A random walk model of fast-phase timing during optokinetic nystagmus

 Most vertebrate animals produce optokinetic nystagmus in response to rotation of their visual surround. Nystagmus consists of an alternation of slow-phase eye rotations, which follow the surround, and fast-phase eye rotations, which quickly reset eye position. The time intervals between fast phases vary stochastically, even during optokinetic nystagmus produced by constant velocity rotation of a uniform surround. The inter-fast-phase interval distribution has a long tail, and intervals that are long relative to the mode become even more likely as constant surround velocity is decreased. This paper provides insight into fast-phase timing by showing that the process of fast-phase generation during constant velocity optokinetic nystagmus is analogous to a random walk with drift toward a threshold. Neurophysiologically, the output of vestibular nucleus neurons, which drive the slow phase, would approximate a random walk with drift because they integrate the noisy, constant surround velocity signal they receive from the visual system. Burst neurons, which fire a burst to drive the fast phase and reset the slow phase, are brought to threshold by the vestibular nucleus neurons. Such a nystagmic process produces stochastically varying inter-fast-phase intervals, and long intervals emerge naturally because, as drift rate (related to surround velocity) decreases, it becomes more likely that any random walk can meander for a long time before it crosses the threshold. The theoretical probability density function of the first threshold crossing times of random walks with drift is known to be that of an inverse Gaussian distribution. This probability density function describes well the distributions of the intervals between fast phases that were either determined experimentally, or simulated using a neurophysiologically plausible neural network model of fast-phase generation, during constant velocity optokinetic nystagmus. Received: 1 June 1995/Accepted in revised form: 15 February 1996     

Time constants of vestibular nuclei neurons in the goldfish: A model with ocular proprioception

A simple model of the vestibular-ocular reflex with a proprioceptive eye velocity feedback loop is used to simulate recent data on the vestibular responses of neurons in the vestibular nuclei of spinal goldfish. The data support the hypothesis that a proprioceptive feedback loop elongates the vestibular nucleus time constant to equal that of the slow phase eye movements of vestibular nystagmus.     

The Mechanism of Saccade Motor Pattern Generation Investigated by a Large-Scale Spiking Neuron Model of the Superior Colliculus

The subcortical saccade-generating system consists of the retina, superior colliculus, cerebellum and brainstem motoneuron areas. The superior colliculus is the site of sensory-motor convergence within this basic visuomotor loop preserved throughout the vertebrates. While the system has been extensively studied, there are still several outstanding questions regarding how and where the saccade eye movement profile is generated and the contribution of respective parts within this system. Here we construct a spiking neuron model of the whole intermediate layer of the superior colliculus based on the latest anatomy and physiology data. The model consists of conductance-based spiking neurons with quasi-visual, burst, buildup, local inhibitory, and deep layer inhibitory neurons. The visual input is given from the superficial superior colliculus and the burst neurons send the output to the brainstem oculomotor nuclei. Gating input from the basal ganglia and an integral feedback from the reticular formation are also included.We implement the model in the NEST simulator and show that the activity profile of bursting neurons can be reproduced by a combination of NMDA-type and cholinergic excitatory synaptic inputs and integrative inhibitory feedback. The model shows that the spreading neural activity observed in vivo can keep track of the collicular output over time and reset the system at the end of a saccade through activation of deep layer inhibitory neurons. We identify the model parameters according to neural recording data and show that the resulting model recreates the saccade size-velocity curves known as the saccadic main sequence in behavioral studies. The present model is consistent with theories that the superior colliculus takes a principal role in generating the temporal profiles of saccadic eye movements, rather than just specifying the end points of eye movements.     

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. Received: 8 August 1997 / Accepted in revised form: 18 June 1998     

A learning network model of the neural integrator of the oculomotor system

Certain premotor neurons of the oculomotor system fire at a rate proportional to desired eye velocity. Their output is integrated by a network of neurons to supply an eye positon command to the motoneurons of the extraocular muscles. This network, known as the neural integrator, is calibrated during infancy and then maintained through development and trauma with remarkable precision. We have modeled this system with a self-organizing neural network that learns to integrate vestibular velocity commands to generate appropriate eye movements. It learns by using current eye movement on any given trial to calculate the amount of retinal image slip and this is used as the error signal. The synaptic weights are then changed using a straightforward algorithm that is independent of the network configuration and does not necessitate backwards propagation of information. Minimization of the error in this fashion causes the network to develop multiple positive feedback loops that enable it to integrate a push-pull signal without integrating the background rate on which it rides. The network is also capable of recovering from various lesions and of generating more complicated signals to simulate induced postsaccadic drift and compensation for eye muscle mechanics.     

Eye movements and brainstem neuronal responses evoked by cerebellar and vestibular stimulation in chicks

Summary The vestibulo-ocular reflex undergoes adaptive changes that require inputs from the cerebellar flocculus onto brainstem vestibular neurons. As a step toward developing an in vitro preparation in chicks for studying the synaptic basis of those changes, we have elucidated the organization of the pathways through which the flocculus influences vestibulo-ocular movements. Electrical stimulation of the vestibular ampulla evoked brief, contralaterally directed movements in both eyes. Although single current pulses to the flocculus elicited no response, conjunctive stimulation of the flocculus and the vestibular apparatus significantly reduced the vestibularly-evoked movement. Trains of current pulses applied to the flocculus and ampulla evoked eye movements directed toward and away from the side of stimulation, respectively. Recordings from the brainstem revealed neurons that were activated by ipsilateral vestibular stimulation and inhibited by ipsilateral floccular stimulation. Our sample included neurons in the lateral vestibular nucleus, the ventrolateral portion of the medial vestibular nucleus, and the superior vestibular nucleus. Similarities between these findings and those of similar studies in mammals indicate that the chick will provide a good model system for cellular studies of adaptive changes in the vestibulo-ocular reflex.Abbreviations FTN flocculus target neuron - VOR vestibuloocular reflex     

A model of the nystagmus induced by off vertical axis rotation

A three-dimensional model is proposed that accounts for a number of phenomena attributed to the otoliths. It is constructed by extending and modifying a model of vestibular velocity storage. It is proposed that the otolith information about the orientation of the head to gravity changes the time constant of vestibular responses by modulating the gain of the velocity storage feedback loop. It is further proposed that the otolith signals, such as those that generate L-nystagmus (linear acceleration induced nystagmus), are partially coupled to the vestibular system via the velocity storage integrator. The combination of these two hypotheses suggests that a vestibular neural mechanism exists that performs correlation in the mathematical sense which is multiplication followed by integration. The multiplication is performed by the otolith modulation of the velocity storage feedback loop gain and the integration is performed by the velocity storage mechanism itself. Correlation allows calculation of the degree to which two signals are related and in this context provides a simple method of determining head angular velocity from the components of linear acceleration induced by off-vertical axis rotation. Correlation accounts for the otolith supplementation of the VOR and the sustained nystagmus generated by off-vertical axis rotation. The model also predicts the cross-coupling of horizontal and vertical optokinetic afternystagmus that occurs in head-lateral positions and the reported effects of tilt on vestibular responses.     

Bifurcation theory explains waveform variability in a congenital eye movement disorder

In dynamical systems, configurations that permit flexible control are also prone to undesirable behavior. We study a bilateral model of the oculomotor pre-motor network that conforms with the neuroanatomical constraint that brainstem neurons project to cerebellar Purkinje cells on both sides, but Purkinje cells project back to brainstem neurons on the same side only. Bifurcation analysis reveals that this network asymmetry enables flexible control by the cerebellum of brainstem network dynamics, but small changes in connection pattern or strength lead to behavior that is unstable, oscillatory, or both. The model produces the full range of waveform types associated with the hereditary eye movement disorder know as congenital nystagmus, and is consistent with findings linking the disorder with abnormal connectivity or limited plasticity in the cerebellum.     

The vestibulo-ocular reflex and velocity storage in spinocerebellar ataxia 8

The autosomal dominant spinocerebellar ataxias (SCAs) are a group of neurodegenerative diseases characterized by progressive instability of posture and gait, incoordination, ocular motor dysfunction, and dysarthria due to degeneration of cerebellar and brainstem neurons. Among the more than 20 genetically distinct subtypes, SCA8 is one of several wherein clinical observations indicate that cerebellar dysfunction is primary, and there is little evidence for other CNS involvement. The aim of the present work was to study the decay of the horizontal vestibulo-ocular reflex (VOR) after a short period of constant acceleration to understand the pathophysiology of the VOR due to cerebellar Purkinje cell degeneration in SCA8. The VOR was recorded in patients with genetically defined SCA8 during rotation in the dark. Moderate to severely affected patients had a qualitatively intact VOR, but there were quantitative differences in the gain and dynamics compared to normal controls. During angular velocity ramp rotations, there was a reversal in the direction of the VOR that was more pronounced in SCA8 compared to controls. Modeling studies indicate that there are significant changes in the velocity storage network, including abnormal feedback of an eye position signal into the network that contributes to this reversal. These and other results will help to identify features that are diagnostic for SCA subtypes and provide new information about selective vulnerability of neurons controlling vestibular reflexes.     

The effect of a long stay under microgravity on the vestibular function and tracking eye movements

Pre-and postflight examinations of cosmonauts participating in missions ISS-3 to ISS-9 on the International Space Station were performed using a computer-aided method of integrated assessment of the oculomotor system. The role and significance of the vestibular system in the eye tracking were determined; the individual and general characteristics of spontaneous oculomotor reactions and oculomotor reactions induced by visual and vestibular stimuli after a long-term stay at zero gravity (126–195 days) were determined; and the changes in the indices of oculomotor reactions were monitored. Studies of the vestibular function, intersensory interactions, and the tracking function of the eyes in the crew members were performed on the second, fifth (sixth), and ninth (tenth) days of the readaptation period. The results of the postflight examinations showed a significant change in the accuracy, velocity, and temporal characteristics of eye tracking and an increase in the vestibular reactivity. It was shown that the structure of visual tracking (the accuracy of fixational eye rotations and smooth tracking) was disturbed (the appearance of correcting saccades, the transition of smooth tracking to saccadic tracking) only in those cosmonauts who, in parallel to an increased reactivity of the vestibular input, also had central changes in the oculomotor system (spontaneous nystagmus, gaze nystagmus). With one exception, recovery of the indices of the accuracy of tracking eye movements in cosmonauts to the background level in the selected period of examination was not observed, although a positive trend was recorded.     

On the predominance of anti-compensatory eye movements in vestibular nystagmus

The predominance of anti-compensatory eye movements in vestibular nystagmus recorded during sinusoidal and post-rotational tests is interpreted in terms of a mathematical model of the vestibulo-ocular system. Namely, a direct pathway between the vestibular nuclei and the saccadic mechanism is assumed. In the range of frequencies of natural head movements this pathway carries on a signal proportional to head angular velocity. Therefore, during active head movements the saccadic mechanism is forced to produce quick eye rotations in the direction of head movement and, thus, to cooperate in the task of picking up visual targets outside the visual field. During passive head movements giving rise to nystagmus the assumed pathway contributes to reduce the error in eye resetting due to the saccadic delay. Analytical considerations and simulation results seem to prove the adequacy of the proposed model.Work supported by the National Research Council (C.N.R.), Rome, Italy     

The unknown but knowable relationship between Presaccadic Accumulation of activity and Saccade initiation

The goal of this short review is to call attention to a yawning gap of knowledge that separates two processes essential for saccade production. On the one hand, knowledge about the saccade generation circuitry within the brainstem is detailed and precise – push-pull interactions between gaze-shifting and gaze-holding processes control the time of saccade initiation, which begins when omnipause neurons are inhibited and brainstem burst neurons are excited. On the other hand, knowledge about the cortical and subcortical premotor circuitry accomplishing saccade initiation has crystalized around the concept of stochastic accumulation – the accumulating activity of saccade neurons reaching a fixed value triggers a saccade. Here is the gap: we do not know how the reaching of a threshold by premotor neurons causes the critical pause and burst of brainstem neurons that initiates saccades. Why this problem matters and how it can be addressed will be discussed. Closing the gap would unify two rich but curiously disconnected empirical and theoretical domains.

     

Simulations of neuromuscular control in lamprey swimming.

The neuronal generation of vertebrate locomotion has been extensively studied in the lamprey. Models at different levels of abstraction are being used to describe this system, from abstract nonlinear oscillators to interconnected model neurons comprising multiple compartments and a Hodgkin-Huxley representation of the most relevant ion channels. To study the role of sensory feedback by simulation, it eventually also becomes necessary to incorporate the mechanical movements in the models. By using simplifying models of muscle activation, body mechanics, counteracting water forces, and sensory feedback through stretch receptors and vestibular organs, we have been able to close the feedback loop to enable studies of the interaction between the neuronal and the mechanical systems. The neuromechanical simulations reveal that the currently known network is sufficient for generating a whole repertoire of swimming patterns. Swimming at different speeds and with different wavelengths, together with the performance of lateral turns can all be achieved by simply varying the brainstem input. The neuronal mechanisms behind pitch and roll manoeuvres are less clear. We have put forward a 'crossed-oscillators' hypothesis where partly separate dorsal and ventral circuits are postulated. Neuromechanical simulations of this system show that it is also capable of generating realistic pitch turns and rolls, and that vestibular signals can stabilize the posture during swimming.     

The interstitial nucleus of Cajal of the cat. I. Neurons activity related to vertical eye movements

Interstitial nucleus of Cajal (INC) neurons activity was studied during vertical optokinetic nystagmus (OKN) and after-nystagmus (OKAN) in awake cats lying on their right side. The activity of one hundred neurons was recorded in the left INC and analysed in relation with the vertical component of OKN and OKAN. The activity of 27 neurons was correlated either to eye position or to both eye velocity and eye position; 18 of these neurons were recorded in their on-direction and their off-direction. The analysis of the 18 neurons showed that the activity of 8 of them was correlated to eye position in the on-direction and in the off-direction and the correlation to eye position was higher than to eye velocity; these neurons are considered as position neurons. Seven other neurons had a higher correlation to eye position that to eye velocity in the on-direction and this relation reversed in the off-direction, these neurons are considered as position-velocity neurons. Thirty two burst-neurons were activated only during quick phases of OKN and OKAN and they were silent during slow phases and periods of fixation. Nine burst neurons had an upward on-direction and 23 neurons a downward on-direction. The eye velocity-average burst frequency (ABF) and quick phase duration-burst duration relationships had low correlations and suggested that INC burst neurons were excitatory premotor neurons. Statistical analysis showed that downward on-direction burst neurons had a higher ABF that upward on-direction burst neurons. Moreover, during OKN and OKAN, the velocity sensitivity of INC burst neurons was the same. The activity of the remaining neurons (41 neurons) was not quantitatively correlated to vertical and horizontal eye movements; they were classified as irregular tonic neurons. This study shows that INC neurons carry an eye position signal which was never reported before. This supports the results of INC lesion studies which showed that INC is involved in the vertical velocity to position integration. Moreover, there is an up versus down asymmetry in the frequency of INC burst neurons.     

Performance of a model for a local neuron population

A model of a local neuron population is considered that contains three subsets of neurons, one main excitatory subset, an auxiliary excitatory subset and an inhibitory subset. They are connected in one positive and one negative feedback loop, each containing linear dynamic and nonlinear static elements. The network also allows for a positive linear feedback loop. The behaviour of this network is studied for sinusoidal and white noise inputs. First steady state conditions are investigated and with this as starting point the linearized network is defined and conditions for stability is discovered. With white noise as input the stable network produces rhythmic activity whose spectral properties are investigated for various input levels. With a mean input of a certain level the network becomes unstable and the characteristics of these limit cycles are investigated in terms of occurence and amplitude. An electronic model has been built to study more closely the waveforms under both stable and unstable conditions. It is shown to produce signals that resemble EEG background activity and certain types of paroxysmal activity, in particular spikes.     

Modeling the interaction among three cerebellar disorders of eye movements: periodic alternating,gaze-evoked and rebound nystagmus

A woman, age 44, with a positive anti-YO paraneoplastic cerebellar syndrome and normal imaging developed an ocular motor disorder including periodic alternating nystagmus (PAN), gaze-evoked nystagmus (GEN) and rebound nystagmus (RN). During fixation there was typical PAN but changes in gaze position evoked complex, time-varying oscillations of GEN and RN. To unravel the pathophysiology of this unusual pattern of nystagmus, we developed a mathematical model of normal function of the circuits mediating the vestibular-ocular reflex and gaze-holding including their adaptive mechanisms. Simulations showed that all the findings of our patient could be explained by two, small, isolated changes in cerebellar circuits: reducing the time constant of the gaze-holding integrator, producing GEN and RN, and increasing the gain of the vestibular velocity-storage positive feedback loop, producing PAN. We conclude that the gaze- and time-varying pattern of nystagmus in our patient can be accounted for by superposition of one model that produces typical PAN and another model that produces typical GEN and RN, without requiring a new oscillator in the gaze-holding system or a more complex, nonlinear interaction between the two models. This analysis suggest a strategy for uncovering gaze-evoked and rebound nystagmus in the setting of a time-varying nystagmus such as PAN. Our results are also consistent with current ideas of compartmentalization of cerebellar functions for the control of the vestibular velocity-storage mechanism (nodulus and ventral uvula) and for holding horizontal gaze steady (the flocculus and tonsil).

     

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 (d ^k x/dt _k 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.     

Modelling Feedback Excitation,Pacemaker Properties and Sensory Switching of Electrically Coupled Brainstem Neurons Controlling Rhythmic Activity

What cellular and network properties allow reliable neuronal rhythm generation or firing that can be started and stopped by brief synaptic inputs? We investigate rhythmic activity in an electrically-coupled population of brainstem neurons driving swimming locomotion in young frog tadpoles, and how activity is switched on and off by brief sensory stimulation. We build a computational model of 30 electrically-coupled conditional pacemaker neurons on one side of the tadpole hindbrain and spinal cord. Based on experimental estimates for neuron properties, population sizes, synapse strengths and connections, we show that: long-lasting, mutual, glutamatergic excitation between the neurons allows the network to sustain rhythmic pacemaker firing at swimming frequencies following brief synaptic excitation; activity persists but rhythm breaks down without electrical coupling; NMDA voltage-dependency doubles the range of synaptic feedback strengths generating sustained rhythm. The network can be switched on and off at short latency by brief synaptic excitation and inhibition. We demonstrate that a population of generic Hodgkin-Huxley type neurons coupled by glutamatergic excitatory feedback can generate sustained asynchronous firing switched on and off synaptically. We conclude that networks of neurons with NMDAR mediated feedback excitation can generate self-sustained activity following brief synaptic excitation. The frequency of activity is limited by the kinetics of the neuron membrane channels and can be stopped by brief inhibitory input. Network activity can be rhythmic at lower frequencies if the neurons are electrically coupled. Our key finding is that excitatory synaptic feedback within a population of neurons can produce switchable, stable, sustained firing without synaptic inhibition.     

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.     

A model of Alexander's law of vestibular nystagmus

The observation that the amplitude of vestibular nystagmus grows as gaze is increased in the direction of the nystagmus fast phase and diminished with gaze in the opposite direction is known as Alexander's law. We have developed an analog computer model to simulate Alexander's law in nystagmus secondary to dysfunction of a semicircular canal. The model utilizes relevant brainstem anatomy and physiology and includes gaze modulation of vestibular signals and push-pull integration to create eye positition commands. When simulating normally functioning semicircular canals, the model produced no nystagmus. When simulating total impairment of the canal on one side with gaze directed maximally in the opposite direction, the model produced a large amplitude nystagmus with linear slow phases directed toward the affected side. As gaze was changed from far contralateral to ipsilateral, the nystagmus gradually diminished to zero. When simulating partial impairment of one canal, the nystagmus was smaller in amplitude and absent in ipsilateral gaze.