Effect of Diverse Recoding of Granule Cells on Optokinetic Response in A Cerebellar Ring Network with Synaptic Plasticity
EEffect of Diverse Recoding of Granule Cells on Optokinetic Response in A CerebellarRing Network with Synaptic Plasticity
Sang-Yoon Kim ∗ and Woochang Lim † Institute for Computational Neuroscience and Department of Science Education,Daegu National University of Education, Daegu 42411, Korea
We consider a cerebellar ring network for the optokinetic response (OKR), and investigate theeffect of diverse recoding of granule (GR) cells on OKR by varying the connection probability p c fromGolgi to GR cells. For an optimal value of p ∗ c (= 0 . L g , corresponding to the modulation gain ratio, increases with increasing the learning cycle, and itsaturates at about the 300th cycle. By varying p c from p ∗ c , we find that a plot of saturated learninggain degree L ∗ g versus p c forms a bell-shaped curve with a peak at p ∗ c (where the diversity degree inspiking patterns of GR cells is also maximum). Consequently, the more diverse in recoding of GRcells, the more effective in motor learning for the OKR adaptation. PACS numbers: 87.19.lw, 87.19.lu, 87.19.lvKeywords: Optokinetic response, Cerebellar ring network, Diverse recoding, Effective long-term depression,Effective motor learning
I. INTRODUCTION
The cerebellum receives information from the sensorysystems, the spinal cord and other parts of the brain andthen regulates motor movements. For a smoothly inte-grated body movement, the cerebellum activates a largeset of spatially separated muscles in a precise order andtiming. Thus, the cerebellum plays an essential role infine motor control (i.e., precise spatial and temporal mo-tor control) for coordinating voluntary movements suchas posture, balance, and locomotion, resulting in smoothand balanced muscular activity [1–3]. Moreover, it isalso involved in higher cognitive functions such as timeperception and language processing [2, 3]. Animals andhumans with damaged cerebella are still able to initiatemovements, but these movements become slow, inexact,and uncoordinated [4, 5].The spatial information of movements (e.g., amplitudeor velocity) is called “gain,” while the temporal infor-mation of movements (e.g., initiation or termination) iscalled “timing” [6]. The goal of cerebellar motor learn-ing is to perform precise gain and temporal control formovements. The cerebellar mechanisms for gain and tim-ing control for eye movements have been studied in thetwo types of experimental paradigms; (1) gain controlfor the optokinetic response (OKR) and the vestibulo- ∗ Electronic address: [email protected] † Electronic address: [email protected] ocular reflex [1, 7] and (2) timing control for the eyeblinkconditioning [8, 9]. Here, we are concerned about gainadaptation of OKR. When the eye tracks a moving objectwith the stationary head, OKR may be seen. When themoving object is out of the field of vision, the eye movesback rapidly to the original position where it first saw.In this way, OKR consists of two consecutive slow andfast phases. Experimental works on OKR in vertebratessuch as rabbits, mice, and zebrafishes have been done indiverse aspects [10–17].In the Marr-Albus-Ito theory for cerebellar computa-tion [1, 18, 19], the cerebellum is considered to act asa simple perceptron (i.e., pattern associator) which as-sociates input [mossy fiber (MF)] patterns with output[Purkinje cell (PC)] patterns. The input patterns becomemore sparse and less similar to each other via recodingprocess in the granular layer, consisting of the granule(GR) and the Golgi (GO) cells. Then, the recoded inputsare fed into the PCs via the parallel fibers (PFs) (i.e., theaxons of GR cells). In addition to the PF recoded signals,the PCs also receive the error-teaching signals throughthe climbing-fiber (CF) from the inferior olive (IO). ThePF-PC synapses are assumed to be the only synapses atwhich motor learning occurs. Thus, synaptic plasticitymay occur at the PF-PC synapses (i.e., their synapticstrengths may be potentiated or depressed). Marr in [18]assumes that a Hebbian type of long-term potentiation(LTP) (i.e., increase in synaptic strengths) occurs at thePF-PC synapses when both the PF and the CF signalsare conjunctively excited [20, 21]. This Marr’s theory(which directly relates the cerebellar function to its struc- a r X i v : . [ q - b i o . N C ] A ug ture) represents a milestone in the history of cerebellum[22]. In contrast to Marr’s learning via LTP, Albus in [19]assumes that synaptic strengths at PF-PC synapses aredepressed [i.e., an anti-Hebbian type of long-term depres-sion (LTD) occurs] in the case of conjunctive excitationsof both the PF and the CF signals. In the case of Albus’learning via LTD, PCs learn when to stop their inhibi-tion (i.e. when to disinhibit) rather than when to fire. Inseveral later experimental works done by Ito et al., clearevidences for LTD were obtained [24, 25, 27]. Thus, LTDbecame established as a unique type of synaptic plasticityfor cerebellar motor learning [26, 28–30].In addition to experimental works on the OKR [10–17],computational works have also been performed [6, 31].The Marr-Albus model of the cerebellum was also re-formulated to incorporate dynamical responses in termsof the adaptive filter model (used in the field of engi-neering control) [32, 33]. The cerebellar structure maybe mapped onto an adaptive filter structure. Throughanalysis-synthesis process of the adaptive filter model.the (time-varying) filter inputs (i.e., MF “context” sig-nals for the post-eye-movement) are analyzed into diversecomponent signals (i.e., diversely recoded PF signals).Then, they are weighted (i.e., synaptic plasticity at PF-PC synapses) and recombined to generate the filter out-put (i.e., firing activity of PCs). The filter is adaptivebecause its weights are adjusted by an error-teaching sig-nal (i.e., CF signal), employing the covariance learningrule [34]. Using this adaptive filter model, gain adapta-tion of OKR was successfully simulated [31]. Recently,Yamazaki and Nagao in [6] employed a spiking networkmodel, which was originally introduced for Pavlovian de-lay eyeblink conditioning [35]. As elements in the spik-ing network, leaky integrate-and-fire neuron models wereused, and parameter values for single cells and synapticcurrents were adopted from physiological data. Througha large-scale computer simulation, some features of OKRadaptation were successfully reproduced.However, the effects of diverse recoding of GR cellson the OKR adaption in previous computational worksare still needed to be more clarified in several dynam-ical aspects. First of all, dynamical classification of di-verse PF signals (corresponding to the recoded outputs ofGR cells) must be completely done for clear understand-ing their association with the error-teaching CF signals.Then, based on such dynamical classification of diversespiking patterns of GR cells, synaptic plasticity at PF-PC synapses and subsequent learning progress could bemore clearly understood. As a result, understanding onthe learning gain and the learning progress for the OKRadaptation is expected to be so much improved.To this end, we consider a cerebellar spiking ring net-work for the OKR adaptation, and first make a dy-namical classification of diverse spiking patterns of GRcells (i.e., diverse PF signals) by changing the connectionprobability p c from GO to GR cells in the granular layer.An instantaneous whole-population spike rate R GR ( t )(which is obtained from the raster plot of spikes of indi- vidual neurons) may well describe collective firing activ-ity in the whole population of GR cells [36–43]. R GR ( t )is in-phase with respect to the sinusoidally-modulatingMF input signal for the post-eye-movement, although ithas a central flattened plateau due to inhibitory inputsfrom GO cells.The whole population of GR cells is divided into GRclusters. These GR clusters show diverse spiking patternswhich are in-phase, anti-phase, and complex out-of-phaserelative to the instantaneous whole-population spike rate R GR ( t ). Each spiking pattern is characterized in termsof the “conjunction” index, denoting the resemblance (orsimilarity) degree between the spiking pattern and theinstantaneous whole-population spike rate R GR ( t ) (cor-responding to the population-averaged firing activity).To quantify the degree of diverse recoding of GR cells,we introduce the diversity degree D , given by the rel-ative standard deviation in the distribution of conjunc-tion indices of all spiking patterns. We mainly consideran optimal case of p ∗ c (= 0 .
06) where the spiking pat-terns of GR clusters are the most diverse. In this case, D ∗ (cid:39) .
613 which is a quantitative measure for diverserecoding of GR cells in the granular layer. We also inves-tigate dynamical origin of these diverse spiking patternsof GR cells. It is thus found that, diverse total synapticinputs (including both the excitatory MF inputs and theinhibitory inputs from the pre-synaptic GO cells) intothe GR clusters result in production of diverse spikingpatterns (i.e. outputs) in the GR clusters.Next, based on dynamical classification of diverse spik-ing patterns of GR clusters, we employ a refined rulefor synaptic plasticity (constructed from the experimen-tal result in [23]) and make an intensive investigationon the effect of diverse recoding of GR cells on synapticplasticity at PF-PC synapses and the subsequent learn-ing process. PCs (corresponding to the output of thecerebellar cortex) receive both the diversely-recoded PFsignals from GR cells and the error-teaching CF signalsfrom the IO neuron. We also note that the CF signalsare in-phase with respect to the instantaneous whole-population spike rate R GR ( t ). In this case, CF signalsmay be regarded as “instructors,” while PF signals canbe considered as “students.” Then, in-phase PF studentsignals are strongly depressed (i.e., their synaptic weightsat PF-PC synapses are greatly decreased through strongLTD) by the in-phase CF instructor signals. On theother hand, out-of-phase PF student signals are weaklydepressed (i.e., their synaptic weights at PF-PC synapsesare a little decreased via weak LTD) due to the phasedifference between the student PF and the instructor CFsignals. In this way, the student PF signals are effectively(i.e., strongly/weakly) depressed by the error-teaching in-structor CF signals.During learning cycles, the “effective” depression (i.e.,strong/weak LTD) at PF-PC synapses may cause a bigmodulation in firing activities of PCs, which then ex-ert effective inhibitory coordination on vestibular nucleus(VN) neuron (which evokes OKR eye-movement). Forthe firing activity of VN neuron, the learning gain de-gree L g , corresponding to the modulation gain ratio (i.e.,normalized modulation divided by that at the 1st cycle),increases with learning cycle, and it eventually becomessaturated.Saturation in the learning progress is clearly shown inthe IO system. During the learning cycle, the IO neuronreceives both the excitatory sensory signal for a desiredeye-movement and the inhibitory signal from the VN neu-ron (representing a realized eye-movement). We intro-duce the learning progress degree L p , given by the ratioof the cycle-averaged inhibitory input from the VN neu-ron to the cycle-averaged excitatory input of the desiredsensory signal. With increasing cycle, the cycle-averagedinhibition (from the VN neuron) increases (i.e., L p in-creases), and converges to the constant cycle-averagedexcitation (through the desired signal). Thus, at aboutthe 300th cycle, the learning progress degree becomessaturated at L p = 1. At this saturated stage, the cycle-averaged excitatory and inhibitory inputs to the IO neu-ron become balanced, and we get the saturated learninggain degree L ∗ g ( (cid:39) . p c from p ∗ c (= 0 . L ∗ g versus p c is found to form a bell-shaped curvewith a peak ( L ∗ g (cid:39) . p ∗ c . With increasing ordecreasing p c from p ∗ c , the diversity degree D in firingactivities of GR cells also forms a bell-shaped curve witha maximum value ( D ∗ (cid:39) . p ∗ c . We note that boththe saturated learning gain degree L ∗ g and the diversitydegree D have a strong correlation with the Pearson’scorrelation coefficient r (cid:39) . p c . Finally,we give summary and discussion in Sec. IV. In AppendixB, glossary for various terms characterizing the cerebellarmodel is given to help readers keep track of them. II. CEREBELLAR RING NETWORK WITHSYNAPTIC PLASTICITY
In this section, we describe our cerebellar ring networkwith synaptic plasticity for the OKR adaptation. Figure1(a) shows OKR which may be seen when the eye trackssuccessive stripe slip with the stationary head. Wheneach moving stripe is out of the field of vision, the eyemoves back quickly to the original position where it first (a)
Pre-cerebellarNucleiRetinal Slip InformationRetina Stripe Slip(Visual Motion) (b1) f M F ( t ) ( H z ) t (msec) (b2) f D S ( t ) ( H z ) t (msec) FIG. 1: (a) Optokinetic response when the eye tracks stripeslip. External sensory signals: (b1) firing rate f MF ( t ) of themossy-fiber (MF) context signal for the post-eye-movementand (b2) firing rate f DS ( t ) of the inferior-olive (IO) desiredsignal (DS) for a desired eye-movement. saw. Thus, OKR is composed of two consecutive slowand fast phases (i.e., slow tracking eye-movement andfast reset saccade). It takes 2 sec (corresponding to 0.5Hz) for one complete slip of each stripe. Slip of the visualimage across large portions of the retina is the stimulusthat stimulates optokinetic eye movements, and also thestimulus that produces the adaptation of the optokineticsystem. A. MF Context Signal and IO Desired Signal
There are two types of sensory signals which transferthe retinal slip information from the retina to their tar-gets by passing intermediate pre-cerebellar nuclei (PCN).In the 1st case, the retinal slip information first passesthe pretectum in the midbrain, then passes the nucleusreticularis tegmentis pontis (NRTP) in the pons, and fi-nally it is transferred to the granular layer (consisting ofGR and GO cells) in the cerebellar cortex via MF sen-sory signal containing “context” for the post-eye-movent.The MF context signals are modeled in terms of Poissonspike trains which modulate sinusoidally at the stripe-slip frequency f s = 0 . f MF of Poisson spike trains for theMF context signal is given by f MF ( t ) = − f MF cos(2 πf s t ) + f MF ; f MF = 15 Hz , (1)which is shown in Fig. 1(b1).In the 2nd case, the retinal slip information passes onlythe pretectum, and then (without passing NRTP) it isdirectly fed into to the IO via a sensory signal for a “de-sired” eye-movement. As in the MF context signals, theIO desired signals are also modeled in terms of the samekind of sinusoidally modulating Poisson spike trains atthe stripe-slip frequency f s = 0 . f DS of Poisson spike trains for the IO desiredsignal (DS) is given by: f DS ( t ) = − f DS cos(2 πf s t ) + f DS ; f DS = 1 . , (2)which is shown in Fig. 1(b2). In this case, the peak firingrate for the IO desired signal is reduced to 3 Hz to satisfylow mean firing rates ( ∼ /
10 of the peak firing rate of theMF signal) [45, 46].
B. Architecture for Cerebellar Ring Network
As in the famous small-world ring network [47, 48], wedevelop a one-dimensional ring network with a simple ar-chitecture, which is in contrast to the two-dimensionalsquare-lattice network [6, 35]. This kind of ring net-work has advantage for computational and analytical ef-ficiency, and its visual representation may also be easilymade.Here, we employ such a cerebellar ring network for theOKR. Figure 2(a) shows the box diagram for the cere-bellar network. The granular layer, corresponding to theinput layer of the cerebellar cortex, consists of the ex-citatory GR cells and the inhibitory GO cells. On theother hand, the Purkinje-molecular layer, correspondingto the output layer of the cerebellar cortex, is composedof the inhibitory PCs and the inhibitory BCs (basketcells). The MF context signal for the post-eye-movementis fed from the PCN (pre-cerebellar nuclei) to the GRcells. They are diversely recoded via inhibitory coor-dination of GO cells on GR cells in the granular layer.Then, these diversely-recoded outputs are fed via PFs tothe PCs and the BCs in the Purkinje-molecular layer.The PCs receive another excitatory error-teaching CFsignals from the IO, along with the inhibitory inputs fromBCs. Then, depending on the type of PF signals (i.e.,in-phase or out-of-phase PF signals), diverse PF (stu-dent) signals are effectively depressed by the (in-phase)error-teaching (instructor) CF signals. Such effective de-pression at PF-PC synapses causes a large modulationin firing activities of PCs (principal output cells in thecerebellar cortex). Then, the VN neuron generates thefinal output of the cerebellum (i.e., it evokes OKR eye-movement) through receiving both the inhibitory inputsfrom the PCs and the excitatory inputs via MFs. ThisVN neuron also provides inhibitory inputs for the real-ized eye-movement to the IO neuron which also receivesthe excitatory desired signals for a desired eye-movementfrom the PCN. Then, the IO neuron supplies excitatoryerror-teaching CF signals to the PCs. (a)
PCNDesired SignalEye MovementMF(context signal )PCN CF (error teaching signal)PF
Cerebellar Cortex
Granular Layer
GRGO BCPC IO
Purkinje-MolecularLayer VN GR cell, Glomeruli, GO cell (b) N C PC, BC (c) N P C FIG. 2: Cerebellar Ring Network. (a) Box diagram for thecerebellar network. Lines with triangles and circles denote ex-citatory and inhibitory synapses, respectively. GR (granulecell), GO (Golgi cell), and PF (parallel fiber) in the gran-ular layer, PC (Purkinje cell) and BC (basket cell) in thePurkinje-molecular layer, and other parts for VN (vestibu-lar nuclei), IO(inferior olive), PCN(pre-cerebellar nuclei), MF(mossy fiber), and CF (climbing fiber). (b) Schematic dia-gram for granular-layer ring network with concentric innerGR and outer GO rings. Numbers represent granular layerzones (bounded by dotted lines) for N C = 32. In each I thzone ( I = 1 , · · · , N C ), there exists the I th GR cluster on theinner GR ring. Each GR cluster consists of GR cells (solidcircles), and it is bounded by 2 glomeruli (stars). On theouter GO ring in the I th zone, there exists the I th GO cell(diamonds). (c) Schematic diagram for Purkinje-molecular-layer ring network with concentric inner PC and outer BCrings. Numbers represent the Purkinje-molecular-layer zones(bounded by dotted lines) for N PC = 16. In each J th zone,there exist the J th PC (solid circle) on the inner PC ring andthe J th BC (solid triangle) on the outer BC ring. Figure 2(b) shows a schematic diagram for thegranular-layer ring network with concentric inner GRand outer GO rings. Numbers represent granular-layerzones (bounded by dotted lines). That is, the numbers1, 2, · · · , and N C denote the 1st, the 2nd, · · · , andthe N C th granular-layer zones, respectively. Hence, thetotal number of granular-layer zones is N C ; Fig. 2(b)shows an example for N C = 32. In each I th zone( I = 1 , · · · , N C ), there exists the I th GR cluster on theinner GR ring. Each GR cluster consists of N GR excita-tory GR cells (solid circles). Then, location of each GRcell may be represented by the two indices ( I, i ) whichrepresent the i th GR cell in the I th GR cluster, where i = 1 , · · · , N GR . Here, we consider the case of N C = 2 and N GR = 50, and hence the total number of GR cellsis 51,200. In this case, the I th zone covers the angularrange of ( I − θ ∗ GR < θ < I θ ∗ GR ( θ ∗ GR = 0 . ◦ ). On theouter GO ring in each I th zone, there exists the I th in-hibitory GO cell (diamond), and hence the total numberof GO cells is N C .We note that each GR cluster is bounded by 2glomeruli (corresponding to the axon terminals of theMFs) (stars). GR cells within each GR cluster share thesame inhibitory and excitatory synaptic inputs throughtheir dendrites which contact the two glomeruli at bothends of the GR cluster. Each glomerulus receives in-hibitory inputs from nearby 81 (clockwise side: 41 andcounter-clockwise side: 40) GO cells with a random con-nection probability p c (= 0 . .
1. Thus, 245 PFs (i.e. GR cell axons) innervatea GO cell.Figure 2(c) shows a schematic diagram for thePurkinje-molecular-layer ring network with concentricinner PC and outer BC rings. Numbers denotethe Purkinje-molecular-layer zones (bounded by dottedlines). In each J th zone ( J = 1 , · · · , N PC ), there existthe J th PC (solid circles) on the inner PC ring and the J th BC (solid triangles) on the outer BC ring. Here, weconsider the case of N PC = 16 , and hence the total num-bers of PC and BC are 16, respectively. In this case, each J th ( J = 1 , · · · , N PC ) zone covers the angular range of( J − θ ∗ PC < θ < J θ ∗ PC , where θ ∗ PC (cid:39) . ◦ (correspond-ing to about 64 zones in the granular-layer ring network).We note that diversely-recoded PFs innervate PCs andBCs. Each PC (BC) in the J th Purkinje-molecular-layerzone receives excitatory synaptic inputs via PFs from allthe GR cells in the 288 GR clusters (clockwise side: 144and counter-clockwise side: 144 when starting from theangle θ = ( J − θ ∗ PC in the granular-layer ring network).Thus, each PC (BC) is synaptically connected via PFs to the 14,400 GR cells (which corresponds to about 28 % ofthe total GR cells). In addition to the PF signals, eachPC also receives inhibitory inputs from nearby 3 BCs(central side: 1, clockwise side: 1 and counter-clockwiseside: 1) and excitatory error-teaching CF signal from theIO.Outside the cerebellar cortex, for simplicity, we con-sider just one VN neuron and one IO neuron. Both ex-citatory inputs via 100 MFs and inhibitory inputs fromall the 16 PCs are fed into the VN neuron. Then, theVN neuron evokes the OKR eye-movement and suppliesinhibitory input for the realized eye-movement to the IOneuron. One additional excitatory desired signal fromthe PCN is also fed into the IO neuron. Then, throughintegration of both excitatory and inhibitory inputs, theIO neuron provides excitatory error-teaching CF signalsto the PCs. C. Leaky Integrate-And-Fire Neuron Model withAfterhyperpolarization Current
As elements of the cerebellar ring network, we chooseleaky integrate-and-fire (LIF) neuron models which in-corporate additional afterhyperpolarization (AHP) cur-rents that determine refractory periods [49]. This LIFneuron model is one of the simplest spiking neuron mod-els. Due to its simplicity, it can be easily analyzed andsimulated. Thus, it has been very popularly used as aneuron model.The following equations govern dynamics of states ofindividual neurons in the X population: C X dv ( X ) i dt = − I ( X ) L,i − I ( X ) AHP,i + I ( X ) ext − I ( X ) syn,i , i = 1 , · · · , N X , (3)where N X is the total number of neurons in the X pop-ulation, X = GR and GO in the granular layer, X = PCand BC in the Purkinje-molecular layer, and in the otherparts X = VN and IO. In Eq. (1), C X (pF) represents themembrane capacitance of the cells in the X population,and the state of the i th neuron in the X population at atime t (msec) is characterized by its membrane potential v ( X ) i (mV). The time-evolution of v ( X ) i ( t ) is governed by4 types of currents (pA) into the i th neuron in the X population; the leakage current I ( X ) L,i , the AHP current I ( X ) AHP,i , the external constant current I ( X ) ext (independentof i ), and the synaptic current I ( X ) syn,i .We note that the equation for a single LIF neuronmodel [without the AHP current and the synaptic currentin Eq. (3)] describes a simple parallel resistor-capacitor(RC) circuit. Here, the leakage term is due to the resistorand the integration of the external current is due to thecapacitor which is in parallel to the resistor. Thus, inEq. (3), the 1st type of leakage current I ( X ) L,i for the i thneuron in the X population is given by: I ( X ) L,i = g ( X ) L ( v ( X ) i − V ( X ) L ) , (4)where g ( X ) L and V ( X ) L are conductance (nS) and reversalpotential for the leakage current, respectively.When the membrane potential v ( X ) i reaches a threshold v ( X ) th at a time t ( X ) f,i , the i th neuron fires a spike. Afterfiring (i.e., t ≥ t ( X ) f,i ), the 2nd type of AHP current I ( X ) AHP,i follows: I ( X ) AHP,i = g ( X ) AHP ( t ) ( v ( X ) i − V ( X ) AHP ) for t ≥ t ( X ) f,i . (5)Here, V ( X ) AHP is the reversal potential for the AHP current,and the conductance g ( X ) AHP ( t ) is given by an exponential-decay function: g ( X ) AHP ( t ) = ¯ g ( X ) AHP e − ( t − t ( X ) f,i ) /τ ( X ) AHP , (6)where ¯ g ( X ) AHP and τ ( X ) AHP are the maximum conductanceand the decay time constant for the AHP current. As τ ( X ) AHP increases, the refractory period becomes longer.The 3rd type of external constant current I ( X ) ext for thecellular spontaneous discharge is supplied to only thePCs and the VN neuron because of their high sponta-neous firing rates [50, 51]. In Appendix A, Table I showsthe parameter values for the capacitance C X , the leakagecurrent I ( X ) L , the AHP current I ( X ) AHP , and the externalconstant current I ( X ) ext . These values are adopted fromphysiological data [6, 35]. D. Synaptic Currents
The 4th type of synaptic current I ( X ) syn,i into the i thneuron in the X population consists of the following 3kinds of synaptic currents: I ( X ) syn,i = I ( X,Y )AMPA ,i + I ( X,Y )NMDA ,i + I ( X,Z )GABA ,i . (7)Here, I ( X,Y )AMPA ,i and I ( X,Y )NMDA ,i are the excitatory AMPA( α -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid)receptor-mediated and NMDA ( N -methyl- D -aspartate)receptor-mediated currents from the pre-synaptic source Y population to the post-synaptic i th neuron in the tar-get X population. On the other hand, I ( X,Z )GABA ,i is the in-hibitory GABA A ( γ -aminobutyric acid type A) receptor-mediated current from the pre-synaptic source Z popu-lation to the post-synaptic i th neuron in the target X population.Similar to the case of the AHP current, the R (=AMPA, NMDA, or GABA) receptor-mediated synapticcurrent I ( T,S ) R,i from the pre-synaptic source S populationto the i th post-synaptic neuron in the target T popula-tion is given by: I ( T,S ) R,i = g ( T,S ) R,i ( t ) ( v ( T ) i − V ( S ) R ) , (8)where g ( T,S )( R,i ) ( t ) and V ( S ) R are synaptic conductance andsynaptic reversal potential (determined by the type of the pre-synaptic source S population), respectively. Weget the synaptic conductance g ( T,S ) R,i ( t ) from: g ( T,S ) R,i ( t ) = ¯ g ( T ) R N S (cid:88) j =1 J ( T,S ) ij w ( T,S ) ij s ( T,S ) j ( t ) , (9)where ¯ g ( T ) R and J ( T,S ) ij are the maximum conductance andthe synaptic weight of the synapse from the j th pre-synaptic neuron in the source S population to the i thpost-synaptic neuron in the target T population, respec-tively. The inter-population synaptic connection fromthe source S population (with N s neurons) to the target T population is given by the connection weight matrix W ( T,S ) (= { w ( T,S ) ij } ) where w ( T,S ) ij = 1 if the j th neu-ron in the source S population is pre-synaptic to the i thneuron in the target T population; otherwise w ( T,S ) ij = 0.The post-synaptic ion channels are opened due to thebinding of neurotransmitters (emitted from the source S population) to receptors in the target T population. Thefraction of open ion channels at time t is represented by s ( T,S ) . The time course of s ( T,S ) j ( t ) of the j th neuron inthe source S population is given by a sum of exponential-decay functions E ( T,S ) R ( t − t ( j ) f ): s ( T,S ) j ( t ) = F ( s ) j (cid:88) f =1 E ( T,S ) R ( t − t ( j ) f ) , (10)where t ( j ) f and F ( s ) j are the f th spike time and the to-tal number of spikes of the j th neuron in the source S population, respectively. The exponential-decay func-tion E ( T,S ) R ( t ) (which corresponds to contribution of apre-synaptic spike occurring at t = 0 in the absence ofsynaptic delay) is given by: E ( T,S ) R ( t ) = e − t/τ ( T ) R Θ( t ) or (11a)= ( A e − t/τ ( T ) R, + A e − t/τ ( T ) R, )Θ( t ) , (11b)where Θ( t ) is the Heaviside step function: Θ( t ) = 1 for t ≥ t <
0. Depending on the source andthe target populations, E ( T,S ) R ( t ) may be a type-1 singleexponential-decay function of Eq. (11a) or a type-2 dualexponential-decay function of Eq. (11b). In the type-1case, there exists one synaptic decay time constant τ ( T ) R (determined by the receptor on the post-synaptic target T population), while in the type-2 case, two synapticdecay time constants, τ ( T ) R, and τ ( T ) R, exist. In most cases,the type-1 single exponential-decay function of Eq. (11a)appears, except for the two synaptic currents I (GR , GO)GABA and I (GO , GR)NMDA .In Appendix A, Table II shows the parameter valuesfor the maximum conductance ¯ g ( T ) R , the synaptic weight J ( T,S ) ij , the synaptic reversal potential V ( S ) R , the synapticdecay time constant τ ( T ) R , and the amplitudes A and A for the type-2 exponential-decay function in the gran-ular layer, the Purkinje-molecular layer, and the otherparts for the VN and IO, respectively. These values areadopted from physiological data [6, 35]. E. Synaptic Plasticity
We use a rule for synaptic plasticity, based on the ex-perimental result in [23]. This rule is a refined one forthe LTD in comparison to the rule used in [6, 35], thedetails of which will be explained below.The coupling strength of the synapse from the pre-synaptic neuron j in the source S population to the post-synaptic neuron i in the target T population is J ( T,S ) ij . Initial synaptic strengths are given in Table II. Here, weassume that learning occurs only at the PF-PC synapses.Hence, only the synaptic strengths J (PC , PF) ij of PF-PCsynapses may be modifiable, while synaptic strengths ofall the other synapses are static. [Here, the index j forthe PFs corresponds to the two indices ( M, m ) for GRcells representing the m th (1 ≤ m ≤
50) cell in the M th (1 ≤ M ≤ ) GR cluster.] Synaptic plasticityat PF-PC synapses have been so much studied in di-verse experimental [23, 25–28, 52–60] and computational[6, 19, 35, 61–67] works.With increasing time t , synaptic strength for each PF-PC synapse is updated with the following multiplicativerule (depending on states) [23]: J (PC , PF) ij ( t ) → J (PC , PF) ij ( t ) + ∆ J (PC , PF) ij ( t ) , (12)where ∆ J (PC , PF) ij ( t ) = ∆LTD (1) ij + ∆LTD (2) ij + ∆LTP ij , (13)∆LTD (1) ij = − δ LT D · J (PC , PF) ij ( t ) · CF i ( t ) · ∆ t ∗ r (cid:88) ∆ t =0 ∆ J LT D (∆ t ) , (14)∆LTD (2) ij = − δ LT D · J (PC , PF) ij ( t ) · [1 − CF i ( t )] · P F ij ( t ) · D i ( t ) · ∆ t ∗ l (cid:88) ∆ t =0 ∆ J LT D (∆ t ) , (15)∆LTP ij = δ LT P · [ J (PC , PF)0 − J (PC , PF) ij ( t )] · [1 − CF i ( t )] · P F ij ( t ) · [1 − D i ( t )] . (16)Here, J (PC , PF)0 is the initial value (=0.006) for the synap-tic strength of PF-PC synapses. Synaptic modification(LTD or LTP) occurs, depending on the relative timedifference ∆ t [= t CF (CF activation time) - t PF (PF ac-tivation time)] between the spiking times of the error-teaching instructor CF and the diversely-recoded studentPFs. In Eqs. (14)-(16), CF i ( t ) represents a spike trainof the CF signal coming into the i th PC. When CF i ( t )activates at a time t , CF i ( t ) = 1; otherwise, CF i ( t ) = 0.This instructor CF firing causes LTD at PF-PC synapsesin conjunction with earlier (∆ t >
0) student PF firings inthe range of t CF − ∆ t ∗ r < t PF < t CF (∆ t ∗ r (cid:39) . CF i ( t ) = 0, correspondingto Eqs. (15) and (16). Here, P F ij ( t ) denotes a spiketrain of the PF signal from the j th pre-synaptic GR cellto the i th post-synaptic PC. When P F ij ( t ) activates attime t , P F ij ( t ) = 1; otherwise, P F ij ( t ) = 0. In thecase of P F ij ( t ) = 1, PF firing may give rise to LTD orLTP, depending on the presence of earlier CF firings inan effective range. If CF firings exist in the range of t PF + ∆ t ∗ l < t CF < t PF (∆ t ∗ l (cid:39) − . D i ( t ) =1; otherwise D i ( t ) = 0. When both P F ij ( t ) = 1 and D i ( t ) = 1, the PF firing causes another LTD at PF-PCsynapses in association with earlier (∆ t <
0) CF firings[see Eq. (15)]. The likelihood for occurrence of earlier CFfirings within the effective range is very low because meanfiring rates of the CF signals (corresponding to outputfirings of individual IO neurons) are ∼ D i ( t ) = 0 (i.e., absence ofearlier associated CF firings), LTP occurs due to the PFfiring alone [see Eq. (16)]. The update rate δ LT D for LTDin Eqs. (14) and (15) is 0.005, while the update rate δ LT P for LTP in Eqs. (16) is 0.0005 (= δ LT D / J LT D (∆ t ) varies depending on the rel-ative time difference ∆ t (= t CF − t P F ). We employthe following time window for the synaptic modification∆ J LT D (∆ t ) [23]:∆ J LT D (∆ t ) = A + B · e − (∆ t − t ) /σ , (17)where A = − . B = 0 . t = 80, and σ = 180. Figure3 shows the time window for ∆ J LT D (∆ t ). As shown wellin Fig. 3, LTD occurs in an effective range of ∆ t ∗ l < ∆ t < ∆ t ∗ r . We note that a peak exists at t = 80 msec, and LTD (1)
LTD (2) -200 0 200 4000.000.150.30 J LT D ( t ) t (msec) FIG. 3: Time window for the LTD at the PF-PC synapse.Plot of synaptic modification ∆ J LTD (∆ t ) for LTD versus ∆ t [see Eq. (17)]. hence peak LTD occurs when PF firing precedes CF firingby 80 msec. A CF firing causes LTD in conjunction withearlier PF firings in the black region (0 < ∆ t < ∆ t ∗ r ), andit also gives rise to another LTD in association with laterPF firings in the gray region (∆ t ∗ l < ∆ t < t > ∆ t ∗ r or < ∆ t ∗ l ), PF firings alone leads to LTP,due to absence of effectively associated CF firings.Finally, we discuss the advantages of our refined rulefor synaptic plasticity in comparison to the synaptic rulein [6, 35]. Our rule is constructed from the experimentalresult in [23]. In the presence of a CF firing, a majorLTD (∆LTD (1) ) takes place in association with earlierPF firings in the range of t CF − ∆ t ∗ r < t PF < t CF (∆ t ∗ r (cid:39) . (2) ) occurs inassociation with later PF firings in the range of t CF Numerical integration of the governing Eq. (3) for thetime-evolution of states of individual neurons, along withthe update rule for synaptic plasticity of Eq. (12), is doneby employing the 2nd-order Runge-Kutta method withthe time step 1 msec. For each realization, we chooserandom initial points v ( X ) i (0) for the i th neuron in the X population with uniform probability in the range of v ( X ) i (0) ∈ ( V ( X ) L − . , V ( X ) L + 5 . V ( X ) L are given in Table I. III. EFFECT OF DIVERSE SPIKINGPATTERNS OF GR CLUSTERS ON MOTORLEARNING FOR THE OKR ADAPTION In this section, we study the effect of diverse recodingof GR cells on motor learning for the OKR adaptationby varying the connection probability p c from the GOto the GR cells. We mainly consider an optimal case of p ∗ c = 0 . 06 where the spiking patterns of GR clusters arethe most diverse. In this case, we first make dynamicalclassification of diverse spiking patterns of the GR clus-ters. Then, we make an intensive investigation on the ef-fect of diverse recoding of GR cells on synaptic plasticityat PF-PC synapses and the subsequent learning processin the PC-VN-IO system. Finally, we vary p c from theoptimal value p ∗ c , and study dependence of the diversitydegree D of spiking patterns and the saturated learninggain degree L ∗ g on p c . Both D and L ∗ g are found to formbell-shaped curves with peaks at p ∗ c , and they have strongcorrelation with the Pearson’s coefficient r (cid:39) . A. Firing Activity in The Whole Population of GRCells As shown in Fig. 2, recoding process is performed inthe granular layer (corresponding to the input layer ofthe cerebellar cortex), consisting of GR and GO cells.In the GR-GO feedback system, GR cells (principal out-put cells in the granular layer) receive excitatory contextsignals for the post-eye-movement via the sinusoidally-modulating MFs [see Fig. 1(b1)] and make recoding ofcontext signals. In this recoding process, GO cells makeeffective inhibitory coordination for diverse recoding ofGR cells. Thus, diversely recoded signals are fed into thePCs (principal output cells in the cerebellar cortex) viaPFs. Due to this type of diverse recoding of GR cells, thecerebellum was recently reinterpreted as a liquid statemachine with powerful discriminating/separating capa-bility (i.e., different input signals are transformed intomore different ones via recoding process) rather than thesimple perceptron in the Marr-Albus-Ito theory [68, 69].We first consider the firing activity in the whole pop-ulation of GR cells for p ∗ c = 0 . 06. Collective firing ac-tivity may be well visualized in the raster plot of spikeswhich is a collection of spike trains of individual neu-rons. Such raster plots of spikes are fundamental datain experimental neuroscience. As a population quan-tity showing collective firing behaviors, we use an instan-taneous whole-population spike rate R GR ( t ) which maybe obtained from the raster plots of spikes [36–43]. Toobtain a smooth instantaneous whole-population spikerate, we employ the kernel density estimation (kernelsmoother) [70]. Each spike in the raster plot is con-voluted (or blurred) with a kernel function K h ( t ) [such (a) i (b) R G R ( t ) ( H z ) t (msec) (c1) A ( t ) t (msec) In-Phase Out-of-Phase(c2) A ( G ) ( t ) t (msec) (d1) f ( i ) G R ( t ) ( H z ) t (msec) (d2) f ( p ) G R ( t ) ( H z ) t (msec) FIG. 4: Firing activity of GR cells in an optimal case of p c (connection probability from GO to GR cells) = 0.06. (a)Raster plots of spikes of 10 randomly chosen GR cells. (b) In-stantaneous whole-population spike rate R GR ( t ) in the wholepopulation of GR cells. Band width for R GR ( t ): h = 10 msec.Plots of the activation degrees (c1) A ( t ) in the whole popu-lation of GR cells and (c2) A ( G ) ( t ) in the G spiking group[ G : in-phase (solid curve) and out-of-phase (dotted curve)].Plots of (d1) instantaneous individual firing rate f ( i )GR ( t ) forthe active GR cells and (d2) instantaneous population spikerate f ( p )GR ( t ) in the whole population of GR cells. Bin size for(c1)-(d2): ∆ t = 10 msec. as a smooth Gaussian function in Eq. (19)], and then asmooth estimate of instantaneous whole-population spikerate R GR ( t ) is obtained by averaging the convoluted ker-nel function over all spikes of GR cells in the whole pop-ulation: R GR ( t ) = 1 N N (cid:88) i =1 n i (cid:88) s =1 K h ( t − t ( i ) s ) , (18)where t ( i ) s is the s th spiking time of the i th GR cell, n i is the total number of spikes for the i th GR cell, and N is the total number of GR cells (i.e., N = N c · N GR =51 , h : K h ( t ) = 1 √ πh e − t / h , − ∞ < t < ∞ . (19)Throughout the paper, the band width h of K h ( t ) is 10msec.Figure 4(a) shows a raster plot of spikes of 10 ran-domly chosen GR cells. At the initial and the final stagesof the cycle, GR cells fire sparse spikes, because the firingrates of Poisson spikes for the MF are low. On the otherhand, at the middle stage, the firing rates for the MFare relatively high, and hence spikes of GR cells becomerelatively dense. Figure 4(b) shows the instantaneouswhole-population spike rate R GR ( t ) in the whole popula-tion of GR cells. R GR ( t ) is basically in proportion to thesinusoidally-modulating inputs via MFs. However, it hasa different waveform with a central plateau. At the ini-tial stage, it rises rapidly, then a broad plateau appearsat the middle stage, and at the final stage, it decreases slowly. In comparison to the MF signal, the top part of R GR ( t ) becomes lowered and flattened, due to the effectof inhibitory GO cells. Thus, a central plateau emerges.We next consider the activation degree of GR cells.To examine it, we divide the whole learning cycle (2000msec) into 200 bins (bin size: 10 msec). Then, we getthe activation degree A i for the active GR cells in the i th bin: A i = N a,i N , (20)where N a,i and N are the number of active GR cells inthe i th bin and the total number of GR cells, respectively.Figure 4(c1) shows a plot of the activation degree A ( t ) inthe whole population of GR cells. It is nearly symmetric,and has double peaks with a central valley at the middlestage; its values at both peaks are about 0.94 and thecentral minimum value is about 0.65.Presence of the central valley in A ( t ) is in contrastto the central plateau in R GR ( t ). Appearance of such acentral valley may be understood as follows. The wholepopulation of GR cells can be decomposed into two typesof in-phase and out-of-phase spiking groups. Spiking pat-terns of in-phase (out-of-phase) GR cells are in-phase(out-of-phase) with respect to R GR ( t ) (representing thepopulation-averaged firing activity in the whole popula-tion of GR cells); details will be given in Figs. 5 and 6.Then, the activation degree A ( G ) i of active GR cells inthe G spiking group in the i th bin is given by: A ( G ) i = N ( G ) a,i N , (21)where N ( G ) a,i is the number of active GR cells in the G spiking group ( G = i and o for the in-phase and the out-of-phase spiking groups, respectively) in the i th bin. Thesum of A ( G ) i ( t ) over the in-phase and the out-of-phasespiking groups is just the activation degree A i ( t ) in thewhole population. Figure 4(c2) shows plots of activationdegree A ( G ) ( t ) in the in-phase (solid line) and the out-of-phase (dotted curve) spiking groups. In the case ofin-phase ( G = i ) spiking group, A ( i ) ( t ) has a centralplateau, while A ( o ) ( t ) has double peaks with a centralvalley in the case of out-of-phase ( G = o ) spiking group.Hence, small contribution of out-of-phase spiking groupat the middle stage leads to emergence of the centralvalley in A ( t ) in the whole population.We note again that, in the whole population the ac-tivation degree A ( t ) with a central valley is in contrastto R GR ( t ) with a central plateau. To understand thisdiscrepancy, we consider the bin-averaged instantaneousindividual firing rates f ( i )GR of active GR cells: f ( i )GR = N s,i N a,i ∆ t , (22)where N s,i is the number of spikes of GR cells in the i th bin, N a,i is the number of active GR cells in the i th0 I=594 C (594) =0.85 (a1) i Complex Out-of-Phase Spiking PatternsAnti-Phase Spiking PatternsIn-Phase Spiking Patterns I=49 , C (49) = 0.57 (b1) i I=543 C (543) =0.48 (a2) R ( I ) G R ( t ) t (msec) R ( I ) G R ( t ) t (msec) I=192 C (192) =0.12 (c1) i R ( I ) G R ( t ) I=91 C (91) =0.10 (c2) I=773 C (773) =0.08 (c3) t (msec) I=382 C (382) =0.05 (c4) I=101, C (101) = 0.23 (b2) I=663 C (663) =0.47 (a3) I=349 C (349) = 0.07 (c6) I=332 C (332) =0.42 (a4) I=705 C (705) = 0.18 (c5) I=399 C (399) =0.40 (a5) FIG. 5: Diverse spiking patterns in the GR clusters in theoptimal case of p ∗ c = 0 . 06. Raster plots of spikes and instanta-neous cluster spike rates R ( I )GR ( t ) for diverse spiking patterns.Five in-phase spiking patterns in the I th GR clusters; I =(a1) 594, (a2) 543, (a3) 663, (a4) 332, and (a5) 399. Twoanti-phase spiking patterns in the I th GR cluster; I = (b1)49 and (b2) 101. Six complex out-of-phase spiking patternsin the I th GR clusters; I = (c1) 192, (c2) 91, (c3) 773, (c4)382, (c5) 705, and (c6) 349. C ( I ) represents the conjunctionindex of the spiking pattern in the I th GR cluster. bin, and the bin size ∆ t is 10 msec. Figure 4(d1) showsa plot of f ( i )GR ( t ) for the active GR cells. We note thatactive GR cells fire spikes at higher firing rates at themiddle stage because f ( i )GR ( t ) has a central peak. Then,the bin-averaged instantaneous population spike rate f ( p )GR is given by the product of the activation degree A i ofEq. (20) and the instantaneous individual firing rate f ( i )GR of Eq. (22): f ( p )GR = A i f ( i )GR = N s,i N ∆ t . (23)The instantaneous population spike rate f ( p )GR ( t ) inFig. 4(d2) has a central plateau, as in the case of R GR ( t ).We note that both f ( p )GR ( t ) and R GR ( t ) correspond to bin-based estimate and kernel-based smooth estimate for theinstantaneous whole-population spike rate for the GRcells, respectively [36]. In this way, although the ac-tivation degree A ( t ) of GR cells are lower at the mid-dle stage, their population spike rate becomes nearlythe same as that in the neighboring parts (i.e., centralplateau is formed), due to the higher individual firingrates. B. Dynamical Classification of Spiking Patterns ofGR Clusters There are N C (= 2 ) GR clusters. N GR (= 50) GRcells in each GR cluster share the same inhibitory andexcitatory inputs via their dendrites which synaptically contact the two glomeruli (i.e., terminals of MFs) at bothends of the GR cluster [see Fig. 2(b)]; nearby inhibitoryGO cell axons innervate the two glomeruli. Hence, GRcells in each GR cluster show similar firing behaviors.Similar to the case of R GR ( t ) in Eq. (18), the firing ac-tivity of the I th GR cluster is characterized in terms of itsinstantaneous cluster spike rate R ( I )GR ( t ) ( I = 1 , · · · , N C ): R ( I )GR ( t ) = 1 N GR N GR (cid:88) i =1 n ( I ) i (cid:88) s =1 K h ( t − t ( I,i ) s ) , (24)where t ( I,i ) s is the s th spiking time of the i th GR cell inthe I th GR cluster and n ( I ) i is the total number of spikesfor the i th GR cell in the I th GR cluster.We introduce the conjunction index C ( I ) of each GRcluster, representing the degree for the conjunction (as-sociation) of the spiking behavior [ R ( I )GR ( t )] of each I thGR cluster with that of the whole population [ R GR ( t )in Fig. 4(b)] [i.e., denoting the degree for the resem-blance (similarity) between R ( I )GR ( t ) and R GR ( t )]. Theconjunction index C ( I ) is given by the cross-correlationat the zero-time lag [i.e., Corr ( I )GR (0)] between R ( I )GR ( t )and R GR ( t ): Corr ( I )GR ( τ ) = ∆ R GR ( t + τ )∆ R ( I )GR ( t ) (cid:113) ∆ R ( t ) (cid:114) ∆ R ( I )GR2 ( t ) , (25)where ∆ R GR ( t ) = R GR ( t ) − R GR ( t ), ∆ R ( I )GR ( t ) = R ( I )GR ( t ) − R ( I )GR ( t ), and the overline denotes the time av-erage over a cycle. We note that C ( I ) represents wellthe phase difference (shift) between the spiking patterns[ R ( I )GR ( t )] of GR clusters and the firing behavior [ R GR ( t )]in the whole population.In all the 2 GR clusters, we obtain their conjunctionindices C ( I ) , make intensive examination of the phase dif-ference of R ( I )GR ( t ) with respect to R GR ( t ), and thus clas-sify the whole GR clusters into the in-phase, the anti-phase, and the complex out-of-phase spiking groups. Fig-ure 5 shows examples for diverse spiking patterns of GRclusters. This type of diversity arises from inhibitory co-ordination of GO cells on the firing activity of GR cellsin the GR-GO feedback system in the granular layer.Five examples for “in-phase” spiking patterns in the I th ( I = 594, 543, 663, 332, and 399) GR clusters aregiven in Figs. 5(a1)-5(a5), respectively. Raster plot ofspikes of N GR (= 50) GR cells and the corresponding in-stantaneous cluster spike rate R ( I )GR ( t ) are shown, alongwith the value of C ( I ) in each case of the I th GR clus-ter. In all these cases, the instantaneous cluster spikerates R ( I )GR ( t ) are in-phase relative to the instantaneouswhole-population spike rate R GR ( t ). Among them, inthe case of I = 594 with the maximum conjunction in-dex C max (= 0 . R (594)GR ( t ) with a central plateau is the1most similar (in-phase) to R GR ( t ). In the next case of I = 543 with C ( I ) = 0 . , R (543)GR ( t ) has a central sharppeak, and hence its similarity degree relative to R GR ( t )decreases. The remaining two cases of I = 663 and 332(with more than one central peaks) may be regarded asones developed from the case of I = 594. With increasingthe number of peaks in the central part, the value of C ( I ) decreases, and hence the resemblance degree relative to R GR ( t ) is reduced. The final case of I = 399 with doublepeaks can be considered as one evolved form the case of I = 543. In this case, the value of C ( I ) is reduced to 0.40.Based on the examples in Figs. 5(a1)-5(a5), spikingpatterns which have central plateau, central sharp peak,and two or more central peaks in the middle part of cycleare considered as in-phase spiking patterns relative to theinstantaneous whole-population spike rate R GR ( t ). Wemake an intensive examination of the instantaneous clus-ter spike rates R ( I )GR of the GR clusters with C ( I ) < . C ∗ h ( (cid:39) . 39) betweenthe in-phase and the complex out-of-phase spiking pat-terns. For C ( I ) > C ∗ h in-phase spiking patterns such asones in Figs. 5(a1)-5(a5) appear. On the other hand,when passing the higher threshold C ∗ h from the above,complex out-of-phase spiking patterns (with C ( I ) < C ∗ h )emerge. These complex out-of-phase spiking patternshave left-skewed (right-skewed) peaks near the 1st (3rd)quartile of cycle [i.e., near t = 500 (1500) msec], explicitexamples of which will be given below in Figs. 5(c1)-5(c4). Thus, all the GR clusters, exhibiting in-phasespiking patterns, constitute the in-phase spiking groupwhere the range of C ( I ) is ( C ∗ h , C max ); C max = 0 . 85 and C ∗ h (cid:39) . I th ( I = 49 and 101) GR clusters are given in Figs. 5(b1)and 5(b2), respectively. We note that, in both cases, theinstantaneous cluster spike rates R ( I )GR ( t ) are anti-phasewith respect to R GR ( t ) in the whole population. In thecase of I = 49 with the minimum conjunction index C min (= − . R (49)GR ( t ) is the most anti-phase relativeto R GR ( t ), and it has double peaks near the 1st and the3rd quartiles and a central deep valley at the middle ofthe cycle. The case of I = 101 with C ( I ) (= − . 23) maybe regarded as evolved from the case of I = 49. It hasan increased (but still negative) value of C ( I ) (= − . R ( I )GR ( t )] which have double peaks near the 1stand the 3rd quartiles and a central valley at the middleof cycle are regarded as anti-phase spiking patterns withrespect to R GR ( t ). Like the case of the above in-phasespiking patterns, through intensive examination of R ( I )GR of the GR clusters with C ( I ) > − . , we determine thelower threshold C ∗ l ( (cid:39) − . 20) between the anti-phase andthe complex out-of-phase spiking patterns. For C ( I ) < C ∗ l anti-phase spiking patterns such as ones in Figs. 5(b1)-5(b2) exist. In contrast, when passing the lower threshold -1 0 10.00.10.2 -1 0 10.00.10.20.00.30.6-1 0 10.00.10.2-1 0 10.00.10.2 In-Phase(c1) F r ac ti on o f C (I) C (I) C (I) ComplexOut-of-Phase ComplexOut-of-Phase(c3) F r ac ti on o f C (I) C (I) ComplexOut-of-PhaseAnti-Phase In-Phase (b) F r ac ti on Spiking Group Anti-Phase(c2) F r ac ti on o f C (I) C (I) (a) F r ac ti on o f C (I) C (I) FIG. 6: Characterization of diverse spiking patterns in theGR clusters in the optimal case of p ∗ c = 0 . 06. (a) Distributionof conjunction indices {C ( I ) } for the GR clusters in the wholepopulation. (b) Fraction of spiking groups. Distribution ofconjunction indices {C ( I ) } for the (c1) in-phase, (c2) anti-phase, and (c3) complex out-of-phase spiking groups. Binsize for the histograms in (a) and in (c1)-(c3) is 0.1. (d) Bardiagram for the ranges of conjunction indices {C ( I ) } for the in-phase, anti-phase, and complex out-of-phase spiking groups. C ∗ l from the below, complex out-of-phase spiking patterns(with C ( I ) > C ∗ l ) appear. These complex out-of-phasespiking patterns have a central peak which is transformedfrom the central valley at the middle of cycle, along withdouble peaks near the 1st and the 3rd quartiles. Anexplicit example will be given below in Fig. 5(c5). Thus,all the GR clusters, showing anti-phase spiking patterns,form the anti-phase spiking group where the range of C ( I ) is ( C min , C ∗ l ); C min = − . 57 and C ∗ l (cid:39) − . C ∗ l ( (cid:39) − . < C ( I ) < C ∗ h ( (cid:39) . I th ( I =192, 91, 773, 382, 705, and 349) GR clus-ters. The cases of I = 192 and 91 seem to be developedfrom the in-phase spiking pattern in the I th ( I = 594 or543) GR cluster. In the case of I = 192, R (192)GR has aleft-skewed peak near the 1st quartile of the cycle, whilein the case of I = 91, R (91)GR has a right-skewed peak nearthe 3rd quartile. Hence, the values of C ( I ) for I = 192and 91 are reduced to 0.12 and 0.10, respectively. Inthe next two cases of I = 773 and 382, they seem tobe developed from the cases of I = 192 and 91, respec-tively. The left-skewed (right-skewed) peak in the case of I = 192 (91) is bifurcated into double peaks, which leadsto more reduction of conjunction indices; C (773) = 0 . C (382) = 0 . 05. In the remaining case of I = 705, it2seems to be evolved from the anti-phase spiking patternin the I = 101 case. The central valley for I = 101 istransformed into a central peak. Thus, R (705)GR has threepeaks, and its value of C (705) is a little increased to -0.18. As C ( I ) is more increased toward the zero, R ( I )GR becomes more complex, as shown in Fig. 5(c6) in thecase of I = 349 with C (349) = − . C ( I ) in the whole GR clus-ters. C ( I ) increases slowly from the negative to the peakat 0.55, and then it decreases rapidly. For this distribu-tion {C ( I ) } , the range is (-0.57, 0.85), the mean is 0.32,and the standard deviation is 0.516. Then, the diversitydegree D for the spiking patterns [ R ( I )GR ( t )] of all the GRclusters is given by: D = Relative Standard Deviationfor the Distribution {C ( I ) } , (26)where the relative standard deviation is just the stan-dard deviation divided by the mean. In the optimal caseof p ∗ c = 0 . D ∗ (cid:39) . D versus p c , D ∗ is just the maximum, and hencespiking patterns of GR clusters at p ∗ c is the most diverse.We decompose the whole GR clusters into the in-phase,anti-phase, and complex out-of-phase spiking groups.Figure 6(b) shows the fraction of spiking groups. The in-phase spiking group with C ∗ h ( (cid:39) . < C ( I ) < C max (=0 . 85) is a major one with fraction 50.2%, while the anti-phase spiking group with C min (= − . < C ( I ) < C ∗ l ( (cid:39)− . 20) is a minor one with fraction 5.8%. Between them( C ∗ l < C ( I ) < C ∗ h ), the complex out-of-phase spiking groupwith fraction 44% exists. In this case, the spiking-groupratio, given by the ratio of the fraction of the in-phasespiking group to that of the out-of-phase spiking group(consisting of both the anti-phase and complex out-of-phase spiking groups), is R ∗ (cid:39) . . Thus, in the opti-mal case of p ∗ c = 0 . 06, the fractions between the in-phaseand the out-of-phase spiking groups are well balanced.Under such good balance between the in-phase and theout-of-phase spiking groups, spiking patterns of the GRclusters are the most diverse.Figures 6(c1)-6(c3) also show the plots of the frac-tions of conjunction indices C ( I ) of the GR clusters inthe in-phase, anti-phase, and complex out-of-phase spik-ing groups, respectively. The ranges for the distribu-tions {C ( I ) } in the three spiking groups are also given inthe bar diagram in Fig. 6(d). In the case of in-phasespiking group, the distribution {C ( I ) } with a peak at0.55 has only positive values in the range of ( C ∗ h , C max )( C max = 0 . 85 and C ∗ h (cid:39) . {C ( I ) } with a peak at -0.25 has only neg-ative values in the range of ( C min , C ∗ l ) ( C min = − . C ∗ l (cid:39) − . {C ( I ) } with a peak at0.35 is ( C ∗ l , C ∗ h ), and the mean and the standard deviationare 0.174 and 0.242, respectively. As will be seen in thenext subsection, these in-phase, anti-phase, and complexout-of-phase spiking groups play their own roles in thesynaptic plasticity at PF-PC synapses, respectively.Finally, we study the dynamical origin of diverse spik-ing patterns in the I th GR clusters. As examples, weconsider two in-phase spiking patterns for I = 594 and543 [see the spiking patterns in Figs. 5(a1) and 5(a2)],one anti-phase spiking pattern for I = 49 [see the spikingpattern Fig. 5(b1)], and two complex out-of-phase spik-ing patterns for I = 192 and 91 [see the spiking patternsin Figs. 5(c1) and 5(c2)]. In Fig. 7, (a1)-(a5) correspondto the cases of I =594, 543, 49, 192, and 91, respectively.Diverse recodings for the MF signals are made in theGR layer, composed of excitatory GR and inhibitory GOcells (i.e., in the GR-GO cell feedback loop). In this case,spiking activities of GR cells are controlled by two typesof synaptic input currents (i.e., excitatory synaptic in-puts through MF signals and inhibitory synaptic inputsfrom randomly connected GO cells). Then, we make in-vestigations on the dynamical origin of diverse spikingpatterns of the GR clusters (shown in Fig. 5) throughanalysis of total synaptic inputs into the GR clusters.Synaptic current is given by the product of synapticconductance g and potential difference [see Eq. (8)].Here, synaptic conductance determines the time-courseof synaptic current. Hence, it is enough to consider thetime-course of synaptic conductance. The synaptic con-ductance g is given by the product of synaptic strengthper synapse, the number of synapses M syn , and the frac-tion s of open (post-synaptic) ion channels [see Eq. (9)].Here, the synaptic strength per synapse is given by theproduct of maximum synaptic conductance ¯ g and synap-tic weight J , and the time-course of s ( t ) is given bya summation for exponential-decay functions over pre-synaptic spikes, as shown in Eqs. (9) and (10).We make an approximation of the fraction s ( t ) of openion channels (i.e., contributions of summed effects of pre-synaptic spikes) by the bin-averaged spike rate f ( I ) X ( t ) ofpre-synaptic neurons ( X = MF and GO); f ( I )MF ( t ) is thebin-averaged spike rate of the MF signals into the I thGR cluster and f ( I )GO ( t ) is the bin-averaged spike rate ofthe pre-synaptic GO cells innervating the I th GR cluster.Then, the conductance g ( I ) X ( t ) of synaptic input from X (=MF or GO) into the I th GR cluster ( I = 1 , · · · , N C )is given by: g ( I ) X ( t ) (cid:39) M ( R ) f · f ( I ) X ( t ) . (27)Here, the multiplication factor M ( R ) f [= maximum synap-3 I= C (594) =0.85 t (msec) i < g ( I ) t o t ( t ) > r I= C (543) =0.48(a2) i t (msec) R ( I ) G R ( t ) (a1) < f ( I ) X ( t ) > r I= C (49) = 0.57(a3) t (msec) i R ( I ) G R ( t ) I= C (192) =0.12(a4) t (msec) R ( I ) G R ( t ) i I= C (91) =0.10(a5) t (msec) i R ( I ) G R ( t ) R ( I ) G R ( t ) < f ( I ) X ( t ) > r < g ( I ) t o t ( t ) > r < f ( I ) X ( t ) > r < g ( I ) t o t ( t ) > r < f ( I ) X ( t ) > r < g ( I ) t o t ( t ) > r Gray: MFBlack: GO < f ( I ) X ( t ) > r < g ( I ) t o t ( t ) > r (b) < g ( w ) t o t ( t ) > r t (msec) (c) F r ac ti on o f C i n ( I ) C in( I ) FIG. 7: Dynamical origin for the diverse spiking patterns inthe GR clusters in the optimal case of p ∗ c = 0 . 06. In-phasespiking patterns for I = (a1) 594 and (a2) 543, anti-phasespiking pattern for (a3) I = 49, and complex out-of-phasespiking patterns for I = (a4) 192 and (a5) 91. In (a1)-(a5),top panel: raster plots of spikes in the sub-population of pre-synaptic GO cells innervating the I th GR cluster, 2nd panel:plots of f ( I ) X ( t ) : bin-averaged instantaneous spike rates ofthe MF signals ( X = MF) into the I th GR cluster (grayline) and bin-averaged instantaneous sub-population of pre-synaptic GO cells ( X = GO) innervating the I th GR cluster(black line); (cid:104)· · · (cid:105) r represents the realization average (numberof realizations is 100), 3rd panel: time course of (cid:104) g ( I ) tot ( t ) (cid:105) r :conductance of total synaptic inputs (including both the ex-citatory and inhibitory inputs) into the I th GR cluster, andbottom panel: plots of R ( I )GR ( t ) : instantaneous cluster spikerate in the I th GR cluster. (b) Plot of cluster-averaged con-ductance (cid:104) g ( w ) tot ( t ) (cid:105) r of total synaptic inputs into the GR clus-ters versus t . (c) Distribution of conjunction indices {C ( I )in } for the conductances of total synaptic inputs into the GRclusters. tic conductance ¯ g R × synaptic weight J (GR ,X ) × numberof synapses M (GR ,X )syn ] varies depending on X and the re-ceptor R on the post-synaptic GR cells. In the case ofexcitatory synaptic currents into the I th GR cluster withAMPA receptors via the MF signal, M (AMPA) f = 2 . g AMPA = 0 . , J (GR , MF) = 8 . , and M (GR)syn = 2. In con-trast, in the case of the I th GR cluster with NMDA recep-tors, ¯ g NMDA = 0 . , and hence M (NMDA) f = 0 . , whichis much less than M (AMPA) f . For the inhibitory synaptic current from pre-synaptic GO cells to the I th GR clusterwith GABA receptors, M (GABA) f = 2 . 72; ¯ g GABA = 0 . ,J (GR , GO) = 10 , and M (GR , GO)syn = 9 . . Then, the conduc-tance g ( I ) tot of total synaptic inputs (including both theexcitatory and the inhibitory inputs) into the I th GRcluster is given by: g ( I ) tot ( t ) = g ( I )MF − g ( I )GO = g ( I )AMPA + g ( I )NMDA − g ( I )GO = 3 . f ( I )MF ( t ) − . f ( I )GO ( t ) . (28)Total synaptic input with conductance g ( I ) tot ( t ) is fed intoGR cells in the I th GR cluster, and then the correspond-ing output, given by the instantaneous cluster spike rate R ( I )GR ( t ) , emerges. Through averaging g ( I ) tot ( t ) over all theGR clusters, we obtain the cluster-averaged conductance g ( w ) tot ( t ) of total synaptic inputs into the GR clusters: g ( w ) tot ( t ) = 1 N C N C (cid:88) I =1 g ( I ) tot ( t ) . (29)The cluster-averaged total synaptic input with g ( w ) tot ( t )gives rise to the cluster-averaged output, given bythe instantaneous whole-population spike rate R w ( t ) [= N C (cid:80) N C I =1 R ( I )GR ( t )].In Figs. 7(a1)-7(a5), the top panels show the rasterplots of spikes in the sub-populations of pre-synaptic GOcells innervating the I th GR clusters. We obtain bin-averaged (sub-population) spike rates f ( I )GO ( t ) from theraster plots. The bin-averaged spike rate of pre-synapticGO cells in the i th bin is given by n ( s ) i N pre ∆ t , where n ( s ) i isthe number of spikes in the i th bin, ∆ t (=10 msec) is thebin size, and N pre (=10) is the number of pre-synapticGO cells. Via an average over 100 realizations, we obtainthe realization-averaged (bin-averaged) spike rate of pre-synaptic GO cells (cid:104) f ( I )GO ( t ) (cid:105) r because N pre (= 10) is small; (cid:104)· · · (cid:105) r represent a realization-average. The 2nd panelsshow (cid:104) f ( I )GO ( t ) (cid:105) r (black line) and (cid:104) f ( I )MF ( t ) (cid:105) r (gray line).We note that (cid:104) f ( I )GO ( t ) (cid:105) changes depending on I , while (cid:104) f ( I )MF ( t ) (cid:105) is independent of I . In contrast to the spikingactivity of GR cells [which exhibit random repetition oftransitions between active (bursting) and inactive (silent)states (see Fig. 4(a))], GO cells exhibit relatively regu-lar spikings, which may be well seen in slightly-modifiedsinusoidal-like bin-averaged spike rate (cid:104) f ( I )GO ( t ) (cid:105) r [35, 71].Then, we may get the realization-averaged conductance (cid:104) g ( I ) tot ( t ) (cid:105) r of total synaptic inputs in Eq. (28), which isshown in the 3rd panels. These conductances (cid:104) g ( I ) tot ( t ) (cid:105) r of total synaptic inputs show diverse patterns dependingon I , although (cid:104) f ( I )GO ( t ) (cid:105) , related to inhibitory synapticinput, exhibits relatively regular patterns.We note that the shapes of (cid:104) g ( I ) tot ( t ) (cid:105) r (corresponding tothe total input into the I th GR cluster) in the 3rd panelsare nearly the same as those of R ( I )GR ( t ) (corresponding to4the output of the I th GR cluster) in the bottom panels.Hence, we expect that in-phase (out-of-phase) inputs intothe GR clusters may result in generation of in-phase (out-of-phase) outputs (i.e., responses) in the GR clusters. Toconfirm this point clearly, similar to case of the spikingpatterns [ R ( I )GR ( t )] (i.e., the outputs) in the GR clusters,we introduce the conjunction index for the total synapticinput into the I th GR cluster between (cid:104) g ( I ) tot ( t ) (cid:105) r (conduc-tance of total synaptic input into the I th GR cluster) andthe cluster-averaged conductance of total synaptic inputs (cid:104) g ( w ) tot ( t ) (cid:105) r . Figure 7(b) shows the plot of (cid:104) g ( w ) tot ( t ) (cid:105) r versus t . We also note that the shape of (cid:104) g ( w ) tot ( t ) (cid:105) r is similar tothe instantaneous whole-population spike rate R GR ( t ) inFig. 4(b).As in the case of the conjunction index C ( I ) for thespiking patterns (i.e. outputs) in the I th GR cluster [seeEq. (25)], the conjunction index C ( I )in for the total synap-tic input is given by the cross-correlation at the zero-timelag (i.e., Corr ( I )in (0)) between (cid:104) g ( I ) tot ( t ) (cid:105) r and (cid:104) g ( w ) tot ( t ) (cid:105) r : Corr ( I )in ( τ ) = ∆ (cid:104) g ( w ) tot ( t + τ ) (cid:105) r ∆ (cid:104) g ( I ) tot ( t ) (cid:105) r (cid:113) ∆ (cid:104) g ( w ) tot ( t ) (cid:105) r (cid:113) ∆ (cid:104) g ( I ) tot ( t ) (cid:105) r , (30)where ∆ (cid:104) g ( w ) tot ( t ) (cid:105) r = (cid:104) g ( w ) tot ( t ) (cid:105) r −(cid:104) g ( w ) tot ( t ) (cid:105) r , ∆ (cid:104) g ( I ) tot ( t ) (cid:105) r = (cid:104) g ( I ) tot ( t ) (cid:105) r −(cid:104) g ( I ) tot ( t ) (cid:105) r , and the overline represents the timeaverage. Thus, we have two types of conjunction indices, C ( I ) [output conjunction index: given by Corr ( I )GR (0)] and C ( I )in [input conjunction index: given by Corr ( I )in (0)] forthe output and the input in the I th GR cluster, respec-tively.Figure 7(c) shows the plot of fraction of input con-junction indices {C ( I )in } in the whole GR clusters. Wenote that the distribution of input conjunction indices inFig. 7(c) is nearly the same as that of output conjunc-tion indices in Fig. 6(a). C ( I )in increases slowly from thenegative value to the peak at 0.55, and then it decreasesrapidly. In this distribution of {C ( I )in } , the range is (-0.57,0.85), the mean is 0.321, and the standard deviation is0.516. Then, we obtain the diversity degree D in for thetotal synaptic inputs {(cid:104) g ( I ) tot ( t ) (cid:105) r } of all the GR clusters: D in = Relative Standard Deviationfor the Distribution of {C ( I )in } . (31)Hence, D in (cid:39) . 607 for the synaptic inputs, which isnearly the same as D ∗ ( (cid:39) . C. Effect of Diverse Recoding in GR Clusters onSynaptic Plasticity at PF-PC Synapses Based on dynamical classification of spiking patternsof GR clusters, we investigate the effect of diverse re- 400 16000.00.51.0 400 1600 400 1600 400 1600 400 1600400 16000.00.51.0 400 1600 400 1600 400 1600 400 16000 200 4000.00.51.0 0 200 4000.000.070.14 ~ (a1) 1st cycle J ( t ) (a2) 100th cycle (a3) 200th cycle t (msec) (a4) 300th cycle (a5) 400th cycle ~ (b1) 1st cycle < J ( t ) > (b2)100th cycle (b3)200th cycle t (msec) (b4)300th cycle (b5)400th cycle ~ (c) < J ( t ) > Cycle (d) M J Cycle FIG. 8: Change in synaptic weights of active PF-PC synapsesduring learning in the optimal case of p ∗ c = 0 . 06. (a1)-(a5) Cycle-evolution of distribution of normalized synapticweights ˜ J of active PF signals. (b1)-(b5) Cycle-evolution ofbin-averaged (normalized) synaptic weights (cid:104) ˜ J (cid:105) of active PFsignals. Bin size: ∆ t = 100 msec. Plots of (c) cycle-averagedmean (cid:104) ˜ J (cid:105) and (d) modulation M J for (cid:104) ˜ J (cid:105) versus cycle. coding in the GR clusters on synaptic plasticity at PF-PC synapses. As shown in the above subsection, MFcontext input signals for the post-eye-movement are di-versely recoded in the granular layer (corresponding tothe input layer of the cerebellar cortex). The diversely-recoded in-phase and out-of-phase PF (student) signals(corresponding to the outputs from the GR cells) are fedinto the PCs (i.e., principal cells of the cerebellar cor-tex) and the BCs in the Purkinje-molecular layer (cor-responding to the output layer of the cerebellar cortex).The PCs also receive in-phase error-teaching CF (instruc-tor) signals from the IO, along with the inhibitory inputsfrom the BCs. Then, the synaptic weights at the PF-PCsynapses vary depending on the relative phase differencebetween the PF (student) signals and the CF (instructor)signals.We first consider the change in normalized synapticweights ˜ J of active PF-PC synapses during learning inthe optimal case of p ∗ c = 0 . J ij ( t ) = J (PC , PF) ij ( t ) J (PC , PF)0 . (32)where the initial synaptic strength ( J (PC , PF)0 = 0 . J ( t ) of active PF-PC synapses. With increasing the learning cycle, normal-ized synaptic weights ˜ J ( t ) are decreased due to LTD atPF-PC synapses, and eventually their distribution seemsto be saturated at about the 300th cycle. We note thatin-phase PF signals are strongly depressed (i.e., strongLTD) by the in-phase CF signals, while out-of-phase PFsignals are weakly depressed (i.e., weak LTD) due to thephase difference between the PF and the CF signals. As5 400 16000.31.0 400 1600 400 1600 400 1600 400 16000.31.00.31.0 400 16000.31.0 400 1600 400 1600 400 1600 400 1600 (b1) (b2) (b3) t (msec) (b4) (b5) G : Out-of- Phase ~ (c1)1st cycle J ( G ) ( t ) (c2)100th cycle (c3)200th cycle (c4)300th cycle (c5) G : Anti- Phase400th cycle ~ (a1)1st cycle J ( G ) ( t ) (a2)100th cycle (a3)200th cycle (a4)300th cycle (a5) G : In-Phase400th cycle (d1) (d2) (d3) t (msec) (d4) (d5) G : Complex Out-of- Phase FIG. 9: Change in synaptic weights of active PF-PC synapsesin each spiking group during learning in the optimal caseof p ∗ c = 0 . 06. Cycle-evolution of distributions of normal-ized synaptic weights ˜ J ( G ) of active PF signals in the G spiking group [G: (a1)-(a5) in-phase and (b1)-(b5) out-of-phase]. Cycle-evolution of distributions of normalized synap-tic weights ˜ J ( G ) of active out-of-phase PF signals in the G spiking group [G: (c1)-(c5) anti-phase and (d1)-(d5) complexout-of-phase]. shown in Fig. 4(c2), the activation degree A ( G ) of thein-phase spiking group ( G = i ) is dominant at the mid-dle stage of the cycle, while at the other parts of thecycle, the activation degrees of the in-phase ( G = i ) andthe out-of-phase ( G = o ) spiking groups are comparable.Consequently, strong LTD occurs at the middle stage,while at the initial and final stages somewhat less LTDtakes place due to contribution of both the out-of-phasespiking group (with weak LTD) and the in-phase spikinggroup.To more clearly examine the above cycle evolutions,we get the bin-averaged (normalized) synaptic weight ineach i th bin (bin size: ∆ t = 100 msec): (cid:104) ˜ J ( t ) (cid:105) i = 1 N s,i N s,i (cid:88) f =1 ˜ J i,f ( t ) , (33)where ˜ J i,f is the normalized synaptic weight of the f thactive PF signal in the i th bin, and N s,i is the total num-ber of active PF signals in the i th bin. Figures 8(b1)-8(b5) show cycle-evolution of bin-averaged (normalized)synaptic weights (cid:104) ˜ J ( t ) (cid:105) of active PF signals. In each cy-cle, (cid:104) ˜ J ( t ) (cid:105) forms a well-shaped curve. With increasing thecycle, the well curve comes down, its modulation [=(max-imum - minimum)/2] increases, and saturation seems tooccur at about the 300th cycle.We also get the cycle-averaged mean (cid:104) ˜ J ( t ) (cid:105) via timeaverage of (cid:104) ˜ J ( t ) (cid:105) over a cycle: (cid:104) ˜ J (cid:105) = 1 N b N b (cid:88) i =1 (cid:104) ˜ J ( t ) (cid:105) i , (34)where N b is the number of bins for cycle averaging, andthe overbar represents the time average over a cycle. Fig- ures 8(c) and 8(d) show plots of the cycle-averaged mean (cid:104) ˜ J ( t ) (cid:105) and the modulation M J for (cid:104) ˜ J ( t ) (cid:105) versus cycle.The cycle-averaged mean (cid:104) ˜ J ( t ) (cid:105) decreases from 1 to 0.372due to LTD at PF-PC synapses. However, strength of theLTD varies depending on the stages of the cycle. At themiddle stage, strong LTD occurs, due to dominant contri-bution of in-phase active PF signals. On the other hand,at the initial and the final stages, somewhat less LTDtakes place, because both the out-of-phase spiking group(with weak LTD) and the in-phase spiking group makecontributions together. As a result, with increasing cy-cle, the middle-stage part comes down more rapidly thanthe initial and final parts, and hence the modulation M J increases from 0 to 0.112.We now decompose the whole active PF signals intothe in-phase and the out-of-phase active PF signals, andmake an intensive investigation on their effect on synap-tic plasticity at PF-PC synapses. Figure 9 shows cycle-evolution of distributions of normalized synaptic weights˜ J ( G ) of active PF signals in the G spiking group [(a1)-(a5)in-phase ( G = i ) and (b1)-(b5) out-of-phase ( G = o )] inthe optimal case of p ∗ c = 0 . 06; the out-of-phase spikinggroup consists of the anti-phase and the complex out-of-phase spiking groups. With increasing learning cycle,normalized synaptic weights ˜ J ( G ) for the in-phase andthe out-of-phase PF signals are decreased, and saturatedat about the 300th cycle.We note that the strength of LTD varies distinctlydepending on the type of spiking group. The stu-dent PF signals (corresponding to the axons of the GRcells) are classified as in-phase or out-of-phase PF sig-nals with respect to the “reference” signal R GR ( t ). Here, R GR ( t ) is the instantaneous whole-population spike rateof Eq. (18)(denoting the population-averaged firing ac-tivity in the whole population of the GR cells). It is ba-sically in proportion to the sinusoidal MF context signal f MF ( t ) of Eq. (1), although its top part becomes loweredand flattened due to inhibitory coordination of GO cells.The PF student signals (coming from the GR clusters)are characterized in terms of their conjunction indices C ( I ) of Eq. (25) between the instantaneous cluster spikerate R ( I )GR ( t ) and the reference signal R GR ( t ). The rangesof {C ( I ) } for the in-phase and the out-of-phase spikinggroups are (0.39, 0.85) and (-0.57, 0.39), respectively,as shown in Fig. 6(d). The range for the out-of-phasespiking group is broader than that for the in-phase spik-ing group. These student PF signals are depressed bythe instructor CF signals (coming from the IO neuron).We also note that the instructor CF signals are in-phasewith the reference signal R GR ( t ), because the CF sig-nals are also basically in proportion to the sinusoidal IOdesired signal f DS ( t ) of Eq. (2) which is in-phase with f MF ( t ). Hence, in-phase student PF signals are stronglydepressed by the instructor CF signals, because they are“well-matched” with the in-phase CF signals. On theother hand, out-of-phase student PF signals are weaklydepressed by the instructor CF signals, because they are6 < W J ( G ) ( t ) >(< J ( t ) >) ~ (e1) (e2) (e3) (e4) (e5) 400 16000.31.0 400 1600 400 1600 t (msec) 400 1600 400 1600 < W J ( G ) ( t ) >(< J ( t ) >) ~ (f1) < W J ( G ) ( t ) >(< J ( t ) >) ~ Initial Gray Stage(g2) < W J ( G ) ( t ) >(< J ( t ) >) ~ Final Gray Stage(g3) < W J ( G ) ( t ) >(< J ( t ) >) ~ G : In-Phase, Out-of-Phase; Whole PopulationMiddle Gray Stage(g1) P C ( G ) ( % ) Cycle (h2) P C ( G ) ( % ) (h3) Cycle P C ( G ) ( % ) Cycle (h1) P C ( G ) ( % ) Cycle (f2) ~ (a1) 1st cycle < J ( G ) ( t ) > (d1) F ( G ) ( t ) (a2)100th cycle (a3)200th cycle (a4)300th cycle (a5) 400th cycle < J ( G ) ( t ) > ~ (b) ( G ) (c) M J Cycle (d2) (d3) (d4) (d5) FIG. 10: Optimal case of p ∗ c = 0 . 06. (a1)-(a5) Cycle-evolution of bin-averaged (normalized) synaptic weights (cid:104) ˜ J ( G ) (cid:105) of activePF signals in the G spiking-group. Plots of (b) cycle-averaged mean (cid:104) ˜ J ( G ) (cid:105) and (c) modulation M ( G ) J for (cid:104) ˜ J ( G ) (cid:105) versus cycle.(d1)-(d5) Cycle-evolution of firing fraction F ( G ) of active PF signals in the G spiking group. (e1)-(e5) Cycle-evolution ofweighted bin-averaged synaptic weights (cid:104) W J ( G ) (cid:105) of active PF signals in the G spiking group; for comparison, bin-averagedsynaptic weights (cid:104) ˜ J (cid:105) in the whole population of GR cells are also given. (f1) Plots of cycle-averaged means (cid:104) W J ( G ) (cid:105) and (cid:104) ˜ J (cid:105) for (cid:104) W J ( G ) (cid:105) and (cid:104) ˜ J (cid:105) versus cycle. (f2) Plots of percentage contributions P C ( G ) of the G spiking group (i.e., (cid:104) W J ( G ) (cid:105) / (cid:104) ˜ J (cid:105) )versus cycle. Left, right, and middle vertical gray bands in each cycle in (e1)-(e5) denote the initial, final, and middle stages,respectively. Plots of stage-averaged values (cid:104) W J ( G ) (cid:105) and (cid:104) ˜ J (cid:105) versus cycle at the (g1) middle,(g2) initial, and (g3) final stagesof a cycle. Plots of percentage contributions P C ( G ) of the G spiking group versus cycle at the (h1) middle, (h2) initial, and(h3) final stages of a cycle. “ill-matched” with the in-phase CF signals.In the case of active in-phase PF signals, the distribu-tion of their normalized synaptic weights ˜ J ( G ) ( t ) ( G = i )forms the bottom dense bands in Figs. 8(a1)-8(a5), due tostrong LTD at the in-phase PF-PC synapses. The (ver-tical) widths of these bottom bands are narrow becauseof the narrow range of { C ( I ) } . On the other hand, in thecase of active out-of-phase PF signals, the distributionof { ˜ J ( G ) ( t ) } ( G = o ) constitutes the upper sparse well-shaped parts in Figs. 8(a1)-8(a5), because of weak LTDat the out-of-phase PF-PC synapses. Due to the broadrange of { C ( I ) } , the heights of the well-shaped parts forthe out-of-phase PF signals are higher than those of thebottom bands for the in-phase PF signals with the nar-row range. Moreover, the shapes for the distributions of { ˜ J ( G ) ( t ) } are consistent with the activation degrees A ( G ) of Eq.(21) of the active PF signals in the G spiking groupwhich is also shown in Fig. 4(c2). The activation degree A ( o ) ( G = o ) for the out-of-phase spiking group has asmall minimum in the middle stage of cycle, which leadsto the well-shaped distributions of { ˜ J ( G ) ( t ) } ( G = o ).Hence, the in-phase PF signals (with strong LTD) makea large contribution in the middle stage of cycle, whileat the initial and the final stages, contributions of boththe out-of-phase PF signals (with weak LTD) and thein-phase PF signals are comparable. In the above way, effective depression (i.e.,strong/weak LTD) at PF-PC synapses occurs, de-pending on the spiking type (in-phase or out-of-phase)of the active PF signals; strong (weak) LTD takes placefor the in-phase (out-of-phase) PF signals. However,contributions of these in-phase and out-of-phase spikinggroups vary depending on the stages of cycle. Figure4(c2) shows the activation degree A ( G ) of the in-phase( G = i ) and the out-of-phase ( G = o ) spiking groups.In the middle stage of cycle, the in-phase spiking grouphas a larger activation degree A ( i ) . On the other hand,at the initial and the final stages, the activation degreesof the in-phase and the out-of-phase spiking groupsare comparable. Hence, strong LTD takes place inthe middle stage, due to a large contribution of thein-phase spiking group. Thus, the in-phase spikinggroup makes a big contribution to formation of theminimum of bin-averaged (normalized) synaptic weights (cid:104) ˜ J ( t ) (cid:105) in Figs. 8(b1)-8(b5). In contrast, at the initialand the final stages of cycle, less LTD occurs becauseof comparable contributions of both the out-of-phasespiking group with weak LTD and the in-phase spikinggroup with strong LTD. Hence, maxima of (cid:104) ˜ J ( t ) (cid:105) appearat the initial and the final stages of cycle becauseboth the (weakly-depressed) out-of-phase and the(strongly-depressed) in-phase spiking groups contribute7together.In this way, the in-phase and the out-of-phase spikinggroups play their own roles in formation of modulationof (cid:104) ˜ J ( t ) (cid:105) . The minimum of (cid:104) ˜ J ( t ) (cid:105) in the middle stage ofcycle is formed via a large contribution of the in-phasespiking group with strong LTD, while formation of themaxima of (cid:104) ˜ J ( t ) (cid:105) in the initial and the final stages ismade via comparable contributions of the out-of-phasespiking group with weak LTD and the in-phase spikinggroup with strong LTD. Consequently, this kind of con-structive interplay between the in-phase (strong LTD)and the out-of-phase (weak LTD) spiking groups leads toa big modulation of (cid:104) ˜ J ( t ) (cid:105) , as shown in Fig. 8(d). (Moredetailed discussion on this point will be given below inFig. 10.)We also make further decomposition of the out-of-phase PF signals into the anti-phase and the complexout-of-phase ones. Figure 9 also shows cycle-evolutionof distributions of normalized synaptic weights ˜ J ( G ) ofactive out-of-phase PF signals in the G spiking group[ G : (c1)-(c5) anti-phase and (d1)-(d5) complex out-of-phase]. As the learning cycle is increased, normalizedsynaptic weights ˜ J ( G ) for the anti-phase and the com-plex out-of-phase PF signals are decreased and saturatedat about the 300th cycle. In the case of anti-phase PFsignals, weak depression occurs, and they constitute thetop part for the out-of-phase PF signals in Figs. 9(b1)-9(b5). On the other hand, in the case of complex out-of-phase PF signals, intermediate LTD takes place, andthey form the bottom part for the out-of-phase PF signalsin Figs. 9(b1)-9(b5). These anti-phase (weak LTD) andcomplex out-of-phase (intermediate LTD) spiking groupsmake contribution to formation of maxima of (cid:104) ˜ J ( t ) (cid:105) atthe initial and the final stages of cycle, together with thein-phase spiking group with strong LTD.Figures 10(a1)-10(a5) show cycle-evolutions of bin-averaged (normalized) synaptic weights (cid:104) ˜ J ( G ) ( t ) (cid:105) of ac-tive PF signals in the G spiking-group [i.e., correspond-ing to the bin-averages for the distributions of ˜ J ( G ) ( t )in Figs. 9(a1)-9(a5) and Figs. 9(b1)-9(b5)]; G : in-phase(solid circles) and out-of-phase (open circles). In thecase of in-phase PF signals, they are strongly depressedwithout modulation. On the other hand, in the case ofout-of-phase PF signals, they are weakly depressed withmodulation; at the initial and the final stages, they aremore weakly depressed in comparison with the case atthe middle stage.The cycle-averaged means (cid:104) ˜ J ( G ) ( t ) (cid:105) and modulations M ( G ) J for (cid:104) ˜ J ( G ) ( t ) (cid:105) are given in Figs. 10(b) and 10(c),respectively. Both cycle-averaged means in the cases ofthe in-phase and the out-of-phase PF signals decrease,and saturations occur at about the 300th cycle. In com-parison with the case of out-of-phase PF signals (opencircles), the cycle-averaged means in the case of in-phasePF signals (solid circles) are more reduced; the saturatedlimit value in the case of in-phase (out-of-phase) PF sig-nals is 0.199 (0.529). In contrast, modulation occurs only for the out-of-phase PF signals (open circles), it increaseswith cycle, and becomes saturated at about the 300th cy-cle where the saturated value is 0.087.In addition to the above bin-averaged (normalized)synaptic weights (cid:104) ˜ J ( G ) ( t ) (cid:105) , we need another informationon firing fraction F ( G ) ( t ) of active PF signals in the G (in-phase or out-of-phase) spiking group to obtain thecontribution of each spiking group to the bin-averagedsynaptic weights (cid:104) ˜ J ( t ) (cid:105) of active PF signals in the wholepopulation. The firing fraction F ( G ) i of active PF signalsin the G spiking group in the i th bin is given by: F ( G ) i = N ( G ) s,i N s,i , (35)where N s,i is the total number of active PF signals in the i th bin and N ( G ) s,i is the number of active PF signals inthe G spiking group in the i th bin. We note that F ( G ) ( t )is the same, independently of the learning cycle, becausefiring activity of PF signals depends only on the GR andGO cells in the granular layer.Figures 10(d1)-10(d5) show the firing fraction F ( G ) ( t )of active PF signals in the G spiking group. The firingfraction F ( G ) ( t ) for the in-phase ( G = i ) active PF signals(solid circles) forms a bell-shaped curve, while F ( G ) ( t ) forthe out-of-phase ( G = o ) active PF signals (open circles)forms a well-shaped curve. The bell-shaped curve for thein-phase PF signal is higher than the well-shaped curvefor the out-of-phase PF signal. For the in-phase PF sig-nals, the firing fraction F ( i ) ( t ) is about 0.94 (i.e., 94%) atthe middle stage, and about 0.51 (i.e., 51%) at the initialand the final stages. On the other hand, for the out-of-phase PF signals, the firing fraction F ( o ) ( t ) is about0.49 (i.e., 49%) at the initial and the final stages, andabout 0.06 (i.e., 6%) at the middle stage. Consequently,the fraction of in-phase active PF signals is dominant atthe middle stage, while at the initial and the final stages,the fractions of both in-phase and out-of-phase active PFsignals are nearly the same.The weighted bin-averaged synaptic weight (cid:104) W ( G ) J (cid:105) i foreach G spiking group in the i th bin is given by the prod-uct of the firing fraction F ( G ) i and the bin-averaged (nor-malized) synaptic weight (cid:104) ˜ J ( G ) (cid:105) i : (cid:104) W ( G ) J (cid:105) i = F ( G ) i (cid:104) ˜ J ( G ) (cid:105) i , (36)where the firing fraction F ( G ) i plays a role of a weightingfunction for (cid:104) ˜ J ( G ) (cid:105) i . Then, the bin-averaged (normal-ized) synaptic weight (cid:104) ˜ J (cid:105) i of active PF signals in thewhole population in the i th bin [see Eq. (33)] is givenby the sum of weighted bin-averaged synaptic weights (cid:104) W ( G ) J (cid:105) i of all spiking groups: (cid:104) ˜ J (cid:105) i = all spiking groups (cid:88) G (cid:104) W ( G ) J (cid:105) i . (37)8Hence, (cid:104) W ( G ) J ( t ) (cid:105) represents contribution of the G spik-ing group to (cid:104) ˜ J ( t ) (cid:105) of active PF signals in the wholepopulation.Figures 10(e1)-10(e5) show cycle-evolution of weightedbin-averaged synaptic weights (cid:104) W ( G ) J ( t ) (cid:105) of active PFsignals in the G spiking group [ G : in-phase (solid cir-cles) and out-of-phase (open circles)]. In the case of in-phase PF signals, the bin-averaged (normalized) synapticweights (cid:104) ˜ J ( G ) ( t ) (cid:105) are straight horizontal lines, the firingfraction F ( G ) ( t ) is a bell-shaped curve, and hence theirproduct leads to a bell-shaped curve for the weightedbin-averaged synaptic weight (cid:104) W ( G ) J ( t ) (cid:105) . With increas-ing the cycle, the horizontal straight lines for (cid:104) ˜ J ( G ) ( t ) (cid:105) come down rapidly, while there is no change with cyclein F ( G ) ( t ). Hence, the bell-shaped curves for (cid:104) W ( G ) J ( t ) (cid:105) also come down quickly, their modulations also are re-duced in a fast way, and they become saturated at aboutthe 300th cycle.On the other hand, in the case of out-of-phase PFsignals, the bin-averaged (normalized) synaptic weights (cid:104) ˜ J ( G ) ( t ) (cid:105) lie on a well-shaped curve, the firing fraction F ( G ) ( t ) also is a well-shaped curve, and then their prod-uct results in a well-shaped curve for the weighted bin-averaged synaptic weight (cid:104) W ( G ) J ( t ) (cid:105) . With increasing thecycle, the well-shaped curves for (cid:104) ˜ J ( G ) ( t ) (cid:105) i come downslowly, while there is no change with cycle in F ( G ) ( t ).Hence, the well-shaped curves for (cid:104) W ( G ) J ( t ) (cid:105) also comedown gradually, their modulations also are reduced littleby little, and eventually they get saturated at about the300th cycle.For comparison, bin-averaged (normalized) synapticweights (cid:104) ˜ J ( t ) (cid:105) of active PF signals in the whole popu-lation (crosses) are also given in Figs. 10(e1)-10(e5), andthey form a well-shaped curve. According to Eq. (37), thesum of the values at the solid circle (in-phase) and theopen circle (out-of-phase) at a time t in each cycle is justthe value at the cross (whole population). At the middlestage of each cycle, contributions of in-phase PF signals(solid circles) are dominant [i.e., contributions of out-of-phase PF signals (open circles) are negligible], while atthe initial and the final stages, contributions of out-of-phase PF signals are larger than those of in-phase PFsignals (both contributions must be considered). Conse-quently, (cid:104) ˜ J ( t ) (cid:105) of active PF signals in the whole popu-lation becomes more reduced at the middle stage thanat the initial and the final stages, due to the dominanteffect (i.e., strong LTD) of in-phase active PF signals atthe middle stage, which results in a well-shaped curve for (cid:104) ˜ J ( t ) (cid:105) in the whole population.We make a quantitative analysis for contribution of (cid:104) W ( G ) J ( t ) (cid:105) in each G spiking group to (cid:104) ˜ J ( t ) (cid:105) in the wholepopulation. Figure 10(f1) shows plots of cycle-averagedweighted synaptic weight (cid:104) W ( G ) J ( t ) (cid:105) (i.e., time averageof (cid:104) W ( G ) J ( t ) (cid:105) over a cycle) in the G spiking-group [ G :in-phase (solid circles) and out-of-phase (open circles)] and cycle-averaged synaptic weight (cid:104) ˜ J ( t ) (cid:105) of Eq. (34) inthe whole population (crosses) versus cycle. The cycle-averaged weighted synaptic weights (cid:104) W ( G ) J ( t ) (cid:105) in the in-phase spiking group (solid circles) are larger than those inthe out-of-phase spiking group (open circles), and theirsums correspond to the cycle-averaged synaptic weight (cid:104) ˜ J ( t ) (cid:105) in the whole population (crosses). With increas-ing cycle, both (cid:104) W ( G ) J ( t ) (cid:105) and (cid:104) ˜ J ( t ) (cid:105) become saturatedat about the 300th cycle. In the in-phase spiking group (cid:104) W ( G ) J ( t ) (cid:105) decreases rapidly from 0.722 to 0.198, while (cid:104) W ( G ) J ( t ) (cid:105) in the out-of-phase spiking group decreasesslowly from 0.273 to 0.174. Thus, the saturated valuesof (cid:104) W ( G ) J ( t ) (cid:105) in both the in-phase and the out-of-phasespiking groups become close.The percentage contribution P C ( G ) of (cid:104) W ( G ) J ( t ) (cid:105) in the G spiking group to (cid:104) ˜ J ( t ) (cid:105) in the whole population is givenby: P C ( G ) (%) = (cid:104) W ( G ) J ( t ) (cid:105)(cid:104) ˜ J ( t ) (cid:105) × . (38)Figure 10(f2) shows a plot of P C ( G ) versus cycle [ G :in-phase (solid circles) and out-of-phase (open circles)]. P C ( G ) of the in-phase spiking group decreases from 72.2% to 53.2 %, while P C ( G ) of the out-of-phase spikinggroup increases from 27.3 % to 46.8 %. Thus, the satu-rated values of P C ( G ) of both the in-phase and the out-of-phase spiking groups get close.We are particularly interested in the left, the right,and the middle vertical gray bands in each cycle inFigs. 10(e1)-10(e5) which denote the initial (0 < t < < t < < t < (cid:104) W ( G ) J ( t ) (cid:105) appears at the middle stage,while the minimum (maximum) occurs at the initial andthe final stages. Figures 10(g1)-10(g3) show plots ofstage-averaged weighted synaptic weight (cid:104) W ( G ) J ( t ) (cid:105) [i.e.,time average of (cid:104) W ( G ) J ( t ) (cid:105) over a stage] in the G spiking-group [ G : in-phase (solid circles) and out-of-phase (opencircles)] and stage-averaged synaptic weight (cid:104) ˜ J ( t ) (cid:105) [i.e.,time average of (cid:104) ˜ J ( t ) (cid:105) over the stage] in the whole pop-ulation (crosses) versus cycle in the middle, the initial,and the final stages, respectively. The sum of the val-ues of (cid:104) W ( G ) J ( t ) (cid:105) at a time t in the in-phase and theout-of-phase spiking groups corresponds to the value of (cid:104) ˜ J ( t ) (cid:105) in the whole population. As the cycle is increased,both (cid:104) W ( G ) J ( t ) (cid:105) and (cid:104) ˜ J ( t ) (cid:105) become saturated at aboutthe 300th cycle. Figures 10(h1)-10(h3) also show plotsof percentage contribution P C ( G ) of the G spiking group(i.e., ratio of the stage-averaged weighted synaptic weight (cid:104) W ( G ) J ( t ) (cid:105) in the G spiking group to the stage-averagedsynaptic weight (cid:104) ˜ J ( t ) (cid:105) in the whole population) in the9middle, the initial, and the final stages, respectively [ G :in-phase (solid circles) and out-of-phase (open circles)].In the case of in-phase spiking group, (cid:104) W ( G ) J ( t ) (cid:105) de-creases rapidly with cycle in all the 3 stages, while in thecase of out-of-phase spiking group, it also decreases ina relatively slow way with cycle. At the middle stage, (cid:104) W ( G ) J ( t ) (cid:105) in the in-phase spiking group (solid circles) ismuch higher than that in the out-of-phase spiking group(open circles), and it decreases rapidly from 0.944 to0.266. In this case, the percentage contribution P C ( G ) of the in-phase spiking group increases from 94.4 % to97.0 %. Consequently, the contribution of in-phase spik-ing group is dominant at the middle stage, which leads tostrong LTD at the PF-PC synapses. On the other hand,at the initial and the final stages, with increasing cy-cle (cid:104) W ( G ) J ( t ) (cid:105) in the out-of-phase spiking group becomeslarger than that in the in-phase spiking group. The per-centage contribution P C ( G ) of the out-of-phase spikinggroup increases from 49.2 % to 70.2 %, while that ofthe in-phase spiking group decreases from 50.8 % to 29.8%. As a result, the contribution of out-of-phase spikinggroup at the initial and the final stages is larger than thatof in-phase spiking group, which results in weak LTD atthe PF-PC synapses.In the above way, under good balance between the in-phase and the out-of-phase spiking groups (i.e., spiking-group ratio R ∗ (cid:39) . p ∗ c = 0 . D. Effect of PF-PC Synaptic Plasticity onSubsequent Learning Process in The PC-VN-IOSystem As a result of diverse recoding in the GR clusters,strong LTD occurs at the middle stage of a cycle dueto dominant contribution of the in-phase spiking group.On the other hand, at the initial and the final stages,somewhat less LTD takes place due to contribution ofboth the out-of-phase spiking group (with weak LTD)and the in-phase spiking group. Due to this kind of ef-fective (strong/weak) LTD at the PF-PC synapses, a bigmodulation in synaptic plasticity at the PF-PC synapses(i.e., big modulation in bin-averaged normalized synapticweight (cid:104) ˜ J ( t ) (cid:105) ) emerges. In this subsection, we investigatethe effect of PF-PC synaptic plasticity with a big modula-tion on the subsequent learning process in the PC-VN-IOsystem. 400 16002080 400 1600 400 1600 400 1600 400 1600313 400 1600050100 400 1600 400 1600 400 1600 400 16000 200 4004070100 0 200 40001530 (b1) (b2) (b3) t (msec) (b4) (b5) i (a1) 1st cycle R P C ( t ) ( H z ) (a2) 100th cycle (a3) 200th cycle (a4) 300th cycle (a5) 400th cycle < R P C ( t ) > r ( H z ) (c3) t (msec) < R P C ( t ) > r ( H z ) (d1) Cycle (d2) M P C Cycle FIG. 11: Change in firing activity of PCs during learningin the optimal case of p ∗ c = 0 . 06. (a1)-(a5) Raster plots ofspikes of PCs and (b1)-(b5) instantaneous population spikerates R PC ( t ). (c1)-(c5) Realization-averaged instantaneouspopulation spike rates (cid:104) R PC ( t ) (cid:105) r ; number of realizations is100. Plots of (d1) cycle-averaged mean (cid:104) R PC ( t ) (cid:105) r and (d2)modulations M PC for (cid:104) R PC ( t ) (cid:105) r versus cycle. Figure 11 shows change in firing activity of PCs dur-ing learning in the optimal case of p ∗ c = 0 . 06. Cycle-evolutions of raster plots of spikes of 16 PCs and the cor-responding instantaneous population spike rates R PC ( t )are shown in Figs. 11(a1)-11(a5) and Figs. 11(b1)-11(b5),respectively. Since the number of PCs is small, R PC ( t )seems to be a little rough. To get a smooth estimate for R PC ( t ), we make 100 realizations. Realization-averagedsmooth instantaneous population spike rates (cid:104) R PC ( t ) (cid:105) r are given in Figs. 11(c1)-11(c5); (cid:104)· · · (cid:105) r denotes realizationaverage and the number of realizations is 100. (cid:104) R PC ( t ) (cid:105) r seems to be saturated at about the 300th cycle.With increasing the learning cycle, raster plots ofspikes of all the 16 PCs become more and more sparseat the middle stage, which may be clearly seen in theinstantaneous population spike rate (cid:104) R PC ( t ) (cid:105) r . Due tothe effect of synaptic plasticity at the PF-PC synapses,the minimum of (cid:104) R PC ( t ) (cid:105) r appears at the middle stage,while the maximum occurs at the initial and the finalstages. Thus, the modulation of (cid:104) R PC ( t ) (cid:105) r increases withincreasing the cycle.In-phase PF (student) signals are strongly depressedby the in-phase CF (instructor) signals, while out-of-phase PF signals are weakly pressed due to the phase dif-ference between the out-of-phase PF signals and the in-phase CF signals. Fraction of in-phase PF signals at themiddle stage of a cycle is dominant. On the other hand,at the initial and the final stages, fraction of out-of-phasePF signals are larger than that of in-phase PF signals.Thus, bin-averaged normalized synaptic weights (cid:104) ˜ J ( t ) (cid:105) form a well-shaped curve, as shown in Figs. 8(b1)-8(b5).That is, strong LTD occurs at the middle stage, while atthe initial and the final stages, weak LTD takes place. Asa result of this kind of effective depression (strong/weak0 400 1600050100 400 1600 400 1600 400 1600 400 1600400 1600070140 400 1600 400 1600 400 1600 400 1600 f VN ( t ) ( H z ) R ea li za ti on t (msec) f VN ( t ) ( H z ) (b3) t (msec) (c1) Cycle (c2) M VN Cycle (d) L g Cycle FIG. 12: Change in firing activity of the VN neuron duringlearning in the optimal case of p ∗ c = 0 . 06. (a1)-(a5) Rasterplots of spikes of the VN neuron (i.e., collection of spike trainsfor all the realizations; number of realizations is 100) and (b1)-(b5) bin-averaged instantaneous individual firing rate f VN ( t );the bin size is ∆ t = 100 msec. Plots of (c1) cycle-averagedmean f VN ( t ) and (c2) modulation M VN for f VN ( t ) versus cy-cle. (d) Plot of learning gain degree L g versus cycle. LTD) at the (excitatory) PF-PC synapses, (cid:104) R PC ( t ) (cid:105) r be-comes lower at the middle stage (strong LTD) than at theinitial and the final stages (weak LTD). Thus, (cid:104) R PC ( t ) (cid:105) r forms a well-shaped curve with a minimum at the middlestage.Figures 11(d1) and 11(d2) show plots of cycle-averagedmean (cid:104) R PC ( t ) (cid:105) r (i.e., time average of (cid:104) R PC ( t ) (cid:105) r over acycle) and modulation M PC of (cid:104) R PC ( t ) (cid:105) r versus cycle,respectively. Due to LTD at the PF-PC synapses, thecycle-averaged mean (cid:104) R PC ( t ) (cid:105) r decreases monotonicallyfrom 86.1 Hz, and it becomes saturated at 51.7 Hz atabout the 300th cycle. On the other hand, the mod-ulation M PC increases monotonically from 2.6 Hz, andit get saturated at 24.1 Hz at about the 300th cycle.Consequently, a big modulation occurs in M PC due tothe effective depression (strong/weak LTD) at the PF-PC synapses. These PCs (principal cells of the cerebellarcortex) exert effective inhibitory coordination on the VNneuron which evokes OKR eye-movement.Figure 12 shows change in firing activity of the VNneuron which produces the final output of the cerebel-lum during learning in the optimal case of p ∗ c = 0 . f VN ( t ) [i.e., the num-ber of spikes of the VN neuron in a bin divided by the binwidth (∆ t = 100 msec)] are shown in Figs. 12(a1)-12(a5) 400 1600050100 400 1600 400 1600 400 1600 400 1600400 1600024 400 1600 400 1600 400 1600 400 16000 200 4005075100 0 200 4000.500.751.000 200 400012 R ea li za ti on t (msec) f I O ( t ) ( H z ) (c3) t (msec) < I (IO,VN)GABA > r , < | I (IO,DS)AMPA | > r (a1) < I (I O , VN ) GA B A > r ( < | I (I O , D S ) A M P A | > r ) Cycle (a2) L p Cycle (d) f I O ( t ) ( H z ) Cycle FIG. 13: Change in firing activity of the IO neuron duringlearning in the optimal case of p ∗ c = 0 . 06. Plots of (a1)realization-average for the cycle-averaged inhibitory synapticcurrent from the VN neuron ( (cid:104) I (IO , VN)GABA (cid:105) r ) (open circles) andrealization-average for the magnitude of the cycle-averagedexcitatory synaptic current through the IO desired signal( (cid:104)| I (IO , DS)AMPA |(cid:105) r ) (crosses) versus cycle; number of realizations (cid:104)· · · (cid:105) r is 100. (a2) Plot of learning progress degree L p versuscycle. (b1)-(b5) Raster plots of spikes of the IO neuron (i.e.,collection of spike trains for all the realizations; number ofrealizations is 100) and (c1)-(c5) bin-averaged instantaneousindividual firing rate f IO ( t ); the bin size is ∆ t = 400 msec.(d) Plot of cycle-averaged individual firing rate f IO ( t ) versuscycle. and Figs. 12(b1)-12(b5), respectively. f VN ( t ) seems to besaturated at about the 300th cycle.In contrast to the case of PCs, as the cycle is increased,raster plots of spikes of the VN neuron become more andmore dense at the middle stage, which may be clearlyseen in the instantaneous individual firing rates f VN ( t ).Due to the effective inhibitory coordinations of PCs onthe VN neuron, the maximum of f VN ( t ) appears at themiddle stage, while the minimum occurs at the initialand the final stages. Thus, f VN ( t ) forms a bell-shapedcurve.Figures 12(c1) and 12(c2) show plots of cycle-averagedmean f VN ( t ) of f VN ( t ) [i.e., time average of f VN ( t ) overa cycle] and modulation M VN of f VN ( t ) versus cycle, re-spectively. Due to the decreased inhibitory inputs fromthe PCs, the cycle-averaged mean f VN ( t ) increases mono-tonically from 44.3 Hz, and it becomes saturated at 71.5Hz at about the 300th cycle. Also, the modulation of f VN ( t ) increases from 20.2 Hz, and it gets saturated at32.5 Hz at about the 300th cycle. As a result of effectiveinhibitory coordination of PCs, a big modulation occurs1in M VN .The learning gain degree L g , corresponding to themodulation gain ratio, is given by the normalized modu-lation of f VN ( t ) divided by that at the 1st cycle: L g = M VN M VN at the 1st cycle , (39)where M VN at the 1st cycle is 20.2 Hz. Figure 12(d)shows a plot of L g versus cycle. L g increases mono-tonically from 1, and it becomes saturated at about the300th cycle. Thus, we get the saturated learning gaindegree L ∗ g ( (cid:39) . L ∗ g ( (cid:39) . p ∗ c = 0 . 06 wherespiking patterns of GR clusters with the diversity degree D ∗ ( (cid:39) . L ∗ g ( (cid:39) . L p , given by the ratio of thecycle-averaged inhibitory input from the VN neuron tothe magnitude of the cycle-averaged excitatory input viathe desired signal: L p = I (IO , VN)GABA | I (IO , DS)AMPA | , (40)where I (IO , VN)GABA is the cycle-averaged inhibitory GABAreceptor-mediated current from the VN neuron into theIO neuron, and I (IO , DS)AMPA is the cycle-averaged excitatoryAMPA receptor-mediated current into the IO neuron viathe desired signal; no (excitatory) NMDA receptors existon the IO neuron. [Note that the 4th term in Eq. (3) isgiven by − I ( X ) syn,i ( t ), because I (IO , CN)GABA > I (IO , US)AMPA < . ]Figure 13(a1) shows plots of I (IO , VN)GABA (open circles)and | I (IO , DS)AMPA | (crosses) versus cycle in the optimal case of p ∗ c = 0 . 06. With increasing the cycle, the cycle-averagedinhibitory input from the VN neuron increases, and con-verges to the constant magnitude of the cycle-averagedexcitatory input through the IO desired signal. Thus,as shown in Fig. 13(a2), L p increases with cycle, and atabout the 300th cycle, it becomes saturated at L p = 1.In this saturated case, the cycle-averaged excitatory andinhibitory inputs into the IO neuron are balanced.We also study the firing activity of IO neuron dur-ing learning process. Figures 13(b1)-13(b5) and Fig-ures 13(c1)-13(c5) show cycle-evolutions of raster plotsof spikes of the IO neuron (i.e., collection of spike trainsfor all the realizations; number of realizations is 100) andthe bin-averaged instantaneous individual firing rates f IO In-Phase(e1) F r ac ti on o f C (I) C (I) C (I) Out-of-Phase Out-of-Phase(e2) Out-of-Phase In-Phase (d) F r ac ti on Spiking Group (c) F r ac ti on o f C (I) C (I) i (b) R G R ( t ) ( H z ) t (msec) FIG. 14: Highly-connected case of p c = 0 . 6. (a) Raster plotof spikes of 10 randomly chosen GR cells. (b) Instantaneouswhole-population spike rate R GR ( t ) in the whole populationof GR cells. Band width for R GR ( t ): h = 10 msec. (c) Distri-bution of conjunction indices {C ( I ) } for the GR clusters in thewhole population. (d) Fraction of spiking groups. Distribu-tions of conjunction indices {C ( I ) } for the (e1) in-phase and(e2) out-of-phase spiking groups. Bin size for the histogramsin (c) and (e1)-(e2) is 0.1. (f) Ranges of {C ( I ) } in the in-phaseand the out-of-phase spiking groups. [i.e., the number of spikes of the IO neuron in a bin di-vided by the bin width (∆ t = 400 msec)], respectively. Inthe 1st cycle, relatively dense spikes appear at the mid-dle stage of the cycle in the raster plot of spikes, due tothe effect of excitatory IO desired signal. However, withincreasing the cycle, spikes at the middle stage becomesparse, because of increased inhibitory input from theVN neuron. In this case, the bin-averaged instantaneousindividual firing rate f IO ( t ) of the IO neuron forms abell-shaped curve due to the sinusoidally-modulating de-sired input signal into the IO neuron. With increasingthe cycle, the amplitude of f IO ( t ) decreases due to theinhibitory input from the VN neuron, and it becomessaturated at about the 300th cycle. Thus, the cycle-averaged individual firing rate f IO ( t ) is decreased from1.51 Hz to 0.09 Hz, as shown in Fig. 13(d). The firingoutput of the IO neuron is fed into the PCs via the CFs.Hence, with increasing the cycle, the error-teaching CFinstructor signals become weaker and saturated at aboutthe 300th cycle.While the saturated CF signals are fed into the PCs,saturation for the cycle-averaged bin-averaged synapticweights (cid:104) ˜ J ( t ) (cid:105) appears [see Fig. 8(c)]. Then, the sub-sequent learning process in the PC-VN system also be-comes saturated, and we get the saturated learning gaindegree L ∗ g ( (cid:39) . E. Dependence of Diversity Degree D and LearningGain Degree L g on p c (Connection Probability fromGO to GR Cells) In the above subsections, we consider only the optimalcase of p ∗ c = 0 . 06 (i.e., 6%) where the spiking patterns ofthe GR clusters are the most diverse. From now on, wevary the connection probability p c from GO to GR cells,and investigate the dependence of the diversity degree D for the spiking patterns of the GR clusters and thelearning gain degree L g on p c .We first consider the highly-connected case of p c =0 . randomly chosen GR cells, and thepopulation-averaged firing activity in the whole popu-lation of GR cells may be well seen in the instantaneouswhole-population spike rate R GR ( t ) in Fig. 14(b). Asshown in Fig. 2(b), each GR cluster is bounded by twoglomeruli (corresponding to the terminals of the MFs)at both ends. Each glomerulus receives inhibitory inputsfrom nearby 81 GO cells with the connection probability p c = 0 . 6. Hence, on average, about 49 GO cell axonsinnervate each glomerulus. Then, each GR cell in a GRcluster receives about 97 inhibitory inputs via two den-drites which contact the two glomeruli at both ends. Dueto the increased inhibitory inputs from the pre-synapticGO cells, spike density in the raster plot is decreased, andthe top part of R GR ( t ) becomes lowered and broadly flat-tened, in comparison with the optimal case in Fig. 4(b).Thus, R GR ( t ) becomes more different from the firing rate f MF for the MF signal in Fig. 1(b1).GR cells in each GR cluster shares the same inhibitoryand the excitatory inputs through their dendrites whichsynaptically contact the two glomeruli at both ends.Thus, GR cells in each GR cluster exhibit similar fir-ing activity. Then, similar to the case of R GR ( t ), thecluster-averaged firing activity in the I th GR cluster( I = 1 , · · · , ) may be well described in terms of itsinstantaneous cluster spike rate R ( I )GR ( t ) of Eq. (24). Inthis case, the conjunction index C ( I ) of the I th GR clus-ter (representing the similarity degree between the spik-ing behavior [ R ( I )GR ( t )] of the I th GR cluster and thatof the whole population [ R GR ( t )]) is given by the cross-correlation at the zero time lag between R ( I )GR ( t ) and R GR ( t ) [see Eq. (25)].Figure 14(c) shows the distribution of conjunction in-dices { C ( I ) } with a peak at 0.65. When compared withthe optimal case in Fig. 6(a) with a peak at 0.55, thewhole distribution is moved to the right, and the val-ues of C ( I ) for all the GR clusters are positive. Thus, allthe anti-phase and complex out-of-phase spiking patternswith negative values of C ( I ) disappear. Only the in-phaseand out-of-phase spiking patterns with positive values of C ( I ) persist. Consequently, the mean of the distribution { C ( I ) } is increased to 0.613, while the standard deviationis decreased to 0.125, in comparison to the optimal casewhere the mean and the standard deviation are 0.320 and 0.516, respectively. Then, the diversity degree D of thespiking patterns { R ( I )GR ( t ) } in all the GR clusters, rep-resenting a quantitative measure for diverse recoding inthe granular layer, is given by the relative standard de-viation for the distribution { C ( I ) } [see Eq. (26)]. In thehighly-connected case of p c = 0 . 6, its diversity degree is D (cid:39) . 204 which is much smaller than D ∗ ( (cid:39) . D for the spiking patterns (corre-sponding to the firing outputs) of the GR clusters arisesdue to decrease in differences between the total synapticinputs into each GR clusters. As the connection prob-ability p c from the GO to GR cells is increased, differ-ences between the total inhibitory synaptic inputs fromthe pre-synaptic GO cells into each GR clusters are de-creased due to increase in the number of pre-synapticGO cells. On the other hand, the excitatory inputs intoeach GR clusters via MFs are Poisson spike trains withthe same firing rates, and hence they are essentially thesame. Thus, differences between the total synaptic in-puts (including both the inhibitory and the excitatoryinputs) into each GR clusters become reduced. Theseless different inputs into the GR clusters produce lessdifferent outputs (i.e. spiking patterns) in the GR clus-ters, which leads to decreases in the diversity degree D in the highly-connected case.We decompose the whole GR clusters into the in-phaseand the out-of-phase spiking groups. Unlike the opti-mal case of p ∗ c = 0 . 06, no anti-phase spiking group ap-pears. Figure 14(d) shows the fraction of spiking groups.The in-phase spiking group is a major one with fraction81.5%, while the out-of-phase spiking group is a minorone with fraction 18.5%. In comparison with the opti-mal case where the fraction of in-phase spiking group is50.2%, the fraction of in-phase spiking group for p c = 0 . R (cid:39) . 405 which is much larger thanthat ( R ∗ (cid:39) . D for the spik-ing patterns in the GR clusters is decreased so much to D (cid:39) . D ∗ ( (cid:39) . p ∗ c = 0 . C ( I ) of the GR clusters inthe in-phase and the out-of-phase spiking groups, respec-tively. The ranges for the distributions {C ( I ) } in thetwo spiking groups are also given in the bar diagramin Fig. 14(f). As in the optimal case of p ∗ c = 0 . 06 inFigs. 5 and 6, we determine the threshold C th ( (cid:39) . R ( I )GR ( t ) of the GR clusters relative to R GR ( t ).For C ( I ) > C th , in-phase spiking patterns with one or3 400 16000.00.51.0 400 1600 400 1600 400 1600 400 16000 200 4000.00.51.0 0 200 4000.000.070.14400 1600050100 400 1600 400 1600 400 1600 400 16000 200 4004070100 0 200 40001530 ~ (a1)1st cycle < J ( t ) > (a2)100th cycle (a3)200th cycle t (msec) (a4)300th cycle p c =0.6, p c =0.06 (a5)400th cycle < J ( t ) > ~ (b1) Cycle p c =0.6 p c =0.06(b2) M J Cycle < R P C ( t ) > r ( H z ) (c3) t (msec) p c =0.6, p c =0.06 400th cycle(c5) < R P C ( t ) > r ( H z ) (d1) Cycle p c =0.6 p c =0.06(d2) M P C Cycle FIG. 15: Change in synaptic weights of active PF-PCsynapses and firing activity of PCs during learning in thehighly-connected case of p c = 0 . 6; for comparison, datain the optimal case of p ∗ c = 0 . 06 are also given. (a1)-(a5) Cycle-evolution of bin-averaged (normalized) synapticweights (cid:104) ˜ J ( t ) (cid:105) of active PF signals (bin size: ∆ t = 100 msec).Plots of (b1) cycle-averaged mean (cid:104) ˜ J ( t ) (cid:105) and (b2) modulation M J for (cid:104) ˜ J ( t ) (cid:105) versus cycle. (c1)-(c5) Realization-averagedinstantaneous population spike rate (cid:104) R PC ( t ) (cid:105) r ; the numberof realizations is 100. Plots of (d1) cycle-averaged mean (cid:104) R PC ( t ) (cid:105) r and (d2) modulations M PC for (cid:104) R PC ( t ) (cid:105) r versuscycle. In (a1)-(a5), (b1)-(b2), and (d1)-(d2), solid (open) cir-cles represent data in the case of p c = 0.6 (0.06). In (c1)-(c5),solid (dotted) lines denote data in the case of p c = 0.6 (0.06). more peaks in the middle part of cycle exist. On theother hand, when passing the threshold C th from theabove (i.e., for C ( I ) < C th ) out-phase spiking patternsappear. These out-of-phase spiking patterns have left-skewed (right-skewed) peaks near the 1st (3rd) quartileof cycle (i.e., near t = 500 (1500) msec).In the case of in-phase spiking group, the distribution {C ( I ) } with a peak at 0.65 has positive values in the rangeof (0 . , . p c = 0 . p c = 0 . p c = 0 . {C ( I ) } with a peak at 0.35 has only positivevalues in the range of (0 . , . C ( I ) . Only out-of-phasespiking patterns which are developed from the in-phase spiking patterns appear, and hence they have just pos-itive values of C ( I ) . Thus, the range for p c = 0 . p c = 0 . p ∗ c = 0 . 06, the in-phase andthe out-of-phase spiking groups are shown to play theirown roles for synaptic plasticity at the PF-PC synapses,respectively. As a result of cooperation in their good-balanced state, effective depression (strong/weak LTD)at the PF-PC synapses occurs, which eventually leads toeffective motor learning for the OKR adaptation in thePC-VN-IO system. On the other hand, in the highly-connected case of p c = 0 . 6, the in-phase spiking groupbecomes a dominant one, and hence good balance be-tween the in-phase and the out-of-phase spiking groupsis broken up. In such an unbalanced state, contributionof the out-of-phase spiking group to the synaptic plastic-ity at the PF-PC synapses is decreased so much, whichis clearly shown below.Figure 15 shows change in synaptic weights of activePF-PC synapses and firing activity of PCs during learn-ing in the highly-connected case of p c = 0 . 6. Cycle-evolution of bin-averaged synaptic weights (cid:104) ˜ J ( t ) (cid:105) (solidcircles) of active PF signals is shown in Figs. 15(a1)-15(a5). For comparison, data for (cid:104) ˜ J ( t ) (cid:105) (open circles)in the optimal case of p ∗ c = 0 . 06 are also given. In com-parison with the optimal case, the bin-averaged synap-tic weights (cid:104) ˜ J ( t ) (cid:105) at the middle stage are less depressed,while at the initial and the final stages, they are moredepressed. Thus, the modulation of (cid:104) ˜ J ( t ) (cid:105) is reduced somuch. This small modulation is distinctly in contrastto the big modulation in the optimal case. (cid:104) ˜ J ( t ) (cid:105) alsobecomes saturated at about the 300th cycle, as in theoptimal case.In this highly-connected case of p c = 0 . , less depres-sion (in the bin-averaged synaptic weights (cid:104) ˜ J ( t ) (cid:105) ) at themiddle stage may be easily understood in the followingway. We note that, at the middle stage, the in-phasespiking group makes a dominant contribution to (cid:104) ˜ J ( t ) (cid:105) (i.e., the contribution of the out-of-phase spiking groupto (cid:104) ˜ J ( t ) (cid:105) may be negligible), as discussed in Fig. 10 inthe optimal case of p c = 0 . . However, the degree ofdepression changes depending on the relative phase dif-ference between the in-phase PF (student) signals andthe error-teaching (instructor) CF signals. The CF sig-nals are in-phase ones with respect to the firing rate f DS ( t ) of the Poisson spike trains for the IO desired signal(for a desired eye-movement) in Fig. 1(b2), and the in-phase PF signals are in-phase ones relative to the instan-taneous whole-population spike rate R GR ( t ). We notethat, depending on p c , the reference signal R GR ( t ) forthe in-phase PF signals has a varying “matching” de-gree relative to the sinusoidally-modulating IO desiredsignal f DS ( t ). Here, the matching degree M d is quan-4titatively given by the cross-correlation at the zero-timelag [ Corr M (0)] between R GR ( t ) and f DS ( t ): Corr M ( τ ) = ∆ f DS ( t + τ )∆ R GR ( t ) (cid:113) ∆ f ( t ) (cid:113) ∆ R ( t ) , (41)where ∆ f DS ( t ) = f DS ( t ) − f DS ( t ), ∆ R GR ( t ) = R GR ( t ) − R GR ( t ), and the overline denotes the time average overa cycle.With increasing p c , the top part of R GR ( t ) becomesmore broadly flattened, and hence its matching degreewith respect to f DS ( t ) is decreased. For p c = 0 . , M d = 0 . , which is smaller than that ( M d = 0 . p ∗ c = 0 . 06. Hence, due to decrease inthe matching degree between R GR ( t ) and f DS ( t ), on av-erage, in-phase PF signals for p c = 0 . p c = 0 . 6, the bin-averagedsynaptic weights (cid:104) ˜ J ( t ) (cid:105) (solid circles) at the middle stage(with a dominant in-phase spiking group) are less de-pressed than those (open circles) in the optimal case. Inthis way. at the middle stage where the contribution ofthe in-phase spiking group is dominant, the depressiondegree for (cid:104) ˜ J ( t ) (cid:105) is determined by the matching degree M d between R GR ( t ) and f DS ( t ).More depression (in the bin-averaged synaptic weights (cid:104) ˜ J ( t ) (cid:105) ) at the initial and the final stages for p c = 0 . F ( G ) ( t ) of the in-phase (51%) and the out-of-phase (49%)spiking groups are nearly the same (see Eq. (35) andFig. 10). Hence, at the initial and the final stages, some-what less LTD occurs at the PF-PC synapses, in contrastto strong LTD at the middle stage, because both theout-of-phase spiking group (with weak LTD) and the in-phase spiking group make contributions together. How-ever, in the highly-connected case of p c = 0 . 6, the frac-tion of the in-phase spiking group is so much increasedto 81 . . F ( G ) ( t ) = 0 . (cid:104) ˜ J ( t ) (cid:105) (solid circles) at the initial and the finalstages are more depressed than those (open circles) inthe optimal case. In this way. at the initial and the finalstages, the depression degree for (cid:104) ˜ J ( t ) (cid:105) is determined bythe relative fractions of the in-phase and the out-of-phasespiking groups; as the fraction of the in-phase spikinggroup is increased, (cid:104) ˜ J ( t ) (cid:105) is more depressed.Figures 15(b1) and 15(b2) show plots of cycle-averagedmean (cid:104) ˜ J ( t ) (cid:105) and modulation M J for (cid:104) ˜ J ( t ) (cid:105) versus cycle,respectively; p c = 0 . p ∗ c = 0 . 06 (opencircles). Both the cycle-averaged mean (cid:104) ˜ J ( t ) (cid:105) and themodulation M J for (cid:104) ˜ J ( t ) (cid:105) become saturated at aboutthe 300th cycle. With increasing the cycle, the cycle-averaged mean (cid:104) ˜ J ( t ) (cid:105) decreases from 1 to 0.374 due to LTD at the PF-PC synapses, which is similar to the op-timal case of p ∗ c = 0 . 06 where (cid:104) ˜ J ( t ) (cid:105) decreases from 1 to0.372. On the other hand, the modulation M J increasesvery slowly from 0 to 0.023, in contrast to the optimalcase with a big modulation where it increases quicklyfrom 0 to 0.112. When compared with the optimal case,bin-averaged synaptic weights (cid:104) ˜ J ( t ) (cid:105) at the initial and thefinal stages come down more rapidly (i.e., they are moredepressed), while at the middle stage, they come downrelatively slowly (i.e., they are less depressed). This kindof less-effective synaptic plasticity at the PF-PC synapsesarises due to the decreased matching degree of Eq. (41)between R GR ( t ) and f DS ( t ) (leading to less depressionat the middle stage) and the much-increased fraction ofin-phase spiking group (resulting in more depression atthe initial and the final stages). As a result, the modu-lation M J makes a slow increase to its saturated value(= 0.023), which is markedly in contrast to the optimalcase with a big modulation.We next consider the effect of PF-PC synaptic plastic-ity with a reduced small modulation on the subsequentlearning process in the PC-VN system. Figures 15(c1)-15(c5) show cycle-evolution of realization-averaged in-stantaneous population spike rate (cid:104) R PC ( t ) (cid:105) r of the PCs(number of realizations: 100); p c = 0 . p ∗ c = 0 . 06 (dotted line). As a result of PF-PC synapticplasticity, (cid:104) R PC ( t ) (cid:105) r becomes lower at the middle stagethan at the initial and the final stages. Thus, like the caseof (cid:104) ˜ J ( t ) (cid:105) , (cid:104) R PC ( t ) (cid:105) r forms a well-shaped curve, and it be-comes saturated at about the 300th cycle. At the middlestage, (cid:104) R PC ( t ) (cid:105) r (solid line) for p c = 0 . p ∗ c = 0 . 06 due toless-depressed (cid:104) ˜ J ( t ) (cid:105) . On the other hand, at the initialand the final stages (cid:104) R PC ( t ) (cid:105) r (solid line) for p c = 0 . p ∗ c = 0 . 06 because of more-depressed (cid:104) ˜ J ( t ) (cid:105) . As a re-sult of such less-effective PF-PC synaptic plasticity, themodulation of (cid:104) R PC ( t ) (cid:105) r for p c = 0 . (cid:104) R PC ( t ) (cid:105) r and modulation M PC for (cid:104) R PC ( t ) (cid:105) r versus cycle, respectively; p c = 0 . p ∗ c = 0 . 06 (open circles). Both (cid:104) R PC ( t ) (cid:105) r and M PC become saturated at about the 300th cycle. With in-creasing the cycle, the cycle-averaged mean (cid:104) R PC ( t ) (cid:105) r decreases from 86.1 Hz to 51.8 Hz due to LTD at thePF-PC synapses, which is similar to the optimal casewhere (cid:104) R PC ( t ) (cid:105) r decreases from 86.1 Hz to 51.7 Hz. Onthe other hand, the modulations M PC increases slowlyfrom 2.6 Hz to 5.6 Hz, which is distinctly in contrast tothe optimal case where it increases rapidly from 2.6 Hzto 24.1 Hz. Such a small modulation in (cid:104) R PC ( t ) (cid:105) r for p c = 0 . 400 1600070140 400 1600 400 1600 400 1600 400 16000 200 400406590 0 200 4002030400 200 4001.01.41.8 f VN ( t ) ( H z ) (a3) t (msec) f VN ( t ) ( H z ) (b) Cycle (c) M VN Cycle p c =0.6, p c =0.06(d) L g Cycle FIG. 16: Change in firing activity of the VN neuron duringlearning in the highly-connected case of p c = 0 . 6; for com-parison, data in the optimal case of p ∗ c = 0 . 06 are also given.(a1)-(a5) Cycle-evolution of bin-averaged instantaneous indi-vidual firing rates f VN ( t ); the bin size is ∆ t = 100 msec. Plotsof (b) cycle-averaged mean f VN ( t ) and (c) modulation M VN for f VN ( t ), and (d) learning gain degree L g versus cycle. In(a1)-(a5) and (b)-(d), solid (open) circles represent data for p c = 0.6 (0.06). the VN neuron which evokes OKR eye-movement.Change in firing activity of the VN neuron duringlearning in the highly-connected case of p c = 0 . p ∗ c = 0 . 06 (open circles) are alsogiven. Figures 16(a1)-16(a5) show cycle-evolution of bin-averaged instantaneous individual firing rate f VN ( t ). Itseems to be saturated at about the 300th cycle. Due tothe inhibitory coordination of PCs on the VN neuron, themaximum of f VN ( t ) occurs at the middle stage, while theminima appear at the initial and the final stages. Thus, f VN ( t ) forms a bell-shaped curve, in contrast to the well-shaped curves of (cid:104) R PC ( t ) (cid:105) r . f VN ( t ) (solid circles) at themiddle stage is smaller than that (open circles) in theoptimal case of p ∗ c = 0 . 06 due to more inhibition of thePCs on the VN neuron, while at the initial and the fi-nal stages, f VN ( t ) (solid circles) is larger than that (opencircles) in the optimal case because of less inhibition ofPCs on the VN neuron. As a result of such less-effectiveinhibitory coordination of the PCs, the modulation of f VN ( t ) becomes smaller than that in the optimal case.Figures 16(b) and 16(c) show plots of cycle-averagedmean f VN ( t ) and modulation M VN for f VN ( t ). Both f VN ( t ) and M VN become saturated at about the 300thcycle. f VN ( t ) increases from 44.3 Hz to 71.4 Hz, whichis nearly the same as that in the optimal case where itincreases from 44.3 Hz to 71.5 Hz. M VN also increasesslowly from 20.2 Hz to 22.6 Hz, which is in contrast tothe optimal case where M VN increases quickly from 20.2Hz to 32.5 Hz. Then, the learning gain degree L g , given (c) F r ac ti on o f C ( I ) C ( I ) (a) i (b) R G R ( t ) ( H z ) t (msec) p c =0.006 p c =0.06(f) L g Cycle (e1) f VN ( t ) ( H z ) (e2) 200th cycle100th cycle (e4)(e3) t (msec) p c =0.006, p c =0.06 (e5)400th cycle (d1) M d p c < J ( t ) > m i n ~ (d2) p c FIG. 17: Lowly-connected case of p c = 0 . randomly chosen GR cells. (b) Instantaneouswhole-population spike rate R GR ( t ) in the whole populationof GR cells. Band width for R GR ( t ): h = 10 msec. (c) Distri-bution of conjunction indices { C ( I ) } for the GR clusters in thewhole population. (d1) Plot of matching degree M d versus p c . (d2) Plot of the minimum (cid:104) ˜ J ( t ) (cid:105) min versus p c . (e1)-(e5)Cycle-evolution of bin-averaged instantaneous individual fir-ing rates f VN ( t ); the bin size is ∆ t = 100 msec. (f) Plot oflearning gain degree L g (solid circles) versus cycle; for com-parison, data (open circles) in the optimal case of p ∗ c = 0 . by the normalized gain ratio, is shown in Fig. 16(d). L g increases from 1 and becomes saturated at L ∗ g (cid:39) . L ∗ g ( (cid:39) . p c = 0 . 6, goodbalance between the in-phase and the out-of-phase spik-ing group in the optimal case ( R ∗ (cid:39) . R (cid:39) . D (cid:39) . D ∗ ( (cid:39) . R GR ( t ) and f DS ( t ), synaptic plasticity atthe PF-PC synapses becomes less effective, which alsoresults in less-effective motor learning for the OKR eye-movement.6We next consider the lowly-connected cases of p c =0 . 006 (i.e. 0.6 %). Figures 17(a) and 17(b) show theraster plot of spikes of 10 randomly-chosen GR cells andthe instantaneous whole-population spike rate R GR ( t ) inthe whole population of GR cells for p c = 0 . R GR ( t ) becomes more similarto the firing rate f MF ( t ) of the Poisson spike train forthe MF signal in Fig. 1(b1), in contrast to the highly-connected case of p c = 0 . R GR ( t ) [see Fig. 14(b)].Due to so much decrease in inhibitory inputs from theGO cells into the GR clusters, only the in-phase spik-ing group [where spiking patterns are similar to f MF ( t )]appears (i.e., all the out-of-phase spiking group disap-pears), which is distinctly in contrast to the optimal caseof p ∗ c = 0 . 06 where diverse spiking groups such as thein-phase, anti-phase, and complex out-of-phase spikinggroups coexist. The distribution of conjunction indices {C ( I ) } of the GR clusters ( C ( I ) : representing the similar-ity degree between the spiking behavior [ R ( I )GR ( t ): instan-taneous cluster spike rate] of the I th GR cluster and that[ R GR ( t )] in the whole population) is shown in Fig. 17(c).In comparison to the optimal case in Fig. 6(a), the distri-bution is shifted to the positive region, due to existence ofonly the in-phase GR clusters with positive values of C ( I ) .Thus, it has a peak at 0.75, and its range is (0 . , . p c = 0 . {C ( I ) } are0.737 and 0.129, respectively, which is in contrast to theoptimal case with the smaller mean (= 0.320) and thelarger standard deviation (= 0.516). Then, the diversitydegree D of the spiking patterns in all the GR clusters,given by the relative standard deviation for the distribu-tion { C ( I ) } [see Eq. (26)], is D (cid:39) . 175 which is muchsmaller than D ∗ ( (cid:39) . p c = 0 . p c =0 . 006 with the highly-connected case of p c = 0 . D ( (cid:39) . p c = 0 . D ∗ ( (cid:39) . D (cid:39) . 175 for p c = 0 . p c = 0 . C ( I ) exists, along with the major in-phase spiking group. Wealso note that, the in-phase spiking patterns in both thelowly- and the highly-connected cases have completelydifferent waveforms. For p c = 0 . 006 the in-phase spikingpatterns are more similar to the sinusoidally-modulatingMF signal f MF ( t ), while those for p c = 0 . f MF ( t ) due to broad flatness in their top part.Thus, the matching degree M d (= 0 . p c = 0 . 006 is larger than that (= 0 . p c = 0 . . In this way, there are two independent ways via increaseor decrease in p c from the optimal value p ∗ c (= 0 . 06) tobreak up the good balance between the in-phase and theout-of-phase spiking groups, which results in decrease inthe diversity degree in recoding of the GR cells.As a result of reduced diversity in recoding of theGR cells, the modulation for the bin-averaged synap-tic weights (cid:104) ˜ J ( t ) (cid:105) is much decreased, in contrast to thebig modulation in the optimal case of p ∗ c = 0 . 06. Fig-ure 17(d1) shows the plot of the matching degree M d between R GR ( t ) and f DS ( t ) versus p c . With decreasing p c , M d increases monotonically. Hence, M d (= 0 . p c = 0 . 006 is larger than that (=0.857) in the opti-mal case of p c = 0 . 06. Due to increase in M d , at themiddle stage in the lowly-connected case of p c = 0 . (cid:104) ˜ J ( t ) (cid:105) is expected to occur, in com-parison to the optimal case. Figure 17(d2) shows theplot of the minimum (cid:104) ˜ J ( t ) (cid:105) min of the well-shaped curvefor (cid:104) ˜ J ( t ) (cid:105) (appearing at t = 1000 msec) versus p c . Wenote that, as p c is decreased from p ∗ c (= 0 . (cid:104) ˜ J ( t ) (cid:105) min decreases so much slowly. Hence, (cid:104) ˜ J ( t ) (cid:105) min (= 0 . p c = 0 . 006 becomes just a little smaller than that(=0.274) in the optimal case. Thus, for p c = 0 . (cid:104) ˜ J ( t ) (cid:105) is only a little more depressed than that in theoptimal case of p c = 0 . 06. At the initial and the finalstages, only in-phase spiking group exists, unlike the op-timal case where both the in-phase and the out-of-phasespiking groups coexist. Hence, much more depression in (cid:104) ˜ J ( t ) (cid:105) occurs in comparison to the optimal case. Thus,modulation in (cid:104) ˜ J ( t ) (cid:105) for p c = 0 . 006 becomes much re-duced, in comparison to the optimal case.To make this point more clear, we divide the distri-bution of conjunction indices {C ( I ) } with a range (0.32,0.98) in Fig. 17(c) into the two parts by taking the mean(=0.737) as a reference. Thus, the 1st in-phase spik-ing sub-group has its conjunction indices higher than themean (=0.737) [i.e., the range is (0.737, 0.98)], while the2nd in-phase spiking sub-group has its conjunction in-dices lower than than the mean [i.e., the range is (0.32,0.737)]. In this way, the whole in-phase spiking groupis decomposed into the two sub-groups. Then, we ob-tain the firing fraction F ( G ) i of active PF signals in the G spiking group [see Eq. (35)], where G corresponds tothe 1st or the 2nd in-phase spiking sub-group. Similar toFigs. 10(d1)-10(d5), the firing fraction F ( G ) ( t ) for the 1st(2nd) in-phase spiking sub-group is found to form a bell-shaped (well-shaped) curve. At the initial and the finalstages, the firing fractions F ( G ) ( t ) for the 1st and the 2ndin-phase spiking sub-groups are equal (i.e., 50%). On theother hand, at the middle stage, the values of F ( G ) ( t ) forthe 1st and the 2nd in-phase spiking sub-groups are 0.61(61%) and 0.39 (39%), respectively. We note that moredepression in (cid:104) ˜ J ( t ) (cid:105) occurs for the 1st in-phase spikinggroup because their conjunction indices are larger thanthose in the 2nd in-phase spiking sub-group. Judgingfrom the ratio of their firing fractions, at the middle stage7(0 . 61 : 0 . (cid:104) ˜ J ( t ) (cid:105) is a little more depressed, in compar-ison to the initial and the final stages (0 . . p c = 0 . , a small modula-tion appears in (cid:104) ˜ J ( t ) (cid:105) , in contrast to the big modulationin the optimal case. The less-effective synaptic plasticityat the PF-PC synapses leads to reduced modulation inthe realization-averaged instantaneous population spikerate (cid:104) R PC ( t ) (cid:105) r of the principal PCs in the cerebellar cor-tex which also exert less-effective inhibitory coordinationon the VN neuron which produces the final output of thecerebellum (i.e., OKR eye-movement). Figures 17(e1)-17(e5) show cycle-evolution of the bin-averaged instanta-neous firing rate f VN ( t ) (solid circles) of the VN neuronfor p c = 0 . p ∗ c = 0 . 06 are also given. With increas-ing the cycle, f VN ( t ) seems to be saturated at about the300th cycle. f VN ( t ) (solid circles) at the middle stage isa little larger than that (open circles) in the optimal casedue to a little less inhibition of the PCs on the VN neu-ron for p c = 0 . f VN ( t ) (solid circles) is much larger thanthat (open circles) in the optimal case because of muchless inhibition of PCs on the VN neuron in comparisonto the optimal case.In this way, as a result of less-effective inhibitory coor-dination of the PCs, the modulation of f VN ( t ) becomesmuch smaller than that in the optimal case. Plots ofthe learning gain degree L g [corresponding to the mod-ulation gain ratio for f VN ( t )] are shown in Fig. 17(f) inboth the lowly-connected case of p c = 0 . 006 (solid cir-cles) and the optimal case of p ∗ c = 0 . 06 (open circles).For p c = 0 . L g increases very slowly and becomessaturated at L ∗ g (cid:39) . 099 at about the 300th cycle. Thissaturated learning gain degree L ∗ g ( (cid:39) . L ∗ g (cid:39) . p c = 0 . 006 results in less-effective motor learning forthe OKR adaptation.Before proceeding to the general variation of p c , wenow summarize the main points obtained in the abovehighly-connected ( p c = 0 . 6) and lowly-connected ( p c =0 . D [=0.204 ( P c = 0 . 6) and0.175 ( p c = 0 . D = 1 . p ∗ c = 0 . 06, because the fractionof the in-phase spiking group is increased from 50.2%to 81.5% ( p c = 0 . 6) and 100% ( p c = 0 . (cid:104) ˜ J ( t ) (cid:105) isdetermined by the firing fractions F ( G ) of Eq. (35) ofthe in-phase and the out-of-phase spiking groups. For p c = 0 . F ( i ) of the in-phase spiking group is0.82 (82%) and 1.0 (100%), respectively, in comparisonto 0.51 (51%) in the optimal case of p ∗ c = 0 . 06. Hence,for p c = 0 . (cid:104) ˜ J ( t ) (cid:105) occurs at the initial and the final stages of cycle, which is in contrast to the optimal case(e.g., see Fig. 15).In the middle stage, the in-phase spiking group make adominant contribution, independently of p c ; F ( i ) = 0.94(94% for p ∗ c = 0 . p c = 0 . p c = 0 . (cid:104) ˜ J ( t ) (cid:105) variesdepending on p c . The degree of depression in (cid:104) ˜ J ( t ) (cid:105) is de-termined by the matching degree M d of Eq. (41) betweenthe instantaneous whole-population spike rate R GR ( t )and the IO desired signal f DS ( t ); R GR ( t ) plays a “ref-erence” signal for the in-phase “student” PF signals andthe error-teaching instructor CF signals are in-phase with f DS ( t ). Depending on p c , the reference signal R GR ( t ) hasa changing matching degree M d with respect to f DS ( t ).With decreasing p c , M d is monotonically increased [seeFig. 17(d1)] [i.e, R GR ( t ) becomes more similar to the si-nusoidal IO desired signal f DS ( t )].In the highly-connected case of p c = 0 . , M d = 0 . M d = 0 . 857 in theoptimal case of p ∗ c = 0 . 06. Due to decrease in M d for p = . 6, in the middle stage (where the in-phase spikinggroup is dominant), less depression in (cid:104) ˜ J ( t ) (cid:105) occurs (seeFig. 15). Consequently, for p c = 0 . (cid:104) ˜ J ( t ) (cid:105) decreases, in comparison to that in the optimalcase, due to more depression at the initial and the finalstages and less depression at the middle stage.In the lowly-connected-connected case of p c = 0 . M d = 0 . M d = 0 . p ∗ c = 0 . 06. However, as shown inFig. 17(d2), the minimum (cid:104) ˜ J ( t ) (cid:105) min of the well-shapedcurve for (cid:104) ˜ J ( t ) (cid:105) decreases so much slowly with decreas-ing p c . Thus, for p c = 0 . 006 only a little more depressionin (cid:104) ˜ J ( t ) (cid:105) in comparison with the optimal case. As ex-plained in the above, at the initial and the final stages,only the in-phase spiking group (with strong LTD) exists,in contrast to the optimal case where both the out-of-phase (with weak LTD) and the in-phase spiking groups.Hence, much more depression takes place in comparisonto the optimal case. As a result, for p c = 0 . 006 the mod-ulation in (cid:104) ˜ J ( t ) (cid:105) is decreased, in comparison with that inthe optimal case, because of much more depression at theinitial and the final stages and a little more depressionat the middle stage.Such decrease in modulation in (cid:104) ˜ J ( t ) (cid:105) for p c = 0 . R PC ( t )of the PCs, which then exerts less-effective inhibitory co-ordination on the VN neuron (which evokes the OKR).Consequently, a small modulation in the firing activity f VN ( t ) of the VN neuron arises [see Fig. 16 for p c = 0 . p c = 0 . p c = 0 . p c = 0 . (b) D p c (c) L g p c In-Phase Out-of-Phase(a) F r ac ti on p c (d) L g D FIG. 18: Strong correlation between the diversity degree D and the saturated learning gain degree L ∗ g . (a) Fraction ofin-phase (solid circles) and out-of-phase (open circles) spikinggroups. (b) Plot of diversity degree D for the spiking patternsof all the GR clusters versus p c . (c) Saturated learning gaindegree L ∗ g versus p c . (d) Plot of L ∗ g versus D . pendence of the diversity degree D for the spiking pat-terns of the GR clusters and the saturated learning gaindegree L ∗ g on p c by varying it from the optimal value( p ∗ c = 0 . p c . The fraction of the in-phase spiking group (solidcircles) forms a well-shaped curve with a minimum at p c = p ∗ c (= 0 . p ∗ c = 0 . 06. Forsufficiently small p c , we have two sample cases where thefraction of in-phase spiking group is 1 (i.e., the fraction ofout-of-phase spiking group is 0). We note that, in the op-timal case of p ∗ c = 0 . 06, fractions of the in-phase (50.2%)and the out-of-phase spiking (49.8%) groups are well bal-anced (i.e., good balance between the in-phase and theout-of-phase spiking groups). As p c is changed (i.e., in-creased or decreased) from p ∗ c , the fraction of the in-phasespiking group increases, and then the spiking-group ratio R (i.e., the ratio of the fraction of the in-phase spikinggroup to that of the out-of-phase spiking group) increasesfrom the golden spiking-group ratio R ∗ ( (cid:39) . D for the spiking patterns of the GR clusters andthe saturated learning gain degree L ∗ g versus p c , respec-tively. The diversity degrees D forms a bell-shaped curvewith a maximum D ∗ ( (cid:39) . p ∗ c = 0 . 06 with the golden spiking-group ratio R ∗ (cid:39) . 008 (i.e., good balance between the in- and the out-of-phasespiking group). We note that, in this optimal case wherethe recoding of the GR cells is the most diverse, the sat-urated learning gain degree L ∗ also has its maximum( L ∗ g (cid:39) . p c is changed (i.e., increased or de-creased) from the optimal value (= 0.06), the spiking-group ratio R is increased, because of increase in thefraction of in-phase spiking group. Then, the diversitydegree D in recodings of the GR cells becomes decreased,which also results in decrease in the saturated learninggain degree L ∗ g from the maximum. Thus, L ∗ g also formsa bell-shaped curve, as in the case of D , and they havetheir maxima at the optimal values ( p ∗ c = 0 . L ∗ g versus D . As shownclearly in Fig. 18(d), both L ∗ g and D have a strong correla-tion with the Pearson’s correlation coefficient r (cid:39) . IV. SUMMARY AND DISCUSSION We are interested in the gain adaptation of OKR forthe eye movement. Various experimental works on theOKR have been done in rabbits, mice, and zebrafishes[10–17]. Moreover, some features of the OKR adaptationwere successfully reproduced through computational sim-ulations in the adaptive filter model [31] and the spikingnetwork model [6]. However, effects of diverse recodingof GR cells on the OKR adaptation in previous compu-tational works are necessary to be more clarified in sev-eral dynamical aspects. Particularly, the previous workslacked complete dynamical classification of diverse spik-ing patterns of GR cells and their association with theerror-teaching CF signals. We note that such dynami-cal classification of diverse recoding of GR cells may bea basis for clear understanding of the synaptic plastic-ity at the PF-PC synapses and the subsequent learningprogress in the PC-VN-IO system.For the effective study of OKR, we first introduced acerebellar ring network. This ring network with a simplearchitecture has advantage for computational and ana-lytical efficiency, and its visual representation may alsobeen easily made. Moreover, we also employed a refinedrule for the synaptic plasticity, based on the experimentalresult in [23]. We note that this one-dimensional cerebel-lar ring network with a refined synaptic plasticity is incontrast to the two-dimensional square-lattice network[6].For the first time, we made complete quantitative clas-sification of diverse spiking patterns in the GR clustersvia introduction of the conjunction index {C ( I ) } and thediversity degree D . Each spiking pattern in the I th GRcluster may be characterized in terms of its conjunctionindex C ( I ) , denoting the degree for the association of thespiking behavior of each I th GR cluster [characterizedby the instantaneous cluster spike rate R ( I )GR ( t )] with the9population-averaged firing activity in the whole popula-tion [given by the instantaneous whole-population spikerate R GR ( t )]. Then, the whole spiking patterns are de-composed into the two types of in-phase and out-of-phasespiking groups which are in-phase and out-of-phase withrespect to the “reference” signal R GR ( t ), respectively.Furthermore, the degree of diverse recoding of the GRcells may be quantified in terms of the diversity degree D , given by the relative standard deviation in the distri-bution of {C ( I ) } . Thus, D gives a quantitative measurefor diverse recoding of GR cells. Dynamical origin forthe appearance of diverse spiking patterns (i.e., outputsof the GR clusters) has also been investigated. It has thusbeen found that diverse total synaptic inputs (includingboth the excitatory inputs via MFs and the inhibitoryinputs from the pre-synaptic GO cells) into the GR clus-ters lead to production of diverse spiking patterns in theGR clusters.Based on the above dynamical classification of diversespiking patterns in the GR clusters, we made intensive in-vestigations on the effect of diverse recoding of GR cellson the OKR adaptation (i.e., its effect on the synap-tic plasticity at the PF-PC synapses and the subsequentlearning process in the PC-VN-IO system). To the bestof our knowledge, this type of approach, based on the in-phase and the out-of-phase spiking groups, is unique forthe study of OKR. Diversely-recoded student PF signalsfrom GR cells and the instructor error-teaching CF sig-nals from the IO neuron are fed into the PCs. We notethat the CF signals are always in-phase with respect tothe reference signal R GR ( t ). During the whole learningprocess, these in-phase and out-of-phase spiking groupshave been found to play their own roles, respectively.The in-phase student PF signals have been found to bestrongly depressed (i.e., strong LTD) by the instructorCF signals, because they are well-matched with the in-phase CF signals. On the other hand, the out-of-phasestudent PF signals have been found to be weakly de-pressed (i.e., weak LTD) by the instructor CF signals,because they are ill-matched with the in-phase CF sig-nals. However, contributions of these in-phase and out-of-phase spiking groups vary depending on the stage ofcycle. In the middle stage of each cycle, strong LTDtakes place through dominant contributions of the in-phase spiking group, which results in emergence of a min-imum of the bin-averaged (normalized) synaptic weight (cid:104) ˜ J (cid:105) of active PF signals. In contrast, at the initial andthe final stages of each cycle, less LTD occurs (i.e., max-ima of (cid:104) ˜ J (cid:105) appear) because both the out-of-phase spikinggroup with weak LTD and the in-phase spiking groupwith strong LTD make contributions together. Hence,the bin-averaged synaptic weights (cid:104) ˜ J (cid:105) have been foundto form a well-shaped curve (i.e., appearance of a mini-mum in the middle stage and maxima at both the initialand final stages). In this way, a big modulation in (cid:104) ˜ J (cid:105) emerges through interplay of the in-phase and the out-of-phase spiking groups.Due to this kind of effective depression (i.e., strong/weak LTD) at the PF-PC synapses, the(realization-averaged) instantaneous population spikerate (cid:104) R PC ( t ) (cid:105) r of PCs (corresponding to the principaloutputs of the cerebellar cortex) has been found to forma well-shaped curve with a minimum in the middle stage.Consequently, a big modulation occurs in (cid:104) R PC ( t ) (cid:105) r .These PCs exert effective inhibitory coordination on theVN neuron (which evokes OKR eye-movement). Thus,the (realization-averaged) instantaneous individual firingrate f VN ( t ) of the VN neuron has been found to form abell-shaped curve with a maximum in the middle stage.The maximum in the middle stage of cycle is formed dueto dominant contribution of the in-phase spiking groupwith strong LTD, while appearance of minima at the ini-tial and the final stages is made via comparable contribu-tions of the in-phase and the out-of-phase spiking groups.In this case, the learning gain degree L g , correspondingto the modulation gain ratio (i.e., normalized modula-tion divided by that at the 1st cycle for f VN ), has beenfound to increase with learning cycle and to be saturatedat about the 300th cycle.By varying p c , we investigated dependence of the di-versity degree D of the spiking patterns and the satu-rated learning gain degree L ∗ g on p c . Both D and L ∗ g have been found to form bell-shaped curves with peaks( D ∗ (cid:39) . 613 and L g ∗ (cid:39) . p ∗ c = 0 . 06. Also, in this optimal case, each GR cellreceives inhibitory inputs from about 10 nearby GO cells.In the references [6, 35] where the parameter values weretaken from physiological data, the average number ofnearby GO cell axons innervating each GR cell is about 8,which is close to that in the optimal case. Hence, we hy-pothesize that the granular layer in the cerebellar cortexhas evolved toward the goal of the most diverse recod-ing. Moreover, Both D and L ∗ g have also been found tohave a strong correlation with the Pearson’s correlationcoefficient r (cid:39) . p c in a given speciesof animals (e.g., a species of rabbit, mouse, or zebrafish)in an experiment seems to be practically difficult, unlikethe case of computational neuroscience where p c can beeasily changed. Instead, we consider an experiment forseveral species of animals (e.g., 3 species of rabbit, mouse,and zebrafish). In each species, we consider a large num-ber of randomly chosen GR cells ( i = 1 , · · · , L ). Then,through many learning cycles, one can obtain the peri-stimulus time histogram (PSTH) for each i th GR cell[i.e., (bin-averaged) instantaneous individual firing rate f ( i )GR ( t ) of the i th GR cell]. GR cells are expected to ex-hibit diverse PSTHs. Then, in the case of each i th GRcell, we obtain its conjunction index C i between its PSTH f ( i )GR ( t ) and the CF signal from the IO neuron [i.e., thePSTH of the IO neuron f IO ( t )]. In this case, the con-junction index C i is given by the cross-correlation at the0zero-time lag between f ( i )GR ( t ) and f IO ( t ). Thus, we getthe diversity degree D of PSTHs of GR cells, given by therelative standard deviation in the distribution of {C i } , forthe species.Besides the PSTHs of GR cells, under the many learn-ing cycles, we can also get a bell-shaped PSTH of a VNneuron [i.e., (bin-averaged) instantaneous individual fir-ing rate f VN ( t ) of the VN neuron]. The normalized mod-ulation of f VN ( t ) (divided by that at the 1st cycle) cor-responds to the learning gain degree L g . Thus, a set of( D , L g ) can be experimentally obtained for each species,and the set of ( D , L g ) may vary depending on the species.Then, for example in the case of 3 different species of rab-bit, mouse, and zebrafish, with the three different datasets for ( D , L g ), one can examine our main result (i.e.,whether more diversity in PSTHs of GR cells results inmore effective motor learning for the OKR).We also discuss our results briefly in comparison withother previous works [72–74]. It was discussed in [72]that various forms of synaptic plasticity occur at differ-ent sites in the cerebellum, and they work synergisticallyto create optimal outputs for behavior. In the case ofsynaptic plasticity at the PF-PC synapse, LTD occurswhen firing of the PF signal is in-phase (in conjunction)with that of the CF signal. On the other hand, in theabsence of the CF signal, the PF signal becomes out-of-phase, and LTP takes place. This mechanism for thesynaptic plasticity at the PF-PC synapse is essentiallysimilar to that in our work. However, in our case, weemployed a refined rule for the synaptic plasticity witha time window for the LTD (see Fig. 3), based on theexperimental result in [23], which has more quantitativeadvantages. In the presence of a CF firing, a major LTD(∆LTD (1) ) occurs in conjunction with earlier PF firings,while a minor LTD (∆LTD (2) ) takes place in associationwith later PF firings. Outside the effective range of LTD,PF firings alone result in LTP. The GR cells receive boththe excitatory synaptic inputs via the MFs and the in-hibitory synaptic inputs from the GO cells. It was shownin [73] that sparse excitatory synaptic connectivity via N syn (= 3 ∼ 5) MFs is crucial for pattern separation ofthe MF inputs. In contrast to the work in [73], we con-trolled the inhibitory synaptic inputs into the GR cellsby changing the connection probability p c from the GOcells to the GR cells, and investigated the effect of diverserecoding of GR cells on the motor learning. It was thusfound that the saturated learning gain degree L ∗ g for theOKR is maximum for an optimal value of p c (= 0 . ∼ 25 Hz may appear in thegranular layer (i.e., GR-GO feedback system) [75]. Ul-trafast motor rhythm of ∼ 200 Hz may also appear in thePurkinje (molecular) layer by adding intra-populationsynaptic connections between PCs (BCs) [76]. In thesystem of IO neurons, ∼ 10 Hz motor rhythm appearsin the presence of electric gap junctions between IO neu-rons [77]. Hence, in a future work, it would be inter-esting to investigate the effect of motor rhythms on di-verse recoding of GR cells and learning process in thePC-VN-IO system by adding intra-population couplings.Beyond the synaptic plasticity at PF-PC synapses (con-sidered in this work), diverse synaptic plasticity occurs atother synapses in the cerebellum [72, 78] such as synap-tic plasticity at PF-BC and BC-PC synapses [79], atMF-cerebellar nucleus and PC-cerebellar nucleus synapse[80], and at MF-GR cells synapses [81]. Therefore, as afuture work, it would be interesting to study the effectof diverse synaptic plasticity at other synapses on cere-bellar motor learning. In addition to variation in p c , an-other possibility to change synaptic inputs into the GRcells is to vary NMDA receptor-mediated maximum con-ductances ¯ g (GR)NMDA and ¯ g (GO)NMDA , associated with persistentlong-lasting firing activities. It would also be interest-ing to investigate the effect of NMDA receptor-mediatedsynaptic inputs on diverse recoding of GR cells and mo-tor learning in the OKR adaptation by changing ¯ g (GR)NMDA and ¯ g (GO)NMDA . This work is beyond the scope of the presentwork, and hence it is left as a future work.We also discuss another interesting future work. Inthe present work, we varied the connection probabil-ity p c (from the GO cells to the GR cells) while thesynaptic weight J (GR , GO) (= 10 . 0) per synapse is un-changed. Hence, with increasing (decreasing) p c , the to-tal inhibitory synaptic strength K (GR , GO) on each GRcell also increases (decreases). Instead, one may consideranother case where p c is changed while maintaining aconstant K (GR , GO) . For keeping the value of K (GR , GO) to be a constant, the synaptic weight J (GR , GO) mustalso change depending on the variation of p c ; when p c is increased (decreased), J (GR , GO) should decrease (in-crease) such that K (GR , GO) is a constant. We made a1preliminary work on this issue by fixing the constantvalue of K (GR , GO) at K (GR , GO) ∗ in the optimal caseof p ∗ c = 0 . 06. Particularly, we were interested in howthe spiking patterns in the GR clusters, characterizedby R ( I )GR ( t ), and the population-averaged firing activity,given by the instantaneous whole-population spike rate R GR ( t ), change when varying p c . We first consideredthe highly-connected case of p c = 0 . K (GR , GO) = K (GR , GO) ∗ ). In comparison with the highly-connectedcase in Fig. 14 where K (GR , GO) > K (GR , GO) ∗ , shapes of R GR ( t ) and R ( I )GR ( t ) in the case of K (GR , GO) = K (GR , GO) ∗ were nearly unchanged, while their amplitudes were in-creased because K (GR , GO) was decreased to K (GR , GO) ∗ .We note that the conjunction index C ( I ) between R ( I )GR ( t )and R GR ( t ) represents the degree for similarity betweentheir shapes. Hence, the distribution of {C ( I ) } in thecase of K (GR , GO) = K (GR , GO) ∗ was found to be nearlythe same as that in Fig. 14(c). We also get the diver-sity degree D ( (cid:39) . D (cid:39) . J formaintaining the value of K (GR , GO) seems to have effectmainly on the amplitudes of spiking patterns withoutmuch alteration of their shapes, and hence the diversityfor the spiking patterns seems to be determined mainlyby the high connectivity from the GO to the GR cells.Consequently, in both cases with and without maintain-ing the constant K (GR , GO) ∗ , the diversity degree D forthe spiking patterns seems to be nearly the same.Next, we considered the lowly-connected case of p c =0 . 006 where K (GR , GO) = K (GR , GO) ∗ . In comparison tothe lowly-connected case in Fig. 17 with K (GR , GO) 06. We note that,in this lowly-connected case, both the low-connectivity p c and the increase in synaptic weight J seem to deter-mine the diversity for the spiking patterns, unlike thehighly-connected case. Mainly due to increased synap-tic weight J , conjunction indices for the spiking patternswere broadly distributed in a range of (-0.17,0.75), incontrast to that in Fig. 17(c). Out-of-phase spiking pat-terns with negative conjunction indices also appear, un-like the case in Fig. 17 where only in-phase spiking pat-terns exist. In this case, we obtain the diversity degree is D ( (cid:39) . D (cid:39) . D ∗ (cid:39) . p ∗ c = 0 . 06 in Fig. 6, and decrease p c from 0.06 to 0.006(lowly-connected case) while maintaining the constant K (GR , GO) ∗ . The range in the distribution of conjunc- TABLE I: Parameter values for LIF neuron models with AHPcurrents for the granule (GR) cell and the Golgi (GO) cellin the granular layer, the Purkinje cell (PC) and the basketcell (BC) in the Purkinje-molecular layer, and the vestibularnucleus (VN) and the inferior olive (IO) neurons.Granular Purkinje X -population Layer -Molecular VN IOLayer neuron neuronGR GO PC BCcell cell C X I ( X ) L g ( X ) L V ( X ) L -58.0 -55.0 -68.0 -68.0 -56.0 -60.0 I ( X ) AHP ¯ g ( X ) AHP τ ( X ) AHP V ( X ) AHP -82.0 -72.7 -70.0 -70.0 -70.0 -75.0 v ( X ) th -35.0 -52.0 -55.0 -55.0 -38.8 -50.0 I ( X ) ext tion indices in the optimal case is (-0.57,0.85). We notethat this range became narrowed in the case of lowly-connected case of p c = 0 . p ∗ c to 0.006, in-phase and anti-phase spiking patternswith higher magnitudes of conjunction indices seems tobe “degraded,” and accordingly their magnitude of con-junction indices seem to be lowered. We conjecture thatthe connection probability of p c = 0 . 006 would be toosmall to warrant the high-degree conjunction for the in-phase and the anti-phase spiking patterns. Base on theresults of this preliminary work, we hypothesize a pos-sibility that the diversity degree D might have its maxi-mum at the same optimal value of p ∗ c = 0 . 06 even in thecase of maintaining the constant K (GR , GO) ∗ ; in the lowly-connected case of p c < p ∗ c , D would decrease in a muchslow way in comparison with the case without maintain-ing the constant K (GR , GO) ∗ . To confirm this hypothesis,more intensive future work is necessary by decreasing p c from p ∗ c in the lowly-connected case with the constant K (GR , GO) ∗ where changes in both p c and J have effectson spiking patterns. Acknowledgments This research was supported by the Basic Science Re-search Program through the National Research Founda-tion of Korea (NRF) funded by the Ministry of Education(Grant No. 20162007688). Appendix A: Parameter Values for The LIF NeuronModels and The Synaptic Currents In this appendix, we list two tables which show pa-rameter values for the LIF neuron models in Subsec. II C2 TABLE II: Parameter values for synaptic currents I ( T,S ) R ( t ) into the granule (GR) and the Golgi (GO) cells in the granularlayer, the Purkinje cells (PCs) and the basket cells (BCs) in the Purkinje-molecular layer, and the vestibular nucleus (VN) andthe inferior olive (IO) neurons in the other parts. In the granular layer, the GR cells receive excitatory inputs via mossy fibers(MFs) and inhibitory inputs from GO cells, and the GO cells receive excitatory inputs via parallel fibers (PFs) from GR cells.In the Purkinje-molecular layer, the PCs receive two types of excitatory inputs via PFs from GR cells and through climbingfibers (CFs) from the IO and one type of inhibitory inputs from the BCs. The BCs receive excitatory inputs via PFs from GRcells. In the other parts, the VN neuron receives excitatory inputs via MFs and inhibitory inputs from PCs, and the IO neuronreceives excitatory input via the desired signal (DS) and inhibitory input from the VN neuron.Granular Layer Purkinje-Molecular Layer Other PartsTarget GR GO PC BC VN IOCells ( T )Source MF MF GO PF PF PF CF BC PF MF MF PC DS VNCells ( S )Receptor AMPA NMDA GABA AMPA NMDA AMPA AMPA GABA AMPA AMPA NMDA GABA AMPA GABA( R )¯ g ( T ) R J ( T,S ) ij V ( S ) R τ ( T ) R A , A A i ( t ): activation degree Fraction of active GR cells A ( G ) i ( t ): activation degree Fraction of active GR cells in the G spiking groupin the G spiking group C ( I ) : output conjunction index The degree for the conjunction of the spiking behavior in each I th GR clusterwith the population-averaged spiking behavior in the whole population C ( I )in : input conjunction index The degree for the conjunction of the total synaptic input into each I th GR clusterwith the cluster-averaged total synaptic input D : output diversity degree Relative standard deviation for the distribution of {C ( I ) }D in : input diversity degree Relative standard deviation for the distribution of {C ( I )in } f ( i )GR ( t ): instantaneous individual Individual firing activity of active GR cellsfiring rate f ( p )GR ( t ): instantaneous population Population firing activity of the whole GR cellsfiring rate f X ( t ): firing rate Firing activity of the X cell ( X = VN and IO) L g : learning gain degree Modulation gain ratio. Normalized modulation of the individual firing rate f VN ( t ) ofthe VN neuron divided by that at the 1st cycle L p : learning progress degree Ratio of the cycle-averaged inhibitory input from the VN neuron to the cycle-averaged excitatory input via the IO desired signal M d : matching degree Matching degree between R GR ( t ) (instantaneous whole-population spike rate in thewhole population of GR cells) and f DS ( t ) (IO desired signal) R ( I )GR ( t ): instantaneous cluster spike rate Spiking behavior in each I th GR cluster R X ( t ): instantaneous whole- Population behavior of X cellspopulation spike rate (cid:104) W ( G ) J ( t ) (cid:105) : weighted synaptic weight Contribution of the G spiking group to (cid:104) ˜ J ( t ) (cid:105) of active PF signals in the wholein the G spiking group population C X , the leakage current I ( X ) L , the AHPcurrent I ( X ) AHP , and the external constant current I ( X ) ext aregiven in Table I.For the synaptic currents, the parameter values for themaximum conductance ¯ g ( T ) R , the synaptic weight J ( T,S ) ij ,the synaptic reversal potential V ( S ) R , the synaptic decaytime constant τ ( T ) R , and the amplitudes A and A for the type-2 exponential-decay function in the granular layer,the Purkinje-molecular layer, and the other parts for theVN and IO are given in Table II. Appendix B: Glossary In this appendix, glossary for various terms character-izing the cerebellar model is given in Table III. [1] M. Ito, The Cerebellum and Neural Control (Raven Press,New York, 1984).[2] M. Ito, Ann. N. Y. Acad. Sci. , 273 (2002).[3] M. Ito, The Cerebellum: Brain for an Implicit Self (Pear-son Education Inc., New Jersey, 2012).[4] S. Gilman, J. Bloedel, and R. Lechtenberg, Disordersof the Cerebellum (F.A. Davis Company, Philadelphia,1981).[5] M. U. Manto, Cerebellar Disorders; A Practical Approachto Diagnosis and Managements (Cambridge UniversityPress, Cambridge, 2010).[6] T. Yamazaki and S. Nagao, PLoS One , e33319 (2012).[7] M. Ito, Trends Cogn. Sci. , 313 (1998).[8] M. D. Mauk and N. H. Donegan, Learn. Mem. , 130(1997).[9] K. M. Christian and R. F. Thompson, Learn. Mem. ,427 (2003).[10] S. Nagao, Exp. Brain Res. , 36 (1983).[11] S. Nagao, Exp. Brain Res. , 489 (1988).[12] R. J. Harvey, C. De’Sperati, and P. Strata, Vision Res. , 1615 (1997).[13] M. Iwashita, R. Kanai, K. Funabiki, K. Matsuda, and T.Hirano, Neurosci. Res. , 299 (2001).[14] Y.-Y. Huang and S. C. F. Neuhauss, Front. Biosci. ,1899 (2008).[15] H. Tabata, N. Shimizu, Y. Wada, K. Miura, and K.Kawano, J. Vis. , 13 (2010).[16] H. Matsuno, M. Kudoh, A. Watakabe, T. Yamamori,R. Shigemoto, and S. Nagao, PLoS ONE , e0164037(2016).[17] S. D. Scheetz, E. Shao, Y. Zhou, C. L. Cario, Q. Bai, andE. A. Burton, J. Neurosci. Meth. , 329 (2017).[18] D. Marr, J. Physiol. , 437 (1969).[19] J. S. Albus, Math. Biosci. , 25 (1971).[20] D. O. Hebb, The Organization of Behavior; A Neuropsy-chological Theory (Wiley & Sons, New York, 1949).[21] G. S. Brindley, IBRO Bull. , 80 (1964).[22] P. Strata, J. Physiol. , 5519 (2009).[23] P. Safo and W. G. Regehr, Neuropharmacology , 213(2008).[24] M. Ito, Brain Res. , 237 (2000).[25] M. Ito, M. Sakurai, and P. Tongroach, J. Physiol. ,113 (1982).[26] M. Ito and M. Kano, Neurosci. Lett. , 253 (1982).[27] M. Sakurai, J. Physiol. , 463 (1987).[28] M. Ito, Ann. Rev. Neurosci. , 85 (1989).[29] M. Ito, Physiol. Rev. , 1143 (2001). [30] M. Ito, Nat. Rev. Neurosci. , 896 (2002).[31] H. Gomi and M. Kawato, Biol. Cybern. , 105 (1992).[32] M. Fujita, Biol. Cybern. , 195 (1982).[33] P. Dean, J. Porrill, C.-F. Ekerot, and H. J¨orntell, Nat.Rev. Neurosci. , 30 (2010).[34] T. J. Sejnowski, J. Math. Biol. , 303 (1977).[35] T. Yamazaki and S. Tanaka, Eur. J. Neurosci. , 2279(2007).[36] S.-Y. Kim and W. Lim, J. Neurosci. Meth. , 161(2014).[37] X.-J. Wang, Physiol. Rev. , 1195 (2010).[38] N. Brunel, J. Comput. Neurosci. , 183 (2000).[39] N. Brunel and V. Hakim, Neural Comput. , 1621(1999).[40] N. Brunel and X.-J. Wang, J. Neurophysiol. , 415(2003).[41] C. Geisler, N. Brunel, and X.-J. Wang, J. Neurophysiol. , 4344 (2005).[42] N. Brunel and D. Hansel, Neural Comput. , 1066(2006).[43] N. Brunel and V. Hakim, Chaos , 015113 (2008).[44] K. Pearson, Proc. Royal Soc. Lond. , 240 (1895).[45] A. Mathy, S. S. N. Ho, J. T. Davie, I. C. Duguid, B. A.Clark, and M. H¨ausser, Neuron , 388 (2009).[46] R. R. Llin´as, Front. Neural Circuit. , 96 (2014).[47] D. J. Watts and S. H. Strogatz, Nature , 440 (1998).[48] S. H. Strogatz, Nature , 268 (2001).[49] W. Gerstner and W. Kistler, Spiking Neuron Models (Cambridge University Press, New York, 2002).[50] W. T. Thach, J. Neurophysiol. , 785 (1968).[51] M. H¨ausser and B. A. Clark, Neuron , 665 (1997).[52] E. De Schutter, Trends Neurosci. , 291 (1995).[53] C. Chen and R. F. Thompson, Learn. Mem. , 185(1995).[54] S.-H. Wang, W. Denk, and M. H¨ausser, Nat. Neurosci. , 1266 (2000).[55] M. Coesmans, J. T. Weber, C. I. De Zeeuw, and C.Hansel, Neuron , 691 (2004).[56] V. Steuber, W. Mittmann, F. E. Hoebeek, R. A. Silver,C. I. De Zeeuw, M. H¨ausser, and E. De Schutter, Neuron , 121 (2007).[57] V. Lev-Ram, S. B. Mehta, D. Kleinfeld, and R. Y. Tsien,Proc. Natl. Acad. Sci. USA , 15989 (2003).[58] E. Moln´ar, J. Neurochem. , 1 (2014).[59] Y. Yang and S. G. Lisberger, Nature , 529 (2014).[60] A. R. Gallimore, T. Kim, K. Tanaka-Yamamoto, and E.De Schutter, Cell Rep. , 722 (2018). [61] W. Gerstner and J. L. van Hemmen, Network , 139(1992).[62] D. V. Buonomano and M. D. Mauk, Neural Comput. ,38 (1994).[63] G. T. Kenyon, J. F. Medina, and M. D. Mauk, J. Com-put. Neurosci. , 17 (1998).[64] J. F. Medina, K. S. Garcia, W. L. Nores, N. M. Taylor,and M. D. Mauk, J. Neurosci. , 5516 (2000).[65] P. D. Roberts, J. Comput. Neurosci. , 283 (2007).[66] P. Achard and E. De Schutter, Front. Comput. Neurosci. , 8 (2008).[67] G. Bouvier, J. Aljadeff, C. Clopath, C. Bimbard, J.Ranft, A. Blot, J.-P. Nadal, N. Brunel, V Hakim, andB. Barbour, eLife , e31599 (2018).[68] T. Yamazaki and S. Tanaka, Neural Netw. , 290(2007).[69] W. Maass, T. Natschl¨ager, and H. Markram, NeuralComput. , 2531 (2002).[70] H. Shimazaki and S. Shinomoto, J. Comput. Neurosci. , 171 (2010).[71] S. A. Heine, S M. Highstein, and P. M. Blazquez, J. Neu- rosci. , 17004 (2010).[72] Z. Gao, B. J. van Beugen, and C. I. De Zeeuw, Nat. Rev.Neurosci. , 619 (2012).[73] N. A. Cayco-Gajic, C. Clopath, and A. Silver, Nat. Com-mun. , 1116 (2017).[74] K. Inagaki and Y. Hirata, Cerebellum , 827 (2017).[75] E. D’Angelo, S. K. E. Koekkoek, P. Lombardo, S. Solinas,E. Ros, J. Garrido, M. Schonewille, and C. I. De Zeeuw,Neuroscience , 805 (2009).[76] C. de Solages, G. Szapiro, N. Brunel, V. Hakim, P. Isope,P. Buisseret, C. Rousseau, B. Barbour, and C. L´ena, Neu-ron , 775 (2008).[77] R. R. Llin´as, J. Physiol. , 3423 (2011).[78] C. Hansel, D. J. Linden, and E. D’Angelo, Nat. Neurosci. , 467 (2001).[79] W. Lennon, T. Yamazaki, and R. Hecht-Nielsen, Front.Comput. Neurosci. , 150 (2015).[80] N. Zheng and I. M. Raman, Cerebellum , 56 (2010).[81] E. D’Angelo and C. I. De Zeeuw, Trends Neurosci.32