Models of vestibular semicircular canal afferent neuron firing activity

Abstract

Semicircular canal afferent neurons transmit information about head rotation to the brain. Mathematical models of how they do this have coevolved with concepts of how brains perceive the world. A 19th-century “camera” metaphor, in which sensory neurons project an image of the world captured by sense organs into the brain, gave way to a 20th-century view of sensory nerves as communication channels providing inputs to dynamical control systems. Now, in the 21st century, brains are being modeled as Bayesian observers who infer what is happening in the world given noisy, incomplete, and distorted sense data. The semicircular canals of the vestibular apparatus provide an experimentally accessible, low-dimensional system for developing and testing dynamical Bayesian generative models of sense data. In this review, we summarize advances in mathematical modeling of information transmission by semicircular canal afferent sensory neurons since the first such model was proposed nearly a century ago. Models of information transmission by vestibular afferent neurons may provide a foundation for developing realistic models of how brains perceive the world by inferring the causes of sense data.

INTRODUCTION

In 1825, Flourens reported that pigeons turn in circles when a horizontal semicircular canal is destroyed, consistent with earlier speculation that these structures sense rotation. Similar experiments through the early 20th century established the contribution of the vestibular apparatus to balance and orientation, and the specific role of the semicircular canals in sensing head rotation (Fritzsch and Straka 2014; Pearce 2009; Precht 1978). More than a century later, Steinhausen (1931) began anatomical studies of microscopic sensing mechanisms in semicircular canals. He developed a simple differential equation, the torsion pendulum model, describing what the semicircular canals tell the brain about head movements (Steinhausen 1933). Then, as now, anatomy and physiology were interpreted via technological metaphors. Sense organs were regarded as lenses projecting images of the world into the brain, as in a camera, and sensory nerves as mere passive channels for this projection (Mach 1959). Modelers initially made no distinction between the dynamics of the peripheral end organ, sensory neuron responses, and perception of head rotation (van Egmond et al. 1949). In that paradigm, the brain simply perceives what the sense organs measure.

Confidence in the torsion pendulum model continued to grow until it began to unravel in the 1970s. In three landmark papers, Goldberg and Fernandez reported an idiosyncratic pattern of statistical and dynamical behavior among afferent neurons in the semicircular canal branch of the vestibular nerve (Fernandez and Goldberg 1971; Goldberg and Fernandez 1971a, 1971b). These papers established a new paradigm for research in subsequent decades: what is the pattern, and why do vestibular sensory neurons transmit information to the brain in such an unexpectedly noisy and complicated way?

From the mid-20th century, models of vestibular semicircular canal afferent firing patterns were developed within a dynamical systems framework, relating kinematic state variables of head movement, angular velocity and angular acceleration, to sensory neuron firing rates. This approach produced models that could describe patterns in the data, including correlations between parameters characterizing spontaneous activity, dynamical responses, and noise statistics, without providing plausible explanations for why those patterns occur. These models can generally be characterized as extensions and modifications of Steinhausen’s torsion pendulum model.

Discrepancies began to emerge between the predictions of dynamical systems models based on the torsion pendulum model and neuronal responses at frequencies above a few cycles per second, where vestibular function is most important during natural behavior (Highstein et al. 1996, 2005; Rabbitt 2019). These discrepancies accumulated, and models were modified until they were no longer recognizable as derivatives of the torsion pendulum model.

The “Bayesian Brain” has emerged as a new paradigm in 21st-century neuroscience. In this new view, sense data are samples from random processes whose parameters are states of the world. Sensory afferent neurons transmit data to the brain in trains of action potentials, and the brain perceives what in the world is causing the sense data by inferring the parameters from the data (Körding and Wolpert 2006; Ma and Jazayeri 2014; Pitkow and Angelaki, 2017; Pouget et al. 2013). The semicircular canals of the vestibular apparatus offer an opportunity to develop rigorous mathematical models of perception as Bayesian inference, because they provide sense data about a simple, low-dimensional dynamical system (head rotation) whose relevant parameters (head angular velocity and acceleration) are well defined and measurable. The theory of stochastic point processes provides a framework for Bayesian models of spike trains, avoiding difficulties associated with defining “firing rate” over intervals that are short compared with intervals between spikes (Brillinger 1992; Paulin and Hoffman, 2001a).

In this article we review the historical development of models of semicircular canal afferent neurons across brain metaphors from cameras to dynamical systems to stochastic point-process models. Models that statistically quantify the relationship between head rotation and vestibular afferent neuron spike train data have the potential not only to explain the statistical and dynamical characteristics of semicircular canal afferent neuron behavior but also to provide a foundation for developing Bayesian spiking neuron models of how brains perceive the world.

THE TORSION PENDULUM MODEL

Steinhausen (1931) examined vestibular semicircular canals in pike. Each canal has a swelling, the ampulla, containing sensory epithelium, the crista, which is innervated by a branch of the vestibular nerve (Fig. 1A, n). He described a gelatinous flap, the cupula, sitting over the crista and extending across the ampulla (Fig. 1B), that swings like a gate in response to motion of the endolymphatic fluid in the canal (Löwenstein and Sand 1940b) (Fig. 1C). This was the accepted view for several decades (Jones and Milsum 1965) until Hillman (1974) showed that the cupula is a membrane that blocks the canal over the crista and flexes rather than swings during head rotation (Fig. 1D). Hillman’s work has been confirmed and is generally accepted by experts, but the swinging gate persists in nonspecialist literature and is still depicted in widely-used textbooks [e.g., Fig. 50.13 in Campbell et al. (2015)].

Fig. 1.

A: membranous labyrinth of an elasmobranch, the New Zealand carpet shark, Cephaloscyllium isabella. Three semicircular canals sense rotation around mutually perpendicular axes. n indicates the branch of the vestibular nerve innervating the ampulla of the horizontal canal. [Redrawn by M. Paulin from a photograph by J. Montgomery (with permission from J. Montgomery)]. B: photograph showing gelatinous cupula extending across the ampulla. The afferent nerve exits toward the bottom left. [Reprinted from Steinhausen (1931) by permission from Springer.] C: Steinhausen described the cupula swinging like a gate (Löwenstein and Sand 1940b). D: Hillman (1974) later established that the cupula seals the canal and flexes rather than swings. E: first report of spiking activity in a canal afferent nerve. Top trace, increased spiking after onset of rotation in excitatory direction; bottom trace, decreased spiking after onset of rotation in inhibitory direction. [Adapted from Löwenstein and Sand (1936) with permission from The Company of Biologists.] F: spiking responses during sinusoidal oscillation. White trace shows the rotation angle, and tick marks are 1 s apart. Responses show that population firing rate is 90° phase advanced relative to angular deviation, i.e., in phase with angular velocity, at 1-Hz stimulus frequency. [Adapted from Fig. 3 in Löwenstein and Sand (1940a) by permission of John Wiley and Sons.]

Steinhausen (1933) derived a second-order, constant-coefficient linear differential equation model to describe the angular deflection of the cupula during head movement, taking into account the inertia and viscosity of fluid flow and elasticity of the cupula. This model could explain his observations that the cupula deflects quickly in response to the onset of a sudden rotational stimulus and relaxes back to its resting position slowly if the stimulus is maintained. He observed that the pike’s eyes rotate with corresponding kinematics when the ampulla is deflected, and he deduced that the differential equation model of canal-cupula dynamics also describes compensatory eye movements during head rotation. This implies that the model also quantifies the signal transmitted by the sensory nerve.

van Egmond et al. (1949) noted that Steinhausen’s differential equation is a model of an overdamped torsion pendulum, a mechanically damped flywheel with a torsional spring that provides a restoring torque toward a rest position. They wrote Steinhausen’s model in the form

$$\ddot{\xi }+\frac{\text{Π}}{\text{Θ}}\dot{\xi }+\frac{\text{Δ}}{\text{Θ}}\xi =\alpha \left(t\right)\text{.}$$ (1)

In this model, ξ is the angular displacement of endolymph in the canal and the overdot represents differentiation with respect to time. Thus $\dot{\xi }$ is angular velocity and $\ddot{\xi }$ is angular acceleration. Π is the drag coefficient of velocity-dependent damping of endolymphatic fluid flow. Δ is the restoring torque proportional to displacement from the rest position, which reflects elasticity of the cupula. Θ is the rotational inertia of endolymph in the canal, and α(t) is the angular acceleration of the head (see Fig. 2A). The system (Eq. 1) is overdamped if the restoring torque (Δ) is small compared with the drag coefficient (Π). In that case, the second-order system can be approximated by two first-order linear systems in series, characterized by two time constants, τ₁ = Π/Δ and τ₂ = Θ/Π.

Fig. 2.

A: torsion pendulum model parameters (Jones and Milsum 1965). Ω is head angular velocity, and ϕ is the deflection angle of the cupula (see Eq. 3). Note that the cupula is depicted incorrectly as a swinging flap. CNS, central nervous system. B: normalized Bode gain (top) and phase (bottom) plots of the model. Frequency is in radians per second. Gain is in log₁₀ units and frequency is on a log₁₀ scale, so that a slope of 1 in the gain plot corresponds to a 10-fold increase in gain over a 10-fold increase in frequency. Corner frequencies corresponding to model parameters are at intersections of asymptotic straight line segments in the gain plot and at midpoints of 90° phase shifts in the phase plot (vertical dotted lines). This model has constant gain, in phase with head velocity, over a frequency band between ~0.1 and 5 Hz (horizontal bar). [Adapted from Jones and Milsum (1965) by permission from IEEE.]

van Egmond et al. (1949) stated that if the behavior of a system can be described by a differential equation model of a torsion pendulum, then it is appropriate to call that system a torsion pendulum. They attributed the torsion pendulum model to Steinhausen, although they were the first to use that metaphor. Van Egmond et al.’s 1949 paper became the standard reference for the torsion pendulum model, and its notation was widely adopted, perhaps because Steinhausen’s prior work published in German was less accessible to later workers.

van Egmond et al. (1949) used self-reported head rotation perception by human subjects to estimate parameters of the torsion pendulum model. They concluded that the torsion pendulum model is very accurate, although “as was to be expected there is a slight difference between [the model] and the experimental results” (van Egmond et al. 1949, p. 15). They further reported that “An unexpected proof of the theory has arisen from the measurements of Löwenstein and Sand (1936, 1940a).”

In their 1936 paper, Löwenstein and Sand reported spiking activity in horizontal canal afferent neurons in the vestibular nerve of dogfish. They showed that these neurons fire spontaneously, that they increase their firing rate during ipsiversive rotation, when the animal’s snout rotates toward the side from which neural activity is being recorded, and that they decrease their firing rate during rotation in the opposite direction. In their later paper, Löwenstein and Sand (1940a) reported dynamic modulation of afferent firing rates during sinusoidal and constant-velocity rotation. Figure 1F is a reproduction from Figure 3 in Löwenstein and Sand (1940a), showing multiunit activity 90° phase-advanced relative to head angular deviation during sinusoidal head oscillation at 1 Hz. This phase advance means that afferent firing rate varies in phase with head angular velocity at this frequency, as predicted by the torsion pendulum model. At lower frequencies, the phase advances another 90° to be in phase with head angular acceleration, also as predicted by the torsion pendulum model.

van Egmond et al. (1949) re-presented Löwenstein and Sand’s (1940a) data, showing not just qualitative agreement between the dynamics of rotation perception in humans and semicircular canal afferent firing rates in dogfish afferents, but a remarkable quantitative correspondence between parameter values for human motion perception and dogfish afferent neuron firing rates. They concluded that “As far as measurements allow us to compare, there is no difference between theory and experiment. […] We may therefore conclude that the ray follows the theory of the torsion pendulum” (van Egmond et al. 1949, p. 14).

Groen et al. (1952) recorded spiking activity of individual semicircular canal afferent neurons in the ray, Raja clavata. They reported “striking agreement” between the behavior of individual sensory neurons in rays and the perceptions and eye movements of human subjects experiencing similar stimuli, consistent with the torsion pendulum model. Although Steinhausen developed the torsion pendulum model by analyzing the mechanics of the end-organ, it is clear that van Egmond et al. (1949) and Groen et al. (1952) regarded it as a model of sensory afferent neuron response and as a model of rotation perception in the brain. In this paradigm, originally due to Mach in 1896 (Mach 1959), a sense organ is analogous to a camera lens projecting an image of the world into the brain, and the sensory nerve is merely a passive channel mediating the projection.

Thus, by the mid-20th century, the torsion pendulum equations were used to model the sense organ, the sensory nerve, the perception of head rotation, and reflexive counter-rotations of the eyes during head movement. Below, we focus on how the torsion pendulum model was applied to model semicircular canal afferent firing rates in response to head rotation.

LINEAR DYNAMICAL SYSTEMS MODELS

Graphical methods for linear model identification and parameter estimation for dynamical systems developed by Hendrik Bode at Bell Laboratories in the late 1930s (Makarov 2015) became a mainstay of vestibular neurophysiological data analysis and modeling in the second half of the 20th century. These methods, based on using the Laplace transform to map linear differential equations to algebraic models called transfer functions, provide a framework for decomposing a dynamical system into components and for analyzing how the components contribute to overall system behavior (Young 1969; Young and Oman 1969). The introduction of these methods marks a turning point at which modelers began to treat peripheral mechanics separately from afferent firing rate dynamics. Mechanics and dynamics of the peripheral vestibular apparatus were recently reviewed by Rabbitt (2019).

Jones and Milsum (1965) introduced Bode analysis to vestibular neurophysiological data analysis and modeling. In their notation, the torsion pendulum model,

$$\ddot{\varphi }\left(t\right)+\frac{\text{Π}}{\text{Θ}}\dot{\varphi }\left(t\right)+\frac{\text{Δ}}{\text{Θ}}\varphi \left(t\right)=\text{Ω}\left(t\right)\text{,}$$ (2)

“is conveniently expressed as a transfer function”

$$T\left(s\right)\triangleq \frac{\varphi \left(s\right)}{\text{Ω}\left(s\right)}=\frac{s}{\left({\tau }_{1}s+1\right)\left({\tau }_{2}s+1\right)}$$ (3)

with time constants τ₁ = Π/Δ and τ₂ = Θ/Π. The transfer function is obtained formally from the differential equation by replacing the differentiation operator with multiplication by the algebraic variable, s, and solving for the ratio of output to input. Note that although Jones and Milsum describe Eq. 3 as the transfer function corresponding to the differential Eq. 2, it is only an approximation, valid for overdamped systems, i.e., when viscous drag is large compared with the elastic restoring force on the cupula.

Bode (1940) showed how the parameters of a linear system model could be estimated by plotting the gain (relative amplitude) and phase shift of responses to sinusoidal stimuli at different frequencies, using a simple ruler-and-pencil procedure. Bode analysis made it possible to identify linear differential equation models for dynamical systems, and fit parameters to them, without using computationally intensive numerical methods (Fig. 2). This was very useful in the days before microprocessor-based laboratory computers became common, when computationally intensive methods were also very time-consuming and expensive.

Figure 2B shows Bode gain and phase plots for a torsion pendulum model (Eqs. 2 and 3) from Jones and Milsum (1965), which they derived using parameters reported by van Egmond et al. (1949) and Jones and Spells (1963). A factor of s has been removed from the numerator and denominator in deriving this transfer function. As indicated in the top plot in Fig. 2B, the Bode gain curve “breaks” at two frequencies. This indicates that the system can be modeled using a second-order differential equation with two time constants. Below the lower break frequency, the gain curve is asymptotic to a line with slope = 1, whereas the phase approaches 90° lead. This indicates that at low frequencies, the model output follows the derivative of the stimulus, angular acceleration in this case. Above the upper break frequency, the gain curve is asymptotic to a line with slope = −1 and the phase shift approaches 90° lag, corresponding to model output in phase with angular deviation of the head. Between the break frequencies, the gain has zero slope and there is no phase lag, indicating that the response tracks the angular velocity of the head over this range of frequencies.

Jones and Milsum (1965) had good reason to remove a factor of s in deriving the transfer function and to draw the Bode plots with angular velocity, rather than angular acceleration, as the input variable. In their words,

“… with a transfer function of this form, essentially constant [gain] and zero phase appear to be maintained … over a large intermediate frequency range 0.1–5 Hz, which probably encompasses the majority of excitation frequencies of a natural environment. Clearly this implies that the canal’s normal role is that of an angular velocity transducer.”

Thus, although an inertial sensor cannot directly sense velocity, the semicircular canals compute head angular velocity over most, if not all, of the bandwidth of natural head movements, by balancing viscous drag in endolymphatic fluid flow against the elastic restoring force of the cupula. This integrates inertial torques over the natural bandwidth of head motion, providing an elegant example of how evolution can exploit mechanics of materials for cheap analog computation in the periphery, to preprocess sense data in a way that may reduce the need for energetically expensive neural computation in the brain (Jones and Spells 1963; Squires 2004).

Jones and Milsum (1970) recorded spiking activity from rotation-sensitive secondary vestibular neurons in the brain stem of cats. They reported that cells in this population are functionally homogenous and that their behavior is consistent with the torsion pendulum model. Thus four decades after the torsion pendulum model was first developed as a description of cupula-endolymph mechanics, when researchers had been able to record spiking activity of rotation-sensitive neurons in the afferent nerve and in the brain and quantify dynamic responses of each component using system identification techniques, the torsion pendulum model remained the state of the art for modeling cupula dynamics, sensory afferent neuron activity, and central vestibular neuron responses.

Young (1969) summarized the status of vestibular system models at that time. The literal torsion pendulum model, a rotating mass with a spring and a damper, had been replaced by more realistic models with hydromechanical components. But it all boiled down to the same second-order linear differential equation model (Eq. 1), which correctly predicted neural responses, perception, and compensatory eye movement responses to head rotation. There was still some uncertainty about the precise values of the time constants, but no doubt that the torsion pendulum model was as a triumph of scientific modeling. It was mathematically elegant, solidly based on mechanistic physical analysis, and quantitatively predicted observable behavior from the microscopic level of receptors and neurons through to the macroscopic level of movement and perception. All this was about to change.

DYNAMIC DIVERSITY

The discovery of systematic variation in response dynamics among semicircular canal afferent neurons was presaged by Löwenstein (1955), who investigated the question of what causes spontaneous activity in these neurons. He found that afferents respond to a steady galvanic stimulus but that their firing rates adapt over time. This indicates that afferent neurons have intrinsic dynamics that filter the signal transduced by the sense organ rather than simply copying it to the brain. He also observed that neurons within the afferent nerve appeared to be anatomically inhomogeneous. He noted that two distinct classes of sensory neurons, with different distributions of fiber diameters and different response properties, had been reported in the related lateral-line system of fish, and he speculated that this may be true in the vestibular nerve also.

Goldberg and Fernandez (1971a) recorded spike trains from semicircular canal afferent neurons in squirrel monkeys. The torsion pendulum model predicts that during prolonged constant acceleration, firing rates should approach a constant rate, above or below the spontaneous rate depending on the direction of motion. Instead, however, the firing rates adapted, drifting slowly back toward their resting rates despite the ongoing constant stimulus. This requires another factor in the transfer function, with an associated adaptation time constant.

Adaptation time constants of different neurons were clearly different. Some returned back to their spontaneous rates faster than others during constant acceleration. Goldberg and Fernandez (1971a) divided responses into three classes with low, medium, and high adaptation rates, respectively. They showed that units in the fast-adapting class tend to be more sensitive to angular acceleration, to have lower spontaneous firing rates, and to have less regular spontaneous firing activity patterns than units in the slowly adapting class (Fig. 3).

Fig. 3.

Regularly and irregularly firing semicircular canal afferent neurons. Traces show two extreme examples of statistical behavior within the population but are easily misinterpreted to suggest that the population contains two different kinds of neurons. [Reproduced from Goldberg and Fernandez (1971a).]

Goldberg and Fernandez (1971a) reported that semicircular canal afferent nerves have detailed functional suborganization, containing neurons whose responses deviate from the torsion pendulum model in various ways. Variation in dynamical properties is correlated with variation in statistical properties. These authors speculated that response specificity in afferent neurons might reflect preprocessing for specific tasks with different dynamics. Some afferents might specialize in providing data to central circuits that control compensatory eye movements, for example, whereas others focus on providing data to circuits involved in maintaining postural stability.

Goldberg and Fernandez’s (1971a) report seems to suggest that there are distinct functional classes of afferent neurons, but as the authors themselves pointed out, they divided neurons in their sample into separate classes as a methodological convenience, to simplify testing for covariation among parameters. They found no evidence for any natural division into different functional classes and did not claim that such classes exist.

In a second paper, Fernandez and Goldberg (1971) extended their observations by using sinusoidally varying stimuli. They recorded amplitude changes and phase shifts in afferent firing rates across a range of frequencies and fitted transfer functions. They found systematic deviations between neuronal responses and the predictions of the torsion pendulum model, requiring at least two extra factors in the transfer function. They determined that the simplest linear transfer function that could approximate their data is of the form

$$H\left(s\right)=\left(1+{\tau }_{H}s\right)\frac{{\tau }_{L}s}{1+{\tau }_{L}s}\frac{1}{\left(1+{\tau }_{1}s\right)\left(1+{\tau }_{2}s\right)}\text{.}$$ (4)

This modifies the torsion pendulum model by adding a high-frequency phase lead and a low-frequency adaptation. These appear as the two factors ahead of the torsion pendulum component in the transfer function (Eq. 4).

The gain and phase data of Fernandez and Goldberg (1971) and the new model that they proposed are compared with the torsion pendulum model of van Egmond et al. (1949) on Bode plots in Fig. 4. Data for all neurons in the sample are pooled, emphasizing that the average response across neurons differs qualitatively from the predictions of the torsion pendulum model. There is a major discrepancy at frequencies above ~0.5 Hz. Although it was already well known that vestibular sense data are important at frequencies well above 1 Hz, which the reader can confirm by noting that one can still see clearly while vigorously shaking one’s head, previous single-unit recordings of afferent responses had been restricted to frequencies below ~0.5 Hz for methodological reasons. This presumably explains why such a large and obvious deviation from the torsion pendulum model had not been noted earlier.

Fig. 4.

Average Bode gain (left) and phase (right) plots of semicircular canal afferent neuron firing rate response to angular acceleration. Plotted points show normalized gain and phase of firing rate during sinusoidal stimulation, plus or minus standard deviation. Solid curves show gain and phase of the best-fitting torsion pendulum model. Dashed line shows gain and phase of the best-fitting transfer function of the form in Eq. 4. The plots show that there is a large systematic discrepancy between the torsion pendulum model and average neuronal responses, and that individual responses are highly variable. [Redrawn from Fig. 5 in Fernandez and Goldberg (1971).]

Fernandez and Goldberg (1971) presented Bode plots for individual neurons, showing that although some units behave more like a torsion pendulum than others do, they are all qualitatively different from the torsion pendulum model, with large relative phase advances at frequencies above ~0.5 Hz. They suggested that this phase advance might compensate for phase lags due to time delays and low-pass dynamics of muscle activation on reflex pathways.

In their third paper in the same year, Goldberg and Fernandez (1971b) focused on the statistical properties of semicircular canal afferent neuron spike trains. They introduced the subject by pointing out that Wersall (1956) had described two kinds of hair cells in the crista of the semicircular canals of guinea pigs, with different innervation patterns. Subsequent anatomical studies confirmed that this is true in birds and mammals, whereas other vertebrates have only one type of hair cell and a uniform pattern of innervation. They state the goal of their paper:

“Neurons differ in the regularity of the spacing of their action potentials, in the level of their resting activity, in their sensitivity to angular accelerations and in their dynamic properties. If these differences reflected differences in innervation patterns of the parent axons, it might be expected that variations in one physiological property would be related to variations in other properties. The purpose of the present paper is to demonstrate that this is the case.”

Having previously noted that variability is correlated with spontaneous firing rate, Goldberg and Fernandez (1971b) recorded single units during rotational acceleration to find out how variability changes as a function of firing rate in individual afferents. Their results, reproduced in Fig. 5, show that standard deviation of interval length, SD, changes as a function of mean interval length, L. They introduced the coefficient of variation, CV = SD/L, and argued that because there is a systematic relationship between spike train variability and mean interval length in spontaneous firing of different neurons, and in individual neurons when the mean interval length is altered by rotational stimulation, variability between neurons should be corrected for differences in mean interval length. The rate-standardized measure of variability was later called CV*. It is convenient to use CV* here, although Goldberg and Fernandez (1971b) simply called it CV at the time.

Goldberg and Fernandez (1971b) arbitrarily divided their data into three equal subsamples: regular, with CV* < 0.0579, intermediate, and irregular, with CV* > 0.3. They then showed, by comparing fitted transfer function models for units in the different variability classes, that irregular neurons have lower resting rates, higher gain, faster adaptation, and larger high-frequency lead terms. In short, regularity of spontaneous activity predicts dynamical parameters of responses to rotation.

In their discussion, Goldberg and Fernandez (1971b) note that it is convenient to speak of regular and irregular units as if they constitute separate physiological groupings, but there is little evidence for segregation into distinct populations. It is, they say “more prudent, if not more correct, to view the neurons as if they were members of a single population.” Nonetheless, they finish the discussion by suggesting that differences in afferent response properties may be related to the two anatomical classes of receptor cells and the associated differences in innervation patterns.

In the same year, Precht et al. (1971) reported wide variation in dynamic responses among rotation-sensitive afferents in the vestibular nerve of frogs. Some units responded in phase with angular velocity, in accordance with the predictions of the torsion pendulum model, whereas others were more sensitive to angular acceleration. They concluded that “individual vestibular afferent fibers show such a variety of response characteristics that the mechanical properties of the viscoelastic cupula-endolymph system cannot be derived from them.” They pointed out that the covariation of statistical and dynamical properties of rotation-sensitive units in frogs is similar to the pattern reported in mammals. However, this cannot be explained by differences in peripheral innervation patterns onto different types of hair cell receptors, because there is only one type of hair cell in the frog’s vestibular system. They proposed that differences between neuronal response properties are due to biophysical differences in the neurons themselves.

Elasmobranchs also have only one type of hair cell in their vestibular apparatus. The horizontal semicircular canal branch of the vestibular nerve of the guitarfish, Rhinobatos productus, separates into six to eight distinct bundles that innervate different parts of the crista. This unique anatomical feature allowed O’Leary, Dunn, and Honrubia (O’Leary et al. 1974) to demonstrate that variations in response properties of afferent neurons are correlated with the spatial distribution of their terminals on the crista. O’Leary and colleagues developed and introduced a new system identification and modeling approach using pseudorandom angular accelerations of a servo-controlled turntable to record spike train responses, using a single presentation of a broadband stimulus waveform instead of a prolonged sequence of sinusoids at different frequencies (O’Leary and Dunn 1976; O’Leary and Honrubia 1975, 1976). Response properties were characterized by cross-correlating the random stimulus waveform with the spike train response, producing unit impulse response functions (Marmarelis and Marmarelis 1978) (Fig. 6). Unit impulse response functions (UIRs) fully characterize linear system dynamics and can be interpreted as a signature of a linear system’s behavior.

Fig. 6.

Unit impulse responses (UIRs) of semicircular canal afferent neurons in the guitarfish, Rhinobatos productus. There are four different shapes, fitted by four different models (A–D). Neurons fitted by different models tended to innervate different regions of the crista. Shading of the icons to the right of each UIR indicates the region of the crista innervated by neurons with that response signature. For example, neurons in model A are scattered over the whole crista, whereas neurons in model B are concentrated along the central ridge. τ represents time. The coefficients a_i are characteristic frequencies, whose reciprocals are time constants of linear models. [Redrawn from data in O’Leary and Honrubia (1976).]

O’Leary, Dunn, and Honrubia’s results are summarized in Fig. 6. The UIR is an estimate of the neuron’s firing rate following an impulsive stimulus, assuming linearity. It displays the contribution of previous stimulus values to the system output at any instant. The amplitude of the UIR at τ is the linear contribution of stimulus intensity at time τ before the present to the current output. It can be used to predict the response to an arbitrary stimulus by convolution, or converted to a transfer function via the Fourier transform (Marmarelis and Marmarelis 1978; O’Leary and Honrubia 1975). O’Leary, Dunn, and Honrubia found a range of UIR shapes, each of which could be fitted by one of four mathematical models, with up to three time constants.

The mathematical models fitted by O’Leary, Dunn, and Honrubia differ qualitatively from each other, but rather than interpret this result as evidence for different response types, the authors suggested from inspecting their data that the real system exhibits continuous diversity of behavior. The appearance of four classes may be an artifact of applying Occam’s razor in numerical model fitting. The three simpler models, in Fig. 6, A–C, could be regarded as special cases of the third-order model (Fig. 6D), with some coefficients set to zero. Model order determination procedures tend to eliminate terms that make a small contribution, creating artifactual discontinuities between models that best approximate the behavior of a system in different regions. Thus the emergence of four qualitatively distinct models in O’Leary, Dunn, and Honrubia’s analysis does not provide evidence for distinct classes of afferents, and indeed the authors argue against that interpretation.

The form of an afferent’s UIR is correlated with the nerve bundle from which it was recorded, and therefore with the location on the crista innervated by that afferent (Fig. 6). O’Leary et al. argued that although some of the variation in response properties may be causally related to differences in the mechanics of hair cell cilia deflection in the central crest region of the crista compared with the edges, the neural responses were too diverse to be explained in terms of hydrodynamics and mechanics of the canal and cupula alone. They concluded that variability in synaptic and/or postsynaptic mechanisms must contribute substantially to the diversity of neural responses.

O’Leary et al. noted that the UIR of a linear system is a time-reversed template for the stimulus waveform that causes the maximal response for a given input power, and suggested that this provides response specificity allowing afferent neurons to signal the occurrence of particular patterns in head movement. In this way, the afferent nerve might provide the brain with specific information predigested for particular purposes; e.g., some afferents may provide signals that can be used to drive compensatory eye movements during slow head movements, whereas others provide high-gain, phase-advanced signals suitable for driving reflex responses to sudden perturbations without requiring intensive computation in the brain.

The seminal work of Goldberg and Fernandez (Fernandez and Goldberg 1971; Goldberg and Fernandez 1971a, 1971b; Goldberg and Fernández 1975) and of O’Leary and his collaborators (O’Leary et al. 1974; O’Leary and Dunn 1976; O’Leary and Honrubia 1975, 1976) revealed a new picture of diversity in semicircular canal afferent neuron dynamics during the 1970s. That picture was in clear contrast to the classical torsion pendulum model, which for the previous four decades had been the basis of a confident view that semicircular canal afferent neurons are a homogenous population whose firing rates faithfully encode head angular velocity measured by the sense organ in the behaviorally relevant frequency band.

FRACTIONAL ORDER DYNAMICS

Schneider and Anderson (1976) noted that the high-frequency phase lead in Bode plots of firing rate responses of gerbil semicircular canal afferents does not show the π/2 phase shift, and the corresponding segment of the Bode gain curve does not have the integer slope, expected for a standard linear dynamical model. They added a fractional order integro-differential operator term to the torsion pendulum model to account for this discrepancy. Thorson and Biederman-Thorson (1974) had recently published an influential article, arguing that processes with fractional order dynamics are widespread in sensory systems.

Fractional order operators model physical processes with power law dynamics, That is, mechanisms that relax after a perturbation not with an exponential time course, of the form e^−t/τ, as linear systems do, but with a time course of the form t^−k. In the time domain, a fractional order derivative operator looks like an ordinary derivative operator, $\frac{d^k}{d{t}^k}\text{,}$ except that k may not be an integer. A fractional order integrator looks the same, but with negative k. A fractional order operator in the Laplace domain is formally no different from the linear dynamical operator, s^k, except that k can assume non-integer values. Thus fractional ordinary differential equations and fractional transfer functions look like classical linear dynamical system models but differ in that k is not restricted to integer values. In both cases, the integro-differential operator s^k shows up as a segment with slope k in the Bode gain plot and a shift kπ/2 in the Bode phase plot; the difference is that the slope can be arbitrary and may not be an integer in the fractional model (Oldham and Spanier 1974).

Schneider and Anderson (1976) showed that the high-frequency phase lead in irregular units could be fitted by a torsion pendulum model with an additional fractional order derivative operator, s^k, with 0 < k < 0.5.

$$H\left(s\right)=\frac{A{s}^{k+1}}{\left(1+as\right)\left(1+bs\right)}\text{.}$$ (5)

The torsion pendulum model is a special case with k = 0. This is a unifying dynamical model that accounts for deviations from the torsion pendulum model reported by Fernandez and Goldberg (1971), but with only three rather than the four dynamical parameters of the earlier model (Eq. 4).

Landolt and Correia (1980) reported fractional order dynamics in semicircular canal afferent neurons in pigeons. Their data for all units fitted a simpler model of the form

$$H\left(s\right)=\frac{A{s}^k}{\tau +1}\text{,}$$ (6)

with 0 < k < 1. These authors interpret the parameter τ as the long time constant of the torsion pendulum model. However, the only similarity between Eq. 6 and the torsion pendulum model is that both have a long time constant, reflecting an additional phase advance at low frequencies relative to angular velocity input. As with the torsion pendulum model, the long time constant affects afferent responses at low frequencies, where vestibular contributions to head motion sensing are relatively unimportant under natural conditions. Although Landolt and Correia suggested that their model resembles the torsion pendulum model, the resemblance is somewhat tenuous.

Landolt and Correia (1980), citing Thorson and Biederman-Thorson (1974), observed that fractional order dynamical systems are self-similar over time and do not have intrinsic time scales, unlike integer-order linear systems, whose dynamics are characterized by time constants. Fractional order dynamics could explain reported discrepancies between estimates of model time constants by earlier workers. Converging estimates of time constants of the torsion pendulum model by different researchers might then reflect a growing consensus about experimental protocols and model fitting procedures, rather than an accumulation of evidence causing parameter estimates to converge toward true parameter values.

Paulin and Hoffman (1999, 2001b) fitted models to spike train data from bullfrog semicircular canal afferent neurons recorded in response to band-limited (0.05–4 Hz) Gaussian white noise angular velocity stimulation. Candidate models contained a dynamical filter with transfer function of the form

$$T\left(s\right)=s^k\frac{p\left(s\right)}{q\left(s\right)}\text{,}$$ (7)

where s^k is a fractional order dynamical operator and $\frac{p\left(s\right)}{q\left(s\right)}$ is a standard, integer-order linear transfer function. This form (Eq. 7) includes Schneider and Anderson’s (1976) model and Landolt and Correia’s (1980) model as special cases. The dynamical element (Eq. 7) was followed by a static time-domain nonlinearity that converted the output x(t) of the dynamic element to a nonnegative, saturating firing rate,

$$r\left(t\right)=r_0+\lambda \hspace{0.17em}tanh\left[\alpha \left(x-\beta \right)\right]\text{.}$$ (8)

Amplitude-dependent nonlinearities in semicircular canal afferent responses were noted when afferent activity was first recorded [see Fig. 5 in Groen et al. (1952)], but there was no practical way to model these nonlinearities at the time. In mammals and birds, whose canal afferents have high spontaneous firing rates, it is possible to obtain spike train data with nearly sinusoidal firing rate modulations in response to sinusoidal stimulus, by using low-amplitude stimuli that do not drive responses close to rectification in one direction or to saturation in the other. This nonlinearity cannot be ignored in bullfrogs, because many afferents have low spontaneous firing rates and high gains and are driven to silence by gentle contralateral rotation. The nonlinearity (Eq. 8) coincidentally resembles the rectifying-saturating static nonlinearities called “neurons” in connectionist neural network models (Gibbons 2019).

Paulin and Hoffman (1999, 2001b) identified models by cross-validation. That is, instead of selecting the model that best fits the data, they selected the fitted model that best fitted an independent spike train recorded from the same neuron. Selecting the best predicting model rather than the best fitting model avoids overfitting: it reduces the probability of selecting models that characterize statistical idiosyncrasies in individual responses rather than models that characterize underlying dynamics of the data source. Despite testing all stable candidate models with linear terms up to fifth order in the denominator, p(s), and the numerator, q(s), a purely fractional order model,

$$T(s)=A{s}^k\text{,}$$ (9)

consistently appeared at or near the top of cross-validation rankings. Candidate models included the torsion pendulum and the torsion pendulum models with adaptation and phase lead terms as had been reported by different investigators, but Paulin and Hoffman found no sign of torsion pendulum dynamics in their data (see Fig. 7). Their purely fractional order model (Eq. 9) is similar to the model (Eq. 6) identified by Landolt and Correia (1980) in pigeons, but without the low-frequency phase advance term. This may reflect the lack of stimulus power at low frequencies; i.e., the data may have been insufficient to characterize very low frequency responses. Another possibility is that adaptation observed during slow turns does not occur during broadband stimulation, which involves rapid back-and-forth turning with a fixed baseline.

Fig. 7.

A: fractional order dynamical model with static nonlinearity relating turntable angular velocity to firing rate of bullfrog semicircular canal afferent neuron. B: short segment of band-limited Gaussian white noise velocity stimulus. C: real (up) and model-fitted (down) spike trains. D: instantaneous rate of real (up) and model (down) spikes. [Adapted from Fig. 2 in Paulin and Hoffman (1999) by permission of Elsevier Science and Technology Journals].

Fractional order dynamical operators are time-scale invariant, which manifests in a tendency to generate a similar pattern of responses to stimulus waveforms that are compressed or stretched in time. In particular, fractional order neurons responding to sinusoidal stimuli should show a tendency to phase lock, that is, to respond near a particular point in the stimulus cycle independent of frequency. Hartmann and Klinke (1980) recorded spike trains from semicircular canal afferent neurons in goldfish during high-frequency sinusoidal oscillation and reported that these neurons tend to phase-lock at frequencies above 4 Hz. This phenomenon also has been reported in toadfish semicircular canal afferent neurons (Boyle and Highstein 1990; Highstein et al. 2005).

High-frequency phase locking is consistent with Paulin and Hoffman’s model (Eq. 9), with a phase-lock angle of ϕ = 2πk if there was no amplitude-dependent nonlinearity present, because the maximum firing probability occurs at this phase. This model’s response to high-frequency sinusoidal stimulation has not been investigated.

The stark discrepancy between the nonlinear fractional order model (Eqs. 8 and 9) and the torsion pendulum model of afferent dynamics may simply reflect the fact that earlier studies were technologically restricted to collecting data at low stimulus frequencies and to using integer-order linear transfer function models that could be fitted using numerical and computational methods readily available in the mid-20th century.

Paulin and Hoffman (2001b) noted that the fractional order operator As^k corresponds to a rotation and change of amplitude of vectors in the phase space whose axes are the stimulus variable and its first derivative. For example, the operator As¹ represents a conventional first-order derivative corresponding to a 90° rotation and gain by a factor A that transforms angular velocity into angular acceleration. Paulin and Hoffman (2001b) showed that this allows an observer to infer the conditional probability density of the kinematic state of the head, in angular velocity-angular acceleration space, given that a particular neuron has fired. This is illustrated in Fig. 8, which shows empirical frequency histograms of points in angular velocity-angular acceleration space at spike times generated by models of the form in Eqs. 8 and 9. Figure 8, center panel, shows the distribution of kinematic states at arbitrary times, computed by constructing histograms of states at random times during simulated head movement. This represents the natural distribution of head kinematic states. The other eight panels of Fig. 8 show conditional distributions of states, computed using fractional order models of the form in Eq. 8 and 9, fitted to spike train data from eight different bullfrog semicircular canal afferent neurons.

Fig. 8.

Conditional probability distributions of head kinematic state in angular velocity-angular acceleration space at spike times, generated by simulating the fractional order model (Eq. 9). Center panel shows the unconditional distribution of head state. An individual spike can be interpreted as a packet of information about the instantaneous kinematic state of the head. [Reproduced from Paulin and Hoffman (2001b) by permission of Elsevier Science and Technology Journals.]

These conditional probability distributions, which formalize the familiar concept of receptive fields in a dynamic probabilistic framework, were called “state-space receptive fields” by Paulin and Hoffman (2001b). State-space receptive fields may be not merely a convenient way to visualize what sensory neurons tell the brain; they may help to model neural computation in terms of spatial transformations between probabilistic maps of relevant state spaces in the brain (Paulin 2004; Paulin et al. 2001, 2004). In a Bayesian framework, the natural distribution of states is the prior distribution, which an observer could learn by extended exposure to natural stimuli. State-space receptive fields, which we have computed from long sequences of observations during simulated natural head motion, could be computed instantaneously at spike times by an observer applying Bayes rule (Paulin et al., 2001).

STATES AND THE SINGLE SPIKE

The state-space receptive field was intended to be a generalization of the old and now familiar concept of a receptive field (Hartline 1938), with important differences. First, receptive fields were originally defined as regions in real space within which a stimulus causes a visual neuron to respond (Hartline 1938), but Paulin and Hoffman (2001b) defined state-space receptive fields of semicircular canal afferents as probability distributions in an abstract space of kinematic state variables. Second, receptive fields were originally defined in terms of change in firing rate as the stimulus moves in the relevant space, but Paulin and Hoffman (2001b) defined the state-space receptive field of a semicircular canal afferent neuron as the probability distribution of head kinematic states conditional on observing a single spike.

The realization that information transmission by canal afferent neurons can be modeled using probability distributions of head kinematic state variables conditional on spike times is potentially important for understanding how these neurons transmit information and how the brain processes that information. In the linear systems framework used to develop the fractional order model (Eq. 9), the output of a neuron is its firing rate. In that framework, firing rate is defined and computed in terms of the number of events in an interval. However, the behaviorally relevant bandwidth of head movements detected by the vestibular system is high relative to afferent firing rates (Dickman and Correia 1989; Highstein et al. 1996, 2005; Rabbitt 2019). In this case firing rate is a poor measure of neuronal response (Paulin and Hoffman 2001a). During high-frequency sinusoidal head oscillation, semicircular canal afferents may phase-lock to the stimulus and/or produce so few spikes during a cycle that firing rate is not well defined (Dickman and Correia 1989; Highstein et al. 2005). It then becomes unclear how to define the amplitude of the response, and how to compute and interpret gain in a Bode plot if a neuron’s response is defined in terms of firing rate.

The difficulties that emerge in the high-frequency limiting case merely accentuate a pervasive problem in defining, measuring, and modeling the output of spiking neurons in terms of firing rate. Firing rate is computed empirically from spike trains by filtering, even if the computation does not look like a filtering operation (Paulin 1992; Paulin and Hoffman 2001a). Thus a transfer function identified and fitted to firing rate data is a model not only of the biological system but also of the dynamics of the filter used to compute the firing rate. The extent to which artifacts from this source may have contributed to fitted transfer function models of semicircular canal afferent responses does not appear to have been investigated. However, it is likely to have some effect, particularly at high frequencies. The introduction of new models that treat spikes themselves as the outputs of spiking neurons obviates concern about possible firing rate artifacts in earlier models (Paulin 2004).

A probability distribution defined over an abstract space can easily be represented in real space. For example, in Fig. 8, pixel shading represents probability density in a two-dimensional map. An analogous representation might be formed in a brain using membrane depolarization or some other neuronal state variable across an array of neurons, or a network representing a higher dimensional space (Paulin et al. 2004). Engineers and applied mathematicians use state-space models to analyze stochastic dynamical systems in a range of applications (Zadeh and Desoer 2008). State-space models allow stochastic dynamic problems to be analyzed using spatial reasoning, by transforming kinematics and statistics into geometry. Nervous systems seem particularly adept at spatial reasoning with uncertainty and may have initially evolved to perform quasi-static spatial inference using noisy sense data from hair cell receptors (Paulin and Cahill-Lane, in press). Thus state-space receptive fields may provide a useful foundation for understanding not only information transmission by vestibular sensory neurons but also mechanisms of neural computation for dynamical inference in the vestibulocerebellar brain stem (Paulin 2005; Paulin et al. 2004).

Figure 8 demonstrates that it is possible to extract information from sensory spike trains instantaneously at spike times. The conditional probability distribution of states at spike times of a given neuron differs from the unconditional or prior distribution of states. An observer armed with this map could infer the posterior distribution of states, given that a particular neuron has fired, immediately upon observing a spike. It is evident from Fig. 8 that the posterior distributions are about the same “size” as the prior. Technically, they have similar entropy. This means that a single spike transmits only a small amount of information and reduces the observer’s uncertainty about head kinematic state, the entropy of the posterior distribution, by a correspondingly small amount.

Evidently, the brain can infer head kinematic states much more accurately than suggested by Fig. 8, as evidenced by the fact that the eyes, driven by the vestibulo-ocular reflex, have a much lower variance of angular velocity than the head does during locomotion (Grossman et al. 1988, 1989). The brain receives spikes from large numbers of sensory neurons that fire repeatedly, even from the semicircular canals, which are low-dimensional dynamical sensors with only one mechanical degree of freedom. It must combine information from spikes of different neurons and from individual neurons across time to compute head kinematic state. The optimal strategy, which minimizes uncertainty about head motion (i.e., it minimizes the entropy of the posterior distribution) given all observations up to the present time, is to use the posterior distribution given all previous spikes as the prior distribution for inferring the posterior given each spike. This is called sequential Bayesian inference (Doucet et al. 2001).

A variety of circumstantial evidence indicates that the vestibulocerebellum, which is the central nervous system target of vestibular sensory afferent neurons, is a neural analog of a Bayesian observer for head kinematic state given the data provided by these neurons (Orban de Xivry et al. 2013; Paulin 1993, 2005). A Bayesian model of neural computation for inference in the vestibulocerebellum requires a generative model, a model that can simulate spiking by vestibular sensory afferent neurons.

Paulin and Hoffman’s (2001b) fractional order model (Eqs. 8 and 9) is a generative model, but it has an obvious shortcoming. The spike generating mechanism is the same in all models, which means in particular that the distribution of interspike intervals at rest is the same for all model neurons. This is qualitatively unrealistic and shows that although the model may be able to describe dynamic diversity among semicircular canal afferent neurons, it is missing something fundamental about spiking statistics. Given the well-established correlation between dynamics and spiking statistics, the failure of Paulin and Hoffman’s (2001b) fractional order model to account for diversity in the spiking statistics of these neurons indicates that it also may be missing something fundamental about their dynamics. It can describe the distribution of dynamical diversity among semicircular canal afferent neurons, but it cannot accurately simulate random spike train samples from individual neurons. It is an accurate dynamical model in a classical dynamical systems framework, but not a generative model of the kind required for implementing (or modeling the implementation of) Bayesian inference.

SPIKE TRAIN STATISTICS

Covariation of statistical and dynamical parameters of vestibular sensory neuron firing behavior was first discussed by Goldberg and Fernandez (1971b) (Fig. 5). They quantified spike train variability in squirrel monkey semicircular canal afferent neurons using the coefficient of variation of interspike interval length, CV, the standard deviation relative to the mean. They found a negative correlation between CV and spontaneous firing rate, and between CV and sensitivity of these neurons. Neurons with more irregular firing patterns tend to have lower spontaneous firing rates and to be more sensitive to rotation. This pattern has subsequently been reported in all species studied. They noted that CV varies systematically between neurons with different mean spontaneous firing rates, and the CV of any particular neuron changes similarly when its mean firing rate is altered by rotational stimulation.

Fig. 5.

Standard deviation of interval length varies with mean interval length at different firing rates of individual neurons. Data are from regular (bottom), intermediate (middle), and irregular (top) semicircular canal afferent neurons. [Reproduced from Goldberg and Fernandez (1971b).]

Goldberg et al. (1982) introduced a “normalized coefficient of variation,” CV*, which adjusts for differences in spontaneous rate in computing the variability of different neurons. CV* is constructed to estimate CV at a standard mean interspike interval, which Goldberg and Fernandez (1971b) set at 15 ms for squirrel monkey afferents. They reported a statistically linear relationship between CV* and sensitivity. CV and CV* are summary statistics that ignore the shape of the interval distribution. Afferent interspike interval distributions generally become more skewed as they become more irregular (Blanks et al. 1975; Correia and Landolt 1977; Estes et al. 1975; Hartmann and Klinke 1980; Honrubia et al. 1981) (see Fig. 9). This is true not only when interval distributions are compared between neurons with different firing rates but also when the firing rate of an individual neuron changes in response to a stimulus (Blanks et al. 1975; Goldberg and Fernandez 1971b).

Fig. 9.

A: strong correlation between coefficient of variation (CV) and skew in spike trains recorded from bullfrog semicircular canal afferent neurons (Honrubia et al. 1981). Distributions at extreme values of CV and skew appear qualitatively different, but there is a continuum of shapes between these extremes. Skew ranges from near 0, characteristic of a Gaussian, to near 2, characteristic of an exponential distribution. [Redrawn from Honrubia et al. (1981) by permission of Taylor and Francis.] B: interspike interval distributions of spike trains recorded from regular, intermediate, and irregular cat semicircular canal afferent neurons (Estes et al. 1975). The regular distribution resembles a Gaussian, whereas the irregular distribution resembles an exponential. Intermediate distributions resemble a mixture of the two extremes. m, Mean.

Figure 9B shows examples of interspike interval distributions from regular, intermediate, and irregular semicircular canal afferent neuron spike trains recorded in cats (Blanks et al. 1975; Estes et al. 1975). The most regular distributions, with small CV, resemble narrow Gaussians, whereas the most irregular distributions, with large CV, resemble exponential distributions. Despite the qualitative difference in shape, there is no evidence for distinct classes based on firing regularity. On the contrary, there appears to be a continuum of response variability.

Correia and Landolt (1977) tested a number of candidate models for pigeon semicircular canal afferent neuron interspike interval distributions. They noted that the Erlang distribution (Cox 1962), which is a gamma distribution whose shape parameter, γ, is restricted to integer values, is an obvious candidate model. The Erlang distribution is exponential for γ = 1 and tends toward a Gaussian for large γ. It is the waiting time for γ events in a Poisson process, and so it is a statistical model of a neuron that fires and then resets after integrating or counting γ random synaptic events. Irregular neurons would be those triggered by one or two postsynaptic potentials that are large relative to the spiking threshold, whereas regular units would require many postsynaptic potentials to accumulate before firing. For intermediate values of γ, the shape of the Erlang distribution appears very similar to the interval distributions of intermediate semicircular canal afferent neuron spike trains. Goldberg (2000) later also proposed the Erlang distribution as a model of spike train variability in these neurons, although he downplayed it as a realistic model of biophysical mechanisms.

Another plausible candidate model is the inverse Gaussian or Wald distribution, the distribution of times for a Brownian motion process to reach a barrier (Wald 1944). This also has the Gaussian and the exponential distribution as limiting cases. It has a similar intuitive interpretation as a statistical model of a simple integrate-and-fire neuron, except that in this case the integrated input is Gaussian noise representing a sum of large numbers of randomly occurring postsynaptic events, rather than a discrete sequence of postsynaptic events as in the Erlang model. Regular spiking with Gaussian variability occurs if noise is integrated to a larger threshold, whereas maximally irregular spiking, a Poisson process with exponential interval distributions, arises if the threshold is low enough that the noise is able to trigger the neuron without integration.

Correia and Landolt (1977) tested Erlang, exponential, exponential-with-delay, and inverse Gaussian (Wald) candidate models. They referred to the Erlang distribution as the gamma distribution with integer values of the shape parameter, which it is, and to the inverse Gaussian or Wald as “the pdf for the first passage times in a Wiener-Levy process,” which it is. Each of these models fitted the empirical distributions well over some subrange of variability but diverged qualitatively elsewhere. Thus, although the Erlang and Wald distributions both look prima facie as if they should fit the empirical variety of interval distributions, neither does.

Correia and Landolt (1977) noticed that in their data, CV scales with the square root of mean interval length. This is also true in the data originally presented by Goldberg and Fernandez (1971b) and re-presented by Goldberg (2000). Goldberg and Fernandez’s data are reproduced in Fig. 10A and replotted on log-log axes in Fig. 10B. The lines in Fig. 10B have slope = ½, corresponding to a square-root relationship on linear axes. Standard deviation of interval length for the regular neuron approaches the sampling interval used to acquire the data, and these data have been redigitized from the figure in Goldberg (2000). Note the logarithmic scale on this plot, and that CVs for the most regular neuron are likely to have very large relative errors, particularly at small interval lengths, because of digitization artifacts. Data from the other units, especially the irregular unit, are clearly consistent with the $CV\propto t^{1/2}$ relationship reported by Correia and Landolt (1977). Correia and Landolt interpreted this simple scaling law to indicate that there must be a simple stochastic process common to the spike generation mechanism of all afferents. Square-root scaling between CV and mean interval is characteristic of an inverse Gaussian process (Wald 1944). However, Correia and Landolt ruled out this model because it did not fit their data.

Fig. 10.

A: relationship between mean interval length and standard deviation (s.d.) of interval length for a regular (bottom), an intermediate (middle), and an irregular semicircular canal afferent neuron (top) as mean interval length changes under unidirectional angular acceleration. [Redrawn using data from Goldberg (2000); cf. Fig. 5.] B: relationship between mean interval length and coefficient of variation (CV), using the data from A, plotted on log-log axes. Lines of slope = ½ have been fitted to the data points for each neuron. Straight lines in B transform to power law models for s.d. as a function of mean interval length, which are overlaid on the data in A.

Hoffman et al. (2015) showed that the Erlang distribution is a good fit to the full range of empirical interval distributions in spike trains recorded from semicircular canal afferent neurons in chinchillas, if an extra time-lag parameter is included that allows the shape of the distribution to change independently of the mean interval length. Thus the intuition that shapes of Erlang distributions ought to be able to mimic the various shapes of the empirical interval distributions is correct, at the expense of an additional ad hoc parameter, a time delay in the order of a few milliseconds.

The simplest stochastic process model that could realistically introduce a time lag is to place a Poisson process in series with the Erlang process. This would imply that spikes are generated by two independent random processes, one producing random events with an exponential interval distribution and the other producing random events with an Erlang interval distribution, which alternate to produce the firing pattern. The exponential-Erlang distribution is able to accurately fit and qualitatively reproduce the full range of shapes of chinchilla semicircular canal afferent neuron spike train interspike intervals (Hoffman et al. 2015; Pullar et al. 2015).

Subsequent systematic search using an information criterion to rank a collection of models, including models previously proposed for semicircular canal afferent spiking statistics and those models in series with a Poisson process, identified an exponential distribution in series with a Wald distribution as consistently the best-fitting model to the chinchilla afferent data (Fig. 11) (Paulin et al. 2017). The exponential-Wald, or ex-Wald, distribution was first identified as a model of human reaction times in forced-choice decision experiments given noisy data (Schwarz 2001).

Fig. 11.

Exponential-Wald (ex-Wald) model of spontaneous interspike interval distribution in chinchilla semicircular canal afferent neurons (Paulin et al. 2017). Despite superficially qualitative differences, a simple 3-parameter statistical model accurately captures all of the observed diversity. A–C show the interspike histograms of spike trains recorded from 3 representative afferents, each overlaid with a fitted ex-Wald model. Note the different scales on each set of axes. The exponential (dashed line) and inverse Gaussian or Wald (dotted line) components of the ex-Wald model are shown in a regular neuron (A), an intermediate neuron (B), and an irregular neuron (C). D shows the ex-Wald models in A–C drawn on a common time axis and rescaled so that they all have the same maximum height. Note that each neuron has a minimum interval length >10ms and rarely fires at intervals shorter than 13 ms. [Redrawn from Paulin et al. (2017).]

The ex-Wald is the distribution of intervals generated by an inverse Gaussian or Wald process in series with a Poisson process. The Wald component can be interpreted as a censoring process, which blocks outputs from the Poisson process for a random interval after each event. Conceptually, and perhaps biophysically, these intervals correspond to refractory periods following a spike. The Wald censoring distributions are all approximately Gaussian with approximately the same mean, around 13 ms. Interval distributions of units whose Poisson parameters are much shorter than 13 ms closely resemble the Wald component alone, because the next event in the Poisson process is likely to occur very soon after the end of the refractory period. These are regular units, whose interval distributions resemble narrow Gaussians with a small positive skew. Interval distributions of units whose Poisson parameters are long compared with 13 ms resemble exponential distributions shifted ~13 ms to the right, because the waiting time for the next event in a Poisson process after a wait of some time has an exponential distribution (Landolt and Correia 1978). These are irregular units. As illustrated in Fig. 11D, the refractory period is essentially the same for all units and variability in the interval distribution is mainly due to variability in the Poisson parameter.

Spike trains generated by an ex-Wald process can be interpreted as random subsamples from a Poisson process, with subsampling implemented by blocking output from the Poisson process for a random interval after each event. Paulin et al. (2017) suggested that selection for acuity and energy efficiency may explain why each semicircular canal branch of the vestibular nerve employs thousands of afferent neurons to transmit information about a single mechanical degree of freedom of head movement. For the most regular spike trains that they recorded from chinchilla semicircular canal afferent neurons, subsamples contained fewer than 0.1% of the spikes of the uncensored process. This means that a population containing more than 1,000 spiking neurons could transmit samples from a Poisson process generating tens of thousands of events per second, without any individual neuron in the population firing with an interspike interval much smaller than ~13 ms or, equivalently, without firing faster than ~80 spikes per second.

Because of overlapping sodium and potassium currents, spiking in myelinated axons becomes prohibitively energetically expensive at average rates exceeding ~80/s (Goldberg et al. 2003). In general, it is cheaper to transmit spikes at low average rates on parallel channels than at higher average rates using fewer neurons (Balasubramanian 2015; Laughlin et al. 1998; Niven 2016; Niven and Laughlin 2008; Sengupta et al. 2010; Sterling and Laughlin 2015). This suggests that the Wald or inverse Gaussian component of the ex-Wald model may reflect a biophysical process that has been selected to increase the energy efficiency of signal transmission on the vestibular nerve (maximize bits per joule per second, or bits per watt), by using refractory censoring to randomly subsample an underlying Poisson process.

The ex-Wald distribution is a generative model that accurately characterizes the statistical variability of semicircular canal afferent neurons. We are currently investigating whether it can be extended and/or combined with the fractional order dynamical model (Eq. 9). Naive interpretation of the ex-Wald model suggests that the underlying source of stochasticity in spontaneous firing activity is Brownian motion of hair cell bundles (Denk et al. 1989; Kozlov et al. 2012). In that case “spontaneous” firing can be understood as a dynamical response to microscopic mechanical stimulation of the hair cells caused by thermal noise energy (Dinis et al. 2012). It should therefore be possible to extend the ex-Wald model to include response dynamics by using a differential equation that models how the parameters of the ex-Wald distribution depend on head kinematic state variables. The observed square-root relationship between mean interval length and CV (Fig. 10) (Correia and Landolt 1977) suggests that the dependent variable in this model may be the mean of the Wald component. We are currently trying to develop and test a model of this kind, and to relate its parameters to real parameters of underlying biophysical processes.

BIOPHYSICAL NEURAL MODELS

Goldberg et al. (1982, 1984) suggested that spike train variability in vestibular afferent neurons can be explained by a generic model of neuronal spiking variability proposed by Stein (1967). The mechanism is illustrated in Fig. 12. Immediately after a spike, the membrane is hyperpolarized below its firing threshold. Smaller neurons have shorter membrane time constants and higher input gains. Their membrane potential returns to its equilibrium level faster and crosses the threshold sooner than in irregular neurons. Thus smaller neurons fire faster than larger ones receiving the same inputs. Because it approaches the threshold on an exponential decay trajectory, the membrane potential of the smaller neuron is changing faster as it approaches the threshold. Thus there is a narrower time window during which the noisy membrane potential may first cross the threshold, causing smaller neurons to have more regular interspike intervals. The theory predicts that neurons with more regular firing patterns should be smaller and fire faster than neurons with less regular firing patterns, as is the case for vestibular afferent neurons. Note that if the membrane were a perfect integrator, rather than a leaky integrator as in real neurons, the resulting intervals would have an inverse Gaussian distribution.

Fig. 12.

A: Stein (1967) model of neuronal spiking variability. Neurons with shorter time constants and/or higher input gains reach threshold sooner and cross it at a faster rate. There is a shorter time window during which a noisy membrane potential may first cross the threshold. The plot illustrates the mechanism using a noisy threshold rather than a noisy membrane potential, but the effect is the same. B: biophysical membrane model implementing Stein’s mechanism (see text for details). C: output of model simulation, showing membrane potential of a regular (top) and an irregular (bottom) model neuron. [Adapted from Smith and Goldberg (1986) by permission of Springer.] H, hyperpolarized potential; O, zero potential (resting potential in model); V_T, threshold potential.

Smith and Goldberg (1986) developed a biophysical membrane model of a vestibular sensory afferent neuron, adapted from an earlier model of repetitive firing in cat spinal motor neurons (Kernell 1969; Kernell and Sjoholm 1971). This model, shown in Fig. 12B, is a Hodgkin-Huxley type (channel conductance based) model, including a membrane time constant, τ (the product of membrane resistance and capacitance), synaptic conductance, g_s, potassium conductance, g_k, and a current source, I_p. V_s and V_k are membrane voltage referenced to the respective equilibrium potentials. The apparent voltage V_p in Smith and Goldberg’s equation has been replaced by a current source, I_p, making the model physically and dimensionally correct. This typo in the original formulation had no effect on their conclusions, since they treated V_p correctly as a current source in their analysis.

$$\tau \frac{\text{d}V}{\text{d}t}+\left(1+g_s+g_k\right)V\left(t\right)=g_s{V}_s+g_k{V}_k+I_p\text{.}$$ (10)

Postspike potassium conductance decays exponentially,

$$g_k\left(t\right)=g_{k0}\text{exp}\left(-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{${\tau }_k$}\right.\right)\text{.}$$ (11)

Noise enters the model via synaptic conductance, which is assumed to result from summation of a large number of random quantal events each with amplitude Δg_s and duration Δt_s, occurring at mean rate λ. This gives broadband Gaussian conductance with mean ${\overline{g}}_s=\lambda \Delta g_s\Delta t_s$ and variance ${\sigma }_s^2={\overline{g}}_s\Delta g_s$.

Smith and Goldberg (1986) argued that the membrane time constant τ is likely to be much shorter than the time constant of the potassium current, τ_k. In that case, membrane dynamics will have little effect and the time course of membrane potential will resemble the equilibrium potential determined by instantaneous conductances. The differential equation (Eq. 10) can be converted into an algebraic equation by setting the membrane time constant τ to zero:

$$V\left(t\right)=\frac{\left(g_s{V}_s+g_k{V}_k+V_p\right)}{\left(1+g_s+g_k\right)}\text{.}$$ (12)

From this equation, Smith and Goldberg (1986) were able to qualitatively reproduce the observed relationship between mean interval length and CV by adjusting model parameters, and to simulate neurons with interval statistics resembling those of real vestibular afferent neurons. They reported that despite the simplicity of the model, it reproduces many features of vestibular afferent spontaneous firing, provided that the firing rate is high, but that systematic errors appear when the mean interval becomes shorter than ~25 ms, corresponding to a firing rate above ~40 s⁻¹.

Smith and Goldberg (1986) concluded that the major determinant of discharge regularity is an after-hyperpolarizing potential caused by slow potassium conductances activated by action potentials. They suggested that the various morphological explanations of response variability in vestibular afferent neurons, related to mechanical differences and/or differences in innervation patterns over the crista, may be correlated with discharge regularity but are unlikely to be causally related to it.

More recent studies have supported the idea that postspike ion conductances contribute to vestibular afferent neuron firing rates and regularity (Eatock et al. 2008; Kalluri et al. 2010). A number of channels seem to be involved. Smith and Goldberg’s model (Eqs. 10 and 12) may be an oversimplification at best. Hight and Kalluri (2016) showed that a computational biophysical model incorporating current knowledge about ion channels in vestibular afferent neurons can produce regular and irregular firing patterns resembling the behavior of the real neurons. However, the model has many unconstrained degrees of freedom and can evidently produce a much wider range of behavior. Behavior resembling the behavior of real vestibular sensory afferent neurons can be reverse-engineered but does not emerge from this model.

The model developed by Hight and Kalluri (2016) reassuringly indicates that current molecular-level understanding of vestibular neuron biophysics is qualitatively consistent with at least some of the behavior of these neurons. However, it does not explain why vestibular afferent neurons have evolved to behave as they do. The biophysical model suggests that selection for statistical and dynamical response properties of semicircular canal afferent neurons has not been constrained by variability, or the lack of it, at the molecular-genetic level. The ex-Wald model, however, suggests that afferent response properties have evolved under strong selection pressure for efficient rapid transmission of a high-bandwidth signal, constrained by the kinetics of voltage-gated sodium and potassium channels, which cause catastrophic energy demands at high firing rates. It would be interesting to include estimates of the energy costs of spiking in the Hight-Kalluri model and then optimize its parameters to minimize the cost of information (bits per second per joule) in spike train responses to high-bandwidth Gaussian noise stimuli. We predict that this will produce biophysically plausible parameter values and spike trains with ex-Wald distributions.

DISCUSSION

Models of semicircular canal afferent neurons have coevolved with scientific understanding of mechanisms of sensation and perception in general. In the early “camera” metaphor, sense organs were thought of as lenses projecting an image of the world into the brain, and sensory neurons as mere passive mediators of this projection. It is now clear that vestibular information is transmitted to the brain in a distributed stochastic representation, in which responses of individual neurons are not simply noisy replicates of measurements made in the sense organ. The emerging framework for modeling vestibular sensory neuron behavior outlined in this review is consistent with the view that nervous system evolution is driven by selection pressure for channel capacity, speed, and efficiency, constrained by available biophysical mechanisms (Laughlin 2001, 2013; Niven and Laughlin 2008; Sterling and Laughlin 2015).

The anatomy of the vestibular apparatus itself (Fig. 1A) illustrates functional decomposition of vestibular information in the periphery. Nineteenth-century anatomists correctly deduced that the mutually perpendicular semicircular canals sense orthogonal rotational components of head movement. This orthogonality should optimize the efficiency of signal transmission to the brain, by reducing redundant mutual information about head movements in different nerve branches during head movements. Evolution has evidently exploited the mechanics of materials to construct a peripheral sensing mechanism that not only reduces the expense of transmitting information about head movements to the brain but presumably also reduces energy costs in the brain by providing that information in a form that simplifies neural computation in central vestibular pathways (Fritzsch et al. 2002; Straka et al. 2014).

Is there a more fine-grained decomposition of sensory information into specific channels within individual branches of in the vestibular nerve? This question has been a focus of data analysis and models of semicircular canal afferent neurons since the seminal papers by Goldberg and Fernandez in the 1970s. The most striking characteristic of the behavior of these neurons, first reported by Löwenstein (1955) and confirmed in every subsequent neurophysiological investigation, is that some have very regular firing patterns whereas others have very irregular firing patterns. These differences are an easily recognized signature that predicts other statistical and dynamical properties. However, as Peterson (1998) pointed out, it is a logical fallacy to infer that distinct groups exist on the basis of correlations between parameters and differences between parameters in different groups. Regular and irregular classes were constructed as a methodological convenience by Goldberg and Fernandez (1971b), and as they noted at the time, these reflect differences across a continuously varying population, not a real division between classes of neurons. Peterson (1998) argued that although investigators usually qualify their use of “regular” and “irregular” by pointing out that this is only a rhetorical and analytical convenience; it is potentially harmful to use these terms because it may distract from investigation of real patterns in data and underlying real causes. However, the terms are still commonly used without qualification (Cullen 2011, 2012; Eatock and Songer 2011; Eatock et al. 2008; Kalluri et al. 2010; Sadeghi et al. 2007).

Different kinds of models developed since Goldberg and Fernandez’s initial characterization of dynamic and statistical diversity among semicircular canal afferents generally support the concept of a distributed representation, in which differences in spiking behavior reflect specificity, not just random variation, in the information transmitted by different neurons. Some kind of efficiency argument, essentially an economic argument for the division of labor, has repeatedly been put forward in discussing this functional heterogeneity. For example, Minor and Goldberg (1991) suggested that regular afferents have suitable response dynamics to drive compensatory eye movements during head movements with minimal central processing, and with minimal contribution from irregular afferents. However, later studies indicated that there is a broad contribution from afferents with heterogeneous response properties to individual reflex pathways (e.g., Boyle et al. 1992). The general economic argument remains plausible, but functional specificity in the nerve is evidently more subtle than assigning subgroups of afferents to specific behaviors.

The energy cost of information is not only a key metric for understanding nervous system design (Sterling and Laughlin 2015), but of life itself in a broader context (England 2013, 2015; Ortega and Braun 2013). Neural tissues are particularly energetically expensive relative to other tissues and appear to have evolved in animals under very strong selection pressure for fast, efficient, high-acuity information processing and decision-making (Paulin and Cahill-Lane, in press). The energy costs of neural spiking generally favor distributed signaling at low firing rates on heterogeneous parallel channels (Balasubramanian 2015; Laughlin et al. 1998). Paulin et al. (2017) showed that spontaneous interspike interval distributions of chinchilla semicircular canal afferent neurons can all be modeled accurately using the ex-Wald distribution (Fig. 11) and that this model can be interpreted as an energy-efficient way to transmit a high-bandwidth signal using spiking neurons that become catastrophically inefficient at high firing rates. However, this does not explain why selection for energy efficiency should play out in the particular pattern of dynamic diversity seen in semicircular canal afferents or whether some other factors may contribute to this pattern.

The functional organization of the vestibular nerve cannot be understood in isolation because it functions as a component in a system. The speed, acuity, and energy efficiency of information transmission in the nerve must be evaluated in the context of the evolutionary value or fitness contribution of that information, as well as the speed, accuracy, and energy efficiency of central neural computations for extracting that value. Other things being equal, Bayesian inference is the most efficient way to extract information from data, i.e., to reduce uncertainty about its causes (Levy 2006; Trimmer et al. 2011). If the vestibulocerebellar brain stem functions as a dynamical filter for Bayesian estimation of head kinematic state, as has been suggested (Kuo 2005; Miall and King 2008; Paulin 1989, 1993), then a full understanding of the functional organization of the vestibular nerve requires modeling how the brain uses spike train data provided by the nerve to infer the conditional probability distribution of head kinematic state—the Bayesian posterior distribution—from sense data. A prerequisite for such a model is a generative model of the data, that is, a model that can generate simulated data as a function of head kinematic state. The ex-Wald distribution may provide a simple generative model of semicircular canal afferent firing suitable for this purpose (Paulin and Hoffman 2011a, 2011b; Paulin et al. 2017; Paulin and Van Schaik 2014).

GRANTS

This work was supported by National Institute of Deafness and Other Communications Disorders Grant DC014368 (to L. F. Hoffman) and a grant from the New Zealand Performance-Based Research Fund (to M. G. Paulin).

DISCLOSURES

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

AUTHOR CONTRIBUTIONS

M.G.P. analyzed data; M.G.P. and L.F.H. interpreted results of experiments; M.G.P. prepared figures; M.G.P. drafted manuscript; M.G.P. and L.F.H. edited and revised manuscript; M.G.P. and L.F.H. approved final version of manuscript.