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A competitive integration model of exogenous and endogenous eye ...

A competitive integration model of exogenous and endogenous eye ...

Biol Cyberncollicular

Biol Cyberncollicular neurons have relatively broad visual fields, which isindeed the case (Dorris et al. 2007; Ottes et al. 1986; Schilleret al. 1980). The spatial shape of the inputs to SC was aGaussian around an SC location receiving maximum inputl max . At location l j of neuron j its strength was:R(l max ,l j ) = e −dist(l max,l j ) 2/ cHere, dist(x,y) is the Euclidian distance in mm (whichtranslates into distance in model neurons divided by 100),and c is a constant equal to 0.245 with unit 1 / mm . This functionis 1 at location l max and drops to zero if l j is far away froml max . Both the equation and the parameter values were takenfrom Trappenberg et al. (2001), who based their Gaussiankernel on electrophysiological recordings.In FEF, all neurons were assumed to correspond to oneposition at which a visual stimulus could appear. R(l max ,l j )was equal to 1 for that neuron, and equal to 0 for all others.1.1.3 Lateral interactionsIn several models, most prominently that of Trappenberget al. (2001), lateral interactions between intermediate layercollicular neurons were assumed to have a Mexican hat function,with short-range excitation and long-range inhibition.Little evidence for such interactions exists, however. In astudy of lateral interactions within the intermediate layers ofthe SC, Munoz and Istvan (1998) found mostly inhibitoryinteractions, with lateral excitation only between contralateralfixation zones (they found two saccade cells excited byipsilateral stimulation, but could not establish a spatial gradientin the interactions). We therefore chose the simplestscheme consistent with the evidence, which is lateral inhibitionbetween all modeled SC neurons with uniform strength.Lateral inhibitory interactions were also assumed in FEF.There is little evidence to date on whether inhibitory interactionsexist in FEF, but such interactions are common in theneocortex (Braitenberg and Schüz 1991).Using t as an index for the time step, lateral inhibition wasa function of the firing rates, f j,t−d , of all other model neuronsin the layer d time steps earlier (d, modeling synapticdelays, was set to 3 ms):i lat = s ∑ j(8)f j,t−d (9)The strength of lateral inhibition s was set to 0.01 in SC,and 0.9 in FEF (as there are many more model neurons inSC lateral inhibition was actually about as strong in SC as inFEF).1.1.4 Saccade initiation and trajectoryTo generate saccades from SC activity, we implemented amodel of the brainstem saccade generator that accounts wellfor both saccade characteristics and brainstem recordings(Becker and Jürgens 1990; Van Gisbergen et al. 1981). TheVan Gisbergen et al. (1981) model describes how a drive fromoutside the brainstem leads to a burst of firing in brainstemburst neurons that, via intermediary steps, control the musclesof the eye. Firing of the burst neurons is reduced via afeedback loop that relies on an efference copy of the saccade,an internal copy of the movement of the eyes computed frombrainstem output. This efference copy is subtracted from theoutside drive, leading to a smaller and smaller drive to theburst neurons the closer the saccade is to completion. Ineffect, burst neurons choke themselves off when the saccadeis near completion.Becker and Jürgens (1990) proposed an extension of thisone-dimensional model to a two-dimensional model that canaccount for oblique saccades. They duplicated the model ofVan Gisbergen et al. (1981) to create a horizontal and verticalcomponents of saccades. These together determine saccadedirection. This is in keeping with evidence that long-leadburst generator neurons (LLBNs), the main recipients of SCinput, code mainly for saccades in the vertical and horizontaldirections (Hepp and Henn 1983), with oblique saccadesresulting from an activation of LLBNs coding for horizontaland for vertical saccades. The horizontal and vertical componentsinteract via inhibitory connections. We implementedthis, using parameter values given by Van Gisbergen et al.(1981) in Table 3 (‘Normal’ line, but with b m = 1300 as intheir Fig. 11D), and the coupling factor c given by Beckerand Jürgens (1990).To link this brainstem model to the SC, we made fiveassumptions:• Activity in the SC sums to a horizontal and a verticaldrives to the brainstem system.• A saccade is generated whenever the drive to either thehorizontal or the vertical EBNs crosses a threshold. 2• The length of the saccade is determined by a weightedaverage of activity in the SC.• Activity within the SC continues to influence the directionof the saccade while it is programmed and executed.• The efference copy is computed in the cerebellum.The first assumption is that activity in the whole SC issummed into a horizontal and a vertical drives. To compute2 The movement may also be achieved by combinations of eye andhead movements. Bergeron et al. (2003); Soetedjo et al. (2002). Weonly implement eye movements because most relevant research hasbeen done on head-fixed participants.123

Biol Cybernthese we made use of the fact that SC neurons were placedon radial axes. All neurons on axis a coded for eye movementsin the same direction a . (there were eight axes percolliculus at regular intervals, so that a could take valuesfrom 0 ◦ to 337.5 ◦ in increments of 22.5 ◦ , where 0 ◦ refers toa straight saccade to the right). We summed activation of allneurons on an axis, then broke it into vertical and horizontalcomponents. These, summed, were the horizontal (H) andvertical (V) drives to brain stem:H = ∑ cos( a ) ∑ f j (10a)aj∈aV = ∑ sin( a ) ∑ f j (10b)aj∈aHere, a ranges over all axes and j over all neurons on axisa.The second assumption follows other models of saccadeinitiation (Findlay and Walker 1999). What seems to determinethe exact onset of a saccade is a sharp decrease in theoutput of brainstem omnipause neurons, which fire at highfrequencies during fixation. What causes this decrease is notknown—one obvious source, a drop in drive from SC fixationneurons, is ruled out by the fact that fixation neuronactivity does not correlate very well with omnipause pausing(Everling et al. 1998), and by evidence that omnipause neuronsare silenced not by a drop in excitation but by glycinemediatedinhibition (Kanda et al. 2007). The saccade-relatedburst in SC is an obvious source of this inhibition, althoughit is as yet unclear how this burst translates into inhibition ofomnipause neurons (for more evidence that the burst in SCis related to saccade initiation see Hanes and Schall 1996;Munoz and Schall 2003; Soetedjo et al. 2002). In the model,a saccade was initiated if either H or V crossed a thresholdof 35 (for rightward or upward movements) or −35 (forleftward or downward movements).The third assumption that a weighted average of activityin the SC determines the length of the saccade was a heuristicone (see Goossens and Van Opstal (2006) for a more sophisticatedproposal for how SC activity could set the length ofsaccades). The model of Becker and Jürgens (1990) takes asinput a desired eye displacement vector with a certain direction,, and length, r. To compute both in a simple way fromactivity in the SC, we computed direction as tan −1 (V/H).The desired length of the saccade, r, was computed from theoptimal eccentricity for each model neuron given its positionon the collicular map. r was set to the average of these eccentricities,weighted by firing rate:∑ [J A √ ]e 2u/B u − 2e u/B u cos(v/B v ) + 1 f jr =∑j f (11)jThe formula between brackets is the inverse given by VanGisbergen et al. (1987)oftheformulaofOttes et al. (1986).It takes as inputs u, the distance in mm of neuron j fromthe rostral pole, and v the distance from the midline of thecolliculus. Constant A, the rostral-to-caudal size of the colliculus,was set to 4 mm, B u to 1.4 mm, and B v to 1.8 mm/ rad(taken from Van Gisbergen et al. 1987).The fourth assumption listed above may be the most controversial:we follow Walton et al. (2005) in assuming thatwhile saccades are executed, the desired saccade vector ascomputed from SC activity is constantly updated. This tookthe form that the desired eye displacement vector (r, ) isupdated on every time step for as long as the saccade lasts.Large shifts in SC activity during a saccade can thus redirectthe saccade to another point. Such an assumption seems justifiedin the face of evidence that saccades can be redirectedmid-flight (Amador et al. 1998), and that saccades can becurved under the influence of for example distractors in thevisual field (Ludwig and Gilchrist 2003; McPeek et al. 2003;Port and Wurtz 2003; Van der Stigchel and Theeuwes 2005;Walker et al. 1997). We do not claim that the SC determinesthe exact trajectory of a saccade, since there is enough evidencethat it does not (Bergeron et al. 2003; Goossens andVan Opstal 2000; Quaia et al. 1998). Instead, we suggestthat through continuous input to the brainstem LLBNs, theSC influences the trajectory. This assumption is consistentwith existing models of the brainstem saccade machinery. Inthose models (Gancarz and Grossberg 1998; Goossens andVan Opstal 2006; Lefevre et al. 1998; Quaia et al. 1998), SCoutputs keep featuring in the computations of the weightedaverage during the generation of a saccade. Thus changesin SC input to the brainstem (Gancarz and Grossberg 1998)or the cerebellum (Lefevre et al. 1998) should result in achanged saccade endpoint.The latency at which the threshold is reached cannot bedirectly compared to saccade latencies, as there is an afferentdelay between burst activity within SC and saccade initiationof about 25 ms (Munoz and Wurtz 1995b). Adding this delayto the time at which the threshold is reached yields latenciesthat can be compared to observed saccade reaction times.1.1.4.1 Role of the cerebellum In Van Gisbergen et al.’s(1981) model an efference copy of the saccade, denoted byE, is subtracted from the drive to the brainstem saccade generator.Such an efference copy was already proposed by vonHelmholtz in the nineteenth century, but it has proven hard tofind neurons computing an efference copy of saccades. Opticanand coworkers (Lefevre et al. 1998; Quaia et al. 1998)have proposed that the cerebellum computes what is in effectan efference copy through a spatial code. Although this ideais by no means uncontroversial, we concur that the cerebellumis a likely location for the efference copy (our fifthassumption above). The cerebellum is assumed to inhibit thebrainstem saccade generator in proportion to E, the part ofthe saccade already made, in this way generating the123

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