Dora E. Angelaki

The algebra of neural response vectors

The concept of a “polarization vector” was introduced in two papers on the coding properties of otolith neurons in order to describe the spatial coding properties of these afferents.’.’ The polarization vector was a spatial vector of unit length oriented in the direction of the stimulus which produced the maximal excitatory response in the afferent. This vector could be used to predict the response to a (constant) stimulus in an arbitrary direction by considering that only the stimulus component in the direction of the polarization vector was effective; this operation of taking a vector component can be realized by taking the scalar, or “dot,” product between the polarization vector and a vector describing the stimulus orientation. Indeed, working backwards to the hair cell, which has long been known to have a morphological polarization vector given by the asymmetrically placed kinocilium, the depolarization of the hair cell is directly proportional to the component of bending of the hair bundle in the direction of the kin~cilium.~ We will develop the concept of a neural response vector as a mathematical descriptor which will allow us to predict the response of a neuron or a reflex system to complex time-varying stimuli. We will also develop rules for combining these response vectors to enable predictions when, for example, multiple inputs are considered, or to describe responses expected from converging outputs. In the descriptions that follow, the main assumption is that we are dealing with linear systems; this allows us to describe the output to several inputs by considering the response to the inputs separately, then adding the responses. The other well-known consequence of linearity is that if we stimulate with a sinusoid, the response will be a sinusoid at the same frequency. The concept of a “spatial response vector” arises directly out of the polarizatio? vector concepts described above. For an afferent, A, we use the polarization vector A to characterize the orientation in space, and use the sensitivity (or gain) g,, the response when stimulated in this optimal orientation, as” the vector length (FIGURE 1A). The response to an arbitrary constant stimulus S can be expressed as the comeonent of the stimulus along A, which is computed by taking the dot product of A and S. If we consider the plane that contains both the A and S vectors, and express the orientations of A and S by 5 and 8, the response, as a function of stimulus orientation 8, can be written as

A model for the characterization of the spatial properties in vestibular neurons

Quantitative study of the static and dynamic response properties of some otolith-sensitive neurons has been difficult in the past partly because their responses to different linear acceleration vectors exhibited no “null” plane and a dependence of phase on stimulus orientation. The theoretical formulation of the response ellipse provides a quantitative way to estimate the spatio-temporal properties of such neurons. Its semi-major axis gives the direction of the polarization vector (i.e., direction of maximal sensitivity) and it estimates the neuronal response for stimulation along that direction. In addition, the semi-minor axis of the ellipse provides an estimate of the neurons maximal sensitivity in the “null” plane. In this paper, extracellular recordings from otolith-sensitive vestibular nuclei neurons in decerebrate rats were used to demonstrate the practical application of the method. The experimentally observed gain and phase dependence on the orientation angle of the acceleration vector in a head-horizontal plane was described and satisfactorily fit by the response ellipse model. In addition, the model satisfactorily fits neuronal responses in three-dimensions and unequivocally demonstrates that the response ellipse formulation is the general approach to describe quantitatively the spatial properties of vestibular neurons.

Spatio-temporal convergence (STC) in otolith neurons

It has been recently demonstrated that some primary otolith afferents and most otolith-related vestibular nuclei neurons encode two spatial dimensions that can be described by two vectors in temporal and spatial quadrature. These cells are called broadly-tuned neurons. They are characterized by a non-zero tuning ratio which is defined as the ratio of the minimum over the maximum sensitivity of the neuron. Broadly-tuned neurons exhibit response gains that do not vary according to the cosine of the angle between the stimulus direction and the cells maximum sensitivity vector and response phase values that depend on stimulus orientation. These responses were observed during stimulation with pure linear acceleration and can be explained by spatio-temporal convergence (STC) of primary otolith afferents and/or otolith hair cells. Simulations of STC of the inputs to primary otolith afferents and vestibular nuclei neurons have revealed interesting characteristics: First, in the case of two narrowly-tuned input signals, the largest tuning ratio is achieved when the input signals are of equal gain. The smaller the phase difference between the input vectors, the larger the orientation differences that are required to produce a certain tuning ratio. Orientation and temporal phase differences of 30–40° create tuning ratios of approximately 0.10–0.15 in target neurons. Second, in the case of multiple input signals, the larger the number of converging inputs, the smaller the tuning ratio of the target neuron. The tuning ratio depends on the number of input units, as long as there are not more than about 10. For more than 10–20 input vectors, the tuning ratio becomes almost independent of the number of inputs. Further, if the inputs comprise two populations (with different gain and phase values at a given stimulus frequency), the largest tuning ratio is obtained when the larger population has a smaller gain. These findings are discussed in the context of known anatomical and physiological characteristics of innervation patterns of primary otolith afferents and their possible convergence onto vestibular nuclei neurons.

Models of membrane resonance in pigeon semicircular canal type II hair cells

Pigeon vestibular semicircular canal type II hair cells often exhibit voltage oscillations following current steps that depolarize the cell membrane from its resting potential. Currents active around the resting membrane potential and most likely responsible for the observed resonant behavior are the Ca++-insensitive, inactivating potassium conductance IA (A-current) and delayed rectifier potassium conductance IK. Several equivalent circuits are considered as representative of the hair cell membrane behavior, sufficient to explain and quantitatively fit the observed voltage oscillations. In addition to the membrane capacitance and frequency-independent parallel conductance, a third parallel element whose admittance function is of second order is necessary to describe and accurately predict all of the experimentally obtained current and voltage responses. Even though most voltage oscillations could be fitted by an equivalent circuit in which the second order admittance term is overdamped (i.e., represents a type of current with two time constants, one of activation and the other of inactivation), the sharpest quality resonance obtained with small current steps (around 20 pA) from the resting potential could be satisfactorily fit only by an underdamped term.

The horizontal vestibulo-ocular reflex during linear acceleration in the frontal plane of the cat

SummaryHorizontal and vertical eye movements were recorded in alert, restrained cats that were subjected to whole-body rotations with the horizontal semicircular canals in the plane of rotation and the body centered on the axis or 45 cm eccentric from the axis of rotation. Changes in the horizontal vestibulo-ocular reflex (HVOR) due to the resultant of the linear forces (i.e., gravity and linear acceleration) acting on the otolith organs were examined during off-axis rotation when there was a centripetal acceleration along the animals interaural axis. The HVOR time constant was slightly shortened when the resultant otolith force was not parallel to the animals vertical axis. This effect was independent of the direction of the otolith force relative to the direction of the slow phase eye velocity. No effect on the HVOR amplitude was observed. In addition to changes in the HVOR dynamics, a significant vertical component of eye velocity was observed during stimulation of the horizontal canals when the resultant otolith force was not parallel with the animals vertical axis. The effect was greater for larger angles between the resultant otolith force and gravity. An upward or downward component was elicited, depending on the direction of the horizontal component of eye velocity and the direction of the resultant otolith force. The vertical component was always in the direction that would tend to align the eye velocity vector with the resultant otolith force and keep the eye movement in a plane that had been rotated by the angle between the resultant otolith force and gravity.

Changes in the dynamics of the vertical vestibulo-ocular reflex due to linear acceleration in the frontal plane of the cat

SummaryThe vertical and horizontal components of the vestibulo-ocular reflex (VOR) were recorded in alert, restrained cats who were placed on their sides and subjected to whole-body rotations in the horizontal plane. The head was either on the axis or 45 cm eccentric from the axis of rotation. During off-axis rotation there was a change in the linear force acting on the otolith organs due to the presence of a centripetal acceleration along the animals vertical axis. Otolith forces (defined to be opposite to the centripetal acceleration) directed ventrally with respect to the animal (negative) decreased both the amplitude and time constant of the first-order approximation to the slow phase eye velocity of the vertical vestibulo-ocular reflex (VVOR). Otolith forces directed dorsally (positive) increased the amplitude and time constant. The effects were greater for the up VOR. The asymmetry in the VVOR time constant also depended on the otolith forces, being less in the presence of negative otolith forces that caused the resultant otolith force to move ventrally, towards the direction along which gravity normally acts when the animal is in the upright position. The effects of otolith forces on the up VVOR were independent of whether the animals were tested in the dark or in the light with a stationary visual surround (i.e., during visual suppression). In contrast, the changes in the time constant of the down VVOR were smaller during visual suppression. Simulations of the eye velocity storage mechanism suggest that the gain of the feedback in the storage integrator was modified by the angle between the resultant otolith force and an animal-fixed reference. This could be the animals vertical, i.e., the direction along which gravity normally acts. For larger angles the feedback was less and the amplitude and time constant of the VVOR increased. The transformation of the otolith input was the same for both the up and down VOR, even though the final effect on the eye velocity was asymmetric (larger for up VOR) due to a separate, asymmetric gain element in the velocity storage feedback pathway.

Quantification of different classes of canal-related vestibular nuclei neuron responses to linear acceleration

A change in otolith activity modifies the dynamic responses of both the and the vertical4 vestibuloocular reflexes (VOR). In response to rotations in vertical planes, dynamic otolith activity is necessary for compensatory eye movements in the rabbit5 and the cat.6 Therefore, significant convergence of otolith and canal information in the VOR pathway must occur. The activity of single vestibular nuclei neurons in the decerebrate rat were recorded extracellularly during sinusoidal linear translation in the horizontal head plane. Details of the experimental procedure are presented elsewhere.’ Neurons from the four groups-(1) type I and (2) type I1 horizontal canal related, (3) vertical canal related, and (4) purely otolith-were systematically tested for their responses to translation at various horizontal head orientations. These responses were then used to describe a response ellipse*JO in which the semimajor axis (S1) defined the cell’s direction of maximum sensitivity and its associated gain and phase and the semiminor axis (S2) defined the minimum sensitivity of the cell in the horizontal head plane. When the magnitude of the S2 vector was zero, the response was referred to as narrowly tuned and was characterized by gain values that were proportional to the cosine of the angle between S1 and the stimulus direction and phase values that were constant with respect to stimulus direction. Whereas, a response with a nonzero magnitude of the S2 vector was referred to as broadly tuned and was characterized by a response phase that varied as a function of stimulus angle. The accuracy with which the response ellipse quantitatively described the data was assessed by comparing the direction, gain, and phase values of the maximum response determined empirically with those calculated from the fitted curves (compare the data points with the curve in FIGURE 1). The calculated and experimentally measured responses had high linear regression coefficients (r = 0.93014.9976) and slopes close to unity. Broadly tuned neurons were observed in each of the four groups of neurons studied. The ratio of S2 and S, response magnitudes (tuning ratio) was calculated for all neurons. The distribution of ratios was similar for all neuron groups. In addition,

Non-uniform temporal scaling of hand and finger kinematics during typing

We examined the manner in which the keystroke kinematics of the hand and the fingers varied with the mean rate of typing by trained typists. We used words and phrases in which only one letter was typed with the right hand and all of the remaining letters were typed using the left hand. We varied the typing rate over a threefold range (intervals between keypresses ranging from 150 ms to 500 ms) with the aid of a metronome. The results from four subjects, and three letters (n, u, and o) were analyzed. We did not find a simple scaling that could account for variations in the velocity profiles with typing rate. For some subjects and some letters, the velocities were independent of typing rate. In other instances, the kinematics did depend on typing rate, but to a much greater extent prior to the time of keypress than afterward. Sometimes the velocity profiles of all of the fingers and of the hand changed in a similar manner as the interval between keypresses was varied. In other instances only the focal movement of the hand and the finger used to press the key depended on the interval, whereas the motions of the other fingers did not. We suggest that the consistencies in the velocity profiles which we observed may simplify the problem of arranging a temporally ordered sequence of goal-directed movements.

THE VESTIBULO-OCULAR REFLEX IN THE CAT DURING LINEAR ACCELERATION IN THE SAGITTAL PLANE

The horizontal and vertical components of the vestibulo-ocular reflex (VOR) were recorded in alert cats that were rotated with their head placed on or 45 cm eccentric from the axis of rotation. During off-axis rotation there was a centripetal acceleration along the animals naso-occipital axis that changed the direction and the magnitude of the resultant otolith force in the animals sagittal plane. When the animal was upright and eccentric from the axis of rotation, the horizontal VOR (HVOR) had a shorter time constant and smaller amplitude compared to the on-axis HVOR. The effect was symmetrical for both directions of the naso-occipital linear acceleration. When the animal was on its side and faced away from the axis of rotation, there was a decrease in the time constant of the down VOR. When the animal faced the opposite direction, the down VOR time constant was increased. No statistically significant effect was found on the amplitude of the VVOR and the time constant of the up VOR.