INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION REFERENCES

It is widely accepted that the cerebellum plays an important role in motor learning and adaptation for a wide range of behaviors. Humans and animals with severe cerebellar lesions cannot adequately learn new movements (Baizer and Glickstein 1974; Baizer et al. 1999; Gauthier et al. 1979; Ito et al. 1979; Lisberger et al. 1984; McElligott et al. 1998; Michnovicz and Bennett 1987; Nagao 1983; Pastor et al. 1994; Robinson 1976; Sanes et al. 1990; Thach et al. 1992; Weiner et al. 1983). Cerebellar Purkinje cells (P cells), the only output neurons of the cerebellar cortex, receive three major synaptic inputs: a large number of granule-cell axon (GCA) inputs, multiple inhibitory cell (IC) inputs, and a single climbing fiber (CF) input. ICs receive GCA inputs and project their axons to P cells. P cells exhibit two kinds of spikes: simple spikes (SS) induced by GCA inputs and complex spikes (CS) induced by CF inputs. The physiologically demonstrated synaptic plasticity of P cells has been suggested to be the cellular mechanism responsible for movement adaptation and learning. The known types of synaptic plasticity include long-term depression (LTD) and long-term potentiation (LTP) for excitatory GCA synapses, and rebound potentiation (RP) for inhibitory synapses from ICs. LTD results in a decrease in the synaptic efficacy of GCA synapses and is induced by the conjunctive stimulation of GCAs and CFs (Ito and Kano 1982; Ito et al. 1982). It has been shown that LTP is induced presynaptically by stimulation of GCA in the absence of CF stimulation (Hirano 1990; Sakurai 1987). RP results in an increase in the synaptic efficacy of IC synapses and is induced by conjunctive activation of IC and P cells (Kano 1996; Kano et al. 1992). Changes in P cell firing that explain behavioral changes during adaptation have been recorded for both SSs and CSs (Dufosse et al. 1978; Gilbert and Thach 1977; Nagao 1989; Watanabe 1985). Here, GCA and IC inputs are generally assumed to convey the sensory-motor context to P cells. On the other hand, the CF input is assumed to carry error signals essential for learning and adaptation (Kitazawa et al. 1998; Kobayashi et al. 1998).

The cellular mechanisms of LTD have been intensively studied, mainly in slice and culture preparations (reviewed in Crepel et al. 1996). Based on these in vitro studies, molecular and genetic techniques have been introduced to examine relationships between LTD and behavioral adaptation. For example, the injection of LTD inhibitors into the cerebellum has been found to abolish behavioral adaptation (Li et al. 1995; Nagao and Ito 1991; Yanagihara and Kondo 1996). Furthermore, transgenic mice with protein kinase C inhibitors expressed in P cells have been found to lack both LTD and motor learning capability (De Zeeuw et al. 1998).

Although there have been extensive experimental studies on P cell plasticity and its possible roles in motor learning, detailed and realistic computational studies that quantitatively reconstruct the temporal dynamics of both P cell firing and movement kinematics during motor learning are scarce. However, such computational studies are essential for understanding the mechanisms underlying how these types of synaptic plasticity achieve motor learning and adaptation.

The objectives of this study were to construct a detailed and quantitative neural circuit model that could reproduce both the control and adaptation of ocular following responses and to investigate the above mechanisms. Specifically, we addressed the following six questions. The first question is whether in vivo motor adaptation can be fully explained by the characteristics of P cell plasticity revealed by in vitro slice and culture preparation studies. The second question concerns whether the time difference between GCA inputs and CF inputs during the induction of LTD is crucial for reproducing behavioral modification. In slice experiments, the amount of change induced in the synaptic efficacy was found to depend on the time interval between CF and GCA stimulations (Chen and Thompson 1995; Karachot et al. 1994). We call this the temporal window of LTD. The temporal window reported by Karachot et al. (1994) was criticized for not being appropriate for motor learning (De Schutter 1995; Lukashin 1996). The third question is whether all reported types of plasticity, i.e., LTD, LTP, and RP, are essential for motor learning (cf. De Shutter 1995). The fourth question concerns the range of motor adaptation and learning, which might be explained by synaptic plasticity in the cerebellar cortex. There are several types of motor learning and adaptation, such as the acquisition of new behaviors, and gain and kinematic adaptation to preexisting movements. The fifth question concerns the learning acquisition of temporal waveforms of SS firing. What proportion of the SS firing characteristics of an individual P cell is determined by the characteristics of the CF input to that P cell through synaptic plasticity? The last question is the most abstract and at the highest level; what does the cerebellar cortex computationally acquire in motor learning with these classes of synaptic plasticity?

To answer these questions adequately, a model must be constructed for a movement whose neural control circuit has been well studied. Second, the model must quantitatively reconstruct SS and CS firings based on experimental data. If the motor control system contains feedback loops, movement kinematics must also be quantitatively reproduced based on experimental data by a model containing such feedback loops. Previous simulation studies have examined the adaptation and learning of several kinds of movements, such as vestibular ocular reflexes, saccadic eye movements, smooth pursuit, and arm movements in relation to cerebellar synaptic plasticity (Fujita 1982; Gomi and Kawato 1992; Kettner et al. 1997; Raymond and Lisberger 1998; Schweighofer et al. 1996, 1998). Unfortunately, some of the preceding prerequisites were not satisfied in these studies.

Ocular following responses (OFRs) are movements for which we can construct a simulation model that satisfies all of the above requirements. OFRs are reflex eye movements induced with short latencies by motion of a large-field visual stimulus. They exhibit several types of behavioral adaptation (Miles and Kawano 1986). Previous physiological studies have revealed that the medial superior temporal area (MST) of the cerebral cortex, the dorsolateral pontine nucleus (DLPN), and the ventral paraflocculus (VPFL) of the cerebellum are important loci for controlling OFR (reviewed in Kawano 1999). The characteristics of the neural activity in the MST (Kawano et al. 1994), DLPN (Kawano et al. 1992), and VPFL (Gomi et al. 1998; Kawano and Shidara 1993; Kobayashi et al. 1998; Shidara and Kawano 1993; Shidara et al. 1993) during OFR have been very intensively studied. In particular, SS firing waveforms have been reconstructed by inverse dynamics models of eye movements (Gomi et al. 1998; Shidara et al. 1993). We have already developed a quantitative model of the preceding neural network that controls OFR as a feedback control system (Yamamoto et al. 1997, 2000). Kobayashi et al. (1998) have also developed a quantitative model of CS firing waveforms based on the physiological data.

Based on these previous studies, a large-scale network model for two-dimensional OFR was developed, and its long-term development and short-term adaptation were simulated. With this realistic and quantitative model, we were able to examine the preceding questions about the computational roles of synaptic plasticity in motor learning.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION REFERENCES

Ocular following responses: behavioral and neurophysiological studies

Behavioral studies of monkeys and humans have shown that sudden movements of a visual scene evoke short-latency OFRs (Gellman et al. 1990; Miles et al. 1986; reviewed in Kawano 1999). The experimental paradigm used to study OFRs is described in the following text (Kawano et al. 1992). A random dot pattern in a full visual field positioned in front of a monkey is suddenly moved upward, downward, leftward, or rightward in a velocity-step, position-ramp manner. We call this kind of stimulus a ramp stimulus. The visual field starts to move 50-300 ms after the end of a saccadic eye movement directed toward the central part of the screen (±10°). The ramp movement of the stimulus usually lasts 150 or 300 ms before a mechanical shutter shuts off the visual field. The eyes of the monkey start to follow the stimulus about 50 ms after the ramp stimulus onset (Miles et al. 1986).

OFRs undergo two types of adaptation: gain adaptation after repeated "speed-step" stimuli, and directional adaptation after repeated "direction-step" stimuli (Miles and Kawano 1986). The "speed-step" stimuli contain a "step-up stimulus" and a "step-down stimulus." For the step-up stimulus, the monkey is given a ramp stimulus slower than 100°/s for the first 150 ms. The velocity of the stimulus is changed to 100°/s for the next 150 ms. For the step-down stimulus, the monkey is given a ramp stimulus slower than 100°/s for the first 150 ms. The velocity of the stimulus is changed to 0°/s for the next 150 ms. After repeating the step-up trials, the OFR gain of the monkey, which is defined as the maximum eye velocity divided by the stimulus velocity, increases adaptively for the test ramp stimuli at the initial speed. After repeating the step-down trials, the OFR gain of the monkey decreases for the test ramp stimuli. In the direction-step trials, the direction of the stimulus is changed to 90° counterclockwise 150 ms after the onset of the ramp stimulus. After repeated direction-step trials, the OFR direction is changed to the counterclockwise direction for the test ramp stimuli in the initial direction.

Many previous studies have investigated neural firing during OFR in several brain loci, as shown in Fig. 1. Movement of the retinal image induces neural activity in the visual cortex, the MST (Kawano et al. 1994), and the DLPN (Boussaoud et al. 1992; Brodal 1978; Glickstein et al. 1980, 1985; Kawano et al. 1992; Maunsell and van Essen 1983; May and Andersen 1986; Tusa and Ungerleider 1988; Ungerleider et al. 1984). The axons of DLPN neurons project to the cerebellar VPFL (Glickstein et al. 1990; Langer et al. 1985; Shidara and Kawano 1993) as mossy fibers (Kawano and Shidara 1993) and innervate granule cells in the VPFL cortex. The axons of granule cells (GCAs) project partly to the P cells of the VPFL as parallel fibers (PFs). These GCAs transmit excitatory firing stimuli to P cells. GCAs also project to ICs, i.e., stellate cells and basket cells, in the molecular layer of the cerebellar cortex. ICs then inhibit P cells when they are activated. The inputs of GCAs and ICs to P cells modulate the SSs of the P cells. The P cells of the VPFL project to the brain stem and influence firing of extraocular motor neurons, thus moving the eyes. Retinal image motion information is also conveyed to the accessory optic tract and to the inferior olivary nucleus. Cell firing in the inferior olivary nucleus is transmitted to P cells by CFs. The CF inputs induce CSs of P cells. It has been suggested that the indirect pathway for slow eye movements (Fuchs and Mustari 1993; Mustari et al. 1994), which contains the accessory optic tract, might control OFRs in addition to the MST-DLPN-VPFL pathway (Inoue et al. 2000). Lesions in the VPFL abolish a large portion of OFRs (Miles et al. 1986), suggesting that the MST-DLPN-VPFL pathway is the major control system for OFRs.

View larger version (29K): [in this window] [in a new window]   Fig. 1. A: a large scale network model for 2-dimensional ocular following response (OFR). MST, medial superior temporal area; DLPN, dorsolateral pontine nucleus; VPFL, ventral paraflocculus. B: a magnified diagram of the cerebellar VPFL shown in A. LTD, long-term depression; LTP, long-term potentiation; RP, rebound potentiation.

The direction, speed, and temporal characteristics of cell firing differ markedly between the mossy fiber inputs and the SS firing outputs of the VPFL (Kawano 1999; Kawato 1999; Takemura et al. 2001). First, the preferred direction changes from a uniform distribution to the horizontal and vertical axes only. Each cell in the MST (Kawano et al. 1994) and DLPN (Kawano et al. 1992) exhibits a truncated cosine-function direction tuning in its firing response to large-field visual motion, and its preferred direction is distributed in wide range of 360° without anisotropy. Similar characteristics were reported on the firings of mossy fibers (Kawano and Shidara 1993). However, for vertical P cells, the direction tuning curves are well fitted by full cosine functions, and the preferred direction of CS responses is upward and that of SS responses is downward (Kobayashi et al. 1998; Shidara and Kawano 1993). For horizontal P cells, the preferred direction of CS responses is contralateral and that of SS responses is ipsilateral. Second, the wide distribution of preferred velocities in VPFL inputs changes to a narrow distribution in VPFL outputs. Each MST cell (Kawano et al. 1994) and DLPN cell (Kawano et al. 1992) exhibits an individual preferred stimulus velocity, which is widely dispersed over 10, 20, 40, 80, or 160°/s. On the other hand, each P cell has a preferred velocity for SSs only at high velocities, such as 80 or 160°/s (Shidara and Kawano 1993). Finally, a wide variety of phasic and tonic temporal waveforms in the input firings change to the stereotypical SS waveform, which is well reconstructed by the inverse dynamics model of eye movements (Gomi et al. 1998; Shidara et al. 1993). Takemura et al. (2001) reconstructed the temporal firing frequency patterns of cells in the MST and DLPN, and SSs in the VPFL, from the acceleration, velocity, and position of eye movements or retinal error. They analyzed the discharges of neurons including open-loop/closed-loop portion without considering the extra-retinal information (efference copy, etc.) and succeeded in quantitatively showing the temporal difference in the neural firing of these areas. They found that the acceleration and velocity coefficients of the MST and DLPN ranged widely, whereas those of the VPFL more compactly.

Previous and present models of OFR control

Our previous OFR control system model (Yamamoto et al. 1997) quantitatively reproduced SS firing and eye movements. The squared correlation coefficients between simulated and experimental data were 0.81 for SSs and 0.99 for the eye velocity. In that model, the velocity and acceleration of retinal error were first delayed by 39 ms, then saturated, filtered, and finally weighted to become SS firings. These SS firings passed through a filter that was estimated by inverse-dynamics analysis (Gomi et al. 1998; Shidara et al. 1993) and were then delayed by 12 ms to become the eye movement.

In this paper, we maintain the basic structure of this previous model but extend it to a new model that consists of the following three compartments (Fig. 1A). The first compartment is the MST-DLPN-mossy-fiber-GCA/IC pathway, which is basically the same as in the previous model but is much more biologically detailed and realistic. Figure 2 enlarges the first compartment within the entire system. The second compartment is new and represents the inferior olivary and CF system. The third is also new and represents the accessory optic system. Furthermore, to simulate cerebellar plasticity, we introduce good quantitative models of GCA, IC, and CF firing. In the following text, we explain these three compartments and the synaptic plasticity model.

View larger version (25K): [in this window] [in a new window]   Fig. 2. A detailed diagram of the model, enlarging the medial superior temporal area (MST)-DLPN-VPFL pathway. The model contains 1,080 MST cells and 40 P cells. MST firing is computed from retinal error by making preferred directions (direction), temporal patterns (waveform), and preferred velocities (velocity). The granule cell axons and inhibitory cells carry the MST firing to the 40 P cells through excitatory and inhibitory synapses with individual weights (synapses). The weighted summation of these inputs becomes the SS firing (simple spike). The SS outputs from the VPFL are added to the output from the indirect pathway before the filter to become the eye movement. The retinal error is the visual stimulus motion minus the eye movement. p denotes a Laplace transformation operator.

Model of MST-DLPN-mossy-fiber-GCA/IC pathway

The model MST cells in the first compartment are divided into 12 groups according to the 12 preferred directions covering the entire 360° range separated by 30° (Fig. 3A). The directional tuning of MST cell firing is described by the following truncated cosine tuning function

(1)

Here, f(), _P(degree), (degree), and f_max denote the firing frequency (spikes/s) of the cell, the preferred direction of the cell, the direction of the stimulus, and the maximum firing frequency, respectively. The width of the half-decay for this truncated cosine tuning model is 60° and agrees well with the experimental data, which show the width of the half-decay as 58° (Kawano et al. 1994).

View larger version (28K): [in this window] [in a new window]   Fig. 3. Directional (A), temporal (B), and velocity (C) tuning characteristics of a model MST firing. A: modeling of the direction selectivity of MST cells. Twelve groups of cells possess their preferred directions every 30°; 0° is defined as rightward, and the angle is measured in a counterclockwise rotation. B: modeling of the temporal firing patterns of MST cells. The temporal patterns were modeled by summing the acceleration and velocity of the retinal error weighted by 3 kinds of ratios, such that the maximum of the phasic and tonic components of the firing exhibited the three ratios 2:1 (a), 1:2 (b), and 1:1 (c). C: modeling the selectivity of MST cells on the stimulus velocity. Thirty curves show saturated firing as a function of stimulus velocity for 30 cells with preferred velocities distributed over 10-300°/s every 10°/s from left to right. The abscissa denotes the stimulus velocity. The ordinate denotes the maximal possible firing-frequency output from each MST cell.

To reproduce the experimentally observed dispersion in temporal waveforms of MST cell firings (Takemura et al. 2001), the model MST cells are classified into three groups possessing three different temporal firing patterns (Fig. 3B). The firing frequency of each group is reconstructed by adding the filtered accelerations and filtered velocities of the retinal error with the weights 0.005:0.5, 0.0025:1, and 0.005:1, respectively. We have used the filters of our previous model (Yamamoto et al. 1997): 1/(0.00001p² + 0.0013p + 1) for the velocity and 1/(0.0001p² + 0.03p + 1) for the acceleration. Here, p denotes a Laplace transformation operator. The time constants of the filters were selected to reproduce recorded firing patterns from filtered velocity and acceleration of retinal error. The filtered acceleration and velocity contribute to the phasic and the tonic components in the firing waveforms, respectively (Yamamoto et al. 1997). The three firing patterns have their maximum initial phases and tonic phases with ratios of 2:1 (Fig. 3Ba), 1:2 (Fig. 3Bb), and 1:1 (Fig. 3Bc), respectively, when the test ramp stimulus is 10°/s.

Regarding the preferred velocities, the MST cells are divided into 30 groups, covering 10-300°/s every 10°/s (Fig. 3C). This assumed distribution of MST model cell preferred velocities corresponds well with experimentally obtained distributions (Kawano et al. 1994). The MST model cell fires at a maximum of 300 spikes/s for the preferred velocity. While a majority of MST cells with low preferred velocities do not fire at high velocities, many of the MST cells with high preferred velocities fire even at low velocities (A. Takemura, personal communication). To accommodate these experimental observations, we prepared 30 velocity-tuning curves (Fig. 3C), which have long tails for lower velocities than the preferred velocity but short tails for higher velocities for higher velocities. For example, when the ramp stimulus is 80°/s, the model cell with a preferred velocity of 80°/s fires at a maximum of 300 spikes/s. In contrast, the model cell with a preferred velocity of 50°/s fires at a maximum of 75 spikes/s with the same stimulus. The model cell with a preferred velocity of 100°/s fires at a maximum of 240 spikes/s. The ensemble mean of the outputs from these 30 models corresponds well with the experimentally observed ensemble mean of MST cell firings (Kawano et al. 1994).

A total of 1,080 models of MST cells were obtained from 12 groups for preferred direction, 3 groups for different temporal patterns of the firing frequency, and 30 groups for different preferred velocities (12 × 3 × 30 equals 1,080 models). No remarkable difference has been reported among the firing characteristics of MST cells, DLPN cells, and mossy fibers (Kawano and Shidara 1993; Kawano et al. 1992, 1994). Accordingly, we used the MST cell firing as the cell firing for both DLPN cells and mossy fibers. In addition, no recording has been attempted from GCAs or ICs during OFR. Therefore the current model assumes that the GCA firing is the same as the mossy fiber firing. The IC firing is assumed to be the same as the GCA firing; therefore the MST-DLPN-mossy-fiber pathway contains 1,080 GCAs and 1,080 ICs, which are connected to 40 P cell models. Therefore 1,080 × 2 × 40 synapses made with a total of 40 P cells.

Model of climbing fiber input

The preferred direction of CSs in the VPFL is either upward or contralateral, and the direction-tuning curve is well modeled by cosine functions (Kobayashi et al. 1998). Accordingly, the right VPFL of the model contains two types of P cells (vertical and horizontal); the CF inputs of the vertical P cells prefer the upward stimulus direction, and the CF inputs of the horizontal P cells prefer the leftward direction. The left VPFL also contains two types of P cells; the CF inputs of the vertical P cells prefer upward stimuli, and the CF inputs of the horizontal P cells prefer rightward stimuli. Using these CS preferred directions and their hemispheres, we classify the model P cells into four groups, two on each side (right h-P cells, right v-P cells, left v-P cells, and left h-P cells from top to bottom in Fig. 4).

View larger version (29K): [in this window] [in a new window]   Fig. 4. A detailed diagram of the block of the model that contains P cells and the downstream control network. 10 P cells within each of the 4 groups (right h, right v, left v, and left h) receive 10 different climbing fiber (CF) inputs from 10 different inferior olive (IO) cells. The SS outputs from these 40 P cells are first averaged, weighted, and summed or subtracted, and finally filtered differently by the 3 different filters, FV1, FV2, or FH, to generate upward, downward, or horizontal movements.

To reproduce observed variability in CF firing characteristics in the temporal domain of individual P cells experimentally, we prepared 10 different CF temporal firing patterns for 10 different P cells in each of the four groups. They were determined by CS recordings from 10 P cells of one monkey reported in Kobayashi et al. (1998). The firing probability of CF at time t, CF(t), was well reproduced by a generalized linear model of the acceleration, velocity, and position of the retinal error (Kobayashi et al. 1998). We calculated each deviance for the reconstruction of the temporal CS pattern by the generalized linear models of various combinations of acceleration, velocity, and position of retinal error, and applied the ² test to the sets of deviance. As a result, we confirmed that the generalized linear model of only the velocity of the retinal error (retinal slip) was statistically sufficient. Accordingly, we reconstructed the temporal pattern of the CF firing frequency with a generalized linear model by only considering the retinal slip as follows: CF(t) = S(B ·  + C).

Here, S(x) = expx/(1 + expx). denotes either the horizontal or vertical component of the retinal slip. Because the SS firing frequency of most P cells remains lower than 3 spikes/s (Kobayashi et al. 1998), a CS firing frequency >3 spikes/s was reduced to 3 spikes/s. Note that the CF cosine tuning was reproduced with this equation. We estimated 10 sets of B, C coefficients by using a maximum-likelihood estimation of the recorded data of CS from 10 P cells during an upward 80°/s ramp stimulus (Kobayashi et al. 1998). For the 10 CF models within each P cell group, we used B and C derived from the data. One P cell model received one CF model input. Therefore the four types of P cell groups and 10 types of temporal CF firing frequencies produced 4 × 10 = 40 different CF firings and, accordingly, a total of 40 model P cells (see Fig. 4).

Model system of downstream of VPFL

Here, we describe the model system downstream from the VPFL (Fig. 4). The SS firing of 10 P cells within each of the four groups are first averaged. These averaged outputs from left and right vertical cell groups are further averaged, are reversed in sign, because P cells are inhibitory neurons, and are finally fed into two filters (FV1 and FV2) to compute the vertical component of a 2-degree-of-freedom eye movement. Here, the upward eye movement is defined as positive. On the other hand, the averaged output from the right horizontal P cells is sign-reversed and halved and is then subtracted from that of the left horizontal cell group outputs (Fig. 4). This subtraction is fed into the horizontal filter (FH) to produce the horizontal component of the eye movement. Here, the leftward eye movement is defined as positive.

For the filters that represent the combined characteristics of the neural system downstream of the VPFL and the oculomotor plant, we used the filters developed in our previous study (Yamamoto et al. 1997). 1/(0.0442p² + 2.20p) represents FV1 in Fig. 4, when the firing frequency of SSs of the vertical P cells decreases below the spontaneous firing level of the vertical P cells. 1/(0.107p² + 2.13p  3.42) represents FV2 in Fig. 4, when the vertical P cells increase their SS firing frequency. 1/(0.107p² + 2.13p  3.42) is FH in Fig. 4 for the horizontal P cells.

Model of the indirect pathway

Here, we describe the model compartment of the indirect pathway that parallels the MST-DLPN-VPFL pathway. The MST-DLPN-VPFL pathway for OFR control is known as the direct pathway for the control of optokinetic responses (OKR) (Fuchs and Mustari 1993; Mustari et al. 1994). The OKR control system has another pathway, the indirect pathway, which contains the accessory optic system (Fuchs and Mustari 1993; Mustari et al. 1994) (Fig. 1). The indirect pathway has a velocity storage system and seems to work well several seconds after the onset of the visual stimulus for OKR (reviewed in Fukushima et al. 1992). The visual stimulus for OKR is not very different from that for OFR. Inoue et al. (2000) reported that cells in the nucleus of the optic tract (NOT) in the indirect pathway change their firing frequency during OFR. On the other hand, it is reported that the destruction of the VPFL (a part of the direct pathway) drastically reduces the OFR magnitude (Miles et al. 1986). These reports suggest that the indirect pathway is somewhat involved, but not significantly, in the control of OFR. The firings of the indirect pathway are fed into the IO nucleus, and are then conveyed to P cells by CFs, which in turn cause CSs. Actually, the firing frequency of the indirect pathway is similar in waveform to the firing frequency of CFs, although the firing frequency is much higher (Inoue et al. 2000). We therefore modeled the firing frequency of the indirect pathway by multiplying the firing frequency of CFs by 16 (Fig. 2). This model induces a 1°/s eye velocity with a 10°/s stimulus without the MST-DLPN-VPFL pathway. The firing frequency of the indirect pathway is added to the SSs of P cells below the VPFL and before the filters described in the preceding text (Fig. 2).

Model of synaptic plasticity in the cerebellar cortex

To quantitatively simulate the changes of synaptic weights according to the physiologically known types of synaptic plasticity summarized in Table 1 (LTD, LTP, and RP), we had to model the "temporal window" explicitly. Two detailed studies on this are available. Chen and Thompson (1995) recorded changes in field potentials after stimulating GCAs and CFs with a wide range of time differences between their activations (from 250 to 250 ms). They reported that LTD was maximally induced when GCA inputs were excited 250 ms before CF activation. Karachot et al. (1994) estimated the LTD size by recording changes in extracellular voltage, while periodically stimulating GCAs and CFs with a fixed time delay (GCAs were always delayed in their description). They reported that LTD occurred almost uniformly when the GCA inputs followed the CF input within a 2-s interval.

Therefore the conclusions of these two reports are in sharp contrast to one another. For our simulation, first, we use the Gaussian temporal window, which has its center 200 ms before the CF input and has a SD of 50 ms. This is similar to the least-square fitting of a Gaussian function to the data of Chen and Thompson (1995), which results in a center at 211 ms before the CF input and an SD of 53 ms. At the end of the simulation, we also attempt to use the temporal window, which is based on the interpretation of Karachot et al. (1994). It is spread uniformly between 0 and 2,000 ms after the CF input. The areas below the two temporal windows are both 1. This means that the LTD change induced by one CS firing is the same between the two temporal windows. There have been no reports on temporal windows for LTP and RP. Accordingly, we use the LTD temporal windows for LTP and RP in our simulations.

The following equation represents the change in synaptic weight of GCA inputs in LTD and LTP during each stimulus presentation:

(2)

Here

(3)

In Eq. 2, _GCA(t) denotes the synaptic weight between GCAs and P cells at time t. Here, the time origin is set at the stimulus motion onset for each trial. The first term is for the change caused by LTD. GCA(t) and CF(t) denote the firing frequency at time t for the GCA inputs and the CF inputs, respectively. CFsp, 1/n_LTD, and G(t) are the spontaneous firing frequencies of the CF inputs, the LTD coefficient, and the temporal window, respectively. The intuitive interpretation of the temporal window is easily understood. Each CF input spike occurring at time t' opens a temporal window described by G(t  t'), within which, if a GCA spike falls at time t, LTD is induced and its magnitude is proportional to the amplitude of G(t  t') at that time difference t  t'. LTD decreases _GCA(t), in proportion to the product of the CF increase from its spontaneous level and GCA inputs. The second term represents the change caused by LTP. 1/n_LTP denotes the coefficient for change caused by LTP. LTP increases _GCA(t), in proportion to the product of the CF decrease from its spontaneous level and GCA inputs. The third term represents the exponential self-decay effect to the initial weight ₀. We estimate the time constant  to be (4.67 × 10⁴) [s] based on an experimental report stating that the experimentally induced OFR gain change diminishes by 1/3 in one night (Miles and Kawano 1986).

The following equation represents the change in the synaptic weight of IC inputs in RP

(4)

In Eq. 3, _IC(t) denotes the synaptic weight between IC and P cells at time t, which is negative. The first term is due to RP. IC(t) denotes the firing frequency at time t for IC. 1/n_RP is the RP coefficient. RP increases the absolute value of _IC(t) (and decreases the value). The second term represents the self decay. Equation 3 does not contain a term corresponding to the second term in Eq. 2 because synaptic weights are not affected by IC stimulation in the absence of CFs (Table 1) (Kano 1996).

	METHODS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION REFERENCES

The initial synaptic weights of 1,080 GCA and 1,080 IC inputs into 40 P cells, i.e., ₀ in Eqs. 2 and 3, were prepared as random numbers in the "inborn state," which simulates the P cells of an inborn monkey. The synaptic weight of each excitatory synapse was randomly chosen from a uniform distribution between 0.02 and 0.04. Similarly, the synaptic weight of each inhibitory synapse was randomly chosen from a uniform distribution between 0.02 and 0.04. With these initial synaptic weights, only a slight SS firing modulation was produced by visual motion because the summation of the excitatory inputs and inhibitory inputs roughly canceled each other out.

The acquisition of normal OFR behaviors and SS firings was simulated by repetitively presenting simple ramp stimuli of various directions and speeds. Adaptations of OFR were simulated by repetitive presentations of velocity- and direction-step stimuli. Data recorded in a physiological experiment (Kobayashi et al. 1998) were used as the ramp stimuli at 10, 20, 40, and 80°/s. The ramp stimuli at other velocities, and the speed-step and direction-step stimuli, were synthesized from the preceding experimental data. A visual stimulus was fed into the model every 1 ms, and the model generated MST, SS, and CS firing frequencies and eye movements every 1 ms. The visual stimulus was given for 300 ms, and the simulation was run for 350 ms because the latency from the visual stimulus to the eye movement was about 50 ms (Miles and Kawano 1986). After simulating the neural firing frequency for one trial with fixed synaptic weights, changes in synaptic weights were computed by Eqs. 2 and 3, and the new values were used for the next trial. There are no experimental reports on n_LTD,LTP,RP in Eqs. 2 and 3. Thus we used n that enables the simulated gain to near "1" after repeating the ramp stimuli: (2.33 × 10¹¹) for LTP, (1.04 × 10¹²) for LTD, and (1.04 × 10¹²) for RP. As a test stimulus to examine changes in firing and eye movements, we used a ramp stimulus for 150 ms at 10°/s, called the test ramp stimulus. Matlab (MathWorks) was used on a Sun workstation.

In behavioral experiments on adaptation, 21,000 OFRs in four directions were elicited over 3 days while eliciting OFR gain changes (Miles and Kawano 1986). To simulate OFR acquisition by visual experiences after birth, a comparable number of repetitive visual stimuli presentations for five days were used, specifically, 36,000 ramp trials in four directions. A total of 900 trials of a 300-ms ramp stimulus were repeated at 10 different velocities, i.e., every 10°/s from 10 to 100°/s. The four stimulus directions were down, left, up, and right. The 900 trials × the 10 velocities × the four directions equals 36,000 trials.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION REFERENCES

Acquisition of the ability to control OFR after birth

After the 36,000 ramp trials, eye movements elicited by test stimuli in any direction increased stably to normal gain OFR in that direction because the synaptic weights asymptotically approached stable equilibrium values. For example, the maximum downward eye velocity elicited by the downward test ramp as a function of the learning trial count is shown in Fig. 5A. The mean modulation in the CF frequency of 20 v-P cells over the 350 ms after the onset of the upward stimulus motion of 10°/s decreased to about one-half the initial level as the trial count increased (Fig. 5B). Therefore the "error-signal" encoded by CF modulation was diminished by learning. However, the CF modulation remained even after sufficient learning because there is no way to suppress the initial retinal error that appears before the eye movements. Figure 5, C-F, shows the average SS temporal waveforms of 10 left v-P cells (C and E) and the corresponding eye movements (D and F) during the downward test stimulus, before (C and D) and after learning (E and F). Before repeating the ramp trials (at birth), the SS firing frequency of none of the v-P cells is well modulated during any of the visual stimuli (Fig. 5C); therefore the eyes did not move much (Fig. 5D). After repeating the ramp trials, the SS firing frequencies of the v-P cells were well modulated (Fig. 5E); the eyes followed the visual motion quite well (Fig. 5F). Figure 5G plots the SS frequency averaged over the time interval between 50 and 150 ms of stimulus motion onset as a function of the vertical stimulus velocity. The figure compares experimental data from adult monkeys (Kobayashi et al. 1998) with the average of simulated results from 10 P cells at the end of OFR acquisition. The simulated results denoted by "*" reproduced the experimental data () well. The preferred velocity of SSs became the high velocity of 80°/s for all of the P cells, in agreement with the experimental data (Shidara and Kawano 1993).

View larger version (19K): [in this window] [in a new window]   Fig. 5. Simulation results of OFR acquisition. A: change in the maximum eye velocity with a downward 10°/s ramp stimulus during repeated ramp trials. The abscissa denotes the trial number of the presented ramp stimuli. B: the mean modulation in the complex spike (CS) firing frequency of 20 v-P cells during repeated ramp trials. The ordinate denotes the average increase in CS firing frequency during the 350 ms after the onset of the 10°/s upward ramp stimulus. The abscissa denotes the trial number of the ramp stimuli. C and D: the SS firing frequency pattern (C) and the temporal pattern of eye velocity (D) with a downward 10°/s test ramp before repeated ramp trials. The negative signs in the ordinates of D and F indicate a downward eye velocity. E and F: the SS firing frequency pattern (E) and the temporal pattern of eye velocity (F) with a downward 10°/s test ramp after repeated ramp trials. G: comparison of the results of the simulations (*) and experiments () (Kobayashi et al. 1998) on the dependence of SS mean firing rates on the stimulus velocity.

Figure 6A shows the acquired temporal firing frequency patterns of SSs and CSs of 10 left v-P cells in response to the upward test ramp. Although the amplitudes of the SS modulation varied greatly, the temporal waveforms of SSs and CSs were very stereotypical. Furthermore, P cells with large CS modulations exhibited large SS modulations. Similarly, P cells with smaller CS modulations exhibited smaller SS modulations. There was a statistically significant cell-to-cell negative correlation (0.92, P = 0.001) between the maximum modulations of SSs and the modulations in the mean firing frequency of CSs over a period of 50-150 ms after stimulus onset (Fig. 6B). This statistically significant negative correlation of individual cell characteristics between SS and CS modulations has also been found in monkeys (Kobayashi et al. 1998).

View larger version (31K): [in this window] [in a new window]   Fig. 6. Reciprocal relationships between simulated SS and CS firing frequencies as a result of the simulation. A: temporal firing frequency patterns of the SSs (top, blue curves) and CSs (bottom, red curves) of 10 v-P cell models on the left side with an upward 10°/s ramp stimulus after repeated ramp trials. The patterns of CSs are indicated as increases from their spontaneous firing frequencies. B: maximum modulation (decrease) of the SS firing frequencies plotted against the average modulation (increase) of CSs during the 50-150 ms after stimulus onset for each of the 10 P cells during the upward 10°/s ramp stimulus. The solid line represents the linear regression of the data (n = 10; r = 0.92).

Figure 7 shows the directional selectivity of the four groups of P cells before and after repeating the 36,000 ramp stimuli. In Fig. 7, the blue and red circles connect points that show the maximum and minimum SS firing rates in response to the test ramp in a given direction on a polar plot, before and after learning, respectively. The red arrows are vectorial sums of all of the red circle points and show the preferred directions for the stimulus motion. Before learning, the SSs were not modulated to any stimulus direction (blue circles); the population vector was almost zero. Therefore blue arrows are not seen. After learning, however, the mean preferred directions of the left and right v-P cells became downward and were 266.87 ± 6.17° and 260.85 ± 14.61° (mean ± SD), respectively (Fig. 7, A and B) when 0° is defined as rightward and the angle is measured in a counterclockwise rotation. The mean preferred direction of the left h-P cells became leftward at 176.98 ± 12.94° (Fig. 7C). The preferred direction of the right h-P cells became rightward at 353.78 ± 8.43° (Fig. 7D). As can be seen from the red circles, the directional tuning curves of all of the P cells after OFR acquisition were well-fitted by full cosine functions in agreement with the experimental data of Shidara and Kawano (1993). The acquired preferred direction of SSs was opposite to that of CSs for each P cell. This agreed well with the monkey data (Kobayashi et al. 1998).

View larger version (43K): [in this window] [in a new window]   Fig. 7. Direction-dependent SS activation and preferred directions of the P cells before (blue circles) and after (red circles and arrows) repeated ramp trials. A-D: the averages of 10 left v-P cells, 10 right v-P cells, 10 left h-P cells, and 10 right h-P cells, respectively. The circles denote the SS frequency averaged over the time interval between 50 and 150 ms after the stimulus motion onset for stimuli in 12 different directions. When the maximum modulation was 100 spikes/s, a plotted point appears on the outer filled circle. The red arrows denote the average preferred directions for each of the 4 groups of P cells. They are defined as 2-dimensional vectorial summations of 12 points and are enlarged by a scale of 5 for clear viewing.

Figure 8, A and B, represents the changes in synaptic weights from GCAs and ICs. The average weights of the synapses are graphically represented as the diameters of the open small circles (excitatory) and filled small circles (inhibitory). The relative position of each small circle with respect to the location of the P cell (shown as an open large circle) represents the preferred direction for that synaptic input. Before repeating the stimuli (Fig. 8A), the average weights of the excitatory and inhibitory synapses were very similar among all of the preferred directions. Although the initial synaptic weight for each individual synapse was chosen randomly, the displayed averages were taken from 900 synapses (10 P cells, 30 preferred velocities, 3 waveforms), and gave similar values. Due to these spatially uniform weights, no P cell exhibited strong SS modulation to any stimulus before learning (Fig. 5C). After learning (Fig. 8B), the excitatory and inhibitory synapses changed their weights. LTP increased the excitatory synaptic weights (graphically, enlarging the open circles) and induced SS enhancement with the stimulus in the preferred direction of the GCA inputs. LTD decreased the excitatory synaptic weights (graphically, shrinking the open circles) and induced SS suppression with the stimulus in the no-preferred direction of GCA. RP increased the inhibitory synaptic weights (graphically, enlarging the filled circles) and induced SS suppression with the stimulus in the nonpreferred direction of IC inputs. Overall, the synaptic weight changes shown in Fig. 8B built up the downward preferred direction for v-P cells, the leftward preferred direction for left h-P cells, and the rightward preferred direction for right h-P cells, which are shown in Fig. 7.

View larger version (20K): [in this window] [in a new window]   Fig. 8. Visualized presentation of synaptic weights on P cells before (A) and after (B) learning and the synaptic plasticity mechanisms involved in this change (C). The large circles denote P cells. The diameters of the open and filled small circles denote the synaptic weights with the transformation described below. For clarity, the diameters of the circles were determined from the corresponding synaptic weights  as follows: diameter = (  0) × 100 + 0. Here, 0 was the initial synaptic weight. The open, small circles are average weights of granule cell axons (GCA) excitatory synapses, and the filled, small circles are average weights of inhibitory inhibitory cell (IC) synapses. The positions of the small circles with respect to the open, large circles indicate the preferred directions of these synaptic inputs. The arrows within the large circles (P cells) show the acquired mean preferred directions of SSs. A: before repeated ramp trials, the average synaptic weights were similar in all directions for all P cells. B: after repeated ramp trials, a-d: the results for the left v-P cell, right v-P cell, left h-P cell, and right h-P cell, respectively. IO, inferior olive, and the 4 arrows within the squares at the center of B denote the preferred directions of the CF inputs. C: simulated GCA, IC, and CF firing frequency patterns and synaptic plasticity mechanisms in the "inborn state" model during downward and upward stimuli. The 1st and 2nd rows show temporal patterns of GCAs and ICs with downward and upward preferred directions, respectively. While 2,160 GCAs and ICs exist in the model, only the 2 with preferred velocities of 10°/s, with a small phasic firing and with an upward or downward preferred direction, are shown. The 3rd row shows the temporal patterns of CF inputs for v-P cells. Left and right: temporal firing frequency patterns with downward and upward stimuli, respectively. The 2 broken-line arrows indicate that the designated synaptic plasticity is induced by the interaction between the GCA/IC inputs and CF inputs. D: changes in synaptic weight for each preferred velocity. The ordinate denotes the average change of synaptic weights from the initial weight. The abscissa denotes preferred velocity of the GCA/IC inputs. The averaging was computed for 2,880 synapses (40 P cells, excitatory and inhibitory, 12 directions, 3 firing patterns) for each preferred velocity.

Figure 8C schematically illustrates how the modifications of the excitatory and inhibitory synaptic weights shown in Fig. 8B were induced by the interaction between the GCA and IC inputs and the CF input for a v-P cell. The temporal firing patterns of the GCA and IC inputs with preferred downward and upward directions are shown in the first and second rows, respectively, in Fig. 8C. The third row shows the temporal pattern of the CF input with an upward preferred direction for the v-P cell. The left and right columns show the temporal firing patterns when the stimulus moved downward and upward, respectively. When the stimulus was downward (left), the CF firing frequency decreased below the spontaneous firing frequency (3rd row, left). Then, the conditions for the induction of LTP (summarized in Table 1) were achieved for the GCA synapses with a downward preferred direction as shown by the broken-line arrow in the left column. Because we used the temporal window that occurs mainly before CF modulation, the broken-line arrow goes from the future in CF time to the past in GCA input time, indicating that the GCA inputs preceding the CF inputs underwent LTP. When the stimulus was upward (right), the CF firing frequency increased (3rd row, right). Then, the conditions for the induction of LTD were satisfied for the GCA synapses with an upward preferred direction. Furthermore, the conditions for the induction of RP were satisfied for the IC synapses with an upward preferred direction. The induction of LTD and RP is schematically shown by the broken-line arrow in the second column. When the stimulus was leftward or rightward, the firing frequency of CFs of the v-P cell was not modulated. As a result, no change in synaptic weight was induced for either the GCA or IC inputs. The synaptic weights of the GCA and IC inputs with leftward or rightward preferred directions were not modified because the CF input was not statistically modulated on average when the GCA and IC inputs were modulated. This is because the 36,000 stimulus motions were statistically uniform in their directional distributions.

Very similar combinations of LTP, LTD, and RP resulted in the synaptic weight distributions shown in Fig. 8Bc for left h-P cells and right h-P cells. For h-P cells, the CF input was modulated only by a horizontal component of visual motion. The CF firing increased with contralateral stimuli and decreased with ipsilateral stimuli. Therefore LTP was induced by synapses from GCAs with preferred ipsilateral directions. LTD and RP were induced by synapses from GCAs and ICs with preferred contralateral directions. Synapses from GCAs and ICs with preferred upward or downward directions were not modified because the CF inputs to the h-P cells were not modulated by vertical visual motion.

Figure 8D shows the average weight of synapses at each preferred velocity. The average change in synaptic weight from the initial values was taken from 1,440 synapses (40 P cells, 12 preferred directions, 3 waveforms). Figure 8D reveals a relationship like an exponential decrease in synaptic weight for the inputs at each preferred velocity. The weights for low preferred velocities were high, and those for high velocities were low. These exponentially learned synaptic weights might be the origin of the high preferred velocity of SSs and the saturation of SSs that is simulated in Fig. 5G. The velocity of retinal error was almost the same as the visual stimulus velocity 50 ms after stimulus onset. Meanwhile, the retinal error velocity decreased 50-150 ms after stimulus onset because of the eye movement. This might mean that firing occurs not only in inputs with preferred velocities the same as the stimulus velocity but also in inputs with preferred velocities smaller than the stimulus velocity. The integration of the exponential decrease from 0 to the stimulus velocity increases logarithmically. Thus the firings might show a logarithmic increase when the synaptic weights show an exponential spread at the preferred velocity. Such logarithmic increases might cause the high preferred velocity of SSs and the saturation of SSs.

After OFR acquisition, simulated SS firing was reconstructed by the inverse dynamics model of simulated eye movement, that is, by a linear combination of acceleration, velocity, and position. This analysis followed previous monkey studies (Gomi et al. 1998; Shidara et al. 1993). The estimated coefficients of acceleration, velocity, and position were 0.112 ± 0.074 (mean ± SD), 2.61 ± 1.442, and 6.97 ± 4.27, respectively. The squared correlation coefficient between the simulated firing and the reconstructed firing was 0.993 ± 0.006. The ratio of the acceleration/velocity coefficients was 0.0441 ± 0.0125. This was in good agreement with the ratio (0.0502) obtained from the experimental data that we used to construct the model (Yamamoto et al. 1997). Consequently, our model succeeded in simulating the acquisition of OFR, given the temporal window of plasticity that spread mainly before CF was used. From this point on, the set of synaptic weights obtained after 36,000 ramp trial repetitions will be called the "adult state," because our model with this set captured well the OFR characteristics and P-cell firings of adult monkeys.

Adaptation of OFR gain

After the simulation of OFR acquisition, the step-up stimuli (upward and rightward) and the step-down stimuli (downward and leftward) were presented to the model, following the protocol of a monkey experiment (Miles and Kawano 1986). For a detailed explanation of the speed-step stimuli, see Ocular following responses: behavioral and neurophysiological studies. A total of 250 trials were repeated for each of seven velocities 10, 20, 30, 40, 60, 80, and 100°/s, in one of four directions. Altogether, 7,000 trials were repeated, just as in the monkey adaptation experiment in one day (Miles and Kawano 1986). We used the final synaptic weights _GCA and _IC obtained in the OFR acquisition simulation, that is, the adult state as the initial values of synaptic weights ₀ in Eqs. 2 and 3.

Figure 9, A and B, shows the behaviors of the model system during the speed-step adaptation simulation to the downward and upward test ramps, respectively. The gain to the downward test ramp decreases from 0.92 to 0.54 (Fig. 9Aa), while the gain to the upward test ramp increases from 0.99 to 1.77 (Fig. 9Ba). In response to the downward test ramp (Fig. 9A), the SSs decrease their firing frequency (from b to c), and the downward eye movement becomes smaller (from d to e). On the other hand, in response to the upward test ramp (Fig. 9B), the SSs increase their firing suppression (from b to c) and the upward eye movement becomes larger (from d to e).

View larger version (32K): [in this window] [in a new window]   Fig. 9. Simulation results for gain adaptation. A: changes observed for a 10°/s downward test ramp. B: changes observed for a 10°/s upward test ramp. a: changes in the OFR gain as a function of the speed-step trial number. b and c: average SS firing frequencies of 20 v-P cells before (b) and after (c) repeated speed-step trials. d and e: temporal patterns of eye velocities before (d) and after (e) repeated speed-step trials. A positive eye velocity indicates upward motion. From b to e, the abscissa denotes the time after the stimulus motion onset in ms. C: simulated GCA, IC, and CF firing frequency patterns and synaptic plasticity mechanisms in the "adult state" model during 40°/s speed-step stimuli. The format of this figure is similar to that of Fig. 8C. The 1st and 2nd rows show temporal patterns of GCAs and ICs with preferred downward and rightward directions, respectively, and a preferred velocity of 100°/s. Left and right: temporal firing frequency patterns with downward step-down and upward step-up stimuli, respectively.

Figure 9C shows the simulated GCA, IC, and CF inputs to a v-P cell of the adult state model at the beginning of speed-step adaptation and schematically illustrates how modifications of the excitatory and inhibitory synaptic weights were induced by the interaction between these inputs. The format of Fig. 9C is similar to that of Fig. 8C. The left and right columns correspond to the downward step-down stimulus and upward step-up stimulus, respectively. For the downward step-down stimulus, the CS firing frequency decreased below its spontaneous level for the first 150 ms. However, it increased over its spontaneous level for the next 150 ms because the downward eye velocity surpassed the reduced stimulus velocity and generated the upward retinal error (3rd row, left). GCAs and ICs with downward preferred directions increased their firing frequencies for the first 150 ms (1st row, left). As summarized in Table 1, LTD and RP were induced at the synapses from GCAs and ICs that had a downward preferred direction and fired in the first 150 ms. The broken-line arrow indicates that LTD and RP were induced between the preceding GCA and IC activities with a subsequent increase in CF input according to the temporal window that occurs before CF modulation. The LTD and RP that occurred on synapses with a downward preferred direction reduced the SS firing frequency in v-P cells for the downward stimulus and decreased the eye movement gain (Fig. 9A). Here, we must note that any synaptic plasticity induced by the CF activity occurring in the first 150 ms need not be considered here for adaptation because its influence has already been incorporated in the equilibrium adult state of synaptic weights. In other words, its effects have already been balanced with the other terms in Eqs. 2 and 3 at the beginning of the adaptation simulation.

For the upward step-up stimulus, the CSs increased their firing frequency above the spontaneous level for 300 ms, especially in the last 150 ms (Fig. 9C, 3rd row, right). This is because the upward stimulus velocity was increased, and a larger retinal error was generated. The GCAs and ICs with upward preferred directions increased their firing frequencies over the entire 300 ms (2nd row, right). As summarized in Table 1, LTD and RP were induced at synapses from these GCAs and ICs (broken-line arrow) because of the temporal window that occurs mainly before CF modulation. These instances of LTD and RP increased the magnitude of the SS suppression induced by the upward test ramp in the v-P cells and increased the eye movement gain (Fig. 9B).

In essentially the same manner, the gain to the leftward test ramp decreased from 0.89 to 0.69, and the gain to the rightward test ramp increased from 0.95 to 1.65 (data not shown). When the stimulus was a leftward step-down stimulus, LTD and RP were induced in left h-P cells on synapses from GCAs and ICs with preferred leftward directions. These instances of LTD and RP decreased the SS firing frequency in the left h-P cells and decreased the OFR gain to the leftward stimulus. When the stimulus was the rightward step-up stimulus, LTP was induced in right h-P cells on synapses from GCAs with a preferred rightward direction. This LTP increased the SS firing frequency in the right h-P cells and increased the OFR gain to the rightward stimulus. Consequently, our model with adult-state initial synaptic weights reproduced the main characteristics of the gain adaptation experiment by Miles and Kawano (1986)