How Peripheral Vestibular Damage Affects Velocity Storage: a Causative Explanation

Abstract

Velocity storage is a centrally-mediated mechanism that processes peripheral vestibular inputs. One prominent aspect of velocity storage is its effect on dynamic responses to yaw rotation. Specifically, when normal human subjects are accelerated to constant angular yaw velocity, horizontal eye movements and perceived angular velocity decay exponentially with a time constant circa 15–30 s, even though the input from the vestibular periphery decays much faster (~ 6 s). Peripheral vestibular damage causes a time constant reduction, which is useful for clinical diagnoses, but a mechanistic explanation for the relationship between vestibular damage and changes in these behavioral dynamics is lacking. It has been hypothesized that Bayesian optimization determines ideal velocity storage dynamics based on statistics of vestibular noise and experienced motion. Specifically, while a longer time constant would make the central estimate of angular head velocity closer to actual head motion, it may also result in the accumulation of neural noise which simultaneously degrades precision. Thus, the brain may balance these two effects by determining the time constant that optimizes behavior. We applied a Bayesian optimal Kalman filter to determine the ideal velocity storage time constant for unilateral damage. Predicted time constants were substantially lower than normal and similar to patients. Building on our past work showing that Bayesian optimization explains age-related changes in velocity storage, we also modeled interactions between age-related hair cell loss and peripheral damage. These results provide a plausible mechanistic explanation for changes in velocity storage after peripheral damage. Results also suggested that even after peripheral damage, noise originating in the periphery or early central processing may remain relevant in neurocomputations. Overall, our findings support the hypothesis that the brain optimizes velocity storage based on the vestibular signal-to-noise ratio.

Appendix

In this appendix, we provide derivations for the equations for each unilateral damage scenario shown in Table 2. We begin by listing the relevant variables, followed by derivations for each scenario.

${w}_{n}$: the sum of age-adjusted process noise, for normal vestibular function, arising from the ipsilateral vestibular system ${w}_{in}$ and the contralateral vestibular system ${w}_{cn}$.

${v}_{n}$: age-adjusted measurement noise and is the combined noise arising from all neural sources, with contributions from the ipsilateral ${v}_{in}$ and the contralateral ${v}_{cn}$ vestibular system.

${Q}_{n}$: the variance of age-adjusted process noise, for normal vestibular function, and ${Q}_{in}$ and ${Q}_{cn}$ represent contributions from the ipsilateral and contralateral vestibular systems, respectively.

${R}_{n}$: the variance of age-adjusted measurement noise, for normal vestibular function, and ${R}_{in}$ and ${R}_{cn}$ represent contributions from the ipsilateral and contralateral vestibular systems, respectively.

${R}_{i1},{R}_{i2}, {R}_{i3}, {R}_{i4}, {R}_{i5}$: the measurement noise in the ipsilateral vestibular system after adjustment for unilateral damage and age for each of the five scenarios.

${R}_{c1},{R}_{c2}, {R}_{c3}, {R}_{c4}, {R}_{c5}$: the measurement noise in the contralateral vestibular system after adjustment for unilateral damage and age for each of the five scenarios.

${R}_{1},{R}_{2}, {R}_{3}, {R}_{4}, {R}_{5}$: the measurement noise after convergence of the ipsilateral and contralateral vestibular inputs after adjustment for unilateral damage and age for each of the five scenarios.

${Q}_{1},{Q}_{2}, {Q}_{3}, {Q}_{4}, {Q}_{5}$: the process noise after convergence of the ipsilateral and contralateral vestibular inputs after adjustment for unilateral damage and age for each of the five scenarios.

$SN{R}_{n}$: the age adjusted signal-to-noise ratio for normal vestibular function (i.e., ${Q}_{n}/{R}_{n}$).

$SN{R}_{1}, SN{R}_{2} , SN{R}_{3} , SN{R}_{4} , SN{R}_{5}$: the signal-to-noise ratios after adjustment for unilateral damage and age for each of the five scenarios

$${w}_{1}={w}_{2}={w}_{3}={w}_{4}={w}_{5}={w}_{cn}=0.5 \cdot {w}_{n}$$

$${w}_{n}=0.5\cdot {w}_{in}+0.5\cdot {w}_{cn}$$

$$\begin{aligned}{Q}_{n}&=var({w}_{n})\\&=var(0.5\cdot {w}_{in}+0.5\cdot {w}_{cn})\\&={0.5}^{2}\cdot var({w}_{in})+{0.5}^{2}\cdot var({w}_{cn})\\&={0.5}^{2}\cdot {Q}_{in}+{0.5}^{2}\cdot {Q}_{cn}\end{aligned}$$

$${R}_{n}={0.5}^{2}{\cdot R}_{in}+{0.5}^{2}{R}_{cn}$$

$${R}_{in}={R}_{cn}=2{\cdot R}_{n}$$

Scenario 1: loses signal, loses noise, no change in contralateral noise

$${R}_{i1}=0\cdot {R}_{in}=0$$

$${R}_{c1}={R}_{cn}=2\cdot {R}_{n}$$

$${Q}_{1}={0.5}^{2}\cdot {Q}_{n}$$

$$\begin{aligned}{R}_{1}&={0.5}^{2}\cdot {R}_{i1}+{0.5}^{2}\cdot{R}_{c1}\\&={0.5}^{2}\cdot 0\cdot {R}_{in}+{0.5}^{2}\cdot 2\cdot{R}_{n}\\&=0+{0.5}^{2}\cdot 2\cdot {R}_{n}\\&={0.5}^{2}\cdot 2\cdot {R}_{n}\end{aligned}$$

$$SN{R}_{1}=\frac{{Q}_{1}}{{R}_{1}}=\frac{{0.5}^{2}{Q}_{n}}{{0.5}^{2}\cdot 2\cdot {R}_{n}}=0.5\cdot \frac{{Q}_{n}}{{R}_{n}}=0.5\cdot SN{R}_{n}$$

Scenario 2: loses signal, loses noise, contralateral noise + 26 %

$${R}_{i2}=0\cdot {R}_{in}=0$$

$${R}_{c2}={1.26}^{2}\cdot {R}_{cn}={1.26}^{2}\cdot 2\cdot {R}_{n}$$

$${Q}_{2}={0.5}^{2}\cdot {Q}_{n}$$

$$\begin{aligned}{R}_{2}&={0.5}^{2}\cdot {R}_{i2}+{0.5}^{2}\cdot {R}_{c2}\\&={0.5}^{2}\cdot 0\cdot {R}_{in}+{0.5}^{2}\cdot {1.26}^{2}\cdot 2\cdot {R}_{n}\\&=0+{0.5}^{2}\cdot {1.26}^{2}\cdot 2\cdot {R}_{n}\\&={0.5}^{2}\cdot {1.26}^{2}\cdot 2\cdot {R}_{n}\end{aligned}$$

$$SN{R}_{2}=\frac{{Q}_{2}}{{R}_{2}}=\frac{{0.5}^{2}\cdot {Q}_{n}}{0.5\cdot {1.26}^{2}\cdot {R}_{n}}=\frac{0.5}{{1.26}^{2}}\cdot \frac{{Q}_{n}}{{R}_{n}}=\frac{0.5}{{1.26}^{2}}\cdot SN{R}_{n}$$

Scenario 3: loses signal, retains noise, no change in contralateral noise

$${R}_{i3}={R}_{in}=2\cdot {R}_{n}$$

$${R}_{c3}={R}_{cn}=2\cdot {R}_{n}$$

$${Q}_{3}={0.5}^{2}\cdot {Q}_{n}$$

$$\begin{aligned}{R}_{3}&={0.5}^{2}\cdot {R}_{i3}+{0.5}^{2}\cdot {R}_{c3}\\&={0.5}^{2}\cdot {R}_{in}+{0.5}^{2}\cdot {R}_{cn}\\&={0.5}^{2}\cdot 2\cdot {R}_{n}+{0.5}^{2}\cdot 2\cdot {R}_{n}\\&={R}_{n}\end{aligned}$$

$$SN{R}_{3}=\frac{{Q}_{3}}{{R}_{3}}=\frac{{0.5}^{2}\cdot {Q}_{n}}{{R}_{n}}={0.5}^{2}\frac{{Q}_{n}}{{R}_{n}}=0.25\cdot SN{R}_{n}$$

Scenario 4: lose signal, retain noise, contralateral noise + 26 %

$${R}_{i4}={R}_{in}=2{R}_{n}$$

$${R}_{c4}={1.26}^{2}{R}_{cn}={1.26}^{2}\cdot 2{R}_{n}$$

$${Q}_{4}={0.5}^{2}\cdot {Q}_{n}$$

$$\begin{aligned}{R}_{4}&={0.5}^{2}{R}_{i4}+{0.5}^{2}{R}_{c4}\\&={0.5}^{2}{R}_{in}+{0.5}^{2}\cdot {1.26}^{2}{R}_{cn}\\&={0.5}^{2}\cdot 2{R}_{n}+{0.5}^{2}\cdot {1.26}^{2}\cdot 2{R}_{n}\\&=0.5(1+{1.26}^{2}){R}_{n}\end{aligned}$$

$$\begin{aligned}SN{R}_{4}&={Q}_{4}{R}_{4}\\&={0.5}^{2}\cdot {Q}_{n}\cdot 0.5(1+{1.26}^{2}){R}_{n}\\&={0.5}^{2}\cdot 0.5(1+{1.26}^{2}){Q}_{n}{R}_{n}\\&=0.5\left(1+{1.26}^{2}\right)SN{R}_{n}\end{aligned}$$

Scenario 5: loses signal, retains noise, contralateral noise + 26 %, additional 32.8 % noise

$${R}_{i5}={R}_{in}=2\cdot {R}_{n}$$

$${R}_{c5}={1.26}^{2}\cdot {R}_{cn}={1.26}^{2}\cdot 2\cdot {R}_{n}$$

$${Q}_{5}={0.5}^{2}\cdot {Q}_{n}$$

$$\begin{aligned}{R}_{5}&={1.328}^{2}\cdot ({0.5}^{2}\cdot {R}_{i1}+{0.5}^{2}{R}_{c1})\\&={1.328}^{2}\cdot ({0.5}^{2}\cdot {R}_{in}+{0.5}^{2}\cdot {1.26}^{2}\cdot {R}_{cn})\\&={1.328}^{2}\cdot ({0.5}^{2}\cdot 2\cdot {R}_{n}+{0.5}^{2}\cdot {1.26}^{2}\cdot 2\cdot {R}_{n})\\&={1.328}^{2}\cdot (0.5\cdot \left(1+{1.26}^{2}\right){R}_{n})\end{aligned}$$

$$\begin{aligned}SN{R}_{5}&=\frac{{Q}_{5}}{{R}_{5}}\\&=\frac{{0.5}^{2}\cdot {Q}_{n}}{{1.328}^{2}\cdot 0.5\cdot \left(1+{1.26}^{2}\right)\cdot {R}_{n}}\\&=\frac{{0.5}^{2}}{{1.328}^{2}\cdot 0.5\cdot \left(1+{1.26}^{2}\right)}\cdot \frac{{Q}_{n}}{{R}_{n}}\\&=\frac{0.5}{{1.328}^{2}\left(1+{1.26}^{2}\right)}\cdot SN{R}_{n}\end{aligned}$$