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An Integrative Neural Network for Detecting Inertial Motion and Head Orientation

Department of Anatomy and Neurobiology, Washington University School of Medicine, St. Louis, Missouri 63110

Submitted 19 December 2003; accepted in final form 23 March 2004

	ABSTRACT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX GRANTS ACKNOWLEDGMENTS REFERENCES

 
The ability to navigate in the world and execute appropriate behavioral responses depends critically on the contribution of the vestibular system to the detection of motion and spatial orientation. A complicating factor is that otolith afferents equivalently encode inertial and gravitational accelerations. Recent studies have demonstrated that the brain can resolve this sensory ambiguity by combining signals from both the otoliths and semicircular canal sensors, although it remains unknown how the brain integrates these sensory contributions to perform the nonlinear vector computations required to accurately detect head movement in space. Here, we illustrate how a physiologically relevant, nonlinear integrative neural network could be used to perform the required computations for inertial motion detection along the interaural head axis. The proposed model not only can simulate recent behavioral observations, including a translational vestibuloocular reflex driven by the semicircular canals, but also accounts for several previously unexplained characteristics of central neural responses such as complex otolith–canal convergence patterns and the prevalence of dynamically processed otolith signals. A key model prediction, implied by the required computations for tilt–translation discrimination, is a coordinate transformation of canal signals from a head-fixed to a spatial reference frame. As a result, cell responses may reflect canal signal contributions that cannot be easily detected or distinguished from otolith signals. New experimental protocols are proposed to characterize these cells and identify their contributions to spatial motion estimation. The proposed theoretical framework makes an essential first link between the computations for inertial acceleration detection derived from the physical laws of motion and the neural response properties predicted in a physiologically realistic network implementation.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX GRANTS ACKNOWLEDGMENTS REFERENCES

 
The vestibular system plays a critical role in spatial perception and motor compensation for head movement. As in any inertial guidance system, vestibular information about the angular and linear components of head motion is provided by distinct types of sensors. The semicircular canals act as integrating angular accelerometers to provide the brain with an estimate of rotational head velocity (Fernandez and Goldberg 1971), whereas the otolith organs behave as linear accelerometers that sense net gravito-inertial force (Angelaki and Dickman 2000; Fernandez and Goldberg 1976a,1976b). Because everyday activities such as locomotion give rise to both translations and rotations within a gravitational field, integration of both sets of vestibular signals is vital for motion detection.

Among the most important and well-studied functions of the vestibular system is its contribution to gaze stabilization. Vestibular stimulation elicits short-latency compensatory ocular responses to head motion known as the vestibuloocular reflexes (VORs) that ensure the ability to maintain ocular stability and thus high visual acuity while moving. Early investigations of the VOR focused mainly on the sensorimotor transformations associated with rotational motion [rotational VOR (RVOR)]. More recently, compensatory responses to translation [translational VOR (TVOR)] have been investigated (Angelaki 1998; Busettini et al. 1994; Paige and Tomko 1991a,1991b; Paige et al. 1998; Schwarz and Miles 1991; Schwarz et al. 1989; Telford et al. 1997).

Linear acceleration information is provided to the brain by primary otolith afferents. However, linear accelerometers (including the otoliths) respond similarly to inertial and gravitational accelerations (Einstein's equivalence principle; Einstein 1908). Thus, otolith afferents provide inherently ambiguous sensory information, given that the encoded acceleration could have been generated during either actual translation or a head reorientation relative to gravity (Angelaki and Dickman 2000; Fernandez and Goldberg 1976a,1976b). Yet, behavioral responses to tilts and translation are different. In the oculomotor system, for example, lateral translation elicits horizontal eye movements (Angelaki 1998; Paige and Tomko 1991a; Schwarz et al. 1991; Telford et al. 1997), whereas roll tilt generates mainly ocular torsion (Angelaki and Hess, 1996; Crawford and Vilis 1991; Haslwanter et al. 1992; Seidman et al. 1995). The integration of available sensory information to ensure that otolith signals are correctly processed to generate appropriate perceptual or motor responses thus represents an essential task for the central nervous system.

It has long been proposed that the brain integrates information from both otolith and semicircular canal afferents to differentiate translation from tilt (Guedry 1974; Mayne 1974; Young 1984). Theoretically, the canals should then also contribute to driving the TVOR (Glasauer and Merfeld 1997; Merfeld 1995; Merfeld and Zupan 2002; Mergner and Glasauer 1999; Zupan et al. 2002). This has been confirmed experimentally by examining oculomotor responses to simultaneous roll tilt and translation stimuli, carefully matched to ensure that the gravitational and translational components of acceleration along the interaural head axis cancelled one another out. Despite the absence of a net lateral acceleration stimulus to the otoliths, horizontal ocular responses appropriately directed to compensate for the translational component of motion were nevertheless elicited (Angelaki et al. 1999; Green and Angelaki 2003). The contribution of semicircular canal cues to the generation of these horizontal eye movements was directly demonstrated by the fact that they were no longer evoked in canal-plugged animals (Angelaki et al. 1999). Recently, it has been shown that these canal-driven responses represent an extra-otolithic TVOR that exhibits dynamic properties and a dependency on viewing distance similar to those of the purely otolith-driven reflex (Green and Angelaki 2003). Quantitative analyses demonstrated that the horizontal eye velocity profile associated with this extra-otolith driven TVOR was best correlated with angular head position, suggesting that angular velocity signals from the semicircular canals are processed by an additional neural integrator in the TVOR pathways. It was proposed that the integrative network known as the "velocity storage integrator" might perform this function (Green and Angelaki 2003).

These experimental results are consistent with the predictions of several theoretical studies that propose that the brain explicitly constructs internal estimates of gravity and translational acceleration (Glasauer et al. 1997; Merfeld 1995; Merfeld and Zupan 2002; Merfeld et al. 1993b; Mergner and Glasauer 1999; Zupan et al. 2002). To do so, the brain must effectively solve a vector differential equation that relies on an estimate of head velocity to calculate the rate of change of gravitational acceleration in a head-fixed reference frame. The solution of any differential equation requires the process of temporal integration. Thus, calculation of the instantaneous gravity vector implies a central neural integration of head angular velocity information, in agreement with experimental observations (Green and Angelaki 2003). This gravity estimate can then be combined with the net gravito-inertial acceleration sensed by the otoliths to extract the translational component of acceleration. Although such models have provided computationally rigorous solutions to the problem that are consistent with many experimentally observed behaviors, they are difficult to relate directly to the response properties of individual neurons. Specifically, these models use 3-component vector representations to perform the calculations required to compute head orientation in 3-dimensions (e.g., vector cross-products), whereas the instantaneous firing rate of an individual neuron is a scalar quantity. Thus, although significant progress has been made in outlining the computational requirements for resolving the tilt–translation ambiguity problem, few predictions have been made regarding the types of neural responses expected from a network that effectively implements these calculations. Specifically, how could cells involved in these nonlinear vector computations be identified and what types of experimental and analytical approaches should be used to interpret their responses?

The goals of the current investigation were twofold: 1) to illustrate how an integrative network within the traditional VOR circuitry (Green and Angelaki 2003) could implement these abstract vector computations to distinguish head translations from reorientations relative to gravity; 2) to investigate the predictions of such a structure at the neural level, with the goal of elucidating experimental paradigms appropriate for identifying and characterizing the physiological response properties of neurons involved in inertial motion detection. Preliminary versions of these results have been presented in abstract form (Green and Angelaki 2002; Green et al. 2002).

	MODEL DEVELOPMENT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX GRANTS ACKNOWLEDGMENTS REFERENCES

 
General framework

The general framework for the current theoretical investigation was previously proposed (Green and Angelaki 2003) and is summarized in the schematic of Fig. 1, which illustrates a feedforward model for the VOR. The classical model for sensorimotor transformations in the RVOR (Robinson 1981; Skavenski and Robinson 1973) consists of a parallel set of pathways (labeled as RVOR in Fig. 1) that convey angular head velocity signals (), sensed by the semicircular canals, C(s), to extraocular motoneurons (Mn) both "directly" (i.e., via the short latency 3–neuron–arc VOR pathways) and "indirectly" via the oculomotor neural integrator NI₂ (e.g., Cannon and Robinson 1987). The bottom projection (labeled as TVOR in Fig. 1) represents a proposal for the dynamic processing of linear acceleration signals () sensed by the otoliths, O(s), during translation (Angelaki et al. 2001a; Green and Galiana 1998; Musallam and Tomlinson 1999). A second neural integrator, NI₁, accounts for the recently established contribution of integrated semicircular canal signals to the TVOR (Angelaki et al. 1999; Green and Angelaki 2003). This integrative network, which could be the so-called velocity storage integrator (Raphan et al. 1977, 1979), computes a dynamic estimate of head orientation relative to gravity that, when combined with otolith sensory signals, could be used to extract the component of linear acceleration due to translation (i.e., on cell VO_T) (Green and Angelaki 2003).

View larger version (15K): [in this window] [in a new window]   FIG. 1. Overview of sensorimotor processing in the vestibuloocular reflexes. Schematic illustrates a classical feedforward diagram for the rotational vestibuloocular reflex (RVOR) (Skavenski and Robinson 1973) extended to incorporate the contributions of both otolith and semicircular canal signals to the translational VOR (TVOR) (Green and Angelaki 2003). Inputs to the model are angular head velocity , sensed by the semicircular canals C(s), and linear acceleration , sensed by the otolith organs O(s). Output of the model is eye position . Boxes in the schematic are dynamic elements that represent either a sensor [C(s) and O(s)], the motor plant [P(s)], or a neural filtering process [NI1(s), NI2(s)]. Two central filtering processes include the "oculomotor integrator" [NI2(s)] and an integrative network for canal signals [NI1(s)] that could be the "velocity storage integrator" (Raphan et al. 1979). Circles are summing junctions used to represent particular cell populations including vestibular-only (VO) cells, VOR and VOT, that mediate sensory signal flow in the RVOR and TVOR pathways, respectively, premotor eye-movement–sensitive (EM) neurons and motoneurons (Mn). Circle labeled with an X in the schematic indicates a proposed head-orientation–dependent modulation in the coupling between semicircular canal sensory signals and the integrative network NI1.

Mathematics of tilt–translation discrimination

Several theoretical studies have proposed models for tilt–translation discrimination that combine otolith and canal sensory information in 3 dimensions (3D) to extract the component of linear acceleration that is due to translation from that due to head reorientation relative to gravity (Angelaki et al. 1999; Glasauer and Merfeld 1997; Merfeld 1995; Merfeld and Zupan 2002; Mergner and Glasauer 1999; Zupan et al. 2002). All are based on the premise that canal information about angular velocity is used to keep track of changes in orientation of the gravity vector relative to the head (in which the vestibular sensors are fixed), as described by the first-order differential equation (Goldstein 1980)

(1)

where and are vector representions of gravity and angular velocity, respectively, in head-fixed coordinates and x denotes a vector cross-product. Using the additional information that the net acceleration sensed by the otoliths is the vectorial difference of translational () and gravitational () components

(2)

the translational acceleration component can easily be computed by solving the vector differential equation resulting from the combination of Eq. 1 and 2 (Angelaki et al. 1999; Hess and Angelaki 1997; Viéville and Faugeras 1990)

(3)

Equivalently, the same basic equations have been used to postulate that the otolith measurement of gravito-inertial force is explicitly resolved into central estimates of gravitational and translational accelerations (e.g., Glasauer and Merfeld 1997; Merfeld and Zupan 2002). Specifically, an internal 3D estimate of the gravity vector in head coordinates is obtained from Eq. 1 (i.e., = – x, assuming a known set of initial conditions). The translational acceleration component can be subsequently obtained from Eq. 2. These implementations are mathematically equivalent (Angelaki et al. 2001b). Both involve a central (nonlinear) neural integration of angular velocity, a theoretical prediction that is consistent with recent experimental observations (Green and Angelaki 2003).

Although vector Eqs 1 and 2 can be used to discriminate tilt from translation in 3D, a key focus of the current study is to predict the responses of central neurons whose firing rates are scalar quantities. Thus, to simplify the problem we have chosen to examine tilt–translation discrimination along only the interaural head axis. In particular, we can expand the vector cross-product in Eq. 1 into components as

(4)

where , , and are unit vectors in a right-handed coordinate system along the x [nasooccipital (NO)], y [interaural (IA)], and z [dorsoventral (DV)] axes, respectively. Integration of each vector component in Eq. 4 yields the gravitational acceleration along the x, y, and z axes, as illustrated in Fig. 2. Because we restrict consideration to tilt–translation discrimination along the interaural axis (i.e., y-axis associated with unit vector ) we focus on the calculation of g_y (shaded region in Fig. 2)

(5)

According to Eq. 5, calculation of the gravity component, along the interaural axis (y-axis) requires estimates of the components of gravitational acceleration along the other 2 axes, illustrating the 3D nature of the problem (i.e., coupled integrative networks in Fig. 2). However, the solution can be simplified if we decouple computation of g_y from its dependency on g_x and g_z by limiting consideration to conditions where x- and z-axis translations are small (i.e., f_x = f_z 0). Under these conditions, where g_x –_x and g_z –_z, g_y can be approximated as

(6)

The preceding equation indicates that, for the restricted conditions we consider here, the component of gravity along the interaural head axis can be computed by integrating estimates of yaw and roll head velocities that have been premultiplied by the net instantaneous accelerations sensed by otolith afferents along the nasooccipital and dorsoventral axes, respectively.

View larger version (33K): [in this window] [in a new window]   FIG. 2. Schematic illustration of the computations necessary to calculate the gravitational component of acceleration in a head-fixed reference frame as the head reorients relative to gravity (i.e., = –x; also see Eq. 4). Computations involve multiplications between the vector components (i.e., x-, y-, and z-axis components) of angular velocity signals (solid lines) and gravitational accelerations (dashed lines), followed by integration. Notice that calculation of the gravitational acceleration along any particular axis depends on the components of gravitational acceleration along the 2 other axes, illustrating the requirement for coupled integrative networks in 3D. Here we consider tilt–translation discrimination along only the interaural axis (i.e., y-axis) and thus focus on the computation of gy (shaded region). To simplify the problem we decouple computation of gy from its dependency on gx and gz by assuming that x- and z-axis translational accelerations are small (i.e., fx = fz 0) such that gx –x and gz –z (see MODEL DEVELOPMENT).

General description.    Figure 3A illustrates one of many possible integrative networks (representing NI₁ expanded from Fig. 1) that could perform the computations of Eq. 2 and 6. Circles in the schematic represent summing junctions that are used to represent different vestibular-only (VO; i.e., eye-movement–insensitive) cell populations, whereas boxes represent dynamic elements or filters. These include first-order dynamic approximations of the semicircular canals, C(s) = T_cs/(T_cs + 1), (Fernandez and Goldberg 1971) and the otolith organs, O(s) = 1/(T_os + 1) (Fernandez and Goldberg 1976b) as well as the neural filter, C_LP(s), which represents a low-pass internal model of the semicircular canals [C_LP(s) = 1/(T_cs + 1)].

View larger version (29K): [in this window] [in a new window]   FIG. 3. Proposed model for detecting head orientation and translation. A: implementation of the integrative network NI1 (illustrated in Fig. 1). Circles labeled VO1 through VO5 represent different populations of VO cells. Box labeled CLP(s) represents a low-pass internal model of the semicircular canals [CLP(s) = 1/(Tcs + 1)]. Inputs to the model include yaw and roll velocities, z and x, sensed by the horizontal and vertical canals [C(s) = Tcs/(Tcs + 1)], respectively, and interaural acceleration y, sensed by the otolith organs [O(s) = 1/(Tos + 1)]. X's indicate a multiplicative modulation in the strengths of the projections from cells VO1 and VO2 onto cell VO4 by nasooccipital and dorsoventral accelerations, x and z, respectively. Note that these multiplications implement the head-orientation–dependent modulation illustrated in Fig. 1 at the input to NI1. Here, this nonlinear signal interaction is incorporated as part of the network. Parameters associated with different pathways represent the strength or weight of the projection. B: previously proposed implementation of the premotor network for the RVOR and TVOR consisting of a feedback loop around an internal model [F(s) = Kf/(Tfs + 1)] of the eye plant [P(s) = Kp/(Tps + 1)] (Angelaki et al. 2001a; Green and Galiana 1998). Semicircular canal signals project directly onto eye-contra–sensitive premotor vestibular neurons (EMC) that represent the key interneurons in the shortest-latency RVOR pathways. Eye-ipsi–sensitive premotor neurons (EMI) are proposed to mediate the shortest-latency TVOR pathways (Angelaki et al. 2001a) and receive projections from cell VO3 in A. Separate but structurally identical premotor networks were assumed to simulate horizontal and torsional eye movements. Model parameters are: pdh = 1, ph = 2.568, pd = 0.56, p = 2.568, q1 = 0.25, q2 = 0.22, q3 = 0.1975, K1o = 0, K2o = 0, K1a = 1, K2a = 1, K1b = 0.1, K2b = 0.1, K3 = 0.061, K3b = 1.6, K3c = 6.25, a = 0.19, b = 0.75, d1 = 0.21, d2 = 1.1, Tc = 6, To = 0.0159, Kf = 2.81, Kp = 1, Tf = 0.25, Tp = 0.25. Parameters qoh = 0.39, eh = 0.03, and qot = 0.04, et = 0.15 are qo and e values for the horizontal and torsional premotor systems, respectively. Inputs z and x are in units of deg/s, y is in units of cm/s2, and z and x are in units of g.

We focus here on an integrative network of VO neurons for two reasons: 1) populations of VO neurons have recently been observed that code mainly for either translation (vestibular and fastigial nuclei; Angelaki et al. 2003) or tilt (vestibular nuclei; Zhou et al. 2000); 2) the ability to distinguish head tilts and translations is important for both perceptual and motor responses. Both observations suggest that a network for distinguishing tilt versus translation occurs upstream of the premotor oculomotor networks of eye-movement–sensitive neurons (i.e., network in Fig 3B). In keeping with a previous proposal that the required integrative network could represent that known as the velocity storage integrator (Green and Angelaki 2003), we assume a model structure that is based on a feedback implementation of this integrative network originally proposed by Robinson (1977). Accordingly, neurons that receive direct vestibular sensory projections (i.e., cells VO1, VO2, and VO3) are each interconnected in a feedback loop with an assumed common internal low-pass canal model, C_LP(s), to form a distributed integrative neural network. Potentially, many other model structures could be used to implement the requirements for tilt–translation discrimination described by Eq. 2 and 6. All such networks, however, will be common in the requirements for 1) performing a central neural integration of canal signals and 2) implementing a head-orientation–dependent coupling between canal and otolith-derived sensory information. The key arguments to be made in this study focus on the implications of these common requirements and therefore are largely independent of the particular model structure. In the example network proposed here, the required integration is a distributed network property implemented by positive feedback loops through the low-pass filter, C_LP(s), whereas the head-orientation–dependent sensory coupling is implemented by the multiplicative interactions, denoted by X's, in Fig. 3A.

The premotor oculomotor network used to simulate compensatory VOR responses is illustrated in Fig. 3B. It represents a feedback implementation of the "neural integration and eye plant compensation network" of Fig. 1 and was previously described in detail (Angelaki et al. 2001a; Green 2000; Green and Galiana 1998). For simplicity, we assume separate, but structurally identical, premotor networks for driving horizontal and torsional eye movements.

Dynamic processing of sensory signals.    The goal in this section is to illustrate the relationship between the dynamic computations performed by the model in Fig. 3A and the necessary computations for tilt–translation discrimination presented above. In particular, we will demonstrate that the network can perform the computations described by Eq. 6 to construct an internal estimate of dynamic head orientation relative to gravity on cell VO5. For descriptive purposes we will express this cell's response in the Laplace domain. Note, however, that because canal-related projection weights onto cell VO4 (i.e., projections from cells VO1 and VO2) modulate as a function of _x(t) or _z(t), the system is nonlinear. Laplace domain descriptions thus cannot generally be used here to predict response trajectories over time. However, they can approximate the dynamic characteristics of the system for small movements about an average static head orientation (i.e., a given operating point) when the system exhibits close to linear performance. In particular, for small head movements around a given pitch orientation we can assume static approximations to the linear accelerations along the NO and DV axes (i.e., _x –sin and _z cos in units of g) where angle describes the pitch angle from upright. At mid-high frequencies (>0.1 Hz), where semicircular canal cues were previously confirmed to play a crucial role in resolving ambiguous otolith sensory information (Angelaki et al. 1999), the response of cell VO5 can then be approximated as

(7)

where G_VOH5 and G_VOV5 are static gain terms (G_VOH5 = p_hK_1aK_3b/T_c and G_VOV5 = pK_2aK_3b/T_c). Equation 7 represents a high-frequency, small-angle approximation to the general expression for cell VO5 that assumes model parameters chosen to ensure close to ideal tilt–translation discrimination (see APPENDIX).

Because 1/s is the Laplace domain description of an integrator, Eq. 7 illustrates that the network integrates angular velocity signals, _z and _x, that have been multiplied by _x or _z, as required to construct an internal estimate of the gravitational acceleration component along the interaural axis (i.e., compare Eq. 7 with Eq. 6). Given an internal (scaled) estimate of g_y on cell VO5, Eq. 2 predicts that the translational component of the acceleration, f_y, can be obtained by combining this estimate with the net interaural acceleration signal, _y. In the model this occurs on cell VO3 [i.e., VO3(s) = q_1y(s) + K_3cVO5(s) q₁[_y(s) + g_y(s)] q₁f_y(s)]. Note that, although the analytical expressions presented in this section are valid only for small dynamic head reorientations relative to gravity, the model can simulate appropriate responses even for large tilts in all pitch head orientations, assuming that any concurrent translational acceleration is mainly directed along the y-axis of the head (i.e., f_x, f_z 0; see above). Further details of the Laplace domain descriptions of cell and motor responses are provided in the APPENDIX.

Model simulations

The proposed model was implemented using the MATLAB simulation toolbox SIMULINK (MathWorks, Natick, MA). All simulations were performed using a fixed-step Runge–Kutta numerical integration routine (ode4 in SIMULINK) with time steps fixed at 0.01 s. The model parameters provided in the caption of Fig. 3 were chosen to satisfy the criteria outlined in the APPENDIX.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX GRANTS ACKNOWLEDGMENTS REFERENCES

 
The basic predictions of the model in Fig. 3 will be explored during pure rotations and translations in different head orientations as well as for combinations of rotational and translation motion stimuli that have recently been used to investigate tilt and translation responses. To begin, we will illustrate the ability of the model to generate appropriate behavioral responses. We will then use the model to predict the neural response properties expected from the cell populations involved in estimating spatial orientation and inertial motion.

Behavioral responses

Frequency response predictions.    Figure 4 illustrates predicted horizontal and torsional ocular responses during yaw rotation, interaural translation, and small angle roll rotation (e.g., <30°) from upright orientation. Both yaw rotation and interaural translation are predicted to elicit large horizontal eye movements (Fig. 4, A and C; solid black curves), whereas head roll generates torsional ocular responses (Fig. 4B, solid gray curves), as required for gaze stabilization. Small torsional responses are also elicited in response to head translation, as observed experimentally (Fig. 4C, gray traces; Angelaki 1998; Paige and Tomko 1991a).

View larger version (28K): [in this window] [in a new window]   FIG. 4. Behavioral predictions of the model. Horizontal (black) and torsional (gray) ocular gains and phases are plotted as a function of frequency during (A) yaw rotation, (B) roll rotation, and (C) interaural translation in the upright orientation. Horizontal eye movement sensitivities to the interaural acceleration sensed by the otoliths during translation (black) and roll tilt (gray) are compared in D (in similar units of deg·s–1·g–1). Model predictions in B and C are superimposed on experimentally observed horizontal (black circles) and torsional (gray triangles) ocular responses in rhesus monkeys (data in B: ±22°, 0.1–0.5 Hz, and ±5° at 1-Hz peak roll amplitudes; Angelaki et al. 1999; data in C: 0.1 g at 0.16, 0.2 Hz, and 0.4 g, 0.3- to 5-Hz peak translational accelerations; Angelaki 1998). Illustrated frequency response plots represent analytical predictions that are equivalent to the simulated model behavior for all amplitudes of head movement in A and C. Theoretically predicted responses in B and D (solid black and gray curves) represent linearized approximations valid for small head reorientations relative to gravity (e.g., <30° where static approximations to x(t) and z(t) can be assumed; see MODEL DEVELOPMENT and APPENDIX). Gray squares and dash-dot curves in B describe response gains and phases obtained by model simulation for large head roll angles (±90°). Dashed black trace in B represents the horizontal gain predicted if the canals were perfect transducers of head velocity at all frequencies.

The deterioration in model performance with decreasing frequency occurs because the semicircular canals cease to provide perfect estimates of head velocity. In fact, when the canals are assumed to be perfect transducers of head velocity [i.e., canal transfer function C(s) = 1] negligible horizontal responses to tilt are predicted across all frequencies (Fig. 4B, black dashed lines). In the absence of an accurate canal estimate of head velocity in the real physiological system, other strategies for distinguishing tilt from translation are required to achieve appropriate behavior at low frequencies (e.g., Mergner and Glasauer 1999; Paige and Tomko 1991a; Telford et al. 1997). In the following, we will focus on simulations of cell responses at frequencies above 0.1 Hz, where the semicircular canals provide reliable estimates of head velocity and the network appropriately discriminates tilts and translations.

Simulated behavioral responses to tilt–translation combinations.    Novel combinations of translational and roll tilt movements have recently been used to investigate semicircular canal and otolith contributions to oculomotor responses (Angelaki et al. 1999; Green and Angelaki 2003). Similar stimulus combinations were used to simulate the performance of the model. Four protocols are illustrated at the top of Fig. 5A that consist of either lateral translation (Translation only), roll tilt (Roll tilt only), or combined lateral translation and roll tilt motion stimuli (Roll tilt + Translation motion and Roll tilt – Translation motion). Because the interaural acceleration (_y) stimulus to the otoliths was matched for Translation and Roll tilt motions (each set to a peak of 0.2 g at 0.5 Hz; Fig. 5A, bottom row), combined motion stimuli result either in a doubling of the interaural acceleration stimulus (Roll tilt + Translation) or zero acceleration (Roll tilt – Translation), depending on the relative directions of the two stimuli.

View larger version (40K): [in this window] [in a new window]   FIG. 5. Simulated behavioral responses. A: horizontal and torsional eye velocity responses (hor, solid black; tor, dashed gray) to 4 tilt–translation stimulus paradigms at 0.5 Hz (i.e., Translation only, Roll tilt only, Roll tilt + Translation, and Roll tilt – Translation; Angelaki et al. 1999; Green and Angelaki 2003). B: ocular responses during Supine Roll, Supine Translation, Supine Yaw, and Upright Yaw motions. Bottom 2 traces in A and B illustrate the stimuli: roll and horizontal angular velocities (x, solid black; z, dashed gray lines); net and translational components of the interaural acceleration (y, solid black; fy, dashed gray lines).

The model can also predict appropriate eye movement responses with the head in supine orientation (Fig. 5B). For example, roll rotation and lateral translation elicit torsional and horizontal responses, respectively (Fig. 5B, columns 1 and 2). More interesting is the case of supine yaw rotation that dynamically stimulates the otoliths along the interaural head axis (Fig. 5B, column 3). This stimulus condition presents a similar ambiguity problem to the case of roll tilt from upright. Specifically, because the otoliths are dynamically stimulated during supine yaw rotation, horizontal eye velocity could potentially reflect a combination of TVOR and RVOR response components. If otolith signals were not appropriately interpreted by the brain, supine yaw rotation would elicit significantly smaller horizontal responses than upright yaw rotation, whereas larger horizontal eye velocities would be predicted in the prone orientation. However, model simulations result in identical horizontal eye movements during both upright and supine yaw rotations (Fig. 5B; compare columns 3 and 4). Thus, just as in the case of roll tilt from upright, interaural accelerations attributed to a head reorientation relative to gravity during supine yaw rotation are appropriately distinguished from translation.

Neural response predictions and simulations

Given a model that predicts appropriate behavioral responses, we may now address the predicted response properties of different average neural populations within such a network. First, we will illustrate the frequency response predictions for each cell type during pure roll tilts and translations, as well as their responses to earth-vertical–axis rotations in different pitch head orientations when the canals are stimulated in isolation. We will show that in the integrative network proposed here, traditional interpretations of the responses to these stimuli embed assumptions that can lead to incorrect conclusions with respect to the signals encoded by central neurons. The goals of this section will be to illustrate this point and then to consider experimental protocols appropriate to reveal the underlying properties of the network.

Frequency response and earth-vertical–axis rotation predictions.    Neurons VO1 and VO2 exhibit the expected characteristics of semicircular-canal–sensitive cells, coding for head rotation in head coordinates (Fig. 6). In both upright and supine orientations, cell VO1 modulates in phase with angular yaw velocity but does not respond during roll rotation (Fig. 6, B and C, black traces), whereas cell VO2 responds exclusively to roll rotations (Fig. 6, B and C, gray traces). Neither cell group modulates during a pure translational stimulus (Fig. 6A). The canal afferent-like behavior of cells VO1 and VO2 also holds for earth-vertical–axis rotations at different static pitch orientations (Fig. 6, D and E). Cell VO1 responds maximally during earth-vertical–axis rotation in the upright orientation (i.e., during yaw rotation for a pitch angle = 0°), exhibiting a sensitivity that drops off with the cosine of pitch angle from upright, as predicted for a dominantly horizontal canal-sensitive cell. Similarly, cell VO2 exhibits no modulation during earth-vertical–axis rotation with the head upright and maximal responses in the prone and supine positions (i.e., pitch angles of ±90°) consistent with this cell receiving mainly vertical semicircular canal inputs (Fig. 6E). Accordingly, the properties of VO1 and VO2 are consistent with cells described as "canal-only" neurons (Dickman and Angelaki 2002).

View larger version (32K): [in this window] [in a new window]   FIG. 6. Predicted responses for cell populations VO1 and VO2. A–C: frequency response predictions for cells VO1 (black) and VO2 (gray) during (A) translation, (B) roll rotation, and (C) yaw rotation in upright (solid curves) and supine (dashed curves) orientations. Cell response sensitivities and phases are expressed relative to head velocity. D–E: predicted responses for cells (D) VO1 and (E) VO2 during earth-vertical–axis (EVA) rotations at 0.5 Hz in different static pitch orientations (0°, upright; 90°, prone; –90°, supine).

View larger version (34K): [in this window] [in a new window]   FIG. 7. Predicted responses for cell populations VO3 and VO4. A–C: frequency response predictions for cells VO3 (black) and VO4 (gray) during (A) translation, (B) roll rotation, and (C) yaw rotation in upright (solid curves) and supine (dashed curves) orientations. Cell response sensitivities and phases are expressed relative to stimulus velocity. D–E: predicted responses for cells (D) VO3 and (E) VO4 during EVA rotations at 0.5 Hz in different static pitch orientations (0°, upright; 90°, prone; –90°, supine).

View larger version (27K): [in this window] [in a new window]   FIG. 8. Predicted responses for cell population VO5. A–C: frequency response predictions during (A) translation, (B) roll rotation, and (C) yaw rotation in upright (solid curves) and supine (dashed curves) head orientations. Cell response sensitivities are expressed relative to translational velocity (A) or angular head position (B–C).

This observation has in fact been made for many central translation-sensitive vestibular neurons (Dickman and Angelaki 2002) and was used to suggest that the activities of such neurons might reflect intermediate processing stages in distinguishing between head tilt and translation. Yet the question of how this distinction is made remained unanswered. The model proposed here provides an explanation. Specifically, the responses of cells VO3, VO4, and VO5 differ from those of sensory otolith signals because their activities do indeed reflect the contribution of semicircular canal inputs (Fig. 3A). However, these cells do not code for rotation in the head-fixed reference frame of the canal sensors. Because of the multiplicative canal–otolith interactions at the input to cell VO4, the activities of these cells instead reflect spatially referenced canal signal contributions aligned with the earth-horizontal axis (e.g., vertical canal signals during tilts from upright and horizontal signals during tilts from supine head orientations). They thus encode only the component of rotation orthogonal to gravity that is not observed during earth-vertical–axis rotations. Such a postulated canal-derived signal would nevertheless be difficult to detect because it contributes only under conditions that typically simultaneously stimulate the otoliths. In the following section, we will address how the proposed "hidden" semicircular canal signal contribution to the responses of these neurons can be isolated.

Simulated responses to combined tilt–translation stimuli.    To isolate the contribution of semicircular canal signals to the neural activities of cells VO3, VO4, and VO5 we may investigate the simulated responses of the model neurons for the same set of stimulus combinations employed to examine behavioral responses (Fig. 9). As expected based on the analytical predictions in Figures 6 through 8, cells VO1 and VO2 code for horizontal and roll head velocity, respectively, regardless of head orientation (Fig. 9, A and B). In contrast, VO3 exhibits negligible responses to roll and yaw rotations, in both upright and supine head orientations (e.g., Fig. 9C, columns 2 and 5) but encodes the translational component of acceleration (Fig. 9C, similar responses in columns 1, 3, and 4). Cell VO5, on the other hand, codes specifically for head tilt relative to gravity, responding during either upright roll (Fig. 9E, columns 2, 3, and 4) or supine yaw (Fig. 9E, column 5) but not during pure translation. Finally, cell VO4 exhibits a more complex behavior, being sensitive to both translation and head reorientation relative to gravity.

View larger version (41K): [in this window] [in a new window]   FIG. 9. Simulations of model cell responses. A–E: firing rate modulation of cells VO1 through VO5 during the 4 tilt–translation protocols in the upright orientation (columns 1–4; see Fig. 5 legend and text) and for supine yaw rotation (column 5). F: angular roll and yaw velocities (x, solid black; z, dashed gray lines). G: net and translational interaural accelerations (y, solid black; fy, dashed gray lines).

Variability in individual cell responses

Using a particular model structure and parameter set chosen to achieve ideal tilt–translation discrimination we have illustrated the response properties of several average cell populations within this network. However, in contrast to these idealized model cells, the majority of experimentally observed translation-sensitive neurons do in fact exhibit responses to earth-vertical–axis rotations that reflect the sensory contribution of signals from multiple orthogonal canals (Dickman and Angelaki 2002). Furthermore, relatively few cells have been isolated that respond exclusively to tilt or translation (Angelaki et al. 2003; Zhou et al. 2000), suggesting that these variables may be encoded mainly as population averages (i.e., as represented by our average model neurons).

In the following sections we will explore the effects of varying particular parametric assumptions in the model with two key goals: 1) to examine a more realistic representation of the properties of individual neurons that contribute to the average population responses; 2) to further explore the implications of the most fundamental properties of the proposed model that must be shared by any neural network that effectively implements Eq. 4. These include its function as a neural integrator and the requirement for a coordinate transformation of canal signal contributions. To illustrate these issues we will consider how changes in the coupling of semicircular canal and otolith signals onto the network impact on the expected properties of individual neurons and their compatibility with experimental observations.

Variable semicircular canal-related projection weights.    To achieve close to ideal tilt–translation discrimination in the proposed model we assumed that the net projections from each canal onto cell VO4 (i.e., indirect projections from cells VO1 and VO2) were equal in strength and entirely head-orientation–dependent (i.e., K_1a = K_2a and K_1o = K_2o = 0 in Fig. 10A and Fig. 3A; see Fig. 3 legend). Average cell populations VO3, VO4, and VO5 were then predicted to exhibit no response during earth-vertical–axis rotations (e.g., solid black trace with zero gain in Fig. 10B; see also Fig. 7, D and E). Using the VO4 cell population as an example, we will now examine the effect of relaxing these parametric assumptions to explore the expected range of responses from individual neurons within this population.

View larger version (22K): [in this window] [in a new window]   FIG. 10. Predicted EVA rotation responses for cell population VO4, allowing for potential variability in individual cell characteristics. A: schematic of a cell (part of the VO4 population, Fig. 3A) that receives both head-orientation–dependent and head-orientation–independent horizontal and vertical canal inputs (weights K1a, K2a and K1o, K2o, respectively). Head-orientation–dependent inputs are shown to be multiplied by nasoocciptal and dorsoventral accelerations, x and z. B: cell VO4 exhibits no response during EVA rotations (solid black line, zero gain) when K1o = K2o = 0 and K1a = K2a. When K1o = K2o = 0.2, the cell exhibits a response consistent with equal sensitivities to horizontal and vertical canal stimuli (solid gray line). For comparison, the response predicted if all inputs to the cell were instead head-orientation–invariant (e.g., K1a = K2a = 0, K1o = 1.2, K2o = 1.2) is illustrated with the dotted line. C: responses predicted when horizontal and vertical canal head-orientation–dependent projections have unequal strengths (solid black line: K1a = 2, K2a = 1 or K1a = 1, K2a = 2 and K1o = K2o = 0; solid gray line: K1a = 3, K2a = 1 or K1a = 1, K2a = 3 and K1o = K2o = 0). For comparison, the response predicted from a neuron with a head-orientation–independent sensitivity only to vertical canal input (K2o = 1, K1o = K1a = K2a = 0) is illustrated with the dotted line. D: response patterns observed for different combinations of head-orientation–independent and -dependent projection strengths. Solid black: K1a = 2, K2a = 1, K1o = K2o = 0; solid gray: K1a = 2, K2a = 1, K1o = 0.4, K2o = 0; dashed black: K1a = 1, K2a = 2, K1o = 0.2, K2o = 0; dashed gray: K1a = 1, K2a = 2, K1o = 0, K2o = –1; dotted gray: K1a = 2, K2a = 1, K1o = 2, K2o = 0. Phases (not shown) are always either 0° or 180°.

More fundamental to the arguments here are the predictions made when the assumption of equal orientation-dependent horizontal and vertical canal-related projections is relaxed (i.e., K_1a K_2a). In this case, the predicted responses no longer reflect simple cosine tuning patterns, but rather exhibit second harmonic spatial tuning properties (e.g., solid black and gray curves in Fig. 10C) indicative of head-orientation–dependent canal signal contributions (see APPENDIX for details). Notably, such spatial tuning might not be apparent for small pitch angles, given that these curves could appear similar to those of a neuron that receives exclusively orientation-independent vertical canal signals (Fig. 10C, compare solid black and dotted traces). Examination of cell response properties over a large range of static tilt angles (e.g., ±90°) is therefore likely to be necessary to reveal the presence of head-orientation–dependent rotational sensitivities.

More generally, individual neurons are likely to receive different combinations of orientation-dependent and -independent canal projections, giving rise to a range of response patterns during earth-vertical–axis rotations that reflect different degrees of convergence from multiple orthogonal canal sensors, as observed experimentally (Fig. 10D; Dickman and Angelaki 2002; Siebold et al. 2001). When examined over a larger range of head orientations than those typically used (30°; e.g., Dickman and Angelaki 2002; Siebold et al. 2001), however, their responses are expected to differ considerably from the simple cosine-tuned behavior that has traditionally been assumed for vestibular neurons. Such response patterns do not simply reflect the particular model structure chosen here but would be expected in any network in which head-orientation–dependent canal- and otolith-derived signals converge to distinguish tilts and translations according to the requirements implied by Eq. 2 and 4. Hence, although these complex tuning properties have yet to be observed experimentally, they represent a fundamental model prediction that remains to be tested.

Variable otolith signal projection weights.    Otolith signals couple onto the model network not only through cell VO3, but also through sensory projections directly onto cell population VO4 (projection with weight q₂) and into the neural filter C_LP(s) (projection with weight q₃). The weights of these parameters were chosen both to set particular cell sensitivities to translation and to ensure that cell VO3 exhibits high-pass filtered responses to otolith stimulation with a minimal response to static head tilt (see APPENDIX). With the chosen parameter set the average neural populations responsive to translation (i.e., cells VO3 and VO4) were predicted to modulate closely in phase with head acceleration (Fig. 7). However, a consistent, yet so far unexplained, experimental observation is that eye-movement–insensitive vestibular neurons exhibit dynamic responses to translation that are highly variable and often quite different from those of otolith afferents (Angelaki and Dickman 2000; Chen-Huang and Peterson 2002; Dickman and Angelaki, 2002; Musallam and Tomlinson 2002; Tomlinson et al. 1996). Next we will illustrate that by changing a single parameter, weight q₃, individual neurons with a wide range of dynamic responses to head translation are predicted in the proposed model. Changes in this single parameter are sufficient to illustrate a large range of response gains and phases during translation without affecting the integrative properties of the network.

Figure 11 illustrates distributions of the predicted gains and phases (relative to translational acceleration) of neural populations VO3, VO4, and VO5 at 0.5 Hz when weight q₃ was randomly varied according to a Gaussian distribution about its nominal value (see Fig. 11 legend). Notice that both populations VO3 and VO4 can exhibit a broad distribution of