Biomimetic race model of the loop between the superior colliculus and the basal ganglia: Subcortical selection of saccade targets
BBiomimetic Race Model of the Loop between the Superior Colliculus and theBasal Ganglia: Subcortical Selection of Saccade Targets
Charles Thurat a,b,1, , Steve N’Guyen a,b,c,d , Benoit Girard a,b a Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7222, ISIR, F-75005, Paris, France. b Institut des Systmes Intelligents et de Robotique, CNRS, UMR 7222, ISIR, F-75005, Paris, France. c Sorbonne Universit´es, Collge de France, UMR 7152, LPPA, F-75005, Paris, France. d Laboratoire de Physiologie de la Perception et de lAction, Collge de France, CNRS, UMR 7152, LPPA, F-75231, Paris, France.
Abstract
The Superior Colliculus, a laminar structure involved in the retinotopic mapping of the visual field, plays a cardinal role in theseveral cortical and subcortical loops of the saccadic system. Although the selection of saccade targets has long been thought tobe the sole product of cortical processes, a growing body of evidence hints at the implication of the Superior Colliculus, firstly bythe lateral connections between the neurons of its maps, and secondly by its interactions with the midbrain Basal Ganglia, alreadyrenowned for their role in decision making.We propose a biomimetic population-coded race model of selection based on a dynamic tecto-basal loop that reproduces theobserved ability of the Superior Colliculus to stochastically select between similar stimuli, the accuracy of this selection dependingon the discriminability of the target and the distractors. Our model also offers an explanation for the phenomenon of RemoteDistractor Effect based on the lateral connectivity within the Basal Ganglia circuitry rather than on lateral inhibitions within thecollicular maps.Finally, we propose a role for the intermediate layers of the Superior Colliculus, as stochastic integrators dynamically gated bythe selective disinhibition of the Basal Ganglia channels that is consistent with the recorded activity profiles of these neurons.
Keywords:
Superior Colliculus, saccades, selection, stochastic race model, biomimetic, Basal Ganglia
1. Introduction
Saccadic eye movements are probably those for which wemake the most decisions. During wakefulness, in ordinary vi-sual conditions, numerous potentially interesting targets con-stantly compete for further examination by the fovea, and as afew saccades are made each second, decisions concerning thetarget to be foveated by the next saccade have to be made morethan 10 times a day. The cortical circuitry handling these se-lection processes is now quite well identified, however an in-creasing number of studies highlight the existence of a possiblyautonomous and purely subcortical circuit, also able to select ∗ Corresponding author. Tel.: +33 (0)1 44 27 63 81
Email address: [email protected] (Charles Thurat) the targets of saccades. We propose here a model of this cir-cuit, and of the neural mechanisms operating the selection andits transformation into an eye motor command.A common and well-known models for general selection isthe race model, in which each competing signal feeds its ownevidence counter and the first counter to reach a specific selec-tion threshold gets selected. Another common selection modelis the drift-diffusion model, in which a single noisy counter in-tegrates evidence for two competing signals from a commonstarting point toward one of two selection boundaries (the rateof accumulation, or drift rate, being potentially different foreach signal).Such models have already been proposed to account for
Preprint submitted to Neural Networks October 1, 2018 a r X i v : . [ q - b i o . N C ] S e p ognitive selection processes (Logan and Cowan (1984)), andparticularly for the selection of saccade targets or for saccadecoutermanding, with good accuracy (see Hanes and Carpenter(1999); Schall (2001); Schall et al. (2011) for race models, thelatter proposing a gated accumulator able to reproduce Reac-tion Time recording in easy and difficult discrimination tasks;see Ratcliff and McKoon (2008) for drift-diffusion models).Nevertheless, these models are mostly phenomenological,and do not propose a precise or complete role to the variousneural structures involved in the specific process of saccade se-lection. Therefore, their validity can be questioned when con-fronted to the demands of explaining behavioral resuts withplausible neuronal correlates.The structures involved in saccade selection processes havebeen extensively studied, as reviewed in Moschovakis et al.(1996), and can be broadly summarized as follows: in a firstlong loop, retinal inputs project to the visual areas of the cortex,and are then processed through the Lateral IntraParietal sulcus(LIP), and the Frontal Eye Fields (FEF). These cortical areasproject to several midbrain structures, notably the Basal Gan-glia (BG), with which it forms loops also including the Thala-mus (Th). The FEF also directly project to the midbrain Su-perior Colliculus (SC), which itself projects to the brainstemSaccade Burst Generators (SBG) in order to generate a motorcommand to the Extra-Ocular Muscles (EOM), and to move theeye towards the designed target.A second shorter loop sees a direct projection from the reti-nal inputs to the SC, which then loops with the BG too. The SCouptut also projects down to the SBG.From this brief overview, the Superior Colliculus emergesas a central structure, a crossroad between both cortical andsubcortical circuits; organized into several dorsoventral layers,its superficial layers receive retinal inputs, while its deeper lay-ers project outputs to orienting motor systems. Between these,intermediary layers combine multisensory integration and pre-motor activities. These layers are organized in retinotopic maps,from the superficial layers that describe the visual space, to themotor representations of the deep layers that produce ocular ori- enting responses (see May (2006) for a full review of the SCanatomy).Despite the pivotal position of the SC in these loops, targetselection in the saccadic system has long been thought to occurat the cortical level - mainly involving the Frontal Eye Fields(Fischer (1987); Schall and Hanes (1998)), while the SC wouldonly serve as a visual mapping structure that relays the selectionsignal and process the saccade metrics for the brainstem.This view has been challenged, firstly by showing the impli-cation of the SC in saccade target selection (Mays and Sparks(1980); Ottes et al. (1987); Schiller et al. (1987)), and secondlyby exposing its active role in the selection process (McPeekand Keller (2002, 2004); McPeek (2008); Carello and Krauzlis(2004), among others). The SC laminar organization in severalretinotopic neural maps of the visual field led to the belief thattarget selection could occur by way of reciprocal lateral short-range excitation and long-range inhibition within each map,as modelled in van Opstal and van Gisbergen (1989) and inother Neural Field models of the SC Trappenberg et al. (2001);Taouali (2012). The winner-takes-all properties of such purelycollicular models conflicts with electophysiological and com-portemental data regarding the production of averaging sac-cades (see Chou et al. (1999) for an overview of experimen-tal results concerning averaging saccades): their predictionsstate that two equal loci of SC activation will merge in a sin-gle locus located between the original two, when experimentaldata shows that the two loci persist (see Edelman and Keller(1998)). Moreover, the very existence of longe-range inhibi-tions has been challenged (Ozen et al. (2004)), as well as theability of short ranged excitation to explain the effects of prox-imal distractors on target selection (Casteau and Vitu (2012)).Furthermore, such models do not take into account the vari-ous connections between the SC and the Basal Ganglia (Hikosakaand Wurtz (1983); McHaffie et al. (2005)), a set of midbrainnuclei renowned for its role in action selection (Mink (1996);Ding and Gold (2013)), especially in the generation of saccades(Hikosaka et al. (2000)). Divided in input structures, the Stria-tum (Str) and SubThalamic Nucleus (STN), and output struc-2 GThalamusEOMSC SBG
M2V1I1D1M1 ThTRNFS Str D2STNGPi/SNr GPeStr D1
EBNMN EBN IBNTN MNIBN
MRF
V2I2
Motor MapDecision MapIntegration MapVisual Input Map
Left Right TN Retinal inputs
SumLLB D2 OPN
Figure 1:
Architecture of the SC-BG model.
Several features have been downscaled for ease of comprehension: only one of the two Colliculi has been represented,and only two neurons in each of the four layer. The architecture of the BG is shown as unidimensionnal when the model represents its channels as a 2D map, andonly the projections from one channel’s neurons are represented, since they are identical for the other channels. Only two of the four SBG circuits have beenrepresented, constituting a pair governing eye movements in the horizontal plane. Open triangles represent excitatory synapses, closed triangles represent inhibitorysynapses - for the SC-SBG projection only, synapse thickness represents connection strenght. Bold lines represent one-to-all connections, when normal linesrepresent one-to-one connections. See text for abbreviations.
2. Materials and Methods
The model is based on a rate population coding, in which allneurons of a given population are assumed to be sensitive to thesame set of inputs and share similar electrophysiological prop-erties. Each of these populations of neurons can therefore bemodeled by only one equation, that returns the mean firing rateof the population. The model is organized in three main mod-ules, the SC, BG and SBG, interconnected as shown in fig. 1.On the mathematical side, in order to unify the theoreti-cal models and equations describing the various neurons of themodel, we used the model of locally Projected Dynamical Net-works (lPDS) neurons proposed by Girard et al. (2008) for allthe neurons of the BG, SC and SBG, unless specifically statedotherwise. The activity of any neuron x under this model, whenupdated with Euler integration, obeys to the following equation type: a x ( t + dt ) = max (0 , min (1 , a ( t ) + dtτ x × ( I x ( t ) − γ × a x ( t )))) (1)With a x ( t ) the activity of the neuron at time t , I x ( t ) theweighted sum of all excitatory and inhibitory inputs at time t , γ the leak, and τ x the time constant in ms (both of which beingidentical for all neurons in the model, unless specified other-wise).Note that all the base unit for integration timestep dt is themillisecond. The SC module of the model is based on the model pro-posed by Tabareau et al. (2007) for the description of the deeplayers of the SC and the process of Spatio-Temporal Transfor-mation that turns the spatial activity of the whole Motor mapof the SC into a temporal signal for the SBG. We also tookinspiration from Tabareau et al. (2007) concerning the gluingmechanism that allows the operation of the two colliculi codingeach for one visual hemifield into one abstract mapping on thewhole plane.Our SC is divided in two elements, Right and Left, eachreceiving and processing inputs from the contralateral Retina.Furthermore, both elements are organized in 4 layers repre-senting retinotopic maps with logarithmic mapping, in order toaccount for the laminar structure of the SC and the functionalproperties of the SC layers (see fig. 1 for the architecture of onecolliculus of the SC module). These layers are composed of
N bCell × N bCell lPDS neurons, and the right and left colli-culi are connected so that the combined activity of their motorlayers can be considered as a single ”abstract” mapping on thewhole plane.As explained by Ottes et al. (1986), the logarithmic map-ping governing the geometry of the SC transforms the retino-topic Cartesian coordinates ( [ az, el ] for azimuth and elevation)of a target’s position in the visual field into coordinates ex-pressed in millimeters ( [ X, Y ] ) onto the SC surface by the fol-4owing equation: XB + i × YB = ln( z + AA ) (2)With z = az + i × el , and A and B experimentally estimatedfor the monkey to be respectivelly π/ and . .The most superficial layer is called the Visual layer (or map),and receives direct projections from the retina (cortical inputsare not taken into account in our strictly subcortical model).The retinal input for each target T arg i is represented as a 2D-Gaussian with standard deviation σ = 2 . neurons (correspond-ing to . mm in the monkey SC) and maximal height equal tothe target’s value, centered on the neuron at coordinates [ x i , y i ] in our discretized maps, as per equation (2). Furthermore, theVisual map receives an inhibitory projection from the Summa-tion neurons of the Motor layer, which allows the progressivereset of the map according to the execution status of the currentsaccade (the details of which are described below).A neuron located at coordinates x, y in the Visual map obeysto equation 1, with the following inputs: I V isx,y = I Retinax,y − ω V isSum × a Sum (3)With I Retinax,y the input from the retina for the same neuron,modulating by a gluing process for vertical or quasi-vertical tar-get positions (details for the calculation of the Retinal input aregiven in Appendix A), and ω V isSum the weight of the inhibitoryconnection from the the Summation neurons. γ for the wholeVisual map is set to .The Visual map projects to the Integration map by one-to-one connections. Each neuron in the Integration map acts as anoisy evidence counter, and integrate the activity of its corre-sponding Visual map neuron over time, accumulating the valueof the target (or targets) exciting said Visual neuron at a variablerate depending on said value. Therefore, the whole Integrationmap is akin to a multitude of stochastic race models, each accu-lumating evidence for the selection of one position in the Visualfield.Furthermore, the Visual map to Integration map connectionis subjected to a modulatory inhibition Γ IntBG from the BG out- put, which acts as a gate for selection, allowing for a boost ofthe integration rate of the selected target and a simultaneousbraking of the integration rate of its competitors. Lastly, the In-tegration map receives the same inhibitory projection from theSummation neurons as the Visual map. Consequently, a neu-ron located at coordinates x, y in the Integration map obeys toequation 1, with the following inputs: I Intx,y = ω IntV is × a V isx,y × Γ IntBG − ω IntSum × a Sum + (cid:112) τdt × N (0 , ω n × (cid:113) a Intx,y + ε ) (4)With ω IntV is the weight of the input from the Visual map neu-ron at coordinates x, y , Γ IntBG the modulatory inhibition exertedby the BG on the connection between the Visual neuron at coor-dinates x, y and the Integration neuron at the same coordinates(see fig. 2 for the details of this connectivity), and ω IntSum theweight of the inhibitory connection from the the Summationneurons. γ for the whole Integration map is set to . , suchlow value allowing for the integration of activity over time.The gaussian white noise applied to this neuron is propor-tional to the square root of its previous activity, modulated by aweight ω n .This dependency to activity is necessary so that targets closeto the vertical have the same probability of being selected thantargets elsewhere (see the gluing description in Appendix A).The modulatory inhibition Γ IntBG is calculated for each SCIntegration map neuron as per equation 5: Γ IntBG = 1 − (cid:88) x,y ∈ N a BGx,y T IntBG (5)With T IntBG the specific threshold for the basal output of theBG to the Integration map, and (cid:88) x,y ∈ N a BGx,y the summed outputsof all channels of coordinates x, y in the submap N of the BG,feeding one given SC Integration map neuron (see fig. 2 for thedetails of this connectivity).The threshold T IntBG is set to 0.359, higher than the BG rest-ing output (mesured at 0.349), with the effect of allowing somecommunication between the Visual and Integration map evenwhen the BG inhibitory output is at its basal firing rate. When5his output changes from its rest level, Γ IntBG will either enhanceor decrease the weight of each individual connection (and there-fore the rate of integration of each Integration neuron) accord-ing to the variation of the output of the channels fed by this neu-ron: BG channels coding for a target that gets little evidencecounted in the integration map will lose selection to channelscoding for a target that gets more evidence counted, and there-fore the ”losing” BG channels output will have a stronger in-hibitory influence on their Integration neurons than at rest level,when ”winning” channels will have a weaker inhibitory influ-ence.Thus, the rate of integration of each Integration neuron willbe enhanced if this neuron codes for a target in the process ofbeing selected, but decreased if the neuron codes for a target inthe process of not being selected.
BGchannelsSC Decisionmap
SC−BG concentration weight mapSC Integration
Figure 2:
Anatomy of the SC-BG connections.
Concentration of the SC inte-gration map output to the BG channels, and deconcentration of the BG outputto the connections between the SC maps. The Integration map is divided insubsections of × neurons overlapping each other. Each SC map subsectionfeeds one specific BG channel with the weight pattern given in the inset: theneuron at the center of the red subsection feeds only one BG channel, while thefour neurons of each of its corners all feed four different channels, and the neu-rons in the middle of the edges of this subsection feed two different channels.BG output to the one-to-one connections between the Integration and Decisionmaps (and the connection between the Visual and Integration maps, not repre-sented here) obey to the reverse projection pattern, with the same weight patternso that all BG channels have the same global weigth to the whole SC map. The Integration map output is projected in two different di-rections: the first one is a one-to-one projection to the Decisionmap located deeper in the SC, and the second one is aimed at theThalamus, in order to feed the BG module of the model. ThisIntegration map to BG projection is subjected to a rescaling pro-cess in order to account for the differences in dimensions be-tween the SC retinotopic maps and the BG channel maps. Thevisual field represented by the retinotopic properties of the Inte-gration map is divided in subsections, and a channel of the BGwill be dedicated to receiving the concentrated outputs of allneurons within each subsection (these outputs are weighted sothat all neurons are used to the same proportions by the wholeBG - see fig. 2 for the details of this connectivity).The resulting competition between the BG channels willdisinhibit only the specific subsection of the retinotopic mapcorresponding to the winning channel, hence the sum term inequ. 5.The Decision map receives one-to-one inputs from the In-tegration map, and this connection is also subjected to a modu-latory inhibition Γ DecBG from the BG (see fig. 2 for the details ofthis connectivity), as per equation 6: Γ DecBG = 1 − (cid:88) x,y ∈ N a BGx,y T DecBG (6)With T DecBG the specific threshold for the basal output of theBG to the Decision map, and (cid:88) x,y ∈ N a BGx,y the summed outputsof all channels of coordinates x, y in the submap N of the BG,feeding one given SC Decision map neuron.his modulatory inhibition is very similar to the one exertedon the connection between the SC Visual and Integration mapwith a single, yet major, difference concerning the tuning of thethreshold T DecBG : it is set to 0.349 (when T IntBG is set to 0.359), atthe same level as that of the BG resting output (0.349), with theeffect of completly shutting down the communication betweenthe Integration and Decision map when the BG activity is at itsrest level.When the Integration map has fed enough evidence to theBG for a channel to be selected, the inhibition on the Integra-6ion map to Decision map connection will be selectively liftedfor the subsection of the retinotopic map corresponding to thewinning BG channel, and only the signal generated by the win-ning target will be transmitted from the Integration map to theDecision map. Thus, the losing targets are erased from this map(while still present in the upper maps), and only the signal re-lated to the winning target is transmitted to the Motor map.Therefore, a neuron located at coordinates x, y in the Deci-sion map obeys to equation 1, with the following inputs: I Decx,y = ( ω DecInt × a Intx,y × Γ DecBG ) (7)With ω DecInt the weight of the input from the Integration mapneuron at coordinates x, y , Γ DecBG the modulatory inhibition ex-erted by the BG on the connection between the Integration neu-ron at coordinates x, y and the Decision neuron at the samecoordinates. γ for this map is set to .Finally, the neurons of the Motor map receive one-to-oneconnections from the Decision map, and two inhibitions inher-ited from the SC model of Tabareau et al. (2007). The firstis exerted by the OPNs of the SBG and ensures that no motoractivity is generated while target selection is ongoing. The sec-ond inhibition comes from the summation-integration neuronsin order to perform the process of Spatio-Temporal Transfor-mation as proposed by Groh (2001), and ensures that no matterthe particularities of the current saccade profile (from a quickand strong burst of motor activity to a longuer but weaker burstof activity), the integration of the whole Motor map activity re-mains constant over the duration of the saccade.A neuron located at coordinates x, y in the Motor map obeysto equation 1, with the following inputs: I Motx,y = ω MotDec × a Decx,y × (1 − ω MotSum × a Sum ) − ω MotOP N × a OP N (8)With ω MotDec the weight of the input from the Decision mapneuron at coordinates x, y , ω IntSum the weight of the inhibitoryconnection from the the Summation neurons, and ω MotOP N theweight of the inhibitory connection from the OPNs of the SBG. γ for the whole Motor map is set to . The Motor map’s whole activity is then transmitted as aweighted sum to the EBNs and IBNs of the SBG in order totransform the SC spatial information about the saccade targetlocation into a Cartesian temporal motor command that will betransmitted to the EOM in order to move the eye towards thetarget. The projection weights from the SC to the SBG is tunedto the position in the visual field coded by each Motor map neu-ron, and also to the horizontal or vertical direction of movementcoded by each SBG circuit.As mentionned earlier, the proper preparation of the motorcommand as supposed by the process of STT dictates that theactivity of the Motor map is bound by two set of neurons, ensur-ing that no motor command is prepared as long as selection isnot achieved, and that the total motor activity over time is con-stant for all saccade metrics, as hypothetized by Groh (2001).The LLBs/cMRF and Summation neurons are the operators forthese two respective functions.The LLB of the SC integrates the activity of some of itsmaps, and excites the cMRF that in turn inhibits the OPNs ofthe SBG in order to lift the OPNs’ basal suppression of theMotor map activity (Wang et al. (2013)). In our model, thisLLB/cMRF / OPN connection is simplified in a direct LLB/OPNinhibitory projection. The LLB are fed by the most superficialmap in the SC on which only the activity related to the target(s)having won selection is displayed, that is the Decision map.The LLBs obey to equation 1, with the following inputs: I LLB = ω LLBDec × (cid:88) x,y ∈ SC a Decx,y − E LLB (9)With ω LLBDec the weight of the Decision map input to theLLB, and E LLB the threshold triggering LLB activity. γ forthe LLBs is set to .The Summation neurons implement the mechanism of sum-mation with saturation as proposed by Groh (2001): they re-ceive the summed activity of the whole Motor in order to grad-ually inhibit the SC output, that is the Motor map activity. Thismechanism of SC output regulation has been extended in ourmodel to the regulation of the activity of the Visual and Integra-tion maps too.7he Summation neurons are perfect resetable integrators,thus γ = 0 for them. They obey to the following equation: I Sum = ω SumMot × (cid:88) x,y ∈ SC a Motx,y (10)With I Sum the global input to the Summation neurons, and (cid:88) x,y ∈ SC a Motx,y the weigted sum of the activity of all Motor mapneurons. he BG module of our model is mostly copied from Girardet al. (2008), with three key features changed:1. All the cortical elements of the original model have beenremoved. Therefore, the Thalamus of our model is onlyregulated by a global inhibition of the Thalamic ReticularNucleus and a channel-specific selective inhibition fromthe BG.2. The Thalamus and TRN elements of our BG module arenot only collecting the output of the BG, as in Girard et al.(2008), but are also feeding the input nuclei of the BGs,as indicated by the anatomy of the subcortical BG loops(McHaffie et al. (2005)). Thus, the Thalamus receivesthe afferences from the SC, and feeds them to the inputnuclei of the BG (that is, the Striatum D1 and D2, the FSneurons and the STN).3. The dimensions and parameters of the BG module havebeen adaptated to the size of its SC inputs. The chan-nel architecture of each BG nucleus is therefore bidimen-sionnal, sized to one third of the dimensions of the SCinputs, and its set of parameters is given in Table B.5, inappendix Appendix B.These changes taken into account, our BG module operatesin a similar way with the BG model proposed by Girard et al.(2008): each channel receives inputs (representing the summedactivity of the whole subsection of the SC map each channelis connected to), and competes with the other channels in anoff-center/on-surround inhibition system that promotes the dis-inhibition of the winning channel, and the overinhibition of the losing channels. The feedback loop to the SC will reflect thischange in the inhibitory output of the BGs, and selectively al-low the transmission of signal from one map of the SC to theother only for the subsections linked to the winning channel ofthe BGs (see fig. 1 for the anatomy of the BG module, andfig. 2 for the detailled description of the connectivity betweenthe BG channels and SC maps).The input nuclus of our BG module is the Th, which is di-vided in channels. It receives projection from the Integrationmap of the SC, as well as a regulatory diffuse inhibition fromthe Thalamic Reticular Nucleus (TRN), and a specific channel-to-channel feedback inhibition from the output nucleus of theBG, the GPi.Each channel i obeys to equation 1, with the following in-puts: I T hi = ω T hSC × (cid:88) x,y ∈ SC a SCx,y − ω T hT RN × a T RN − ω T hBG × a GP ii + E T h (11)With ω T hSC the weight of the input from the subsection ofthe SC Integration map of coordinates x, y corresponding tochannel i , ω T hT RN the weight of the TRN input to the Th, ω T hBG the weight of the GPi input from channel i , and E T h the basalactivity of the Th. γ for each channel of the Th is set to .The TRN is not organised into channels, but aggregates ac-tivity from all Th channels in order to exert to a global feedbackinhibition to the Th. It obeys to equation 1, with the followinginputs: I T RN = ω T RNT h × (cid:88) i ∈ BG a T hi (12)With ω T RNT h the weight of the Th input to the TRN. γ forthe TRN is set to (the TRN time constant is τ small rather thanthe standard τ ).The BGs themselves are composed of several nuclei: theStriatum, with its D1 and D2 Medium Spiny Neuron popula-tions plus the Fast Spiking (FS) interneurons, and the STN - thetwo of them constituting the input structures of the BGs, theintermediary GPe, and the GPi/SNr, the output structure of theBGs.8he FS neurons of the Striatum, like the TRN, are repre-sented by one single population of neurons rather than beingdivided in channels. They exert a feedforward inhibition on theMSN. They are fed by the summed output of the Th, and regu-lated by the summed inhibitory output of the GPe. They obeyto equation 1, with the following inputs: I F S = ω F ST h × (cid:88) i ∈ BG a T hi − ω F SGP e × (cid:88) i ∈ BG a GP ei (13)With ω F ST h the weight of the Th input to the FS, and ω F SGP e the weight of the GPe input to the FS. γ for the FS is set to ,and its time constant is τ small rather than the standard τ .The other striatal neuron populations represented in the modelare the D1 and D2 MSNs. They are divided in channels, andobey both to the same global equation 1, with channel-to-channelTh excitation, GPe inhibitory feedback and global FS inhibitoryfeedback; D1 and D2 differ only in the effect of the dopaminelevel: excitatory for D1 and inhibitory for D2. I D i = (1 + λ ) × ( ω D × T h a T hi − ω D GP e × a GP ei ) − ω D F S × a F S + E D (14) I D i = (1 − λ ) × ( ω D T h × a T hi − ω D GP e × a GP ei ) − ω D F S × a F S + E D (15)With λ the dopamine level modulating the inputs to thedendritic tree of either MSN population (kept constant in thisstudy), ω D /D T h and ω D /D GP e the weights of the inputs from theTh and the GPe respectively, ω D /D F S the weight of the FS in-put to the MSNs. Finally, the negative constant inputs E D /D keep the neurons silent when the thalamic inputs are not strongenough, and represent the up-state/down-state property of theseneurons. γ for both D1 and D2 MSN populations is set to .The STN is the second input structure of the BGs, and isalso divided in channels. It receives channel-to-channel inputsfrom the Th, as well as a global GPe inhibitory feedback. Eachof its channels i obeys to equ. 1, with the following inputs: I ST Ni = ω ST NT h × a T hi − ω ST NGP e × (cid:88) i ∈ BG a GP ei + E ST N (16) With ω ST NT h the weight of the Th input to the STN, ω ST NGP e the weight of the summed GPe input to the STN, and E ST N thebasal activity of the STN. γ for the STN is set to , and its timeconstant is τ small rather than the standard τ .The GPe is an intermediary inhibitory nucleus of the BGs,that receives channel-to-channel inhibitory inputs from the D1and D2 populations of the Striatum, and a diffuse excitationfrom the STN. Each channel i of the GPe obeys to equation 1,with the following inputs: I GP ei = ω GP eST N × (cid:88) i ∈ BG a ST Ni − ω GP eD × a D i − ω GP eD × a D i + E GP e (17)With ω GP eST N the weight of the summed STN input to theGPe, ω GP eD /D the weight of the input from either D1 or D2MSNs to the GPe, and E GP e the basal activity of the GPe. γ for each channel of the GPe is set to .he GPi/SNr is the output nucleus of the BGs. It receivesa diffuse excitation from the STN, a diffuse inhibition formthe GPe, and channel-to-channel inhibitory inputs from the D1and D2 populations of the Striatum. Each channel i of the GPiobeys to equation 1, with the following inputs: I GP ii = ω GP iST N × (cid:88) i ∈ BG a ST Ni − ω GP iGP e × (cid:88) i ∈ BG a GP ei − ω GP iD × a D i − ω GP iD × a D i + E GP i (18)With ω GP iST N the weight of the summed STN input to the GPi, ω GP iGP e the weight of the summed GPe input to the GPi, ω GP iD /D the weight of the input from either D1 or D2 MSNs to the GPi,and E GP i the basal activity of the GPi. γ for each channel ofthe GPi is set to . The SBG module of the model is reproduced from Tabareauet al. (2007), with some minor parameters adjustments in orderto account for the changes in the activity profiles of the Motorlayer of our model.The SBG is composed of four identical circuits, each re-sponsible for the rotation of the eye in either the Rightward,Leftward, Upward or Downward direction ( dir ∈ [ R, L, U, D ] ).9hese circuits are coordinated in pairs by crossed projectionsfrom some of their input neurons to the output neurons of thecoordinated circuit, operating in opposed directions dir and dir opp (see fig. 1 for the details of the SBG anatomy and theconnectivity between the circuits in one pair).Each circuit is constituted of a population of Excitatory BurstNeurons, a population of Inhibitory Burst Neurons, a popula-tion of Tonic Neurons and a population of MotoNeurons. Thereis only one set of Omni-Pause Neurons for the whole SBG, thatprojects to each of the four circuits and gates their activity.In each circuit, movements are encoded by bursts of activityrepresenting the vectorial components of the desired rotation,produced as follows:The OPNs are tonically active, and exert a basal inhibitionthat completely shuts down the activity of the input neurons ofthe SBG, the EBNs and IBNs as well as the whole Motor mapof the SC. This inhibition prevents the generation or transmis-sion of unwanted signal through the SBG, and the generation ofparasitic eye movements. The OPNs are themselves inhibitedby the LLBs of the SC when the activity in the Decision layerof the SC is high enough to cross the LLBs activation threshold: I OP N = − ω OP NLLB × a LLB + E OP N (19)With ω OP NLLB the weight of the LLB input to the OPNs, and E OP N the basal activity of the OPNs. γ for the OPNs is set to . The EBNs and IBNs are basally shut down by the OPNs.They also receive the output of the whole Motor layer of theSC, weighted accordingly to the horizontal or vertical directionof movement relevant to each circuit.Therefore, the inputs for the EBNs of the SBG circuit cod-ing for direction dir answer to the following equation: I dirEBN = (cid:88) x,y ∈ SC ( ω SBG − dirSC × a Motx,y ) − ω BNOP N × a OP N (20)With ω SBG − dirSC the weight pattern of the SC-SBG projec-tion for direction dir , as defined by equ.3 of Tabareau et al.(2007) and represented in supplementary fig. B.16, and ω BNOP N the weight of the OPN inhibition on the EBNs. γ for the EBNsis set to .The IBNs have exactly the same inputs, characteristics andmathematical properties as the EBNs, and differ only in the ef-fects of their outputs on their targets : the IBNs of the SBGcircuit coding for direction dir will inhibit the TN and MN ofthe SBG circuit coding for the opposed direction dir opp : theIBNs of the Upward SBG circuit will feed the TNs and MNs ofthe Downward circuit, while the IBNs of the Downward SBGcircuit will feed the TNs and MNs of the Upward circuit. Thesame scheme occurs between the Rightward and Leftward cir-cuits.The TNs of the SBG circuit coding for direction dir have anon-zero resting activity, and integrate the difference betweenthe output of EBN from the same SBG circuit and the outputof the IBN of the SBG circuit coding for the opposed direction dir opp : I dirT N = ω T NBN × ( a dirEBN − a dir opp IBN ) (21)With ω T NBN the weight of the connection from the EBNsand IBNs to the TNs , a dirEBN the activity of the EBNs of theSBG circuit coding for direction dir , and a dir opp IBN the activity ofthe IBNs of the SBG circuit coding for the opposed direction dir opp . The TNs are perfect integrators, thus γ = 0 for them.The MNs of the SBG circuit coding for direction dir re-ceive inputs from the EBNs and TNs of their own SBG circuit,and from the IBNs of the SBG circuit coding for the opposeddirection dir opp . They have a non-zero basal activity due to thenon-zero resting TN component of their input, and therefore al-ways produce a motor command towards the EOM, that allowsfor the stability of gaze between movements. I dirMN = ω MNBN × ( a dirEBN − a dir opp IBN ) + ω MNT N × a dirT N ) (22)With ω MNBN the weight of the connection from the EBNsand IBNs to the MNs , a dirEBN the activity of the EBNs of theSBG circuit coding for direction dir , and a dir opp IBN the activity ofthe IBNs of the SBG circuit coding for the opposed direction dir opp , and ω MNT N the weight of the TNs projection to the MNs. γ for the MNs is set to .10inally, when activity in the SC Decision layer decreasesunder the LLBs activation threshold, the OPNs suppression willprogressively be lifted and the OPN will start inhibiting theEBNs and IBNs again, thus ensuring that the motor commandto the EOM is limited in time.The biomecanics accounting for the eye movement are givenby a standard second-order differential equation that link themovement of the eye in the horizontal or vertical directions tothe difference between the firing rates of the MNs from the SBGcircuits coding for this direction and the opposed one: ¨ θ + ω θ acc θ vit × ˙ θ + ω θ acc θ pos × θ = ω θ acc MN × ( a dirMN − a dir opp MN ) (23)With θ the position of the eye, ˙ θ its movement speed, ¨ θ its movement acceleration, a dirMN and a dir opp MN the activity of theMNs of the SBG circuit coding for direction dir and its oppo-site dir opp , and ω θ acc MN the weight of the connection between theMNs of the SBG and the eye plant. All modules (SC, BG and SBG) of the model have beenhand-tuned individually in order to assess their proper function-ing before being linked together. Then, further tuning was madeso that the whole model would perform correctly.The SBG module kept the original parameters from Tabareauet al. (2007), except for the weight ω θ acc MN , which sets the gain ofthe saccades. This parameter was hand-tuned in order to pro-duce saccades with correct amplitudes.Since the number of channels in the BG module is muchhigher here than in Girard et al. (2008) (121 compared to 6),the strength of all diffuse connections within the module hadto be tuned down in order to prevent diffuse connections fromalways shutting down any one-to-one connection, and to allowthe selective disinhibition of one channel. To reach this goal,the isolated BG module was fed with ”targets” modelled by 2DGaussian inputs similar to those used in the tasks, with var-ied values. The parameters were adjusted until the selection ofa single target with an value between 0.6 and 1 was restored.Finer adjustments were then made so that one or two distrac-tors of inferior values would not disturb the selection process, and that the simultaneous selection of multiple targets occurredonly when they have very close values.The SC module has no significant changes in parametersfor all features directly reproduced from Tabareau et al. (2007),such as the motor layer and integration-saturation mechanisms.The parameters for the added or heavily modified layers, suchas the Visual, Integration and Decision map are mostly similarwith those of the Motor layer, being based upon the same equa-tion types. This module contains three critical parameters thathad to be tuned with care, in order to have an optimal compro-mize between the duration of the selection process (whether thecompeting targets’ values are similar or not), the production ofaccurate saccades, the minimization of the average-to-normalsaccades ratio, and the production of realistic activity patternsfor the various neurons and maps they affect: • the noise weight ω n (cf. equ 4). This parameter has to betset low enought so that the noise level remains low com-pared to the target’s value, but high enough so that theduration of stochastic discrimination between two identi-cal targets by the BG remains compatible with the normallatency for saccades (under 100ms for fast saccades with”easy” selection choices, and up to 200ms for saccadeswith harder selection choices). Thus, it was defined us-ing a grid search on the [0 , interval, by steps of 0.05. • the threshold T IntBG (described in equ 5) for the BG feed-back to the Visual-to-Integration maps connection. Thisparameter has to be higher that the BG resting output toallow the Visual map to Integration map connection, butnot so high that it would take too long for the integratorsto feed enough evidence to the BG to reach this thresholdwithin the realistic timeframes mentionned earlier. Thiswas achieved by grid-searching for the value yielding thebest results within the [0 . , . interval, by steps of0.001. • the threshold T DecBG (described in equ 6) for the BG feed-back to the Integration-to-Decision maps connection, thatallows the BG to basally inhibit all connections between11he Integration map and the Decision map but, once se-lection is achieved, this threshold allows one-to-one con-nections to the Decision map only from the neurons of theIntegration map coding for the winning target’s location;therefore, this parameter needs to be lower than the BGresting output, but still high enough so as to minimize theoccurence of cases where more than one target location isgated from the Integration map to the Decision map. Thiswas achieved by grid-searching for the value yielding thebest results within the [0 . , . interval, by steps of0.001. Targets are characterized by three parameters: their coor-dinates, given by their azimuth and elevation in the Cartesiancoordinates of the visual field projecting on the superficial vi-sual layer of the SC; and their value, indicating the height of the2D Gaussian representing the target in the model.Azimuth and elevation both vary in the [ − ◦ , ◦ ] rangein Cartesian coordinates, and target value is scaled on [0 , .Unless specified otherwise, any simulation occurs as fol-lows: • the model is initiated for 30 ms so that all of its compo-nents reach stable basal activity levels. • targets are presented accordingly to each task protocol. • the simulation runs its course until either 250 ms after asaccade has been produced, or 750 ms after targets pre-sentation if no saccade has been produced. In this task, the model produces saccades towards a singletarget, the position of which varies over the full breadth of thesimulated visual field. This will allow for the assessment of theprecision of saccade generation and the production of controldatasets for all targets positions when no selection occurs.Another goal of this task is to ensure that the activity pro-files of the various neurons of the model respect the electro- physiological properties of their in-vivo counterparts, as recordedin the litterature.Target value is always maximal, and a hundred tests arelaunched for each set of target coordinates, which both varyby increment of ◦ in the [ − ◦ , ◦ ] range. This task aims at reproducing the task of discriminationbetween a target and a various number of distractors used byMcPeek and Keller (2002, 2004); McPeek (2008). In this task,a monkey is trained to discriminate between one target and sev-eral distractors, and to produce a saccade towards the target.This is tested with or without the injection of Lidocane or Mus-cimol in the SC at the location coding for the position of thetarget, in order to study the role of the SC in saccade targetselection.We reproduce the effect of training the monkey for the taskby setting the target’s value at twice that of the distractors (withtarget value varying between 0 and 1 by increments of 0.05,and therefore discriminators value varying between 0 and 0.5by increments of 0.025), and we test the ability of the modelto perform accurate selection between one target and a variablenumber of distractors. We test four visual field setups with avarying number of points in the Retinal input, as described inTable 1, four hundred times each, with the target always locatedat the coordinates given for position 1. This control setup iscalled the ”target-and-distractors” setup.The test element of the experimental task uses the samesetup as the control task, but with the addition an injection ofLidocane/Muscimol at the SC location coding for the targets’scoordinates, dosed in such way as to decrease local neuronalactivity without completely shutting it down. Thus, it is hopedthat the increased value the monkey learned to place on the tar-get will be offset by the reduced neuronal activity at the targetlocus in the SC, and discrimination between target and distrac-tors should be more difficult, the target being more or less con-sidered as another distractor.12ondition position 1 position 2 position 3 position 4 position 5 position 61 point [ ◦ , ◦ ] - - - - -2 points [ ◦ , ◦ ] [ − ◦ , − ◦ ] - - - -4 points [ ◦ , ◦ ] [ ◦ , − ◦ ] [ − ◦ , − ◦ ] [ − ◦ , ◦ ] - -6 points [ ◦ , ◦ ] [ ◦ , − ∗√ ◦ ] [ − ◦ , − ∗√ ◦ ] [ − ◦ , ◦ ] [ − ◦ , ∗√ ◦ ] [ ◦ , ∗√ ◦ ] Table 1:
Point positions in the visual field the various conditions of task 3.
The total number of points in the visual field for each test condition is given in thefirst column. Coordinates are given in [ azimuth, elevation ] in the following columns. Eccentricity is constant across all setups for all elements, and all elementsare equidistant from each other in each setup. The target is always located at position 1 in the target-and-distractors setup. We reproduce the effects of the drug-injection by loweringthe maximal value reachable by neurons at the injection site tothe level of the value of distractors, all points in each conditionhaving therefore the exact same value varying between 0 and 1by increments of 0.05, in the same four visual fields conditionsas the control condition - each being tested four hundred timesas well. This test setup is called the ”distractors-only” setup.The difficulty for the model in this setup lies in its abilityin selecting only one distractor among many identical ones, ina finite time, and without resulting in too many average sac-cades which are the symptoms of the simultaneous selection ofmultiple distractors.
This task is derived from the 2 elements condition of thedistractors-only setup of task 2: it aims at characterizing thelimits of the model in discriminating between two competingtargets of similar value, considering the separation between thetargets. We test the model’s ability to produce a good balanceof correct and average saccades when presented with one targetT1 at a fixed position and another target T2 of similar value butvariable position.The two targets are displayed simultaneously after modelinitialization, with the following parameters: • T1 has a value of 1 and is located at coordinates [20 ◦ , ◦ ] . • T2 also has a value of 1; its azimuth is set to ◦ and itselevation varies in the [ − ◦ , ◦ ] range by incrementsof ◦ (see fig. 3). Figure 3:
Sample of target positions for task 3.
Target 1 (black circle) isalways located at coordinates [20 ◦ , ◦ ] , and target 2 (red circle) is located ata variable distance from target 1 on the same vertical plane. Also noted is thedistance between the two targets. Each condition is tested two hundred times.
3. Results
Before anything else, two criteria concerning the analysisof our model’s results must be clarified.First of all is the evaluation of the accuracy of saccades: sev-eral different criteria assessing the precision of in vivo saccadeshave been proposed in the litterature. Among them, McPeekand Keller (2002) propose that accurate saccades land within aradius of up to one fifth of the target’s eccentricity of the target’sposition, while McPeek (2006) propose that accurate saccadesland within a radius of ◦ , or 15% of the saccade’s amplitude,of the target’s position.Secondly, we must note that all timings plotted from themodel are counted from the appearance of the target in the Vi-13ual map of the SC. This does not take into account the latencybetween the presentation of a target in the visual field of ananimal and the transmission of the retinal information to the su-perficial layers of the SC, which is estimated around 40ms fortype I neurons (which respond well to the kind of stimuli usu-ally used in experimental tasks, as measured by Rizzolatti et al.(1980)).Thus, these 40ms should be added to all timings given in theresults below, when comparing the model’s simulated eventsrelated to target appearance in the visual field with similar in-vivo experimental results. When only one target is displayed, the model is able to per-form saccades with good accuracy across the whole range ofthe simulated visual field: the maximal error between the de-sired and obtained eye endpoints is inferior to 6% of the desiredamplitude of the saccade (see fig. 4), well under the accuracycriteria evoked before.
Figure 4:
Saccades endpoint error map for Task 1. the error is calculated asthe ratio of the difference between the final eye position and the target position,and the amplitude of the desired saccade.
Mean eye speed profiles (as examplified in fig. 5-A) showthat saccade latency is close to 55ms after the onset of activityin the Visual map - that is 95ms after target presentation in thevisual field, when retinal-to-SC input latency is taken into ac- count, which is compatible with express saccade latencies (cf.Fischer and Boch (1983)).The appearance of a target in the Visual map (fig. 5-B, pur-ple curve) feeds a build-up integration by the Integration map(fig. 5-B, light blue curve). This Integration map activity isthen fed to the BG, which results in a progressive desinhibi-tion of the BG output selectively for the channels coding forthe target’s location (fig. 5-B, orange curve for the BG chan-nel coding for the target’s location, vs grey curve for a channelcoding for a section of SC integration map not stimulated by thetarget), starting around 30ms after the onset of Visual map ac-tivity. Consequently, the integration rate in the Integration mapis increased specifically for the neurons coding the target’s po-sition, thus generating a burst of activity that speeds up the fulldisinhibition of the corresponding BG channel. Furthermore,the Integration map to Decision map connection is disinhibitedspecifically for the neurons excited by the target, thus allowingthe burst component of the Integration map activity to evokea burst in the Decision map centered on the target’s location(fig. 5-B, yellow curve).The parametrization of the LLB circuitry, a combination ofhigh activation threshold and weak input weight, ensures thatthe LLB summation of this Decision map burst is quite slow(see fig. 5-B, dark blue curve), and explains the 20ms-delaybetween the onset of the Decision map burst and that of theMotor map burst (fig. 5-B, black curve).The Motor map burst elicits activity in the E-IBNs of thosecircuits of the SBG, concerned with movement towards the po-sition of the target, around 50ms after target presentation, gatedby the OPNs inhibition (fig. 5-C, no BN activity is elicited forthe leftward and downward circuits), which in turn generate aburst of MN activity for the corresponding SBG circuits (fig. 5-D, the MN of the up- and right-ward circuits are activated whenthose of the left- and down-ward circuits are inhibited, in orderto contract or relax the corresponding EOM) and produce theeye movement evoked earlier. The MNs will revert to a tonicactivity, but on a different level than before target presentation,in order to keep the eyes fixed on their final position.14 a c t i v i t y ( a . u ) time (ms) C) OPNBN upBN rightBN downBN left a c t i v i t y ( a . u ) time (ms) C) D)
MN upMN rightMN downMN left s peed ( ° / s ) C) D)A) horizontalverticalaverage a c t i v i t y ( a . u ) C) D)A) B)
OPNSumLLBVisInIntegDecisMotorBG channel 1BG channel 2
Figure 5:
Time course of SC maps activity with regards to target presentation.
Neurons activity profiles during a saccade towards a target located at coordinates [20 ◦ , ◦ ] , aligned with target presentation. A), eye horizontal, vertical and mean speed profiles. B), BG outputs for channels 1 (selected) and 2 (unselected), SBGOPNs and SC neurons coding for the center of the target’s position in each SC map . C), OPNs and E-IBN for all four SBG circuits. D), MNs for all four SBGcircuits. See text for abbreviations. The Motor map burst also feeds the summation neuron re-sponsible for the STT (fig. 5-B, green curve), which exert aninhibitory feedback on the Motor map, as well as on the Vi-sual and Integration maps. The Motor burst has the shortestduration, and disappears around the same time the Visual mapactivity is completely shut down. The bursting part of the In-tegration map activity is sustained by the low leak of this mapwell after the Visual map extinction. As a consequence, the de-cision map burst is the last to be extinguished, causing the LLBinhibition of the OPNs to lift.Fig. 6 summarizes the time courses of the activity of theneuron coding for the target’s position in the Visual, Integration,Decision and Motor maps when aligned to mean saccade onset: the precise timings of these activity profiles can be manipulatedby parameter modifications (such as the LLB threshold, or theweight of the STT inhibitions), but their order is intrinsic to themodel architecture.These characteristics can be compared with the results shownin fig. 1 of McPeek and Keller (2002), that expose electrophysi-ological recordings of the motor-related activity of various neu-rons types : they identify several classes of SC neurons, espe-cially the Visuo-Movement Burst Neurons (VBN), the Visuo-movement Prelude Neurons (VPN) and the Movement Neurons(Mvt.N). The motor-related activity of the VPNs shows a longbuild-up followed by a burst component centered on saccadeonset. The VBN show no such build-up, but only a burst cen-15 a c t i v i t y ( a . u ) time (ms) VisInIntegDecisMotor
Figure 6:
Time course of SC maps activity with regards to saccade onset.
Neurons activity profiles for the Visual, Integration, Decision and Motor mapsaligned with saccade onset, for a saccade towards a target located at coordinates [20 ◦ , ◦ ] . tered on saccade onset. Finally, the Movement Neurons’ dis-charge pattern is also a burst centered on saccade onset, butwith a narrower width than the burst of the VBNs.Setting aside the burst of activity correlated with target pre-sentation exhibited by the VPN and VBN, which has not beenmodelled in our work, the discharge patterns of the VPN isstrongly similar to that of our Integration Map neurons activityprofile, when the VBN are akin to our Decision Map neuronsand the Mvt.N are akin to our Motor Map neurons. Therefore,we hypothesize that the role of the Integration Map, the Deci-sion Map and the Motor map of our model is played in-vivoby the VPN, VBN and Mvt.N populations respectively. Ourmodel is set so that the Motor map burst is temporally con-tained within the Decision map burst, which starts itself afterthe Integration map burst, a feature that the various experimen-tal recordings of the VBN, VPN and Mvt.N by McPeek andKeller (2002) do not allow to verify, but this prediction couldnevertheless be tested in-vivo on these neuron populations, aswell as all our model’s assumptions and predictions concerningthe connections and operations of the various SC maps. azimuth ( ° ) e l e v a t i on ( ° ) azimuth ( ° ) e l e v a t i on ( ° ) Figure 7:
Saccade endpoints distribution for the target-and-distractorssetup of Task 2.
Endpoints distribution when the target is presented alone(top-left), with one distractors (top-right), with three distractors (bottom-left)or with five distractors (bottom-right), when target value is maximal. Red cir-cle indicates target position, black circles distractors positions.
Fig. 7 shows that the model is able to produce perfect selec-tion in all conditions of the target-and-distractors setup, sinceno distractor is ever selected no matter their number, and sac-cade accuracy is as good as in task 1.In the distractors-only setup, selection also occurs, and isaccurate saccades made towards one stimulus (as opposed toaverage saccades made when multiple stimuli are selected) arestochastically distributed between all distractors, as experimen-tally shown in McPeek and Keller (2004) (compare fig. 8-Cwith their fig. 2-e). When accurate saccade are made towardsany distractor, saccade accuracy seems lower in this setup thanin the target-and-distractors setup, since saccade endpoints areless tightly grouped around the chosen distractor (comparedwith the similar condition of the target-and-distractors setup).It also appears that, in the distractors-only setup, selectionis not always independant of the value of the distractors: whentwo distractors compete, the proportion of average saccades(which are a symptom of the simultaneous disinhibition of morethan one competitor by the BG) is much higher when the dis-16 distractors | value 1 C) 0 5 10 15 20 25 30 azimuth ( ° ) e l e v a t i on ( ° ) azimuth ( ° ) e l e v a t i on ( ° ) Figure 8:
Saccade endpoints distribution for the distractors-only setup ofTask 2.
Endpoints distribution when two distractors of value 0.7 compete (top-left), when two distractors of maximal value compete (top-right), when fourdistractors of maximal value compete (bottom-left) or when six distractors ofvalue 0.6 compete (bottom-right). Black circles indicate distractors positions. tractors value is maximal (compare fig. 8-A for results with 2distractors of value 0.7, and fig. 8-B for results with 2 distrac-tors of maximal value).Furthermore, there is a threshold under which the value ofa target is not sufficient to elicit a saccade (fig. 9-A, red curve).Interestingly, this threshold increases with the number of dis-tractors presented simultaneously to the target (fig. 9-A), mean-ing that the more distractors compete with a target of intermedi-ary value, the more difficult it is for the model to elicit saccadestoward this target. This tendency remains moderate, and mightthus be difficult to establish experimentally.The same tendency is also noticeable in the distractors-onlysetup: the value threshold for saccade elicitation increases withthe number of competing distractors when 1, 2 or 4 of themcompete (fig. 9-B, red, green and blue curves). Results in thesix-distractors condition (fig. 9-B, pink curve) contradict thistendency: in this condition, the value threshold for saccade elic-itation is lower than in the 4-distractors condition.In the 6-elements condition, the points competing in the vi-sual field are close enough from each-other for the neurons ofthe Visual map stimulated by any given point of the visual field % o f s a cc ade s l aun c hed target value A) % o f s a cc ade s l aun c hed distractors value A)B)
Figure 9:
Effects of target value on saccade initiation in Task 2.
A), in thetarget-and-distractors setup. B), in the distractors-only setup. to be also stimulated by this point’s neighbours. Thus, the ef-fective value coded by the neuron coding for the center of thetargets in the Visual Map is higher than their value coded by thecorresponding neuron in the retinal input. This value increasebetween the retinal input and its Visual map representation isof course reflected over the whole 2D-Gaussian representationof each competing target, and is proportionnaly bigger for theneurons at the periphery of the Gaussian than for the neuronat its center, thus greatly increasing the total value encoded bysaid Gaussian.In the target-and-distractors setup, the effective value in theVisual map of the element of the visual field which will be se-lected - here the target - is around 2% higher for the center ofany point’s representation in the 1 target and 5 distractors con-dition (the 6-elements conditions of this setup) than in all otherconditions, in the [20 , ms interval after target onset where17ompetition is always ongoing whatever the saccade latency.The effective value in the same map for the element which willbe selected in the distractors-only setup - that is any distractor -is around 4% higher for the center of any point’s representationin the 6-distractors conditions than in the any other condition(1, 2 and 4 distractors), in this same timeframe.This is similar to each distractor of the 6-elements conditionof the distractors-only setup having more value than expected,hence the discrepancy in value threshold for saccade elicitationnoted for this condition in fig. ?? , bottom.In the target vs. distractors protocol, the VPNs recorded byMcPeek and Keller (2002) exhibit a ramping activity followedby either a burst (target case) or a gradual decrease of activity(distractor case). They show (their Fig. 10) that in these twodifferent cases, the ramps are at first undistinguishable, and be-come distinguishable only after integration has been going onfor a while. In our simulations, the neurons of the integrationmap also have a ramping activity (fig. 10-A, continuous anddashed green traces), followed by either a decrease (distractor,dashed trace) or a burst (target, continuous trace). During thisramping phase, the average slope depends on whether the neu-ron codes for a distractor or for the target. However, becauseof noise, the variability of these average firing rates is so large(Fig. 10-A, light blue and light green areas represent the stan-dard deviation around these means), that the divergence of theseramping activities can be only assessed late in the integrationprocess. We thus assume that our Integration map neurons area model for the VPNs recorded in McPeek and Keller (2002).In the same protocol, our Decision neurons (Fig. 10-A, redtraces) can be likened to the activity profiles of the Visuo-MotorBurst neurons (VBN) lacking a strong second visual burst (Fig.7 of McPeek and Keller (2002)): in both cases, the neuronshave no activity until selection is reached, and only the neuroncoding for the selected target exhibit a burst centered on saccadeonset.The model’s Motor map neurons exhibit the same pattern ofactivity when coding for either a selected or unselected targetas the Movement neurons recorded in fig. 13 of McPeek and Keller (2002), emitting a burst only when the selected target isin their receptive field (fig. 10-A, black curves).Finally, the activity profiles obtained for the Integration mapneurons in the simulation of the distractors-only task (Fig. 10-B, continuous and dashed green traces) predict that in such acase, the discriminability between a distractor about to be se-lected for a saccade and another distractor should be delayedeven longer (see in fig. 10-B the overlap in the green and bluearea representing the variability of the integration for each dis-tractor), up to the beginning of the burst associated to move-ment itself. After the identification of VPNs, this conditioncould easily be tested experimentally.Concerning saccade latency in this task (that is the delaybetween the presentation of an element in the visual field andthe beginning of the eye movement - also called Saccadic Re-action Time, or SRT), our model presents two results: firstly,the SRT decreases when the value of the elements in the vi-sual field increases (see the decrease of mean latencies as wellas latency dispersions in each of the four panels of fig. 11 forthe target-and-distractors setup, and the same in fig. 12 for thedistractors-only setup). This is consistent with experimental re-sults in tasks where the discriminability between the stimuli andthe background of the visual panel is low.Secondly, the SRT increases with the total number of ele-ments displayed in the visual field. Mean latency as well aslatency dispersion around the median value increase for a givenvalue with the number of points in competition, whether theybe target and distractors or distractors only: see the comparisonof the four panels of fig. 11 to each-other, as well as those offig. 12, which is resumed in fig. 13.This phenomenon, called Remote Distractor Effect or RDE,seems less obvious in the distractors-only setup than in the target-and distractors setup when the distractor value is low (com-pare fig. 13-A and bottom, blue curves), but saccade latencyis markedly higher in both setups for any given value when thenumber of competing points increases (red and black curves).In this case, the RDE is more pronounced for the distractors-only setup than for the target-and-distractors setup: the increase18 a c t i v i t y ( a . u ) time (ms)A) Figure 10:
Variation of the activity profiles in the SC maps with regards to the difficulty of the selection.
Activity profiles of Visual (blue), Integration (green),Decision (red) and Motor (black) map neurons coding for the positions of two competing points in the visual field. A), averaged over 500 trials in the one target(maximal value) and one distractor condition of the target-and-distractors setup. B), averaged over the 130 trials where only distractor 1 was selected in the twodistractors (maximal value) condition of the distractors-only setup. The green area represents the standard deviation around the mean activity of the integrationneuron of the winner of the competition, and the blue area the same for the activity of the integration neuron of the loser of the competition. t i m e ( m s ) target value SRT dispersionmean SRT target value
SRT dispersionmean SRT t i m e ( m s ) SRT dispersionmean SRT
SRT dispersionmean SRT
Figure 11:
Effects of target value on saccade latency in the target-and-distractors setup of Task 2 . Mean latency and latency dispersion with regards to targetvalue when the target is presented alone (top-left), with one distractors (top-right), with three distractors (bottom-left) or with five distractors (bottom-right) in saccade latency is much larger when going from 1 to 6 dis-tractors of maximal value (fig. 13-B, black curve), than whengoing from 1 target to 1 target and 5 distractors (fig. 13-A, blackcurve).In our case, the source of the RDE lies in the diffuse pro-jections within the basal ganglia loop, which are the substrate of competition between options. Indeed, while each channel istrying to disinhibit its output in the GPi through the Striatumto GPi focused inhibitions, it also tries to inhibit its neighboursthrough the diffuse inhibitions of the TRN over the Th and ofthe FS over the Striatum, as well as through diffuse excitationsof the STN over the GPi. As a consequence, when the number19 t i m e ( m s ) distractor value SRT dispersionmean SRT distractor value
SRT dispersionmean SRT t i m e ( m s ) SRT dispersionmean SRT
SRT dispersionmean SRT
Figure 12:
Effects of distractors value on saccade latency in the distractors-only setup of Task 2.
Mean latency and latency dispersion with regards to distractorvalue when only one distractor is presented (top-left), two distractors (top-right), four distractors (bottom-left) or six distractors (bottom-right) of competitors is increased, the effects of this competition isstronger: each competitor, even the one with the highest input,sees its representation in the TRN and the Striatum attenuatedby the additional inhibitions.This diminishes the Striatum to GPi self-inhibition, whichis the source of the selection, while it additionally receives moreactivity from its neighbors in the GPi from the STN, where theGPe inhibitory feedback is not sufficient to compensate for thisincrease. In the end, the resulting attenuation of the potentialwinner delays the moment when selection is complete.setup/condition 1T - 1D 1T-3D 1T-5Dtarget - distractors +38% +58% +86%2D 4D 6Ddistractors +80% +105% +123%
Table 2:
Total GPi input increase in conditions with multiple distractors,when compared to the 1 target or 1 distractor only reference.
T: target,D: distractor. Values calculated over at least 400 trials where accurate sac-cades have been made towards the target for each condition of the target-and-distractors setup, and over at least 200 trials whre accurate saccades have beenmade towards any distractor in the distractors-only setup
We measured this attenuation by measuring the total inputsto the GPi in the [10 , ms interval after target presentation,where we are sure that the competition is still running, even inthe 1 target case. The increased activity in the GPi, caused bythe additional inputs from competitors, and causing the RDE, isquite strong when one target of maximal value is confronted toan increasing number of distractors (the distractors have half ofthe taget value, Table 2, first line), and even stronger when onlydistractors of maximal value are present (i.e. all inputs have thesame value, Table 2, second line). For any given distance separating the targets, the proportionof average saccades increases with the value of the targets (seefig. 14-A, all curves). Furthermore, this proportion seems alsodependant with target separation: the proportion of average sac-cades is bigger for distances under the ◦ threshold (fig. 14-A,red and green curves respectively representing the proportion ofaverage saccade averaged over the to ◦ separation inter-val, and the to ◦ separation interval) than for separations20 t i m e ( m s ) number of distractors Target-and-distractors setupA) value 0.2value 0.6value 1 t i m e ( m s ) number of distractors Distractors-only setupA)B) value 0.2value 0.6value 1
Figure 13:
Evolution of saccade latencies with the number of distractorsand the value of the target in Task 2.
A), target-and-distractors setup. B),distractors-only setup. greater than ◦ (fig. 14-A, other curves).Fig. 14-B, reveals that the Visual map neurons located di-rectly between the two targets positions are more stimulated forsmall target separations, since in these cases the gaussians gen-erated by each target significantly overlap.When target separation is smaller than ◦ , this overlapgenerates enough activity between the targets representationsin the Visual map to seriously compete with the targets for se-lection, and thus regularly causes average saccades. Targetsseparated by a greater distance suffer less from this effect, andonly when they have high values.This result can be compared with the effects of the quantityof competing elements in the visual field in Task 2, in which wesaw that the representation of each element in the Visual map a v e r age s a cc ade s p r opo r t i on ( % ) targets value (a.u) A) ° -19 ° separation20 ° -22 ° separation23 ° -25 ° separation26 ° -28 ° separation 29 ° -31 ° separation32 ° -34 ° separation35 ° -37 ° separation V i s ua l M ap neu r on a c t i v i t y ( a . u ) normalized distance between the targets (a.u) A)B) ° separation20 ° separation23 ° separation26 ° separation 29 ° separation32 ° separation35 ° separation Figure 14:
Effects of target separation in Task 3 on the production of sac-cade and the overlap of activity loci in the Visual map of the SC.
A), propor-tion of average saccade with regards to targets value, averaged over separationintervals of ◦ for ease of representation. B), peak activity for all Visual mapneurons located directly between the positions of the competing targets of value1, for the same target separations intervals sampled for graph A). can receive additionnal stimulation because of the proximity ofits neighbours. The coordinates given in Table 1 allow to calcu-late the separation between all points in each conditions: . ◦ for the 2-elements condition, ◦ for the 4-elements condition,and ◦ for the 6-elements condition. This ◦ separation isprecisely the threshold after which two neighbouring points sig-nificantly overlap each-other, with the effects seen earlier.In our model, the precise value of this distance thresholdis dictated by parameter σ , which governs the size of the tar-get’s representation in the Visual map. Thus, this threshold canbe adjusted by the tuning of σ . However the prediction of the21xistence of such a threshold is inherent to the model structure.
4. Discussion
In this work, we proposed a biologically constrained modelof the SC-BG loops that accounts for saccade target selectionby way of the race-like competition of visual targets represen-tations in the retinotopic maps of the superficial layers of the SC; the BG themselves act as a threshold detector for the evidenceaccumulated by each target during competition.We showed that the activity profiles of the neurons of thevarious maps modelled in our SC are comparable to in-vivorecordings in the Monkey SC performed by McPeek and Keller(2002), and we therefore proposed a role for the Visuo-MotorPrelude and Burst neurons of the intermediary layers of the SC.We also propose a novel interpretation for the colliculu-baso-collicular loops based on convergent afferences from the SCmaps to the BG channels, on divergent symetrical inhibitory ef-ferences from the BG channels to the projection between theSC superficial and intermediary maps to one another, and ondifferent roles for these efferences, one modulating connectionsweights to amplifiy contrasts between stimuli, and the other gat-ing the transmission of signal to the deeper layers of the SConly for the stimulus having won the competition. Furthermore,we were able to reproduce and propose an explanation for spe-cific experimental data gathered by McPeek and Keller (2004)regarding the effects of local SC inactivation on the selectionbetween one target and multiple distractors.We predict a specific order and general shape for the activ-ity in the SC maps in various selection tasks that can be testedin vivo; we also predict that the occurence of average saccadesis linked to both the separation between the points in competi-tion in the visual field, as well as the value of their representa-tions in the SC Visual map. The quantitative aspects of thesepredictions can be adjusted by the fine tuning of the model’sparameters in order to fit optimally with experimental data, butthese predictions are qualitatively intrinsic to our model’s ar-chitecture. Furthemore, we propose a mechanism for the well-knownRemote Distractor Effect, based on balance of diffuse and spe-cific excitations and inhibitions exerted on each channels ofBG’s output nuclei, that increase the level of inhibition the win-ning target must lift on itself when more points compete in theVisual field.We also predict that the coupling of the BG characteristicswith the properties of the SC mapping can turn the RDE into afacilitating effect on saccade initiation in the specific case of thecompetition between an increasing number of similar stimuli,when the separation between stimuli reaches a critical minimalthreshold.The saccades produced towards single targets by our modelare accurate, as exposed in section 3.1, and this accuracy ismaintained when multiple stimuli of distinct value compete (seethe results of the target-and-distractors setup in section 3.2).Yet, saccade accuracy degrades when multiples stimuli of simi-lar value compete, with more average saccades being produced,and accurate saccades being more dispersed around their cho-sen stimulus. The absence of a Cerebellum module in the modelcould link this to the observation that cerebellar inactivationleads to a degradation of saccade accuracy (Iwamoto and Yoshida(2002)). The addition of a Cerebellum module to the model,that would receive an efferent copy of the SC output to the SBGand correct it on-the-fly would increase saccade accuracy byallowing minute modifications of saccade trajectory when thesaccade endpoint proposed by the SC is outside the confidenceinterval of the selected stimulus by a defined margin (whichwould have to be tuned so as to still allow for average saccades,and not suppress them completely).Another feature lacking in the model is the implementa-tion of a mechanism modeling the variation of saccade durationwith target eccentricity, in order to respect the parameters of themain sequence of in-vivo saccades (as defined by Bahill et al.(1975)). Such a mechanism could be based on the proposalsof Goossens and van Opstal (2006); van Opstal and Goossens(2008), which modulates the maximal activity of neurons in22he SC motor map according to the metrics of the saccade theycode.Several models of selection use a principle of stochasticgated accumulators very similar to our model (Purcell et al.(2010); Schall et al. (2011); Purcell et al. (2012) amongst oth-ers) applied in the context of the cortical loops of the saccadicsystem, but do not model the source of the thresholds they useto gate the selection process. Nevertheless, they propose var-ious operators for this gating mechanism, amongst which theBG. Our model developps this proposal by providing a biolog-ically compatible loop between the SC and its evidence inte-grators, and the BG which inhibitory output dynamically reactswith the accumulation of evidence in order to gate the winningand losing stimuli. The use of BG output in order to gate theSC activity for the selection was already proposed by other SCmodels, such as Arai and Keller (2004, 2005), but these modelsused only a static gating output from the SNr to the SC mapswhich, contrary to our model, is not produced by a real-timedisinhibition process in the BG fed by the SC maps, and there-fore is not able to take into account the effects of any visualinput variation on selection.Furthermore, both cortical (Purcell et al. (2010); Schall et al.(2011); Purcell et al. (2012)) and subcortical (Arai and Keller(2004, 2005), but also the stochastic accumulator model of theSC by Ludwig et al. (2007)) models of saccade target selectionimplement the minimal amout of neurons needed to obtain se-lection, often in the form of two layers (or even just two setsof neurons, with only one neuron for each competing stimulusin each set), a visual one serving as an input-receiver that oper-ates the race to selection, and a motor one where the movementcommand is generated. Our model propose a more accurate (al-though not complete, as exposed before) representation of theanatomical constaints of the biological structures involved inselection (the SC and its connections to the BG), with severalnew loops and intermediate layers in the SC for which we pro-pose roles and biological substrates not previously accountedfor, and with a biologically plausible generation of the motor command by a dedicated structure (the SBG).Lastly, the cortical models of saccade target selection canonly account for some of the saccade timings observed in-vivo:the latency of the signal processed through the visual cortex tothe FEF being on par with the latency of the express saccades,the selection processes for these saccades cannot be explainedby cortical models. The saccades latencies generated by ourmodel range from to ms in cases where no selection oran easy selection is made, to more than ms in cases wheredifficult selection must be resolved before initiating movement.This first range of simulated latencies is consistent with theexpress saccades observed in-vivo (Fischer and Boch (1983)),even though the model does not feature the fixation cells in therostral SC commonly associated with the production of expresssaccades Munoz and Wurtz (1992, 1993a,b). The second rangeof simulated latencies reaches values where one would expectthe intervention of the cortical actors of the saccadic system -most notably the FEF, which are not modelled here, and there-fore bridge the gap between purely subcortical and purely cor-tical models.This addition could modifiy the results of selection by mod-ulating the values of the various stimuli represented in the SCmaps (for example favouring the selection of a color over thelocation of a specific set of coordinates in the visual field), fea-tures and problems partially addressed under the more specificlight of reinforcement learning by the model of N’Guyen et al.(2014) Acknowledgments
This work was partially funded by the HABOT project fromthe Emergence(s) program of the Ville de Paris (France).
Appendix A. Calculation of the Retinal input to the Visualmap and procedure of gluing
Wathever the specific conditions of any given task protocol,the retinal input fed to the SC Visual Map is calculated as such:23 omplete SC Input Map
Left Visualhemifield Right Visualhemifield
Temp1 L Temp1 RTemporary map 1Gluing R = 0%Gluing L = 100% f Temporary map 2 Temp2 RGluing R = 80% f N3N1N2 N4
Gluing L = 20%Fusion of all temporary mapsTarget 1
Azimuth
Temp2 LTarget 2
Elevation
Figure A.15:
Gluing scheme for the projection of the Retinal input to theVisual map.
The green stimulus is far enough from the vertical axis separatingthe two colliculi to be represented on only one collicular map; the red stimu-lus is close enough from the vertical axis to be represented on both collicularmaps, so each of these representations has to be weighted to give both greenand red targets the same total activity. The total activity generated within theboundaries of the Visual field in the left collicular map (hatched colored area inmap Temp2L, containing neuron N2) is inversely modulated by the proportionof the total activity generated in the contralateral map within the boundaries ofthe Visual field (hatched colored area in map Temp2R), and vice-versa for theactivity generated in the right collicular map. Therefore, the complete Visualinput to the SC has a stronger component for the red target in the right colliculusthan in the left colliculus
1. Each point P i presented in the visual field is character-ized by its position in azimuth and elevation [ az i , el i andits value val i . Its position is translated into coordinatesthat will represent the position [ x i , y i ] in the SC maps asper equation (2). Fig. A.15 features two targets, one nearthe vertical axis (red dot) and one away from it (greendot).2. A temporary retinotopic map temp i is created for eachpoint P i , which activity is represented by a 2D-Gaussianwith standard deviation σ = 2 . neurons ( . mm ina Monkey SC), centered on the neuron at coordinates [ x i , y i ] in our discretized maps , and of maximal heightequal to sal i .(a) If P i is far enough from the vertical axis to be rep-resented by a Gaussian fully located in one of thecolliculi (the one contralateral to the visual hemi-field where P i lies), nothing special occurs (cf. thegreen target in fig. A.15, which is fully located inthe right visual hemifield, and whose representationin the retinal input is restricted to the left SC).(b) If P i is close enough to vertical axis (cf. the redtarget in fig. A.15), its Gaussian representation maycross the boundaries of the SC contralateral to thevisual hemifield where P i lies (cf. red target repre-sentation in temporary map Temp2 R in fig.A.15).In such cases, another gaussian is calculated in or-der to represent the portion of the activity generatedby this target in the other SC (cf. red target repre-sentation in temporary map Temp2 L in fig. A.15).These two Gaussians have to be weighted down inorder to keep the total activity generated by the tar-get constant, whether it is represented by one gaus-sian in one colliculus, or two gaussians in the twocolliculi. Thus, the maximum level of activity ofeach gaussian is modulated by a gluing factor calcu-lated by the transfer function described in equationA.1. The gluing for each Gaussian is linked to thedifference between the summed activity generatedby the target within the boundaries of the collicu-lus contralateral to the Gaussian in question, andthe summed activity generated by the target withinthe colliculus ipsilateral to the Gaussian in ques-tion. This solution is similar to the motor gluingproposed in Tabareau et al. (2007).3. Once temporary maps have been calculated for all thepoints displayed in the visual field, all temporary mapsare then merged in order to produce one single and com-plete map of all targets, that will serve as the input to theSC Visual map.24 luing ipsi = 1 − (A.1)
11 + exp(0 . × min ( ϕ, ( I totcontra − I totipsi )) With
Gluing ipsi the gluing factor to be applied to the por-tion I totipsi of the representation of any point of the Visual fieldspecific to the ipsilateral colliculus, I totcontra the equivalent of I totipsi for the contralateral colliculus, and ϕ a minimal thresholdfor the difference between the two summed activities. Appendix B. Model parameters
N bCell ω DecInt N oise . N bChan ω LLBDec . T IntBG . A π/ ω MotDec T DecBG . B . ω MotOP N ε − τ ms ω SumMot . M ax . τ small ms ω V isSum . ϕ L . ω IntSum . σ . G ω MotSum ω IntV is . E LLB . Table B.3:
Parameters of the SC model E OP N . ω MNBN . ω θ acc MN ω BNOP N ω T NBN . ω θ acc θ vit . ω OP NLLB E T N . ω θ acc θ pos τ small ms ω MNT N Table B.4:
Parameters of the SBG model
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The superior colliculus: new approachesfor studying sensorimotor integration. CRC Press. chapter The intracollicu-lar neuronal network. pp. 147–158.Purcell, B., Heitz, R., Cohen, J., Schall, J., Logan, G., Palmeri, T., 2010. Neu-rally constrained modeling of perceptual decision making. PsychologicalReview 117, 1113–1143. doi:10.1037/a0020311.Purcell, B., Schall, J., Logan, G., Palmeri, T., 2012. From salienceto saccades: Multiple-alternative gated stochastic accumulator modelof visual search. The Journal of Neuroscience 32(10), 3433–3446.doi:10.1523/JNEUROSCI.4622-11.2012.Ratcliff, R., McKoon, G., 2008. The diffusion decision model: Theory anddata for two-choice decision tasks. Neural Computation 20(4), 873–922.doi:10.1162/neco.2008.12-06-420.Rizzolatti, G., Butchel, H., Camarda, R., Scandolara, C., 1980. Neurons withcomplex visual properties in the superior colliculus of the macaque monkey.Experimental Brain Research 38(1), 37–42. doi:10.1007/BF00237928.Schall, J., 2001. Neural basis of deciding, choosing and acting. Nature Reviews2(1), 33–42. doi:10.1038/35049054.Schall, J., Hanes, D., 1998. Neural mechanisms of selection and controlof visually guided eye movements. Neural Networks 11(7), 1241–1251.doi:10.1016/S0893-6080(98)00059-8.Schall, J., Purcell, B., Heitz, R., Logan, G., Palmeri, T., 2011. Neural mecha-nisms of saccade target selection: gated accumulator model of the visual-motor cascade. European Journal of Neuroscience 33(11), 1991–2002.doi:10.1111/j.1460-9568.2011.07715.x.Schiller, P., Sandell, J., Maunsell, J., 1987. The effect of frontal eye field andsuperior colliculus lesions on saccadic latencies in the rhesus monkey. Jour-nal of Neurophysiology 57(4), 1033–1049.Shires, J., Joshi, S., Basso, M., 2010. Shedding new light on the role of the basalganglia-superior colliculus pathway in eye movements. Current opinion inNeurobiology 20(6), 717–725. doi:10.1016/j.conb.2010.08.008.Tabareau, N., Bennequin, D., Berthoz, A., Slotine, J., Girard, B., 2007. Geome-try of the superior colliculus mapping and efficient oculomotor computation.Biological Cybernetics 97(4), 279–292. doi:10.1007/s00422-007-0172-2.Taouali, W., 2012. Modlisation de populations neuronales pour l’intgrationvisuo-motrice : Dynamiques et dcisions. Ph.D. thesis. Universit de LOR-RAINE.Trappenberg, T., Dorris, M., Munoz, D., Klein, R., 2001. A model of saccadeinitiation based on the competitive integration of exogenous and endogenoussignals in the superior colliculus. Journal of Cognitive Neuroscience 13(2),256–271. doi:10.1162/089892901564306.Wang, N., Perkins, E., Zhou, L., Warren, S., May, P., 2013. Anatomical evi-dence that the superior colliculus controls saccades through central mesen-cephalic reticular formation gating of omnipause neuron activity. Journal ofNeuroscience 33(41), 16285–16296.