Input minimization: a model of cerebellar learning without climbing fiber error signals.

The cerebellum is critical for motor learning. Current cerebellar learning models follow the Marr/Albus paradigm, in which climbing fibers provide error signals that shape plastic synapses between parallel fibers and Purkinje cells. However, climbing fibers have slow and largely random discharge, and seem unlikely to provide error signals with resolution sufficient to guide cerebellar learning. Parallel fibers carry error signals and could direct the plasticity of their own synapses, but the error signals are carried along with other signals. This report presents the new input minimization (InMin) model, in which Purkinje cells reduce error by minimizing their overall parallel fiber input. The slowly, randomly firing climbing fiber provides only synchronization pulses. InMin offers an alternative that can unify cerebellar findings.

A pattern correlation model of vestibulo-ocular reflex habituation.

Through the process of habituation, the response of the vestibulo-ocular reflex (VOR) is decreased by prolonged, sinusoidal stimulation at lower frequencies (< or =0.1 Hz). Research on goldfish has uncovered frequency-specific and nonlinear behaviors associated with habituation of the goldfish VOR, which include phenomena that cannot be explained using dynamic linear and static nonlinear models. The unexplained phenomena are abrupt decreases at peak response, gain decreases far in excess of linear predictions based on phase, and violation of superposition. Their existence was attributed to a hypothetical switch that closed in the appropriate context. The pattern correlation model provides a new perspective on the process of VOR habituation. Rather than treat the stimulus as a continuous sinusoid, the pattern correlation model breaks it up into a number of discontinuous patterns. The pattern most closely correlated with the current stimulus then decreases the VOR response by the amount of that correlation times a pre-assigned weight. The pattern correlation model explains how the frequency-specific and the nonlinear behaviors may be related, and how the apparent switching phenomena may occur.

Using Bayes' rule to model multisensory enhancement in the superior colliculus.

The deep layers of the superior colliculus (SC) integrate multisensory inputs and initiate an orienting response toward the source of stimulation (target). Multisensory enhancement, which occurs in the deep SC, is the augmentation of a neural response to sensory input of one modality by input of another modality. Multisensory enhancement appears to underlie the behavioral observation that an animal is more likely to orient toward weak stimuli if a stimulus of one modality is paired with a stimulus of another modality. Yet not all deep SC neurons are multisensory. Those that are exhibit the property of inverse effectiveness: combinations of weaker unimodal responses produce larger amounts of enhancement. We show that these neurophysiological findings support the hypothesis that deep SC neurons use their sensory inputs to compute the probability that a target is present. We model multimodal sensory inputs to the deep SC as random variables and cast the computation function in terms of Bayes' rule. Our analysis suggests that multisensory deep SC neurons are those that combine unimodal inputs that would be more uncertain by themselves. It also suggests that inverse effectiveness results because the increase in target probability due to the integration of multisensory inputs is larger when the unimodal responses are weaker.

Analysis and modeling of frequency-specific habituation of the goldfish vestibulo-ocular reflex.

Modification of the vestibulo-ocular reflex (VOR) by vestibular habituation is an important paradigm in the study of neural plasticity. The VOR is responsible for rotating the eyes to maintain the direction of gaze during head rotation. The response of the VOR to sinusoidal rotation is quantified by its gain (eye rotational velocity/head rotational velocity) and phase difference (eye velocity phase--inverted head velocity phase). The frequency response of the VOR in naïve animals has been previously modeled as a high-pass filter (HPF). A HPF passes signals above its corner frequency with gain 1 and phase 0 but decreases gain and increases phase lead (positive phase difference) as signal frequency decreases below its corner frequency. Modification of the VOR by habituation occurs after prolonged low-frequency rotation in the dark. Habituation causes a reduction in low-frequency VOR gain and has been simulated by increasing the corner frequency of the HPF model. This decreases gain not only at the habituating frequency but further decreases gain at all frequencies below the new corner frequency. It also causes phase lead to increase at all frequencies below the new corner frequency (up to some asymptotic value). We show that habituation of the goldfish VOR is not a broad frequency phenomena but is frequency specific. A decrease in VOR gain is produced primarily at the habituating frequency, and there is an increase in phase lead at nearby higher frequencies and a decrease in phase lead at nearby lower frequencies (phase crossover). Both the phase crossover and the frequency specific gain decrease make it impossible to simulate habituation of the VOR simply by increasing the corner frequency of the HPF model. The simplest way to simulate our data is to subtract the output of a band-pass filter (BPF) from the output of the HPF model of the naïve VOR. A BPF passes signals over a limited frequency range only. A BPF decreases gain and imparts a phase lag and lead, respectively, as frequency increases and decreases outside this range. Our model produces both the specific decrease in gain at the habituating frequency, and the phase crossover centered on the frequency of habituation. Our results suggest that VOR habituation may be similar to VOR adaptation (in which VOR modification is produced by visual-vestibular mismatch) in that both are frequency-specific phenomena.

Dual-frequency habituation and dishabituation of the goldfish vestibulo-ocular reflex.

Through the process of habituation, the eye rotational response of the vestibulo-ocular reflex (VOR) can be reduced by prolonged exposure to a head rotational stimulus. In previous work, the goldfish VOR habituated at a single, low frequency (< or = 0.1 Hz) showed frequency specific effects at and near that frequency, and could be dishabituated when combined with a higher frequency rotation. Here we show that the goldfish VOR exposed to prolonged rotation at two frequencies in combination will still produce habituation at low frequency, and can exhibit effects specific to both frequencies. The VOR at a low frequency can be dishabituated if the combined component is switched to a different frequency. These results demonstrate dual-frequency and context specificity of VOR habituation.

Violation of homogeneity by the vestibulo-ocular reflex of the goldfish.

The vestibulo-ocular reflex (VOR) allows animals to maintain stable gaze during head rotations by generating compensatory eye rotations. The VOR is typically tested using sinusoidal head rotation, and VOR gain is calculated as the ratio of the amplitude of eye to head rotational velocity. Through habituation, prolonged exposure to lower frequency sinusoidal head rotation in the dark decreases VOR gain. The VOR has been treated and modeled as a linear system. If it is linear, then the VOR must obey the principle of homogeneity: VOR gain at a particular frequency should be the same regardless of head velocity. We examined the habituated goldfish VOR for homogeneity. We found that it violated this basic principle of linear systems and is therefore non-linear.

Nonuniformity in the linear network model of the oculomotor integrator produces approximately fractional-order dynamics and more realistic neuron behavior.

The oculomotor integrator is a network that is composed of neurons in the medial vestibular nuclei and nuclei prepositus hypoglossi in the brainstem. Those neurons act approximately as fractional integrators of various orders, converting eye velocity commands into signals that are intermediate between velocity and position. The oculomotor integrator has been modeled as a network of linear neural elements, the time constants of which are lengthened by positive feedback through reciprocal inhibition. In this model, in which each neuron reciprocally inhibits its neighbors with the same Gaussian profile, all model neurons behave as identical, first-order, low-pass filters with dynamics that do not match the variable, approximately fractional-order dynamics of the neurons that compose the actual oculomotor integrator. Fractional-order integrators can be approximated by weighted sums of first-order, low-pass filters with diverse, broadly distributed time constants. Dynamic systems analysis reveals that the model integrator indeed has many broadly distributed time constants. However, only one time constant is expressed in the model due to the uniformity of its network connections. If the model network is made nonuniform by removing the reciprocal connections to and from a small number of neurons, then many more time constants are expressed. The dynamics of the neurons in the nonuniform network model are variable, approximately fractional-order, and resemble those of the neurons that compose the actual oculomotor integrator. Completely removing the connections to and from a neuron is equivalent to eliminating it, an operation done previously to demonstrate the robustness of the integrator network model. Ironically, the resulting nonuniform network model, previously supposed to represent a pathological integrator, may in fact represent a healthy integrator containing neurons with realistically variable, approximately fractional-order dynamics.

Analysis and neural network modeling of the nonlinear correlates of habituation in the vestibulo-ocular reflex.

Through the process of habituation, continued exposure to low-frequency (0.01 Hz) rotation in the dark produced suppression of the low-frequency response of the vestibulo-ocular reflex (VOR) in goldfish. The response did not decay gradually, as might be expected from an error-driven learning process, but displayed several nonlinear and nonstationary features. They included asymmetrical response suppression, magnitude-dependent suppression for lower- but not higher-magnitude head rotations, and abrupt-onset suppressions suggestive of a switching mechanism. Microinjection of lidocaine into the vestibulocerebellum of habituated goldfish resulted in a temporary dishabituation. This suggests that the vestibulocerebellum mediates habituation, presumably through Purkinje cell inhibition of vestibular nuclei neurons. The habituated VOR data were simulated with a feed-forward, nonlinear neural network model of the VOR in which only Purkinje cell inhibition of vestibular nuclei neurons was varied. The model suggests that Purkinje cell inhibition may switch in to introduce nonstationarities, and cause asymmetry and magnitude-dependency in the VOR to emerge from the essential nonlinearity of vestibular nuclei neurons.

Induction of periodic alternating nystagmus in intact goldfish by sinusoidal rotation.

During movement, the vestibulo-ocular reflex (VOR) normally maintains retinal image stability by making slow-phase eye rotations that counterbalance head rotations. These eye rotations are nystagmic when the on-going slow-phases are interrupted by fast-phase eye rotations that reset eye position. Periodic alternating nystagmus (PAN) is an eye-movement disorder characterized by uncontrollable nystagmus that alternates direction roughly sinusoidally (at about 0.005 Hz). PAN has been observed only in humans with cerebellar disorders and in monkeys with lesions to the cerebellar nodulus and uvula. We show experimentally in intact goldfish that prolonged rotation in darkness for 1 h at specific frequencies (0.05-0.1 Hz) induces PAN, upon which the normal VOR response is superimposed. We show computationally that rotation-induced PAN may result from decreased cerebellar inhibition of VOR brain stem neural pathways.

The horizontal optokinetic response of the goldfish.

This report describes the dynamics of the horizontal optokinetic response of the goldfish, and compares them with those of other species. Eye rotational velocity in response to step and sinusoidal rotations of the visual surround was tested using goldfish that had both eyes free to view the surround and to rotate with it. The step response was tested by switching on a visual surround display that was rotating at constant velocity, and then switching off the display, leaving the goldfish in the dark. The step-onset response was characterized by rapid and gradual components; the latter rose with an almost linear trajectory for higher surround velocities. The response was more rapid at step-offset than at step-onset. The step-offset response overshot baseline eye velocity for most goldfish and was oscillatory for the others. The steady-state response increased with constant velocity surround rotation within the range +/- 40 deg/sec but saturated outside that range. Steady-state response gain was higher for nasally-directed that for temporally-directed surround rotations. The frequency response was essentially low-pass, with gain decreasing from about 0.9 and phase lag increasing from zero to 90 deg as surround rotational frequency increased from 0.01 to 3.0 Hz. Sinusoidal response gain decreased as a function of surround peak acceleration. The results indicate that the horizontal optokinetic response of the goldfish is nonlinear and resembles in many respects that of mammals. Models developed to simulate the dynamics of the optokinetic response of mammals can be applied to that of goldfish and reproduce its nonlinear features.

Violation of superposition by the vestibulo-ocular reflex of the goldfish.

The vestibulo-ocular reflex (VOR) allows animals to move and see simultaneously by stabilizing the eyes in space. Previous experiments have largely confirmed the assumption that the VOR behaves as a linear system. However, a linear system must obey the superposition principle: its response to a periodic stimulus at any frequency must be the same whether that stimulus is presented alone or combined with stimuli at other frequencies. We first habituated the VOR in goldfish to low-frequency stimulation, which reduced low-frequency VOR gain. We then observed that the low-frequency gain increased almost 20-fold when the low-frequency stimulus was combined with a higher-frequency stimulus. This demonstrated for the first time a violation of superposition by the VOR. The VOR dishabituated and obeyed superposition following removal of the cerebellum.

Vestibular compensation in the horizontal vestibulo-ocular reflex of the goldfish.

Vestibular compensation is the process whereby vestibular system function is restored following unilateral removal of the vestibular receptors (hemilabyrinthectomy). Vestibular compensation was studied in the horizontal vestibulo-ocular reflex (VOR) of the goldfish. Spontaneous VOR (spontaneous nystagmus) was not observed in the goldfish following recovery from the surgery for hemilabyrinthectomy (a period of about 30 min). However, hemilabyrinthectomy resulted in an acute decrease in the gain of the horizontal VOR to approx. 50% of normal, and an increase in phase lead for mid-range frequencies (0.05 to 0.5 Hz). After 1 week of compensation, VOR gain had increased toward normal, and phase lead had returned to normal levels for mid-range frequencies, but increased above normal at low frequencies. After 1 month of compensation, horizontal VOR gain had recovered its normal value for head rotational velocity up to 60 deg/s, but it appeared to saturate for higher head velocity, and phase lead had decreased to normal, and even slightly below normal, at low frequencies. The results suggest that the goldfish is capable of almost completely recovering both the gain and phase of the horizontal VOR following 1 month of compensation for hemilabyrinthectomy. The extent of compensation in the horizontal VOR of the goldfish is greater than that which has been reported for mammals.

Evidence for a cooperative learning mechanism in the vestibulo-ocular reflex.

The vestibulo-ocular reflex (VOR) stabilizes vision by producing eye rotations that counterbalance head rotations. It receives input from a pair of vestibular receptors, and becomes unbalanced when input from one side is removed, producing spontaneous eye rotation in the absence of head rotation (SVOR). SVOR can be eliminated by the process of vestibular compensation. In mammals, SVOR elimination occurs over a period of days, and follows an exponential time course. We show that SVOR elimination in goldfish occurs in a matter of minutes, and its time course is not exponential but is characterized by a sigmoid function suggesting a cooperative mechanism. This time course is reproduced in a stochastic neural network model that has nonspecific reinforcement related to the level of VOR imbalance.

The fractional-order dynamics of brainstem vestibulo-oculomotor neurons.

The vestibulo-ocular reflex (VOR) and other oculomotor subsystems such as pursuit and saccades are ultimately mediated in the brainstem by premotor neurons in the vestibular and prepositus nuclei that relay eye movement commands to extraocular motoneurons. The premotor neurons receive vestibular signals from canal afferents. Canal afferent frequency responses have a component that can be characterized as a fractional-order differentiation (dkx/dtk where k is a nonnegative real number). This article extends the use of fractional calculus to describe the dynamics of motor and premotor neurons. It suggests that the oculomotor integrator, which converts eye velocity into eye position commands, may be of fractional order. This order is less than one, and the velocity commands have order one or greater, so the resulting net output of motor and premotor neurons can be described as fractional differentiation relative to eye position. The fractional derivative dynamics of motor and premotor neurons may serve to compensate fractional integral dynamics of the eye. Fractional differentiation can be used to account for the constant phase shift across frequencies, and the apparent decrease in time constant as VOR and pursuit frequency increases, that are observed for motor and premotor neurons. Fractional integration can reproduce the time course of motor and premotor neuron saccade-related activity, and the complex dynamics of the eye. Insight into the nature of fractional dynamics can be gained through simulations in which fractional-order differentiators and integrators are approximated by sums of integer-order high-pass and low-pass filters, respectively. Fractional dynamics may be applicable not only to the oculomotor system, but to motor control systems in general.

"Velocity leakage" in the pigeon vestibulo-ocular reflex.

The transfer characteristics of the vestibulo-ocular reflex (VOR), and of the semicircular canal primary afferents (SCPAs) that drive it, have been studied in several species. In monkeys and cats, the dominant time constant describing horizontal VOR dynamics (tau hv) is longer than that (tau c) of horizontal SCPAs. This lengthening of the time constant has been attributed to a "velocity storage" mechanism that has been modeled as a positive feedback loop in the VOR pathways. We have studied the transfer characteristics of horizontal and vertical VOR and SCPAs in unanesthetized pigeons. In this species the dominant time constants of both the horizontal and vertical VOR (tau hv and tau vv) are shorter that tau c. This finding indicates that time constants characterizing the lower frequency response of the VOR can be lengthened or shortened depending on the species. We propose that in the pigeon the "velocity leakage" mechanism can be modeled by substituting negative feedback for positive feedback in the model of the VOR pathways. Negative feedback can also account for the further shortening of tau hv and tau vv as VOR gain increases with arousal. Additionally, making the negative feedback loop nonlinear can model the dependency of lower frequency VOR phase on amplitude, and skew in VOR waveforms. Pigeon VOR and SCPA dynamics also differ in their adaptive properties and higher frequency behavior. A predominance of input from highly adaptive SCPAs is proposed to account for the increased adaptation of the vertical VOR as compared with SCPAs overall. A pure time-delay associated with VOR operation can explain the phase lag of the VOR relative to SCPAs at higher frequencies.

Simulating vestibular compensation using recurrent back-propagation.

Vestibular compensation is simulated as learning in a dynamic neural network model of the horizontal vestibulo-ocular reflex (VOR). The bilateral, three-layered VOR model consists of nonlinear units representing horizontal canal afferents, vestibular nuclei (VN) neurons and eye muscle motoneurons. Dynamic processing takes place via commissural connections that link the VN bilaterally. The intact network is trained, using recurrent back-propagation, to produce the VOR with velocity storage integration. Compensation is simulated by removing vestibular afferent input from one side and retraining the network. The time course of simulated compensation matches that observed experimentally. The behavior of model VN neurons in the compensated network also matches real data, but only if connections at the motoneurons, as well as at the VN, are allowed to be plastic. The dynamic properties of real VN neurons in compensated and normal animals are found to differ when tested with sinusoidal but not with step stimuli. The model reproduces these conflicting data, and suggests that the disagreement may be due to VN neuron nonlinearity.

Failure of the oculomotor neural integrator from a discrete midline lesion between the abducens nuclei in the monkey.

Recent anatomical studies indicate that axons of neurons in the vestibular nuclei, projecting to the contralateral abducens nuclei, cross the midline at the abducens level. These axons then give off collaterals to the contralateral vestibular and prepositus nuclei that may be important for the neural integrator that converts eye-velocity to eye-position signals. We disrupted a subset of these commissural projections by making a small midline lesion between the abducens nuclei in a monkey. The vestibulo-ocular reflex and saccades were still present post-lesion, indicating that premotor drive was intact, but the lesion produced severe post-saccadic drift, indicating failure of the neural integrator. We conclude that commissural projections crossing at the abducens level may be important for oculomotor integration.

Neural network models of velocity storage in the horizontal vestibulo-ocular reflex.

The vestibulo-ocular reflex (VOR) produces compensatory eye movements by utilizing head rotational velocity signals from the semicircular canals to control contractions of the extraocular muscles. In mammals, the time course of horizontal VOR is longer than that of the canal signals driving it, revealing the presence of a central integrator known as velocity storage. Although the neurons mediating VOR have been described neurophysiologically, their properties, and the mechanism of velocity storage itself, remain unexplained. Recent models of integration in VOR are based on systems of linear elements, interconnected in arbitrary ways. The present study extends this work by modeling horizontal VOR as a learning network composed of nonlinear model neurons. Network architectures are based on the VOR arc (canal afferents, vestibular nucleus (VN) neurons and extraocular motoneurons) and have both forward and lateral connections. The networks learn to produce velocity storage integration by forming lateral (commissural) inhibitory feedback loops between VN neurons. These loops overlap and interact in a complex way, forming both fast and slow VN pathways. The networks exhibit some of the nonlinear properties of the actual VOR, such as dependency of decay rate and phase lag upon input magnitude, and skewing of the response to higher magnitude sinusoidal inputs. Model VN neurons resemble their real counterparts. Both have increased time constant and gain, and decreased spontaneous rate as compared to canal afferents. Also, both model and real VN neurons exhibit rectification and skew. The results suggest that lateral inhibitory interactions produce velocity storage and also determine the properties of neurons mediating VOR. The neural network models demonstrate how commissural inhibition may be organized along the VOR pathway.