P},$ and $\hat{\varphi}_{Y} $ were extended for continuity.

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Authors and Affiliations

S. B. Bortolami
- 1
Email author
S. Rocca
- 1
S. Daros
- 1
P. DiZio
- 1
- 2
J. R. Lackner
- 1
- 2

1.Ashton Graybiel Spatial Orientation Laboratory MS 033Brandeis UniversityWalthamUSA
2.Volen Center for Complex SystemsBrandeis UniversityWalthamUSA

Mechanisms of human static spatial orientation

Abstract

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Acknowledgments

References

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