Bayesian optimal adaptation explains age-related human sensorimotor changes

Abstract

The brain uses information from different sensory systems to guide motor behavior, and aging is associated with simultaneous decline in the quality of sensory information provided to the brain and deterioration in motor control. Correlations between age-dependent decline in sensory anatomical structures and behavior have been demonstrated in many sensorimotor systems, and it has recently been suggested that a Bayesian framework could explain these relationships. Here we show that age-dependent changes in a human sensorimotor reflex, the vestibuloocular reflex, are explained by a Bayesian optimal adaptation in the brain occurring in response to death of motion-sensing hair cells. Specifically, we found that the temporal dynamics of the reflex as a function of age emerge from (r = 0.93, P < 0.001) a Kalman filter model that determines the optimal behavioral output when the sensory signal-to-noise characteristics are degraded by death of the transducers. These findings demonstrate that the aging brain is capable of generating the ideal and statistically optimal behavioral response when provided with deteriorating sensory information. While the Bayesian framework has been shown to be a general neural principle for multimodal sensory integration and dynamic sensory estimation, these findings provide evidence of longitudinal Bayesian processing over the human life span. These results illuminate how the aging brain strives to optimize motor behavior when faced with deterioration in the peripheral and central nervous systems and have implications in the field of vestibular and balance disorders, as they will likely provide guidance for physical therapy and for prosthetic aids that aim to reduce falls in the elderly.

NEW & NOTEWORTHY We showed that age-dependent changes in the vestibuloocular reflex are explained by a Bayesian optimal adaptation in the brain that occurs in response to age-dependent sensory anatomical changes. This demonstrates that the brain can longitudinally respond to age-related sensory loss in an ideal and statistically optimal way. This has implications for understanding and treating vestibular disorders caused by aging and provides insight into the structure-function relationship during aging.

INTRODUCTION

The brain uses information from different sensory systems to guide motor behavior, and aging is associated with simultaneous decline in the quality of sensory information provided to the brain and deterioration in motor control. For example, substantial loss of sensory organ cells occurs in parallel with changes in sensorimotor reflex dynamics in a number of systems, including the reflexes responding to the motion-sensing vestibular organs located in the inner ear (Lopez et al. 2005; Merchant et al. 2000; Paige 1992; Rosenhall 1973). While correlations between age-dependent decline in sensory anatomical structures and behavior have been demonstrated in most sensorimotor systems (Baezner et al. 2008), robust conceptual frameworks explaining these relationships are still lacking. Furthermore, the relative contribution of age-dependent changes in sensory structures vs. brain function leading to the decline in behavioral performance is unknown. In this study, we use the vestibular system to examine whether a Bayesian (Berniker and Kording 2011) framework explains these relationships.

The “Bayesian brain” includes the notion of statistically optimal processing that reduces sensory estimation errors to the minimum possible when performing linear processing of noisy (Faisal et al. 2008) signals, including for dynamic sensory estimation (Borah et al. 1988; Orban de Xivry et al. 2013; Paulin et al. 1989; Wolpert et al. 1995) and multisensory integration (Ernst and Banks 2002; Landy et al. 1995). A Bayesian optimal model has been used to suggest that differences between a group of young subjects and a group of elderly subjects in a visuomotor tracking task may arise from adaptation to differences in sensory noise (Sherback et al. 2010). Another study (Moran et al. 2014) has shown that age-related attenuation of evoked responses measured by magnetoencephalography (MEG) can be explained by an attenuation of sensory learning effects through the use of dynamic causal modeling (Garrido et al. 2008; Kiebel et al. 2006), which is qualitatively consistent with the Bayesian brain hypothesis via the free energy principle (Friston and Stephan 2007). We determined whether a Bayesian optimal adaptation could explain age-related changes in the behavior of the vestibuloocular reflex (VOR), which stabilizes images on the retina during head motion, as a response to age-related death of hair cells in the angular motion-sensing semicircular canals.

We examined the dynamic properties of the VOR that result in a diminishing response during body rotation, which can be characterized by a time constant describing an exponential decay of eye velocity after body rotation achieves constant angular velocity. Although the VOR time constant in normal humans is ~10–30 s (Dimitri et al. 2001), the vestibular afferent signal decays more quickly (Jones and Milsum 1971), and the brain uses an integration process called “velocity storage” to elongate the signal and make it more accurate (Raphan et al. 1977; Robinson 1977). However, this integration process presumably amplifies neural noise, resulting in erroneous eye motions (Karmali and Merfeld 2012; Laurens and Angelaki 2011; Laurens and Droulez 2007; MacNeilage et al. 2008). The VOR time constant is controlled by the cerebellum (Waespe et al. 1985), and the goal may be to balance this trade-off between inaccuracy and noise. Bayesian approaches such as the Kalman filter (Kalman 1960; Wolpert et al. 1995), which aim to minimize motion estimation errors based on sensory dynamics and the noise properties of afferents and behavior (Borah et al. 1988; Paulin et al. 1989), predict VOR time constants similar to experimental values (Karmali and Merfeld 2012). Of particular relevance to this study, they predict that the optimal VOR time constant is lower when afferent noise is higher. Similar predictions arise from other Bayesian, non-Kalman filter models of velocity storage (Karmali and Merfeld 2012; Laurens and Angelaki 2011; Laurens and Droulez 2007). Experimental studies support this prediction; individuals with high VOR thresholds, which suggest higher vestibular noise, also have lower VOR time constants (Cousins et al. 2013).

It has frequently been postulated that age-related VOR changes, especially decreasing time constant, can be explained by age-related anatomical changes such as hair cell death. This connection is supported by an observed drop in VOR time constant after hair cells are killed by ototoxic drugs (Baloh et al. 1984; Meyers 1970). Furthermore, patients with unilateral vestibular lesions have smaller VOR time constants and higher VOR thresholds compared with normal subjects (Cousins et al. 2013). This study also found that the change in VOR thresholds caused by the lesion was consistent with the prediction of a maximum likelihood model, although the relationship between VOR thresholds and VOR time constant was not modeled. Since the exact mechanism remains unknown, we compared experimental data with predictions of a Bayesian model made with the following elements. First, we assumed that afferent signal-to-noise ratio (SNR) declines proportionally to hair cell density decline, since afferent noise changes little after hair cell loss (Hirvonen et al. 2005). Second, the Bayesian optimal VOR time constant was determined, with a Kalman filter, for the decreased SNR. We also incorporated central adaptation to amplify diminished afferent signals to preserve robust and compensatory eye movements in response to head movements (Allum et al. 1988; Curthoys and Halmagyi 1999; Smith and Curthoys 1989), although its inclusion did not alter our conclusions. Agreement between model predictions and experimental results (Dimitri et al. 2001; Merchant et al. 2000) demonstrate that the brain can longitudinally respond to age-related sensory loss in an ideal and statistically optimal way.

METHODS

Experimental data for age-related changes in VOR behavior and vestibular hair cell population.

We used published data that characterized age-related changes in both the VOR and sensory anatomy. Behavioral data were measured for yaw rotation in the horizontal plane with subjects upright in the dark, and anatomical data were for the vestibular sensory organ that is most sensitive to yaw rotation, the lateral semicircular canal (SCC). To characterize the dynamics of the VOR with age, we chose one study (Dimitri et al. 2001) as the basis of our modeling because 1) it focused on dynamics by characterizing the time constant; 2) the time constant was more precisely determined than in other studies because it was fit for each subject across a set of measurements, rather than using phase at a single frequency to determine time constant; 3) subjects were screened for well-known clinical vestibular disorders or head trauma by vestibular clinical testing and a history; 4) subjects spanned a large age range; and 5) the study included a substantial number of subjects (91 subjects). The authors found that the relationship between VOR time constant τ and age was fit well by the model

$$ln\left(\tau \right)=2.19\text{·}\left(2-e^{-age/27.2}\right)\text{·}\left(e^{-age/172.3}\right)$$ (1)

which is the relationship we use when presenting experimental data (Fig. 1A and Fig. 1 in Dimitri et al. 2001).

Fig. 1.

Age-related behavioral and anatomical changes from published experimental data. A: yaw VOR time constant changes with age, becoming smaller in the elderly (Dimitri et al. 2001). Insets: examples of eye responses to velocity steps for, respectively, a long time constant for a typical 35 yr old and a short time constant for a typical 95 yr old. B: yaw VOR gain, the ratio of eye velocity to head velocity, decreases slightly with age (Dimitri et al. 1996). C: lateral SCC hair cell density declines with age (Merchant et al. 2000).

Another important assay of the angular VOR is angular VOR gain, the ratio of angular eye velocity to angular head velocity for high-frequency motion, which can also be described by the amplitude of the high-frequency plateau of the transfer function describing the VOR. An earlier study from the same authors found a statistically significant decline in experimental VOR gain (V_exp) with age, with a slope of −0.0017/yr (Dimitri et al. 1996), and used many of the same subjects. Based on the first figure in Dimitri et al., we used an intercept VOR gain of 0.83.

Our predictions used published tabular data from a study of hair cell density with aging in the lateral SCC cristae (Fig. 1C) that included a large number of human subjects with ages up to 100 yr and examined temporal bones with modern histology techniques (Merchant et al. 2000). Since there is no statistically significant change in the ratio of type I to type II hair cells with age (Lopez et al. 2005; Merchant et al. 2000), we do not make separate predictions for type I and type II hair cells. Since predictions were based on changes in hair cell density, the fraction of hair cells living (h) was determined relative to infancy; the results show that normalizing with other age ranges had little impact. Since hair cell density was reported grouped over 1-decade ranges (e.g., 41–50 yr), the corresponding VOR time constant was determined by using the center age for each group (e.g., 45 yr).

Predicting Bayesian optimal VOR dynamics in response to hair cell death.

To test the hypothesis that VOR dynamics are a Bayesian optimal adaptation to hair cell death, we made predictions for VOR time constant to compare with experimental VOR time constants. Predictions were based on three interrelated processes (Fig. 2), all of which are grounded in established experimental results. In the following sections, we describe first the expected change in afferent signaling in response to changes in hair cell density, followed by the central compensation process that occurs in response to changes in afferent signaling to maintain VOR function, and finally how these processes combine to generate the changes in the Bayesian optimal VOR time constant predicted by a Kalman filter (Kalman 1960; Kalman and Bucy 1961). Our predictions are based on a conceptual model that is based on known physiological processes and is not meant to be a model of a neuronal network. This is the same as for other Bayesian dynamic vestibular models (Borah et al. 1988; Karmali and Merfeld 2012; Laurens and Angelaki 2011; Laurens and Droulez 2007; MacNeilage et al. 2008; Selva and Oman 2012). When we make reference to afferents, this refers to the entire collective signaling. Guided by the principle of parsimony, we intentionally selected the simplest possible model (Robinson 1977) that could be used to test our hypothesis, since adding additional components would increase the number of free parameters, with insufficient corresponding experimental data to constrain those values. This could have resulted in the Bayesian hypothesis being confirmed because of tuning of additional free parameters.

Fig. 2.

Schematic of the vestibuloocular reflex and predicted changes in behavioral responses in response to hair cell death. A: the vestibuloocular reflex model includes noisy afferents, central compensation, and a velocity storage mechanism modeled with an optimal Kalman filter. B: we illustrate the model predictions in response to head velocity rapidly increasing from rest to constant velocity rotation. Two sets of simulations are shown, the baseline simulation with all hair cells intact (“younger,” C–E) and the condition in which 37% of hair cells have been lost (“older,” F–H). C: representation of SCC afferent firing rate for “younger” case with 100% of hair cells present. D: in this case central compensation for hair cell loss in unneeded, so c = 1 and there is no amplification of the afferent signal. E: the Kalman filter model of velocity storage predicts a VOR time constant of 22 s. F: afferent firing rate is attenuated in the “older” case and is proportional to hair cell density. G: the model determines that a compensation of c = 1.78 makes VOR gain correct for an 85-yr-old subject. (Intuitively it may seem like c should be 1/0.63 = 1.59; results describes why this is not the case.) Compensation also amplifies afferent noise. H: eye velocity decays more quickly than in the “younger” case to optimally account for increased noise, with a time constant of 14 s.

Prediction element 1: impact of hair cell death on afferent signaling.

We assumed that the amplitude of the afferent signals declines proportionally to reduction in hair cell density with age, which is supported by experimental data showing a reduction in VOR gain after aminoglycosides in humans (Black et al. 1987) and chinchilla (Hirvonen et al. 2005) as well as the roughly 50% reduction in VOR gain caused by an acute unilateral lesion (Fetter and Zee 1988; Paige 1983). We included “noise” on afferent signaling, where noise is simply any part of a transmission not related to the signal of interest, such as firing rate variability on an afferent introduced by quantal synaptic release (Faisal et al. 2008). Since there is little change in afferent noise after significant vestibular hair cell death caused by aminoglycosides (Hirvonen et al. 2005; Li and Correia 1998), our predictions kept noise constant with aging. Since the later assumption may seem counterintuitive, we examine it at length in discussion. While we use the term “noise,” variability, firing rate irregularity, and imprecision also describe the same phenomenon. Metrics for quantifying noise include variance, interspike interval (ISI) variability, coefficient of variation, precision (the inverse of variance), reliability (analogous to precision), and threshold. Threshold is related to noise through signal detection theory (Faisal et al. 2008; Green and Swets 1966; Merfeld 2011). (Note that this differs from some work that defines threshold as the required change in stimulus to elicit a fixed change in firing rate.)

Prediction element 2: central compensation to recover VOR gain.

The VOR stabilizes gaze (Leigh and Zee 2006) to keep clear vision by moving the eyes in the opposite direction of the head. To do so accurately, the VOR gain is adaptively tuned (Gauthier and Robinson 1975; Gonshor and Jones 1976), such as when wearing spectacles that require a different gain. It is well established that central compensation improves VOR gain in pathologies such as unilateral loss of the vestibular organs and damage to the vestibular organs from ototoxic drugs (Allum et al. 1988; Black et al. 1987; Curthoys and Halmagyi 1999; Fetter and Zee 1988; Paige 1983; Smith and Curthoys 1989). Similarly, one study found that torsional VOR responses to afferent stimulation using galvanic vestibular stimulation increases between 20 and 59 yr of age, which suggests the presence of central compensation for age-related hair cell loss (Jahn et al. 2003). As well, the VOR gain (Fig. 1B) changes only modestly with age (Dimitri et al. 1996), suggesting a compensatory correction. Thus we assume that the central compensation adaptive mechanism amplifies the afferent signal to compensate for the loss of signal due to hair cell loss during aging, and specifically that the mechanism amplifies the afferent signal to make the VOR gain equal to published experimental data for that age (Dimitri et al. 1996). We implement this by multiplying the afferent signal by a compensation scalar c, using a method described in Implementation. While this amplification would not change SNR, it would result in amplification of noise.

Prediction element 3: changes in Bayesian optimal VOR time constant in response to changes in afferent signaling and adaptation.

This element is based on our model that links VOR dynamics to the signal and noise characteristics of the vestibular periphery (Karmali and Merfeld 2012). In this section we briefly summarize the overall approach and architecture.

As described in introduction, the velocity storage mechanism that prolongs the SCC time constant of roughly 5 s to the VOR time constant of 10–30 s effectively includes an integration process. Since integration of noise can cause drift and increased noise, velocity storage presumably worsens precision, with greater variability (i.e., lower precision) as the VOR time constant gets larger. On the other hand, the brain’s rotation estimate is more accurate when the VOR time constant is larger. Thus there is a trade-off between being accurate and being precise, and the time constant adjusts this balance. We hypothesized that the brain selects a VOR time constant to minimize total error (specifically the root-mean-square error, which combines precision and accuracy) in the estimate of angular velocity (Karmali and Merfeld 2012; Laurens and Angelaki 2011; Laurens and Droulez 2007). Furthermore, we used optimal estimation to show that the velocity storage time constant is “linearly optimal,” meaning that it dynamically minimizes error in the estimated angular motion under the assumption that the system is linear. We determined the optimal responses using a Kalman filter (Kalman 1960; Kalman and Bucy 1961), which has previously been used to study Bayesian processing in vestibular (Borah et al. 1988; Karmali and Merfeld 2012; Oman 1982; Paulin et al. 1989; Selva and Oman 2012), proprioceptive (Wolpert et al. 1995), and visual (Orban de Xivry et al. 2013) sensorimotor responses. Specifically, we used our Kalman filter model (Karmali and Merfeld 2012) that determines the optimal trade-off between reducing noise in the estimate and how quickly the estimate responds, by minimizing the root-mean-square difference between the actual and estimated state. The Bayesian nature of the Kalman filter arises (Berniker and Kording 2011) because at each point in time the Kalman filter (Kalman 1960; Kalman and Bucy 1961) integrates the internal model’s estimate of the sensory input (the prior) with the noisy afferent signal (the likelihood) to update the prediction of the current sensory input (the posterior). This posterior becomes the prior for the next point in time.

The equations below can be briefly summarized as follows: if significant measurement noise is added to the sensory signal, more reliance is placed on the internal model and the VOR time constant gets shorter. If measurement noise added to the sensory signal is small, the estimate is more dependent on the sensory input and the VOR time constant gets longer. This decreased and increased reliance on the sensory input is implemented as a decrease and increase in a parameter called the Kalman filter gain K. Thus there is an explicit link between the velocity storage time constant and noise characteristics. The correct balance is determined (Kalman 1960; Kalman and Bucy 1961) by considering the relationship between the measurement noise and the distribution of the input motion, called process noise. Measurement noise represents errors introduced in the sensory pathway, including the peripheral organ, hair cells, and afferent noise, and is modeled as being normally distributed with variance R; process noise represents system perturbations such as body motions not commanded by the brain, both external (e.g., unstable surfaces, wind gust) and internal (e.g., muscle noise causing body sway), and is modeled as being normally distributed with variance Q. The signal-to-noise ratio (SNR) is the ratio of process noise to measurement noise, i.e.,

$$SNR=Q\text{/}R$$ (2)

The dynamics of the sensory organ, the SCC, are described as a band-pass filter using the transfer function

$$\frac{y}{d}=\frac{{\tau }_{1}s}{{\tau }_{1}s+1}\cdot \frac{1}{{\tau }_{2}s+1}$$ (3)

where d(t) and y(t) are time-varying scalars representing the input angular velocity motion disturbance and output afferent information, respectively. The time constants reflecting SCC fluid mechanics are τ₁ = 5.7 s (Fernandez and Goldberg 1971; Jones and Milsum 1971) and τ₂ = 0.005 s (Fernandez and Goldberg 1971; Groen 1957; Oman et al. 1987), although the specific value of τ₂ does not change the results (Karmali and Merfeld 2012). The Kalman filter (Kalman 1960; Kalman and Bucy 1961) requires the sensor characteristics to be formulated as state space equations

$$\overrightarrow{\dot{x}}\left(t\right)=A\overrightarrow{x}\left(t\right)+Bd\left(t\right)\hspace{0.17em}and\hspace{0.17em}y\left(t\right)=C\overrightarrow{x}\left(t\right)$$ (4)

which for the SCC can be formulated (Karmali and Merfeld 2012) as

$$\overrightarrow{x}\left(t\right)=\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]\text{,}\hspace{0.17em}A=\left[\begin{array}{cc}0& 1\\ -\frac{1}{{\tau }_1{\tau }_2}& -\left(\frac{1}{{\tau }_1}+\frac{1}{{\tau }_2}\right)\end{array}\right]\text{,}B=\left[\begin{array}{c}0\\ 1\end{array}\right]\text{,}\hspace{0.17em}C=\left[\begin{array}{cc}0& \text{1}/{\tau }_2\end{array}\right]$$

where $\overrightarrow{x}\left(t\right)$ is an internal state vector and $\overrightarrow{\dot{x}}\left(t\right)$ is its derivate with respect to time. (Note that matrix C and scalar c are distinct variables). The Kalman filter gain K was determined by solving the standard Kalman equations

$$\dot{P}(t)=AP(t)+P(t)A^T+BQ{B}^T-K(t)RK{(t)}^T$$ (5)

and

$$K(t)=P(t)C^T{R}^{-1}$$ (6)

where Q, R, A, B, and C have been described and P(t) is an estimate of the covariance of the difference between the states of the actual system and the internal model. Since our system dynamics do not change over short periods of time and therefore K₂₁(t) is at steady state (Karmali and Merfeld 2012) and since K₁₁(t) is effectively 0, we refer only to K = K₂₁(t) in the following sections.

The Kalman gain K determines both the dynamics and amplification (Karmali and Merfeld 2012; Merfeld et al. 1993) of the estimated angular velocity relative to the input: the velocity storage time constant is

$$\tau =(K+1){\tau }_1$$ (7)

[i.e., (K + 1) times the SCC time constant], and the overall amplification of the Kalman filter is K/(K + 1). For example, we used K = 3 in our previous work, since it corresponds to a velocity storage time constant of 23 s; this also yields an amplification of 0.75.

We considered other model topologies, and we now provide the justification for the structure chosen. We chose a conceptual model that was rooted in known physiology that allowed to us make dynamic Bayesian predictions. We emphasize that we are not modeling a single afferent together with one or more hair cells. We considered making predictions based on models with multiple neuronlike channels and hair cells, but because there are limited experimental data on the number of neurons involved in velocity storage, on the structure of the network, on the distribution of hair cells innervating each neuron, and on the fraction of regular and irregular afferent neurons, we were unable to develop a model that did not have a large number of unconstrained parameters. In comparison, the model we used had a single free parameter (c) that was uniquely constrained by experimental data describing experimental VOR gain (V_exp). Furthermore, the age-related change in VOR time constant was an emergent property of the model, since it relied on zero free parameters. This approach makes an assumption that, averaged across the population, the network is uniform enough to be approximated by the model, as is the case for other Bayesian models.

Implementation.

To implement these predictions (Fig. 2), for each age group and its corresponding normalized hair cell density h, c was determined so that the predicted VOR gain V equaled the corresponding experimental VOR gain V_exp (Dimitri et al. 1996). The predicted VOR gain is affected by each of the three prediction elements, including reduced afferent signal due to hair cell loss (h), central compensation (c) and the amplification of the Kalman filter [K/(K + 1)], resulting in the equation

$$V=h·c\frac{K}{K+1}$$ (8)

This also resulted in an elevated measurement noise, which we define as

$$R=c^2{R}_{\mathrm{O}}$$ (9)

where R_O is the baseline measurement noise. Note that c is squared because R is variance rather than standard deviation. It also resulted in modified process noise, which we define as

$$Q=c^2{h}^2{Q}_{\mathrm{O}}$$ (10)

where Q_O is the baseline process noise. This resulted in a signal-to-noise ratio

$$SNR=\frac{Q}{R}=\frac{c^2{h}^2{Q}_{\mathrm{O}}}{c^2{R}_{\mathrm{O}}}=h^2\text{·}\frac{Q_{\mathrm{O}}}{R_{\mathrm{O}}}=h^2\text{·}SN{R}_{\mathrm{O}}$$ (11)

where SNR_O is the baseline signal-to-noise ratio. Importantly, the change in SNR is independent of c and depends only on h. Intuitively, this makes sense since any compensation c should equally affect signal and noise. This also means that the Kalman filter prediction varies with h but not c.

We found that the Kalman filter solutions varies only with the SNR rather than the individual values of Q and R. For example, the same solution was found for R = 1(°/s)² and Q = 13.6(°/s)², for R = 10(°/s)² and Q = 136(°/s)², and for R = 0.1(°/s)² and Q = 1.36(°/s)². This simplified the modeling by allowing us to focus on one parameter (SNR) rather than the absolute values of Q and R.

The primary modeling goal was to relate the change in VOR time constant with changing hair cell density. As in previous work, the baseline parameters were constrained using experimental data. The 31–40 yr age group was used as the baseline. h was normalized based on the hair cell density for this age group, and SNR_O was iteratively adjusted to intercept the VOR time constant for this group. This yielded SNR_O = 13.6. We chose to set R_O = 1(°/s)², which resulted in Q_O = 13.6(°/s)². Simulations were also performed using the 21–30 yr age group as the baseline, using SNR_O = 12.2, R_O = 1(°/s)², and Q_O = 12.2(°/s)².

We ran two cases of the model, one that included central compensation and one that did not. In the first, we iteratively determined a solution for c corresponding to each h by minimizing the difference between predicted VOR gain V and experimental VOR gain V_exp. This was implemented with the fminsearch function in Octave 3.8.1 (https://www.gnu.org/software/octave) running on a Dell Inspiron computer under Windows 7. Each iteration included these steps: 1) the Kalman filter equations were used to determine the Kalman filter gain K for Q and R by iteratively solving the equations to minimize norm(P) using fminsearch; 2) the predicted VOR gain was computed with Eq. 8; and 3) the cost function (V − V_exp)² was determined. fminsearch then adjusted c in an attempt to minimize the cost function, and the process was repeated until it converged. In the second case, only the Kalman filter equations were used to determine the Kalman filter gain K for Q and R for each hair cell density h, and with c = 1.

Simulations were also performed to depict the estimated head velocity for the parameters used, which used Euler integration.

RESULTS

Figure 1 summarizes published aging data on VOR behavior and hair cell density. VOR data are assessed for yaw rotation about an Earth-vertical axis, which primarily activates the lateral SCC. Figure 1A shows the changes in VOR time constant with age, determined with a published equation fit to the results of a large group of subjects (Dimitri et al. 2001), and shows a rapid increase in VOR time constant up to roughly 35 yr followed by a gradual decline during adulthood. The insets in Fig. 1A depict the expected eye velocity (gray line) produced by a rapid increase in angular head velocity (dashed line) from zero to a constant velocity. The VOR time constant defines the rate of decay in the eye velocity response, with older individuals having a faster decay. For simplicity of presentation we depict time constants using the response to constant velocity rotation, although the Kalman filter determines the VOR time constant without considering the input motion and other depictions would be equally valid, such as the response to sinusoids or velocity ramps. Figure 1B shows VOR gain, the ratio of eye velocity to head velocity that quantifies how compensatory the VOR is; specifically, given that the VOR gain exhibits high-pass filter characteristics that result in increasing gain with increasing frequency, it is defined as the gain of the high-frequency plateau corresponding to relatively transient stimuli. VOR gain decreases slightly with age (Dimitri et al. 1996). Figure 1C shows the decline in hair cell density with age in the lateral SCC (Merchant et al. 2000).

Figure 2 demonstrates the principles of our predictions. Figure 2A shows a block diagram of the VOR that includes components used to implement the three elements of our predictions. These are briefly described here, with details provided in methods. First, head velocity is sensed by the SCC, which provides an afferent signal that is attenuated proportionally to the fraction of hair cells living (h). This afferent signal is corrupted by noise that changes little with hair cell loss (Hirvonen et al. 2005), with a Gaussian distribution with variance R. Since the latter assumption may seem counterintuitive, we examine it further in discussion. The second element is central compensation (Allum et al. 1988; Curthoys and Halmagyi 1999; Smith and Curthoys 1989), which amplifies the afferent signal by a factor c, which is computed for each h so that the predicted VOR gain V matches the experimental values. Although the mechanism of this learning process is not pertinent to this work, this is likely implemented by attempting to reduce retinal slip (Gauthier and Robinson 1975; Gonshor and Jones 1976). The third element is the velocity storage mechanism, which is modeled with a Bayesian optimal Kalman filter (Kalman 1960; Kalman and Bucy 1961). Based on the signal-to-noise properties of the amplified afferents, the Kalman filter gain K is determined to balance between the VOR being accurate and precise, which directly determines both the time constant and amplification of velocity storage. Specifically, the Kalman filter gain K determines, at each point in time, how much the estimate of velocity relies on the noisy sensory input vs. a filtered estimate of the sensory input and is selected to minimize error by taking into account the statistics of the motion and the measurement noise. For example, when hair cells die, SNR decreases and the Kalman filter responds by reducing K, which shortens the VOR time constant.

The remainder of Fig. 2 illustrates the predictions using two sets of simulations, the baseline simulation with all hair cells intact (“younger,” Fig. 2, C–E) and the condition in which 37% of hair cells have been lost (“older,” Fig. 2, F–H). Figure 2B shows an input to the SCC when head velocity rapidly increases from 0°/s to a constant velocity of 60°/s over 1 s. Figure 2, C and F, show representations of SCC afferent firing rate that initially follow the input and then decay with a time constant of 5.7 s. They exhibit noise from a range of sources including transduction and primary afferents, which are referred to in the Kalman framework as measurement noise. While Fig. 2C shows the baseline case for a “younger” case with 100% of hair cells present (i.e., h = 100%) and peak amplitude is proportional to peak head velocity, Fig. 2F shows the case in which 37% of hair cells have been lost and thus the afferent signal is 37% smaller (i.e., h = 63%), although the noise amplitude is similar. Figure 2, D and G, show the amplified afferent signal after central compensation is applied. For the “younger” case, compensation is unneeded and thus c = 1 and thus there is no amplification, but for the “older” case, compensation c = 1.78 and there is substantial amplification. Importantly, afferent noise is amplified for the “older” case but not for the “younger” case. Figure 2, E and H, show the eye velocity predicted by the optimal Kalman filter model of velocity storage, and in particular an elongation of the decay in eye velocity. Figure 2E shows a predicted response for the “younger” case in which the Kalman filter determined a time constant of 22 s. The eye velocity predicted by the Kalman filter for the “older” case (Fig. 2H) has less elongation. This is because the optimal solution in the presence of higher noise in the amplified afferent signal is a smaller time constant of 14 s. The predicted VOR gain is also ~20% lower for the “older” case, consistent with the age-related decline in VOR gain (Fig. 1B). While intuitively it may seem that compensation c should just be 1/h = 1/0.63 = 1.59 rather than 1.78, reducing the Kalman filter gain K also changes the amplification of the Kalman filter, so an additional increase in c is required to make the VOR gain appropriate. We explicitly note that the goal of determining compensation that makes the VOR gain appropriate for the corresponding age yields a solution different from determining compensation to make afferent responses the same as for the “younger” case. This is why Eq. 8 yields c = 1.78 rather than c = 1.59.

Figure 3 shows simulation results for each age group and compares the results of two cases, one in which central compensation c is adjusted so that the predicted VOR gain matches experimental VOR gain and the second in which no central compensation occurs. It shows the value of c and K predicted for the hair cell density h of each age group, the SNR for each h, and the resulting VOR time constant and VOR gain V. Importantly, the SNR predictions are the same for both simulation cases, consistent with Eq. 11 showing that SNR changes are dependent only on h and not on c. The resulting K and VOR time constant are also the same in both cases.

Fig. 3.

Values of c and K predicted for the hair density h of each age group, the SNR for each h, as well as the resulting VOR time constant and VOR gain V. Results are shown for simulations with VOR gain compensated (+) and with no compensation (◊). For comparison, the experimental VOR gain is also shown (gray circle).

Figure 4 compares experimental results with the Bayesian optimal predictions. Figure 4A depicts the changing hair cell density with age, and Fig. 4B illustrates how age-related changes in VOR time constant impact the dynamics of the VOR response to a rapid increase in head velocity to a constant velocity. For each age group, the rate of decay of eye velocity for experimental (Dimitri et al. 2001) data is compared to the response predicted by our Bayesian optimal estimator based on the hair cell densities depicted in Fig. 4A. Throughout adulthood, the Bayesian optimal prediction is very close to the experimental results. These findings are summarized in Fig. 5, which shows hair cell density for each age group plotted against the yaw VOR time constant for that age group. Each circle in Fig. 5 represents 1 decade of age, with horizontal error bars showing the standard deviation (Merchant et al. 2000) of hair cell density across subjects. These data suggest a relatively monotonic relationship between VOR time constant and hair cell density for most of adulthood, with childhood exhibiting a different relationship that is examined in discussion. The experimental data have a remarkably close correspondence (r = 0.93, P < 0.001) to the Bayesian optimal prediction. We emphasize that the dashed line in Fig. 5 is not a fit to the experimental data but rather the prediction based on hair cell density only. These results are consistent with the brain exhibiting a Bayesian optimal adaptation to determine the appropriate VOR dynamics based on peripheral properties to minimize error in the estimate of angular velocity. Small differences between experimental data and predictions may have resulted from experimental results being derived from two separate subject groups and the small number of subjects in certain groups for the hair cell study. Our results show, however, that the VOR in children is not consistent with Bayesian processing, which is discussed further below.

Fig. 4.

Age-related changes in hair cell densities and behavioral dynamics. A: depiction of lateral SCC hair cell density based on experimental data (Merchant et al. 2000). B: dynamic responses to a rapid increase to constant-velocity angular rotation (dashed line), showing average responses determined experimentally (Dimitri et al. 2001) for each age group (gray line) and predicted Bayesian optimal response based on hair cell densities (dotted black line). Responses become elongated during early adulthood, then less prolonged later in adulthood, as indicated by the time constant labeled by gray vertical bar. Throughout adulthood, there is a strong correspondence between the experimental decay and the Bayesian optimal prediction based on hair cell density.

Fig. 5.

Age causes a decline in both hair cell density and VOR time constant, a measure of sensorimotor dynamics. During adulthood, there is a relatively linear relationship as both decline with advancing age, although childhood exhibits a very different relationship. The Bayesian optimal prediction closely matches the experimental data. We emphasize that this is not a fit, despite its close correspondence to the data. Horizontal bars indicate intersubject standard deviation in lateral hair cell density, and numbers indicate the age range of subjects included in each data point. Predicted VOR time constants are also shown for simulations when no compensation occurred for VOR gain (gray line) and when SNR was anchored using age 21–30 yr VOR time constant (gray dashed line).

To confirm that our predictions are robust, we tested the sensitivity of predictions to changes in model structure and parameters. For example, predictions were identical when no compensation occurred for VOR gain (Fig. 5), which reiterates the comparison in Fig. 3. While SNR was anchored to fit the VOR time constant for ages 31–40 yr, predictions changed slightly when SNR was anchored using any age between 21 and 90 yr (Fig. 5; gray dashed line shows predictions when anchored at age 21–30 yr, r = 0.93, P < 0.001). These results suggest that the consistency between experimental data and predictions arises from Bayesian optimal processing and not an artifact of model design.

DISCUSSION

Our results provide a conceptual framework suggesting that age-related behavioral changes observed in humans are the result of Bayesian optimal adaptation to age-related anatomical changes in neuronal structures. Specifically, we found that changes in VOR dynamics are consistent with the brain functioning in an ideal way in response to changing SNRs of peripheral signals that are proportionate to hair cell density. This relationship was an emergent property of the model, since no free parameters were needed to model the relationship between changes in hair cell density and changes in VOR time constant. These findings demonstrate that the brain is capable of generating the ideal behavioral response when provided with a deteriorating sensory input over a timescale of many years. This also implies that age-related change in the VOR can be explained by hair cell death rather than decline of other structures (Park et al. 2001; Richter 1980; Tang et al. 2001–2002). This is consistent with evidence that ototoxic drugs also cause VOR time constant to decrease when hair cells are killed (Baloh et al. 1984; Black et al. 1987; Ishiyama et al. 2006; Meyers 1970; Tsuji et al. 2000), without affecting other structures such as afferent neurons (Ishiyama et al. 2006), and with cause and effect being temporally related. Although it may be argued that the VOR time constant may be selected in a more functional way—i.e., 25 s is sufficient for any task in an ecological context—we note that this is not at odds with the hypothesis that Bayesian optimal adaptation explains age-related changes in the VOR time constant. Although adult VOR responses are consistent with Bayesian optimal processing, child responses are not. While a number of developmental processes could explain this difference, this is consistent with a study showing that ferret visual cortical activity is not optimal early in development but becomes optimal in adults (Berkes et al. 2011). This study complements our results because it did not examine the relationship between optimality and sensory changes. Indeed, increased variability in child responses could theoretically aid the exploration and learning process (Wu et al. 2014), which would present in our model as increased process noise. We also note that the exact age at which our results suggest optimal processing begins is unknown, since VOR time constants are relatively constant and highly variable between the third and fifth decades. According to Fig. 1 in Dimitri et al. (2001), VOR time constant increases slightly before 35 yr of age, and thus experimental data may be seen as incompatible with the decreasing time constant predicted by our model for 21–30 yr.

Related studies.

Our results build on a number of studies supporting the Bayesian brain hypothesis, for both dynamic and static processing, including a number of studies of vestibular processing. In particular, two studies have asked whether age-related changes are consistent with the Bayesian brain hypothesis. Sherback et al. (2010) developed a linear quadratic Gaussian model of visuomotor tracking. The authors found that changing the model estimate of sensory endogenous noise changed predicted dynamic response in a way that was consistent with the differences in experimental results between younger (Sherback and D’Andrea 2008) and older (Sherback et al. 2010) subjects. Our work builds on this study by combining both sensory and behavioral experimental measurements with modeling. Moran et al. (2014) studied evoked responses measured with MEG during an auditory oddball paradigm. They found that age-related attenuation of evoked responses was explained by an attenuation of sensory learning effects through the use of dynamic causal modeling (Garrido et al. 2008; Kiebel et al. 2006). They suggested that this result is qualitatively consistent with the Bayesian brain hypothesis via the free energy principle (Friston and Stephan 2007); our work complements this work by comparing quantitative predictions with experimental results. Specific to the vestibular system, studies have used models to argue that dynamic Bayesian processing underlies velocity storage (Karmali and Merfeld 2012; Laurens and Angelaki 2011; MacNeilage et al. 2008), the VOR and self-motion perception more generally (Borah et al. 1988; Laurens and Droulez 2007; Paulin et al. 1989; Selva and Oman 2012), as well as nonvestibular responses (Orban de Xivry et al. 2013; Wolpert et al. 1995). These models complement and build on dynamic, non-Bayesian models of vestibular estimation (Angelaki et al. 2004; Angelaki and Cullen 2008; Bos and Bles 2002; Glasauer 1992; Green et al. 2005; Merfeld et al. 1993, 1999; Merfeld and Zupan 2002; Mergner and Glasauer 1999; Newman 2009; Oman 1982; Young 2011; Zupan et al. 2002). Bayesian processing has been demonstrated for perceptual integration of threshold-level static visual cues (Landy et al. 1995), static visual and tactile cues (Ernst and Banks 2002), dynamic visual and vestibular cues (Butler et al. 2010, 2011; De Vrijer et al. 2009; Fetsch et al. 2009, 2011; Gu et al. 2008; Jürgens and Becker 2006; Karmali et al. 2014), and dynamic SCC and otolith tilt cues (Lim et al. 2017). These studies also show that Bayesian cue weighting adjusts to changes in sensory stimulus reliability on a trial-by-trial basis (Fetsch et al. 2011) and that Bayesian priors can be estimated in roughly a dozen trials (Prsa et al. 2015). Bayesian integration also explains how integration of visual, vestibular, neck, and somatosensory cues with priors results in tilt angle-dependent errors in subjective visual vertical and subjective body vertical for large static roll tilts (Alberts et al. 2016a, 2016b; Clemens et al. 2011; De Vrijer et al. 2009). Bayesian models explain how vision can disambiguate otolith tilt-translation cues (MacNeilage et al. 2007).

One study in particular provides a few complementary insights to our work (Cousins et al. 2013). Those authors studied patients with acute unilateral vestibular lesions and followed them serially over time. In addition to measuring VOR gain, VOR time constant, and perceptual time constant in response to 90°/s yaw rotation velocity steps, they measured perceptual and VOR thresholds for detection of yaw rotation, which assays vestibular noise. They found that both perceptual and VOR thresholds were higher in patients than in normal control subjects. They developed a static Bayesian maximum likelihood model to predict the change in thresholds that would be induced by a 50% reduction in signal and found that the experimental results were consistent with the prediction. They also found that VOR gain reduced ~50% and VOR time constant decreased. Furthermore, during the acute stage there was a statistically significant inverse correlation between VOR threshold and VOR time constant, suggesting that higher noise results in a lower VOR time constant. This dynamic relationship was not modeled and this correlation did not hold during the recovery phase or for the relationship between perceptual time constant and perceptual thresholds, so further investigation is required to explore this relationship. While we are not assuming that a unilateral lesion induces the same physiological changes as hair cell loss, this work supports a few of our assumptions: first, that peripheral loss changes VOR signal-to-noise properties and VOR gain; second, that changes in vestibular signal-to-noise properties can alter VOR time constant; and finally, that the brain responds in a Bayesian manner to peripheral vestibular loss, at least in terms of static processing.

Model robustness.

Confidence in our results is strengthened because the model has only one age-related model parameter (c) that is uniquely and completely constrained by experimental data. Furthermore, results are unchanged when no central compensation is applied (i.e., c = 1), showing that the close correspondence between the experimental data and our model is an emergent property of the Kalman filter optimal structure. As with all these previous demonstrations of Bayesian optimal processing by the brain, the validity of predictions relies on their underlying assumptions, the most important of which is that the brain uses linear processing. For example, suboptimal and nonlinear processing could create an artifact of optimal processing, although it is unlikely that these would balance perfectly across the range of our data. Our model also requires that the brain have knowledge of peripheral noise. While the mechanism by which this occurs is unknown, neuronal and behavioral studies have shown that the brain quickly estimates sensory cue reliability (Fetsch et al. 2011), lending support to this assumption. The control simulations that we performed ensured that the results were not an artifact of model assumptions. While changes in baseline SNR_O could shift the prediction curve up or down, they would minimally impact the slope, meaning that the correspondence in age-related change between the prediction and the data is robust. While the structure of the neuronal network and the relative contribution of regular and irregular neurons are important questions, we chose not to address them because there is not enough experimental data to constrain the model, and this would have added additional free parameters. Additional free parameters would have made the model less constrained and increased the likelihood of the prediction erroneously fitting the hypothesis.

Afferent noise after hair cell loss.

Since death of hair cells almost certainly reduces synaptic activity, a reasonable expectation would be a decrease in afferent noise following hair cell death. In fact, the limited published data suggest that, in the vestibular system, afferent noise changes little after hair cell death. In anesthetized chinchilla, there was no significant difference in afferent noise on regular vestibular afferents innervating the ear treated with ototoxic gentamicin compared with afferents innervating the normal, untreated ear (Hirvonen et al. 2005). For irregular afferents, noise was not significantly different except for an increase at a single measurement interval immediately after gentamicin application. However, the authors noted an increase in injury discharge patterns on the treated side, and after removal of suspect recordings there was no significant effect on afferent noise related to hair cell death. This injury discharge pattern may also explain the slightly higher afferent coefficient of variation in anesthetized pigeons that received another ototoxic drug, streptomycin, vs. control pigeons that did not (Li and Correia 1998). Given these results, it is reasonable to make the simplest assumption that hair cell death reduces afferent signal without changing afferent noise. To further explore how this assumption impacts the results, we conducted simulations in which afferent noise either increased or decreased with hair cell loss (Fig. 6). In one simulation we assumed that afferent noise decreases proportional to hair cell loss, and therefore SNR remains unchanged with hair cell loss, and thus there is no change in predicted VOR time constant. In a second simulation, we assumed that afferent noise increases with hair cell loss at the same rate that signal decreases with hair cell loss, and therefore SNR decreases with hair cell loss more quickly than when noise remains constant, and thus the predicted VOR time constant also decreases more quickly than when noise remains constant. In a third simulation, we assumed that afferent noise decreases with hair cell loss, but at half the rate of change of the decrease in signal, and therefore SNR decreases with hair cell loss less quickly than when noise remains constant, and thus the predicted VOR time constant also decreases less quickly than when noise remains constant. These simulations show that the assumption that noise remains constant leads to predicted VOR time constants that most closely fit experimental data.

Fig. 6.

Model predictions when the assumption about afferent noise is varied. Model predictions are shown for no change in afferent noise with hair cell loss (thick dashed line), for a decrease in afferent noise that is proportional to hair cell loss (thin dashed line), for an increase in afferent noise that is proportional to hair cell loss (thin solid line), and for a decrease in afferent noise at half the rate of hair cell loss (thin dot-dashed line). These simulations show that the assumptions that noise remains constant leads to predicted VOR time constants that most closely fit experimental data (thick dashed line).

Nonvestibular contributions.

Our study focused only on vestibular contributions to velocity storage. The experimental VOR results upon which our work was based were determined with en bloc rotation, so subjects had no neck cues and little nonvestibular stimulation, except for perhaps tactile and wind cues. Thus the potential for sensory substitution was minimal. Furthermore, two studies have suggested that neck cues do not contribute to some aspects of velocity storage (Koizuka et al. 1996; Pettorossi et al. 2013). Nonetheless, it is an interesting question about how multisensory integration (e.g., vestibular, proprioceptive, as well as motor efferent) may affect velocity storage processing in a way that would be detected during en bloc rotation. Vestibular and proprioceptive cues follow Bayes’ rule for static processing (Clemens et al. 2011), and thus the processing of each of the cues may change depending on the characteristics of the other. How this bears on our study is unclear, for the following reasons. First, we do not know how nonvestibular cues change with age; for example, if they declined at the same rate, then their optimal weights would not change. Second, it is unknown whether processing modes would be the same for the “natural” and en bloc cases.

Neural implementation.

While we have previously considered how optimal processing could be realized in brainlike parallel architectures (Karmali and Merfeld 2012), this study is agnostic to the implementation by which the brain achieves optimal processing, and uses the Kalman filter simply as a tool to predict optimal responses. The mechanism by which the brain achieves Bayesian processing is not known, although recent studies provide some insight for Bayesian multimodal sensory integration. Modeling work has demonstrated that, for noisy, population-coded stimuli, Bayesian processing arises from simply linear combinations of populations (Ma et al. 2006). Furthermore, neural correlates of stimulus weighting in visual-vestibular integration have been found in the dorsal medial superior temporal area (Fetsch et al. 2011). Although the work could be extended to use our optimal parallel model, this would add one free parameter (the number of parallel pathways), which would have increased the likelihood of overfitting.

Impact of findings.

In addition to these results having the potential to illuminate how the aging brain strives to optimize cognition and motor behavior when faced with deterioration in the peripheral and central nervous systems, they also have substantial clinical implications in the field of vestibular and balance disorders. There is a catastrophic increase in the prevalence of falls with age, and individuals with symptomatic vestibular dysfunction have 12 times the odds of falling (Agrawal et al. 2009). A better understanding of the peripheral and central processes that underlie changes in sensorimotor function in the vestibular system during aging could provide guidance for physical therapy and for prosthetic aids that aim to reduce falls in the elderly (Gryfe et al. 1977).

GRANTS

This research was supported by National Institute on Deafness and Other Communication Disorders Grants R01 DC-013069 (R. F. Lewis) and R03 DC-013635 (F. Karmali).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

F.K. conceived and designed research; F.K. and G.T.W. analyzed data; F.K., G.T.W., and R.F.L. interpreted results of experiments; F.K. prepared figures; F.K. drafted manuscript; F.K., G.T.W., and R.F.L. edited and revised manuscript; F.K., G.T.W., and R.F.L. approved final version of manuscript.

Supplemental Data

Supplemental File 1

Matlab code that implements simulations 1

Supplemental File 2

Matlab code that implements simulations 2

Supplemental File 3

Matlab code that implements simulations 3