Motion processing with wide-ﬁeld neurons in the retino ...

J Comput Neurosci (2010) 28:47–64DOI 10.1007/s10827-009-0186-y

Motion processing with wide-field neuronsin the retino-tecto-rotundal pathway

Babette Dellen · Ralf Wessel · John W. Clark ·Florentin Wörgötter

Received: 3 June 2009 / Revised: 2 July 2009 / Accepted: 9 September 2009 / Published online: 1 October 2009© The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract The retino-tecto-rotundal pathway is themain visual pathway in non-mammalian vertebratesand has been found to be highly involved in visualprocessing. Despite the extensive receptive fields oftectal and rotundal wide-field neurons, pattern dis-crimination tasks suggest a system with high spatialresolution. In this paper, we address the problem ofhow global processing performed by motion-sensitivewide-field neurons can be brought into agreement withthe concept of a local analysis of visual stimuli. Asa solution to this problem, we propose a firing-ratemodel of the retino-tecto-rotundal pathway which de-scribes how spatiotemporal information can be orga-nized and retained by tectal and rotundal wide-fieldneurons while processing Fourier-based motion in ab-sence of periodic receptive-field structures. The model

Action Editor: Jonathan David Victor

B. Dellen (B)Bernstein Center for Computational Neuroscience,Max-Planck-Institute for Dynamics and Self-Organization,Bunsenstrasse 10, 37073 Göttingen, Germanye-mail: [email protected]

B. DellenInstitut de Robòtica i Informàtica Industrial (CSIC-UPC),Llorens i Artigas 4-6, 08028 Barcelona, Spain

R. Wessel · J. W. ClarkPhysics Department, CB 1105, Washington Universityin St. Louis, One Brookings Drive,St. Louis, MO 63130-4899, USA

F. WörgötterBernstein Center for Computational NeuroscienceIII. Physikalisches Institut-Biophysik, Georg-AugustUniversität Göttingen, Friedrich-Hund Platz 1,37077 Göttingen, Germany

incorporates anatomical and electrophysiological ex-perimental data on tectal and rotundal neurons, andthe basic response characteristics of tectal and rotundalneurons to moving stimuli are captured by the modelcells. We show that local velocity estimates may be de-rived from rotundal-cell responses via superposition ina subsequent processing step. Experimentally testablepredictions which are both specific and characteristic tothe model are provided. Thus, a conclusive explanationcan be given of how the retino-tecto-rotundal pathwayenables the animal to detect and localize moving ob-jects or to estimate its self-motion parameters.

Keywords Visual motion · Retino-tecto-rotundalpathway · Optic tectum · Nucleus rotundus ·Optic flow

1 Introduction

Visual motion is one of the most important cues en-abling the animal to interact with its environment ina meaningful way. The computation of local velocityis the basis for such important tasks as the detectionand location of independently moving objects and theestimation of self-motion parameters. Not surprisingly,neurons sensitive to visual motion have been found inmany brain areas (Nakayama 1985; Dellen and Wessel2008). However, computational models of motionprocessing have focused mainly on neurons in the thala-mocortical pathway (Adelson and Bergen 1985; Heeger1988; Wörgötter and Koch 1991; Hennig et al. 2002)and are specific to the small and periodic receptivefields of V1 neurons. The situation is different formotion-sensitive neurons in the retino-tecto-rotundal

48 J Comput Neurosci (2010) 28:47–64

system, which constitutes the main visual pathway innon-mammalian vertebrates. Their response propertiesare generally incompatible with these kinds of modelsdue to their entirely different organization (see Fig. 1).

Retinal axons belonging to the retino-tecto-rotundalpathway project in a precise retinotopical manner tothe optic tectum (TO) (Mpodozis et al. 1996; Kartenet al. 1997). The tectofugal projection arises exclusivelyfrom cells in the tectal layer 13 or stratum griseum cen-trale (SGC) and targets the thalamic nucleus rotundus(Rt) without maintaining a retinotopic organization(Fig. 1(a, b)) (Benowitz and Karten 1976; Engelageand Bischof 1993; Mpodozis et al. 1996; Marin et al.2003). Nevertheless, deficits in pattern-discriminationtasks and the dramatic postlesional threshold vari-ations in acuity measurements point to the exis-tence of a system with high spatial resolution (Hodosand Karten 1966; Hodos 1969; Hodos and Bonbright1974; Mulvanny 1979; Hodos et al. 1984; Macko andHodos 1984; Bessete and Hodos 1989; Watanabe 1991;Güntürkün and Hahmann 1999; Laverghetta andShimizu 1999; Nguyen et al. 2004). Hodos and Karten(1966) conducted behavioral experiments with pigeonswhich were trained to peck one of two discs on which vi-sual stimuli were projected. They observed that lesionsin the nucleus rotundus caused severe deficits in perfor-mance in brightness- and pattern-discrimination tasks.Later on, Laverghetta and Shimizu (1999) showed thatlesions in the nucleus rotundus impaired the detectionof small moving stimuli. Furthermore, lesions in thecaudal ectostriatum, which is the telencephalic target

RGC

tectalSGC-I

RGC

Rt

TORt

(a) (b)

(c) (d)

200 µm

Fig. 1 (a) Retino-tecto-rotundal pathway. (b) Schematic of net-work connectivity of the retino-tecto-rotundal pathway. (c) Re-construction of a tectal neuron (SGC-I) (Luksch et al. 1998).(d) Reconstructed dendritic field of a tectal neuron (Mahani et al.2006)

of the tectofugal visual pathway, also caused severe tomoderate deficits in visual acuity and motion processingtasks (Hodos et al. 1984; Nguyen et al. 2004).

Tectal neurons with somata in tectal layer 13 havelarge circular receptive fields spanning ≈ 10 − 60 de-grees of the visual field (Luksch et al. 1998; Wu et al.2005; Schmidt and Bischof 2001). A reconstruction ofa representative neuron in the optic tectum is shownin Fig. 1(c). The distribution of dendritic endings issparse, such that the summed receptive fields of thedendritic endings fill less than 1% of the total receptivefield (Mahani et al. 2006). The anatomical organizationcorresponds to a spotty receptive-field fine structure(Troje and Frost 1998; Letelier et al. 2002; Mahaniet al. 2006; Schmidt and Bischof 2001) (Fig. 1(d)).Tectal neurons respond vigorously to small movingstimuli, but they are only weakly selective for the ori-entation or direction of motion of the stimulus (Frostand Nakayama 1983; Sun et al. 2002). In the tecto-rotundal projection, rotundal neurons receive inputfrom tectal neurons distributed throughout the entiretectum (Fig. 1(b)), whereby the precise point-to-pointtopography of the retino-tectal projection is completelylost (Benowitz and Karten 1976; Ngo et al. 1994; Kartenet al. 1997; Deng and Rogers 1998; Hellmann andGüntürkün 2001; Marin et al. 2003). The tecto-rotundalprojection is currently interpreted as implementing atransformation from a retinotopically-organized mapinto a functionally-organized map (Hellmann andGüntürkün 2001). The several anatomical subdivisionsof the Rt correlate with neural populations that respondspecifically to different visual modalities, such as two-dimensional motion and in-depth motion (Revzin 1970;Wang and Frost 1990; Wang et al. 1993).

From a theoretical point of view, the following ques-tions arise: (i) How is spatial information organizedin the retino-tecto-rotundal pathway, in view of thesparse but extensive receptive and dendritic fields of theneurons? (ii) How is sensitivity to direction of motiongenerated in the rotundus largely in the absence ofperiodically arranged subunits that account for motionsensitivity in neural models of other brain areas suchas V1 or MT? (iii) How can local velocity estimates beretrieved from motion-sensitive neurons, i.e. rotundalneurons, that have receptive fields spanning up to 120deg of visual angle?

The paper is structured as follows. In Section 2, wepropose a model of the retino-tecto-rotundal pathwayand investigate theoretically the spatial organizationof this pathway. We also propose a neural networkfor the extraction of local-velocity fields from the ro-tundal neural population. In Section 3, we establishby means of computer simulations that the proposed

J Comput Neurosci (2010) 28:47–64 49

model accounts for motion-sensitive responses of neu-rons in the optic tectum and the nucleus rotundus. Wefurther provide experimentally testable predictions anddemonstrate that local-velocity fields can be computedfrom the responses of rotundal model neurons. Finally,in Section 4, the results of the model are discussed anddirections for future research are indicated.

2 Model

In this section, the retino-tectal-rotundal-pathwaymodel is defined and its properties are analyzedtheoretically. The basic organization is described inSection 2.1, then, motion-sensitive mechanisms of ro-tundal model neurons are described in Section 2.2. InSection 2.3 we present a neural model for extractinglocal-velocity fields from rotundal neurons, serving asa proof of concept. In Section 2.4, the connectivityconstraints of the model at the tecto-rotundal pro-jection are explored theoretically. In Sections 2.5–2.7,a preprocessing filter is described, model parametersused in computer simulations are characterized, and anerror measure for performance evaluation is defined,respectively.

2.1 Basic organization of the retino-tecto-rotundalpathway

We propose a firing-rate model of the retino-tecto-rotundal pathway. The tectal and rotundal neurons aremodelled as summing units that integrate the responsesof input neurons, followed by a rectification of thesignal. The receptive/dendritic field properties of tectaland rotundal neurons are constructed from anatomicaland electrophysiological data, and as such provide pa-rameters to the model. Sensitivity to stimulus velocityis introduced into the model by including temporalfilters at the stage of the tecto-rotundal projection inSection 2.2.

Let us model the response to a 2D stimulus I(x, t) ofa tectal neuron i through a continuous, time-dependentfiring-rate function

ritc(t) =

[∫A

Ritc(x)I(x, t)dx

]+

, (1)

where A is the area of the visual field, Ritc(x) is the

receptive field, x = (x, y) is the position vector, and t isthe temporal dimension of the visual input. The visualinput, integrated over the receptive field, is rectified(rectification is being symbolized by [ ]+). Accordingto the rectification model, [a]+ = a if a > τ , and zero

otherwise, where τ is a threshold parameter (Granitet al. 1963). The functional form of the model neuronsis chosen according to standard firing-rate models(Dayan and Abbott 2005). A schematic of the sparseand random connectivity of a tectal neuron is presentedin Fig. 2 (colored in red). The receptive fields of themodel tectal neurons are assumed to have a spotty andrandom fine structure in accordance with experimentaldata. The parameters of the model are specified inSection 2.6 and Fig. 4.

We compute the Fourier transform of the recep-tive field Ri

tc(x) of each model tectal cell i. Multiply-ing the Fourier transforms of the tectal-cell receptivefields of ntc tectal neurons by their respective firing-rate functions and summing, we obtain a representation(or map) of tectal responses in Fourier space

Mtc(k, t) =ntc∑i=1

Ritc(k)ri

tc(t), (2)

where Ritc(k) = F[Ri

tc(x)] is the spatial Fourier transfor-mation of Ri

tc(x) and k = (kx, ky) is a wave vector withspatial frequencies kx and ky. It is important to notethat the map Mtc(k, t) is only implicitly defined throughthe population of tectal neurons. This map serves as amental construct in tracking the functional processingpath of the neural system being modeled. In Section 3.1,

TC

RC

RGC

T IRU

Fig. 2 Schematic of a rotundal model neuron. Tectal cells (TC)in the deep layers of the optic tectum integrate the responses ofretinotopically organized retinal ganglion cells (RGC) in a sparseand random fashion. A representative tectal cell is depicted inred. The spatial integration is followed by a rectification. At thenucleus rotundus, the responses of subpopulations of tectal cellsare summed via intermediate rotundal units (IRU), which couldbe subsets of synapses, or dendritic branches. A representativeintermediate rotundal unit is depicted in green. The rotundal cellresponse is modelled by summing over the responses of the in-termediate rotundal units, followed by a rectification. Sensitivityto stimulus velocity is introduced into the model by temporallyfiltering the subpopulation responses


we will employ computer simulations to show that forlarge numbers of tectal neurons, the approximation

Mtc(k, t) ≈ F [I(x, t)] (3)

is applicable (up to a scaling factor).Each spatial-frequency component of Mtc(k, t) cor-

responds to a subpopulation of tectal neurons, whileeach tectal neuron can be a member of more thanone subpopulation. We assume that rotundal neuronsreceive input from these subpopulations via an interme-diate rotundal unit (depicted in green in Fig. 2). Thesemediating units are merely constructs to schematize thespatial-frequency processing of the rotundal neurons.We model the response of a rotundal neuron j by ran-domly sampling the responses of tectal supopulationsto obtain

r jrc(t) =

[∫A

R jrc(k)Mtc(k, t)dk

]+

, (4)

where R jrc(k) is the spatial Fourier transform of a func-

tion in real space Rrc(x), assuring that r jrc(t) is real

valued. Inserting Eq. (2) in Eq. (4) yields

r jrc(t) =

[∫A

R jrc(k)

(ntc∑i=1

Ritc(k)ri

tc(t)

)dk

]

+(5)

=[

ntc∑i=1

(∫A

R jrc(k)Ri

tc(k)dk)

ritc(t)

]

+(6)

=[

ntc∑i=1

wijritc(t)

]

+(7)

where

wij =∫

AR j

rc(k)Ritc(k)dk (8)

is the connection strength of tectal neuron i and ro-tundal neuron j. Hence, according to our model, theconnectivity pattern at the tecto-rotundal projectionis determined by the receptive-field structure of tectaland rotundal neurons via Eq. (8). Consequently, withinour model, function—expressed in neuronal responseproperties—is directly related to network connectivity.This is a characteristic feature of the model whichmay provide the opportunity in the future to test theunderlying assumptions of the model directly.

The choice of the functional form of the projectioncan be motivated as follows. First, reconstruction ofthe stimulus at the tecto-rotundal projection ensuresthat at each layer of the pathway the stimulus can beencoded using the same number of neurons, as shown inSection 2.4, instead of requiring an increasingly growingnumber of neurons along the pathway. Second, spatialfrequencies are “exposed” at the projection, allowingspatiotemporal filtering to be employed, e.g. to obtainvelocity sensitivity, without requiring periodic receptivefields in accordance with experimental observation.

We reconstruct the visual input through linear super-position of the responses of nrc rotundal neurons, giving

Mrc(x, t) =nrc∑j=1

R jrc(x)r j

rc(t), (9)

where R jrc(x) is the inverse Fourier transform of R j

rc(k).In the Section 3.1 we show by means of computer sim-ulations that for large numbers of tectal and rotundalcells

Mrc(x, t) ≈ I(x, t) (up to a scaling factor). (10)

2.2 Motion processing with tectaland rotundal neurons

So far, we have described how spatial visual data isorganized in our model of the retino-tecto-rotundalpathway. We have defined tectal subpopulations rep-resenting global Fourier components of the visual in-put. However, tectal and rotundal neurons also showselectivity for motion attributes. For example, tectalneurons have been shown to be selective for movingstimuli, while they are only weakly selective for direc-tion of motion (Troje and Frost 1998). This propertyof tectal neurons might have its origin in the synapticproperties of tectal neurons that promote suppressionfor static stimuli (Luksch et al. 2004; Khanbabaie et al.2007), and/or in retinal preprocessing that enhancesstimulus contrast. Hence we preprocess the visual inputwith a high-pass filter (see Section 2.5) to account forspatiotemporal contrast enhancement effects, withoutgoing into more detail here.

In the retino-tecto-rotundal pathway, pronouncedselectivity for direction of motion is observed for ro-tundal neurons. In previous work, it was shown thatmotion processing can be performed in global Fourierspace using spatial-frequency-dependent temporal fil-ters (Dellen et al. 2007). We assume that directionaltemporal filtering takes places at the interface between


tectal and rotundal neurons. In the model, intermediaterotundal units temporally filter the input from the asso-ciated tectal subpopulation. We can write the responseof a rotundal neuron j selective for a velocity v as

r jrc,v(t) =

[∫A

R jrc(k)Mtc,v(k, t)dk

]+

, (11)

with

Mtc,v(k, t) =∫

Tv,k(t − t′)Mtc(k, t′)dt′, (12)

where Tv,k(t) is a temporal filter selective for a tempo-ral frequency ω = k · v, which implements the motionconstraint equation (Adelson and Bergen 1985; Barronet al. 1994). This equation states that all the nonzeropower associated with a translating 2D pattern lies ona plane through the origin in Fourier space, whoseorientation is determined by the pattern velocity vector.The pattern velocity itself can be derived from thenonzero Fourier components by finding the velocity forwhich the constraint lines of the Fourier componentsintersect.

Inserting Eq. (2) in Eq. (11) yields

r jrc,v(t)=

[∫A

R jrc(k)

(∫Tv,k(t − t′)

×(

ntc∑i=1

Ritc(k)ri

tc(t′)

)dt′

)dk

]

+(13)

=[

ntc∑i=1

∫ (∫A

R jrc(k)Tv,k(t−t′)Ri

tc(k)dk)

ritc(t

′)dt′]

+(14)

=ntc∑i=1

∫wij(t − t′)ri

tc(t′)dt′, (15)

where

wij(t) =∫

AR j

rc(k)Tv,k(t)Ritc(k)dk (16)

is a changing effective (functional) connectivity spec-ifying the connection of tectal neuron i and rotundalneuron j.

To date, experimental evidence is insufficient to de-termine the precise properties of the temporal filtering.In this paper, we use the function

Tv,k(t)=∫

exp[iω′t

]exp

[−(ω′ − k · v)2/ξ |k|2] dω′, (17)

if t ≥ 0 and zero otherwise. The function exp[−(ω −k · v)2/ξ |k|2] is a Gaussian of width ξ . This functional

form allows us to adjust how strictly the motion con-straint equation is enforced (by varying ξ). The pa-rameter ξ has dimensions deg2/s2. The temporal filtercontains a spatial-frequency-dependent weighting term,which ensures that the same number of cycles is sam-pled for each spatial frequency.

According to our model, a population of rotundalneurons selective for velocity v segments the part ofthe image moving with the corresponding velocity. Re-construction of the segmented entity in real space canbe achieved by summing over all rotundal responsesselective for the velocity v, giving

Mrc,v(x, t) =∑

j

R jrc(x)r j

rc,v(t). (18)

2.3 Extracting local velocity from rotundal responses

In Section 2.2, we proposed that moving entities canbe segmented by means of velocity-selective rotundalmodel neurons based on Eq. (18). In Section 3.4, we willsupport this proposition with computer simulations. Ofcourse, real image sequences exhibit complex patternsof motion involving accelerated motions, including ro-tation. It is thus desirable to compute a local-velocityfield (or optic-flow field) of the visual input, in whicha velocity estimate is assigned to each point in theimage sequence. In this subsection, we formulate aneural model that permits local-velocity estimates to bederived from populations of velocity-selective rotundalneurons, following up with an algorithmic implementa-tion of the model.

The computation of local velocity requires a jointrepresentation of velocity and position. However, ro-tundal neurons are selective for velocity but not forposition. In the Section 3.1, it is demonstrated thatposition can be retrieved from the rotundal responsesby integrating the responses of certain subpopulations(see also Eqs. (9–10) and Eq. (18)). We assume thatthis operation is implemented by neurons positioned ata higher level in the visual pathway, which we call C1neurons. These neurons might be located in the caudalectostriatum (Gu et al. 2002; Nguyen et al. 2004).

The response of a C1 neuron o jointly selective forvelocity v and position x is defined by

roc1,x,v(t) =

⎡⎣∑

j

R jrc(x)r j

rc,v(t)

⎤⎦

+/−(19)

= [Mrc,v(x, t)

]+/− , (20)

where the idea expressed in Eq. (18) has been adaptedto C1 neurons. Here, we use a full rectification with


[a]+/− = [a]+ if a ≥ 0 and [−a]+ otherwise. Construc-tive interference, measured by taking the absolutevalue, i.e. full rectification, of the superposed weightedresponses of velocity-selective rotundal neurons, leadsto a joint selectivity for position and velocity.

We further implement a spatiotemporal smoothingby combining C1 responses at a secondary stage C2.Thus we obtain

roc2,x,v(t) =

∫Rc2(x − x′)ro

c1,x′,v(t)dx′, (21)

with

Rc2(x′) = exp(−x′2/α2

)(22)

where α is a smoothing parameter with dimension deg.This computational scheme is illustrated in Fig. 3. Thesmoothing operation at the secondary stage could al-ternatively be implemented by horizontal interactionsbetween C1 neurons instead of a two-stage neuralnetwork.

We now can assign a velocity ve(x, t) to each pointof the input sequence I(x, t) by finding the C2 neuronamong the C2 subpopulation selective for x that showsthe strongest response, thereby providing the local ve-locity estimate

ve(x, t) = arg{

maxv

[rc2,x,v

]}. (23)

Note that this step is only introduced to obtain a rep-resentation of local velocity, which allows evaluatingthe performance of the motion-processing system bothqualitatively and quantitively. In the animal brain, simi-lar operations may be implemented in order to generatemotor output.

Fig. 3 Extracting local velocity from rotundal responses. C1neurons integrate the responses of rotundal subpopulations ac-cording to Eq. (20), generating joint selectivity for position andvelocity (depicted in red). C2 neurons average the responses ofC1 neurons of similar selectivity

In the limit of large tectal and rotundal populations,we make the substitutions

Mtc(x, t) = F [I(x, t)] (24)

and

Mrc,v(x, t) = F[Tv,k(t) ∗t F[I(x, t)]] , (25)

in Eqs. (11) and (20), respectively. The symbol ∗t

denotes a temporal convolution and F is the inversespatial Fourier transformation. Keeping in mind thatthe total rectification performed in Eq. (20) is math-ematically equivalent to taking the absolute value ofMrc,v(x, t), Eq. (23) can then be replaced by

ve(x, t)

=arg{

maxv

[∣∣∣F [Tv,k(t) ∗t F[I(x, t)]

]∣∣∣∗x exp(−x2/α2

)]},

(26)

which constitutes an algorithm for the computation ofoptic flow utilizing global spatiotemporal filters. Thesymbol ∗x denotes a spatial convolution. It has beendemonstrated recently that algorithms utilizing globalFourier transformations for velocity estimation are notimpaired by uncertainties arising in algorithms thatutilize a local measurement window (Dellen andWörgötter 2008). Furthermore, a confidence measurecan be defined as

c(x, t)=∣∣∣F [

Tve,k(t) ∗t F[I(x, t)]]∣∣∣ ∗x exp(−x2/α2

), (27)

upon which a threshold τr can be applied to select onlythe more reliable velocity estimates—which is a com-mon strategy in computer vision (Barron et al. 1994).

2.4 Connectivity of the tecto-rotundal projection

The weight (connection strength) of a connection be-tween a tectal and rotundal neuron is defined viaEq. (8). If the weight is zero or sufficiently close tozero, the connection can be considered as non-existing.Hence, according to our model, the number of connec-tions is influenced by the choice of the model parame-ters (see also Section 2.6).

The total number of (non-zero) connections is alsobounded by the upper limit ntc of the sum of Eq. (7).To assure proper transmission of the spatial informa-tion of the stimulus, ntc is assumed to be large, whichimplies that a rotundal neuron makes connections withmany tectal cells. Experiments (in the cerebral cortex)


indicate that a neuron receives input from about 104

neurons (Koch 1999; Pakkenberg et al. 2003). The largedendritic fields of rotundal neurons may suggest aneven higher number for the tecto-rotundal projection.However, we will show in the following that the numberof required connections can be decreased by creatingrotundal subpopulations which receive input from anexclusive subset of tectal neurons, distributed through-out the entire tectum.

Let us assume a rotundal neuron j receives only inputfrom a subpopulation Ps of nc tectal neurons. Hence,we write

r j,src =

[∫R j,s

rc (k)As(k)dk]

+(28)

with

As(k) =∑l∈Ps

Rltc(k)rl

tc. (29)

For large number ntc of tectal neurons, the followingequations hold

F[I(x, t)] =ntc∑i=1

Ritc(k)ri

tc (30)

=nq∑

s=1

∑l∈Ps

Rltc(k)rl

tc (31)

=nq∑

s=1

As(k), (32)

if the subpopulations {P1, P2, ..., Pnq} are mutually ex-clusive and ncnq = ntc is fulfilled. Here, nq denotes thenumber of subpopulations. It can be shown by com-puter simulations that for large numbers of na of rotun-dal neurons, one can write (in some approximation)

As(k) =na∑j=1

R j,src (k)r j,s

rc . (33)

Inserting Eq. (33) in Eq. (32), we obtain

F[I(x, t)] =nq∑

s=1

∑j∈P′

s

R j,src (k)r j,s

rc , (34)

where P′s is a subpopulation of na rotundal neurons

receiving exclusively input from tectal subpopulationPs. Hence, we have a total number of nrc = nanq ro-tundal neurons. According to this derivation, the num-ber of connections can be decreased without impairingfunction if a proportional amount of rotundal subpopu-

lations is created, each subpopulation receiving inputfrom the same group of tectal neurons. Assuming afixed total number of tectal neurons ntc and na constant,then system performance is constant if

ncnrc = nantc = const, (35)

i.e. the number of required tectal neurons within eachsubpopulation can be decreased by increasing the to-tal number of rotundal neurons (by creating moresubpopulations). Using double-tracer injections in thenucleus rotundus, Marin et al. (2003) discovered thatthe tecto-rotundal projection is interdigitating, i.e. in-termingled sets of tectal neurons terminate in separateregions within one subdivision of the nucleus rotun-dus. According to our model, this architecture mighthave its origin in connectivity constraints which haveto be compensated by creating rotundal subpopulations(within a subdivision). These results further demon-strate that incomplete implicit stimulus reconstructionat the tecto-rotundal projection has to be compensatedat the rotundal level by increasing the total numberof neurons if the entire stimulus is to be transmitted,favoring an architecture as proposed by Eq. (8) forefficiency arguments.

2.5 Preprocessing

We preprocess the image sequence with a high-passfilter. This step accounts for change sensitivity at thetectal level. The high-pass (Butterworth) filter is de-fined in Fourier space as

� f = 1/[1 + τ f /

(k2

x + k2y + fsk2

t

)], (36)

where τ f is a threshold parameter with dimensiondeg−2 and kx, ky, and kt are the frequencies of theimage sequence, and fs = s2/deg2. This filter enhancesthe spatiotemporal contrast of the image sequence.

2.6 Tectal and rotundal receptive-field parameters

The parameters of the model are specified by definingthe receptive fields of tectal and rotundal model neu-rons. The receptive field of a tectal neuron is generatedby first creating points in the area A with a uniformprobability of 0.1 points/deg2. This is done by tilingthe potential receptive-field area into small surfaceelements of 1 deg by 1 deg. With probability 0.1, welocate a point (corresponding to a dendritic ending) inthat element. This is repeated for all surface elements.Ideally, this should be done for elements of 1/n deg by1/n deg and probability 0.1/n with n → ∞. In Fig. 4(a),the histogram of nearest neighbor distances between


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

-0.4 -0.2 0 0.2 0.40

2

4

6

8

10

12 x 104

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3 x 104

-1 -0.5 0 0.5 10

2

4

6

8

10

12

14 x 10 4

Num

ber

Num

ber

Num

ber

Num

ber

WeightNearest neighbor distance [deg]

Weight

Weight

(a) (b)

(c)

(d)

(e)

(f)

(g)

(h)

x 104

Fig. 4 Receptive-field properties of model neurons. (a) His-togram of nearest-neighbor distances between points definingthe receptive field of tectal neurons. (b) Histogram of the corre-sponding weight distribution. (c) Spatial pattern of points in realspace used to generate the real-space receptive field of tectal androtundal model neurons (only a quarter of the pattern is shownfor better display). (d) Power spectrum of the spatial Fourierspace of the receptive field shown in (c). (e) Histogram of theweight distribution of tecto-rotundal connections according toEq. (8). (f) Alternative choice of spatial pattern of tectal androtundal receptive fields in real space assuming sparseness inFourier space (only a quarter of the pattern is shown for betterdisplay). (g) Power spectrum of the spatial Fourier space ofthe receptive field shown in (f). (h) Histogram of the weightdistribution of tecto-rotundal connections according to Eq. (8).Most weights are zero

the created points is shown, representing the distribu-tion of dendritic endings of tectal cells. A visual angleof 1 deg corresponds approximately to 100 μm on thetectal surface (Mahani et al. 2006). The resulting distri-bution of points (corresponding to dendritic endings)is thus in a realistic range. Having specified the pointdistribution (or dendritic endings), the correspondingweights to each point are determined by sampling auniform distribution between −0.5 and 0.5, as depicted

in Fig. 4(b). The corresponding spatial pattern is pre-sented in Fig. 4(c). The corresponding energies of thespatial Fourier components are shown in Fig. 4(d).

The receptive fields of rotundal neurons are definedin spatial Fourier space and are generated by takingthe spatial Fourier transform of a point distributionidentical to the one used to generate tectal receptivefields (Fig. 4(c)). To our knowledge, there is no conclu-sive data available about the receptive-field structure ofrotundal neurons. Using Eq. (8), we compute the distri-bution of weights representing the connection strengthbetween a tectal and rotundal neuron. The histogramof the weights is shown in Fig. 4(e). Weights rangebetween −0.5 and 0.5 with a maximum at zero. Hence,for the chosen receptive fields, only a small fractionof connections are strong. This is in accordance withexperimental data (Marin et al. 2003).

Our knowledge about the precise receptive-fieldproperties of tectal neurons is limited. Tectal dendritic-field properties only provide an approximation of tectalreceptive fields (Troje and Frost 1998; Mahani et al.2006; Schmidt and Bischof 2001). However, potentialhidden structures in tectal and rotundal receptive fieldsmay have a considerable impact on tecto-rotundal con-nectivity. Experimental data by Schmidt and Bischof(2001) indicates that the sparse and spotty receptivefields of tectal neurons contain substructures, which arehowever not yet sufficiently quantified to draw furtherconclusions with respect to our model. Receptive-fieldparameters have also been measured for neurons in thesuperior colliculus (Prevost et al. 2007; Mooney et al.1988), the mammalian homolog of the optic tectum,but mainly in superficial layers. Future measurementsof responses of deep tectal neurons to grating stimuli,i.e. spatial frequency, would allow us to further con-strain the model. For example, sparseness in the spatialFourier space (instead of sparseness in real space) oftectal and rotundal receptive fields has a strong impacton the connectivity pattern predicted by the model. InFig. 4(f, g) the corresponding spatial patterns of thereceptive fields are shown in real space and in Fourierspace, respectively. Most weights defining the tecto-rotundal projection are zero, resulting in a sharp peakin the histogram (Fig. 4(h)).

It is important to keep in mind that our modelof the retino-tecto-rotundal pathway is completely de-fined through Eqs. (1)–(18). Experimentally measur-able quantities, such as the receptive-field structureof tectal and rotundal neurons, are parameters to themodel that can be adjusted according to current knowl-edge. Predictions of the model such as the distributionof weights of the tecto-rotundal connection are neces-sarily influenced by the parameter values.


2.7 Error measurements

When the true velocities of a sequence of images ofsizes m×n are given, error measures can be computedto quantify the performance of the algorithm. Accord-ing to Barron et al. (1994), the angular error is defined as

E =∑

x

E(x)/nm, (37)

with

E(x) = arccos{(|v(x) · ve(x)|2+1

)−1/2

/[(|v(x)|2+1

) · (|ve(x)|2+1)]−1/2

}, (38)

where ve is the estimated velocity and v is the truevelocity.

3 Results

In this section, response properties of tectal and rotun-dal model neurons are computed using computer sim-ulations using the model of the retino-tecto-rotundalpathway introduced in Section 2. We choose parame-ters τ f = 0.2 deg2 and ξ = 0.6 frame2/deg2 for the spa-tiotemporal filters of the retino-tecto-rotundal pathway(see Eqs. (17) and (36)). The 2D spatial functions,defining the receptive fields of tectal and rotundal neu-rons, Rtc(x) and Rrc(x), respectively, are specified inSection 2.6. To obtain an intensity distribution witha zero mean for each image sequence, we subtractthe mean intensity value from I(x, t). The rectificationthreshold τ (see Eq. (1)) is set to zero. We further showin Section 3.6 that local velocity fields can be computedfrom the output of model of the retino-tecto-rotundalpathway using the computational scheme derived inSection 2.3. For the C2 neurons, we choose a smooth-ing parameter α = 10 deg (see Eqs. (26) and (22)).The model parameters are not altered unless indicatedotherwise.

3.1 Organization of spatial information in the tectaland rotundal cell populations

Computer simulations are used to predict the responsesof the model tectal and rotundal cell populations. Firstwe demonstrate that the tectal responses approximatethe Fourier transform of an input image of size 95 × 128pixels2, assuming that each pixel corresponds to 1 deg ofvisual angle. For computational reasons, the resolutionis constrained to a maximum of 1 deg, which is belowthe resolution of the avian visual system. The receptive

field of each model tectal cell is generated accordingto Section 2.6 assuming sparseness in real space (seeFig. 5(a–c)). For practicality, the receptive field size waschosen such that the visual field is tiled into equallylarge parts. This detail has no effect on the results of thecomputation. For the tectal-cell populations, we com-puted the representation of the population response inFourier space (Eq. (2)) and calculated the correlationcoefficient of the Fourier transform F[I(x)] of the inputimage and Mtc(k). In Fig. 5(a), the correlation coef-ficient is plotted as function of the number of tectalcells. For large numbers of cells, a correlation coeffi-cient above 0.95 is obtained. The squared correlationcoefficient measures the correlation (or amount of vari-ance reconstructed) and has a value larger than 0.9,demonstrating that the original image can be largelyretrieved from the tectal populations. The input image,shown as the left panel of the inset in Fig. 5(a) andalso shown in Fig. 10(b), is a snapshot of the so-calledtaxi sequence. The reconstructed image from the tectalresponses is juxtaposed as the right panel of the inset.

The receptive field of rotundal neurons is generatedaccording to Section 2.6 assuming sparseness in realspace (see Fig. 4(c–e)) having a size of 95 × 128 deg.The representation of a representative rotundal recep-tive field in Fourier space is shown in Fig. 4(d). Wecan now reconstruct the visual input from the rotundalresponses by computing Mrc(x, t), but replacing Mtc(k)

by F[I(x)]. The correlation coefficient of the inputimage I(x) and Mrc(x) is then calculated and plotted inFig. 5(b) as a function of the number of rotundal cells.For large numbers of cells, a correlation coefficientabove 0.9 is obtained, demonstrating that the origi-nal image can be largely retrieved from the rotundalpopulations.

We investigate how noise in the connectivity be-tween tectal and rotundal neurons affects the qualityof reconstruction. According to Eq. (2), each tectalcell contributes to each supopulation k with a weightRtc(k). To each of these weights, we add a noise termfnnr|Rtc(k)| where fn is the noise factor and nr is ran-dom number drawn from a Gaussian distribution witha standard deviation of 1. The correlation coefficientof the reconstructed image and the original are plottedin Fig. 5(c). Noise in the connectivity impairs image-reconstruction performance for noise terms being inthe range of the average absolute connection strength,approximated by |Rtc(k)|. However, for noise level of50% of the average absolute connection strength, i.e.fn = 0.5, performance drops only by about 10%. Fora noise level of 400%, i.e. fn = 4, performance de-creases by about 40%. These values suggest robustnessand graceful degradation of performance with network


0 0.5 1 1.5 2 2.5 3x 105

0.2

0.4

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1

0 2 4 6 8 10 12x 10 5

0

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0.8C

orre

latio

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ient

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orre

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effic

ient

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(b)

0 2 4 60

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0.6

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1

fn = 0.5

fn = 2

fn = 4

fn = 0

Cor

rela

tion

coef

ficie

nt

(c)

Fig. 5 Stimulus reconstruction from tectal and rotundal popula-tion responses. (a) Correlation coefficient of the Fourier spacerepresentation of the tectal neuronal populations (Eq. (2)) andFourier transform of the visual input plotted for increasing num-bers of tectal neurons. For large numbers of cells, a correlationcoefficient above 0.95 is obtained, corresponding to a correlationlarger than 0.9 (see text). The reconstructed image from the tectalpopulation, presented in the inset (right panel), may be comparedwith the original image (left panel of inset). (b) Correlation coef-ficient of the real-space representation of the rotundal neuronalpopulations (Eq. (6)) and the visual input, plotted against thenumber of rotundal neurons. For large numbers, we achieve cor-relation coefficients larger than 0.95. (c) Correlation coefficientof the original and the reconstructed image for different levels ofnoise fn in the connectivity (see text)

damage, which is typical for coarse-coding schemes(Hinton et al. 1986).

3.2 Response properties of tectal neurons

We also simulate the response of tectal neurons tovarious stimulus attributes, i.e. spatial frequency, orien-tation, and speed. The same parameters are used for thetectal neuron as in the previous subsection. The imagesequence contains 20 frames of size 100 × 100 deg2. Wechoose the size of tectal receptive fields to be 50 × 50deg2. The remaining receptive field parameters are notaltered. The size of the rotundal-cell receptive fields ischosen as 100 × 100 deg2. In the following, velocitiesare defined in deg/frame for convenience, and typicalspeeds of objects in this paper are chosen to be in therange of 0 to 5 deg/frame. The frame rate of the motionsequence allows translating the velocity units to deg/s.Typical frame rates are 24 frames/s. For example, aspeed of 1 deg/frame corresponds to a speed of 24 deg/s.

First, we investigate the response of model tectalneurons to a grating that moves in the x direction with1 deg/frame as a function of the spatial frequency of thegrating. While the spatial-frequency tuning curves ofindividual model neurons exhibit multiple but randompeaks (Fig. 6(a), left panel, blue lines). The mean re-sponse of the population (averaged over 200 neurons)shows a slight preference for high spatial frequencies(Fig. 6b, thick red line). This result can be attributedto the preprocessing of the image sequence with ahigh-pass filter and conforms with the observation thattectal responses are suppressed by static stimuli. Wecalculate the number of peaks and the correspondingpeak heights (above the mean) of the individual tuningcurves. The histogram of the number of peaks and peakheights are plotted in the right upper and lower panel.Multiple peaks are commonly observed.

Next, we calculate the response of tectal neurons to agrating of a spatial frequency k = 0.2 cycles/deg movingwith 1 deg/frame for different orientations of the grat-ing. While the tuning curves of individual neurons showmultiple peaks (Fig. 6(b), blue lines), the averagedtuning curve does not show selectivity for orientation(Fig. 6(b), thick red line). The histogram of the numberof peaks and the peak heights of the correspondingtuning curves are given in the right upper and lowerpanel, respectively.

Lastly, we compute the response of tectal modelneurons to a solid square of size 10 × 10 deg2 movingalong the x axis for different constant velocities. Indi-vidual tuning curves show sensitivity to stimulus speedand occasionally weak directional selectivity (Fig. 6(c),blue lines). The averaged tuning curve exhibits strong


Fig. 6 Response propertiesof tectal model neurons. (a)Spatial frequency tuning and(b) orientation tuning eachsummed over a time intervalof 20 frames and plotted for 9neurons (blue curves). Theaverage tuning curveobtained from 200 modelcells is plotted in thick red.The correspondinghistograms of peak numberand peak height (abovemean) are depicted in theupper and lower right panels,respectively. (c) Responses toa square moving along thex-axis with different constantspeed plotted as a function ofx-velocity for 9 modelneurons in blue and averagetuning curve from 200 modelin thick red. The histogram ofthe speed tuning index isdepicted in the right panel

-200 -100 0 100 2000

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ts]

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(c)

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]

(a)

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sensitivity to stimulus speed, but does not show anydirectional selectivity (Fig. 6(c), thick red line). Thisresult is in accordance with experimental data (Trojeand Frost 1998). We calculated the speed tuning indexof a tectal neuron by computing [r1 − r0]/r1 where r1

and r0 are the respective values of the tuning curve atvx =1 deg/frame and vx =0 deg/frame. All speed tuningindices smaller than zero were set to zero. The resulting

histogram of the speed tuning index is presented in theright panel of Fig. 6(c). Most neurons of the populationhave a speed-tuning index close to 1.

To our knowledge, no experimental tuning curvesto frequency and orientation of spatial Fourier com-ponents are available for deep tectal neurons of thetectofugal pathway. Spatial-frequency tuning has beenmeasured only for neurons in superficial layers of the


optic tectum. However, there is experimental evidencethat deep tectal neurons are only weakly selective fordirection of motion, but strongly tuned to speed (Trojeand Frost 1998; Letelier et al. 2002).

3.3 Response properties of rotundalmodel neurons

Computer simulation is applied to study the responseof a motion-sensitive rotundal model neuron selectivefor a velocity v = (1, 0) deg/frame to a solid square ofsize 10 × 10 deg2 moving along the x axis at differ-ent constant velocities. The parameters of the tectal

neurons are chosen as in the previous subsection. Therotundal-cell receptive fields are again of size 100 × 100deg2. For the temporal filter of the rotundal neurons,we take ξ = 0.6 frame2/deg2. The individual velocitytuning curves are presented in Fig. 7(a) (blue lines).Most of the tuning curves exhibit a pronounced sen-sitivity to direction of motion. This is reflected in theaverage tuning curve (based on 200 neurons), depictedin thick red. Directional selectivity has been observedfor a certain class of rotundal neurons (Wang and Frost1990; Wang et al. 1993).

For comparison, we compute the tuning curves forrotundal neurons without including the preprocessing

Fig. 7 Response propertiesof rotundal model neurons.(a) Velocity tuning curvesof individual rotundal modelneurons, selective for avelocity in the x-directionof 1 deg/frame, are depictedin blue. The average tuningcurve is plotted in thick red.Here, preprocessing isincluded in the model. (b)Same as a, but with nopreprocessing.(c) Histogram of the peakheight for the tuning curvesof rotundal neurons (withpreprocessing).(d) Histogram of the peaknumber obtained from thetuning curves of rotundalneurons (with preprocessing).(e) Histogram of thedirection-tuning indexobtained from the tuningcurves of rotundal neurons(with preprocessing)

Velocity in x-direction [deg/f] Velocity in x-direction [deg/f]

Res

pons

e [a

rb. u

nits

]

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pons

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rb. u

nits

]

(a) (b)

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of n

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of n

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(e)


step at the tectal level (Fig. 7(b)). The resulting tuningcurves show strong selectivity for direction of motion,demonstrating that this rotundal property is not a con-sequence of the preprocessing operation. The selec-tivity for direction stems instead from the temporalfiltering taking place at the tecto-rotundal projection.

To date, there is not sufficient data available to com-pare our results quantitatively to real velocity-tuningcurves. Experiments however have shown that neuronsin the ventral subdivision of the nucleus rotundus aresensitive to the direction of motion (Wang et al. 1993;Wang and Frost 1990).

For the rotundal population of Fig. 7(a), we com-puted the peak height and the number of peaks of thetuning curves. The histograms are plotted in Fig. 7(c, d).Most tuning curves exhibit only a single peak, how-ever double peaks are observed as well. We furthercalculated the direction tuning index of each rotundalneuron as [r1 − r−1]/r1 where r1 and r−1 are the respec-tive values of the tuning curve at vx = 1 and vx = −1deg/frame. Most neurons of the population are direc-tionally selective, with a direction tuning index largerthan 0.5 deg/frame (see Fig. 7(e)). A population of12 × 104 tectal neurons was chosen for this simulation.

3.4 Motion segmentation through motion-sensitiverotundal subpopulations

Simulations are carried out to explore the responsesof rotundal model neurons, that are selective for avelocity v=(1, 0) deg/frame, to a stimulus consisting ofa camouflaged random-dot square moving with a speedof 1 deg/frame to the right, in front of a random-dotbackground pattern moving with a speed of 1 deg/frameto the left. A schematic of the stimulus is given inFig. 8(a). Motion segmentation in the proposed modelis simulated by computing Mrc,v(x, t) for the input im-age sequence based on a tectal cell population of 12 ×104 neurons and a rotundal cell population of 8 × 104

neurons. The response Mr,v(x, t) of the rotundal popu-lation in real space is depicted in Fig. 8(b). The movingcamouflaged random-dot square has been segmentedwith sharp boundaries and precise spatiotemporaldetail.

3.5 Local velocity computation and the apertureproblem

We first investigate the response pattern of a C2 modelneuron population to a moving square stimulus of 3 × 3deg size. The mean response values of C2 model neu-rons at the time point where the stimulus crosses theirreceptive-field center is plotted as a function of the

(a)

(b)

Fig. 8 Motion segmentation. (a) Input image sequence showinga camouflaged random-dot square moving to the right in front ofa random-dot background that moves to the left. (b) Segmentedrandom-dot square reconstructed from a velocity-selective rotun-dal subpopulation response

preferred velocity of the C2 model neurons in Fig. 9(a).The curve peaks at vx = 2 deg/frame, which is thevelocity of the stimulus. The position tuning curve ofa C2 neuron to the same stimulus with respect to itsreceptive-field center is shown in Fig. 9(b). The re-sponsive region of the C2 neurons is approximately20 deg in diameter. If using a larger stimulus of 12 × 12deg size, population tuning becomes less pronounced(see Fig. 9(c), solid line), however, the correct velocityestimate can still be extracted from the population


Fig. 9 Response properties of C2 neurons. (a) Mean responsevalue of C2 model neurons as a function of their preferredvelocity to a small square of 3 × 3 deg size moving along withvx = 2 deg/frame in the x-direction. The correct stimulus velocitycan be derived form the population response by finding the C2model neuron with the largest response. (b) Mean response valueof a C2 model neuron as a function of the position (with respectto the receptive-field center) of the same stimulus. (c) For alarger square of 12 × 12 deg size, the mean response value ofthe C2 model neurons still peaks at the correct stimulus velocity(solid line), even though the square approximates the size of theresponsive area of the C2 model neurons (see b). If the area out-side the responsive region of the model neuron of approximately20 deg in diameter is masked, the population response sug-gests an incorrect stimulus velocity of vx = −2 deg/frame in thex-direction (dashed line)

response. If the area outside the responsive region ofthe model neuron is masked, the responses of the C2neuron suggest an incorrect stimulus velocity of vx =−2 deg/frame in the x-direction (see Fig. 9(c), dashedline). This demonstrates that distributed global process-

ing of velocity and subsequent reconstruction of posi-tion allow velocities to be reconstructed locally with-out introducing an aperture. The parameter choicesof the C2 model neurons have been τ f = 0.2 deg2,ξ = 0.6 frame2/deg2, and α = 3 deg.

3.6 Local-velocity fields from rotundal responses

In this section, we demonstrate that the proposedmodel enables the animal to compute optic flow fromreal image sequences. Using the algorithmic implemen-tation of Eq. (26), we compute the local-velocity fieldsof four image sequences for parameter choices τ f =0.2 deg2, ξ = 0.6 frame2/deg2, α = 10 deg, and τr = 5.For practical implementation reasons, the temporal fil-ter here has been chosen to be non-causal, which isexpected to have only a minor effect on the results.The image sequences selected are benchmark exam-ples commonly used in the machine-vision community(Barron et al. 1994).

In the SRI Sequence, a camera moves parallel to theground plane along the x-axis (horizontal direction) infront of several trees. The velocities are as large as twopixels/frame, and the sequence contains 20 frames. Asnapshot is depicted in the left panel of Fig. 10(a), whilethe right panel shows the estimated velocity field in thex-direction. The optic-flow field captures the predomi-nant velocity pattern of the sequence and segments theimages into foreground and background.

In the Hamburg taxi sequence, a street scene isshown with four moving objects: a taxi turning thecorner, a car in the lower left driving from the left tothe right, a van in the lower right driving from the rightto the left, and a person walking in upper left. Imagespeeds of the four moving objects are approximately1.0, 3.0, 3.0, and 0.3 pixels/frame, respectively. The se-quence contains 20 frames. A snapshot is shown in theright panel of Fig. 10(b). Adopting the same parametersas for the SRI-sequence, our algorithm returns an optic-flow field in which the moving objects are clearly visible(Fig. 10(b), right panel). The velocity estimates areclose to the true velocities of the objects.

The translating and diverging tree sequence are cre-ated by moving a camera sideways and towards animage of a tree, respectively. The algorithm returns flowfields with 97% density and angular errors of 1.19 dega

for the translating-tree sequence and 3.83 dega for thediverging-tree sequence (Fig. 10(c)).

We also apply the algorithm to the well-knownYosemite sequence (Fig. 10(d)). Each frame of theYosemite sequence has been generated by mappingaerial photography onto a digital-terrain map. Speedsin the lower left corner go up to four pixels/frame,


Fig. 10 Estimatedlocal-velocity fields. (a) SRIsequence. (b) Hamburg taxisequence. (c) Translating/diverging tree sequence.(d) Yosemite sequence

0

0.5

1

1.5

2(a)

(b)

(c)

(d)

while the clouds translate with about one pixel/frameto the right. The algorithm achieves an angular error of7.73 dega everywhere except for the area of the clouds.In the cloud area, the true motion is unknown, since theclouds are undergoing Brownian motion and changingshape.

4 Discussion

We have presented a firing-rate model of the retino-tecto-rotundal pathway for the processing of Fourier-based motion. In this model, responses of tectalneurons are obtained by integrating the visual space


over the receptive field of the neuron, which, in ac-cordance with experimental data, is assumed to con-sist of random dots sparsely distributed over a largearea of the visual space. We have established that de-spite of the lack of periodic structures, motion signalscan be generated, giving rise to directionally-selectiveresponses of neurons in the nucleus rotundus. Usingbiologically plausible model parameters, a characteris-tic distribution of direction-tuning indices for the ro-tundal population is predicted. Furthermore, spatialinformation is retained in the population response andcan be retrieved at any stage of the processing stream.As a proof of concept, we showed that local velocityestimates may be derived from responses of the rotun-dal model neuron population through superposition ofrotundal responses by a neural network. This includesthe prediction of neurons jointly selective for positionand velocity, potentially located in the caudal ectostria-tum (Nguyen et al. 2004). Motion-sensitive neurons inthe caudal ectostriatum receive input from the nucleusrotundus and have large receptive fields (Nguyen et al.2004; Gu et al. 2002). The emergence of so-called hotspots within the excitatory receptive field of ectostriatalneurons might indicate the onset of position reconstruc-tion (Gu et al. 2002). Using an algorithmic equivalent ofthe model, local-velocity fields of four real sequencesfeaturing complex motions have been computed for afixed set of parameters, demonstrating the feasibility ofthe approach.

Considering the large receptive fields of tectaland rotundal neurons, a distributed representationof spatiotemporal information is considered to be aplausible choice to describe motion processing inthe retino-tecto-rotundal pathway. The model resultsdemonstrate that high spatial acuity is indeed in agree-ment with the specific properties of this pathway. Thisis also in agreement with work on coarse coding byHinton et al. (1986), who showed that a stimulus can berepresented more accurately by a collection of neuronswith broad response functions than by a collection ofneurons with more finely-tuned response functions.

So far, we confined the analysis and modelingto feedforward processes. However, neurons in thetecto-rotundal pathway exhibit contextual effects. Forexample, tectal responses to a moving stimulus aresuppressed by a background moving in the same direc-tion, and vice versa (Frost and Nakayama 1983; Sunet al. 2002). Contextual influences are thought to bemediated by lateral connections or through feedbackfrom brain areas at a later processing stage (Nakayama1985; Dellen and Wessel 2008). Our model of theretino-tecto-rotundal pathway might be extended toallow for such interactions. In the future, we aim to

investigate the role of isthmo-tectal feedback on mo-tion processing (Meyer et al. 2008). Further, there isevidence that certain classes of neurons in the nucleusrotundus compute various optical variables of loomingobjects (Wang et al. 1993; Sun and Frost 1998). Theneuronal responses of these neurons could be modelledadequately by sampling over tectal subpopulations thatencode radial spatial frequencies at the tecto-rotundalprojection.

Representing the stimulus by distributed representa-tions and performing motion processing in this globalspace offers specific advantages compared to local rep-resentations, such as representational efficiency, i.e.more entities can be encoded by the same numberof neurons, and graceful degradation of performancein response to network damage or noise. For mo-tion processing tasks, distributed representations havethe advantage that local velocity responses obtainedvia superposition of responses of wide-field neuronsare not constrained by the (measurement-window-induced) aperture problem (see Fig. 9), which is typ-ically introduced when utilizing a small measurementwindow, i.e. small receptive fields. Hence, the pro-posed model and the respective optic-flow algorithmare fundamentally different from theories of motionprocessing culminating in the computation of optic flowthat have been developed for the geniculocortical path-way in mammals (Adelson and Bergen 1985; Heeger1988). In these models, velocity estimates are derivedfrom simple and complex cells that feature periodicallyarranged on and off subunits, with each cell coveringa local patch of the visual space. In our model, localvelocity estimates arise from constructive interferenceeffects in distributed representations. Constructive in-terference allows joint selectivity for position and ve-locity to arise, even though at previous processingsteps velocity-sensitive rotundal neurons have receivedinput from the entire visual field. The model showsthat global transformations are not in conflict with thecomputation of local velocities.

Acknowledgements The work has received support from theGerman Ministry for Education and Research (BMBF) viathe Bernstein Center for Computational Neuroscience (BCCN)Göttingen under Grant No. 01GQ0430, the NIH/NEI, ROIEY015678, and the EU Project Drivsco under Contract No.016276-2. JWC acknowledges support from Fundação para aCiência e a Tecnologia of the Portuguese Ministério da Ciência,Tecnologia e Ensino Superior and Fundação Luso-Americana.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.


References

Adelson, E. H., & Bergen, J. R. (1985). Spatiotemporal energymodels for the perception of motion. Journal of the OpticalSociety of America. A, 2, 284–299.

Barron, J. L., Fleet, D. J., Beauchemin, S., & Burkitt, T.(1994). Performance of optical flow techniques. InternationalJournal of Computer Vision, 12, 43–77.

Benowitz, L. I., & Karten, H. J. (1976). Organization of thetectofugal visual pathway in the pigeon: A retrogradetransport study. Journal of Comparative Neurology, 167,503–520.

Bessete, B., & Hodos, W. (1989). Intensity, color, and pat-tern discrimination deficits after lesions of the core andthe belt regions of the ectostriatum. Visual Neuroscience, 2,27–34.

Dayan, P., & Abbott, L. (2005). Theoretical neuroscience: Com-putational and mathematical modeling of neural systems.Cambridge: MIT.

Dellen, B., Clark, J., & Wessel, R. (2007). The brain’s view of thenatural world in motion: Computing structure from functionusing directional fourier transformations. International Jour-nal of Modern Physics B, 21, 2493–2504.

Dellen, B., & Wessel, R. (2008). Visual motion detection. InL. Squire, et al. (Ed.), The new encyclopedia of neuroscience(pp. 291–295). Amsterdam: Elsevier.

Dellen, B., & Wörgötter, F. (2008). A local algorithm for the com-putation of optic flow via constructive interference of globalfourier components. In Proceedings of the British machinevision conference 2008 (pp. 795–804).

Deng, C., & Rogers, L. (1998). Organization of the tecto-rotundaland sp/ips rotundal projection in the chick. Journal ofComparative Neurology, 394, 171–185.

Engelage, J., & Bischof, H. (1993). The organization of thetectofugal pathway in birds: A comparative review. InH. Zeigler, & H. Bischof (Eds.), Vision, brain, and behaviorin birds (pp. 137–158). Cambridge: MIT.

Frost, B., & Nakayama, K. (1983). Single visual neurons codeopposing motion independent of direction. Science, 220,744–745.

Granit, R., Kernell, D., & Shortess, G. K. (1963). Quantitativeaspects of repetitive firing of mammalian motorneurons,caused by injected currents. Journal of Neurophysiology(London), 168, 911–931.

Gu, Y., Wang, Y., Zhang, T., & Wang, S. (2002). Stimulus sizeselectivity and receptive field organization of ectostriatalneurons in the pigeon. Journal of Comparative Physiology.A, 188, 173–178.

Güntürkün, O., & Hahmann, U. (1999). Functional subdivisionsof the ascending visual pathways in the pigeon. BehaviouralBrain Research, 98, 193–201.

Heeger, D. (1988). Optical flow using spatiotemporal filters.International Journal of Computer Vision, 1, 279–302.

Hellmann, B., & Güntürkün, O. (2001). Structural organiza-tion of parallel information processing within the tectofugalvisual system of the pigeon. Journal of Comparative Neurol-ogy, 429, 94–112.

Hennig, M., Funke, K., & Wörgötter, F. (2002). The influ-ence of different retinal sub-circuits on the non-linearity ofganglion cell behavior. Journal of Neuroscience, 22, 8726–8738.

Hinton, G. E., McClelland, J. L., & Rumelhart, D. E. (1986).Distributed representations. In D. E. Rumelhart, J. L.McClelland, & the PDP research group (Eds.), Parallel dis-tributed processing: Explorations in the microstructure ofcognition (Vol. 1, pp. 77–109). Cambridge: MIT.

Hodos, W. (1969). Color-discrimination deficits after lesions ofthe nucleus rotundus in pigeons. Brain, Behavior and Evolu-tion, 2, 185–200.

Hodos, W., & Bonbright, J. (1974). Intensity difference thresh-olds in pigeons after lesions of the tectofugal and thalamofu-gal visual pathways. Journal of Comparative & PhysiologicalPsychology, 87, 1013–1031.

Hodos, W., & Karten, H. (1966). Brightness and patterndiscrimination deficits after lesions of nucleus rotundusin the pigeon. Experimental Brain Research, 2, 151–167.

Hodos, W., Macko, K., & Besette, B. (1984). Near-field acuitychanges after visual system lesions in pigeons. II. Telen-cephalon. Behavioural Brain Research, 13, 15–30.

Karten, H., Cox, K., & Mpodozis, J. (1997). Two distinct popu-lations of tectal neurons have unique connections within theretinotectorotundal pathway of the pigeon (Columbia livia).Journal of Comparative Neurology, 387, 449–465.

Khanbabaie, R., Mahani, A., & Wessel, R. (2007). Contextualinteraction of gabaergic circuitry with dynamic synapses.Journal of Neurophysiology, 97(4), 2802–2811.

Koch, C. (1999). Biophysics of computation: Information process-ing in single neurons. New York: Oxford UniversityPress.

Laverghetta, A. V., & Shimizu, T. (1999). Visual discriminationin the pigeon (columba livia): Effects of selective lesions ofthe nucleus rotundus. Neuroreport, 10, 981–985.

Letelier, J., Marin, G., Karten, H., Fredes, F., Sentis, E.,Weber, P., et al. (2002). Tectal ganglion cells in the pi-geon (columbia livia): Microstructure of their motion sen-sitive receptive fields. Society for Neuroscience Abstract, 28,761.17.

Luksch, H., Cox, K., & Karten, H. (1998). Bottlebrush dendriticendings and large dendritic fields: Motion-detecting neuronsin the tectofugal pathway. Journal of Comparative Neurol-ogy, 396, 399–414.

Luksch, H., Khanbabaie, R., & Wessel, R. (2004). Synapticdynamics mediate sensitivity to motion independent of stim-ulus details. Nature Neuroscience, 7(4), 380–388.

Macko, K., & Hodos, W. (1984). Near-field acuity after visualsystem lesions in pigeons. I. Thalamus. Behavioural BrainResearch, 13, 1–14.

Mahani, A., Khanbabaie, R., Luksch, H., & Wessel, R. (2006).Sparse spatial sampling for the computation of motion inmultiple stages. Biological Cybernetics, 10, 1–12.

Marin, G., Letelier, J., Henny, P., Sentis, E., Farfan, G.,Fredes, F., et al. (2003). Spatial organization of the pi-geon tectorotundal pathway: An interdigitating topographicarrangement. Journal of Comparative Neurology, 458(4),361–380.

Meyer, U., Shao, J., Chakrabarty, S., Brandt, S., Luksch,H., & Wessel, R. (2008). Distributed delays stabilizeneural feedback systems. Biological Cybernetics, 99(1), 79–87.

Mooney, R. D., Nikoletseas, M. M., Ruiz, S. A., & Rhoades,R. W. (1988). Receptive-field properties and morphologicalcharacteristics of the superior colliculus neurons that projectto the lateral posterior and dorsal lateral geniculate nucleiin the hamster. Journal of Neurophysiology, 59(4), 1333–1351.

Mpodozis, J., Cox, K., Shimizu, T., Bischof, H., Woodson, W., &Karten, H. (1996). Gabaergic inputs to the nucleus rotundus(pulvinar inferior) of the pigeon (Columbia livia). Journal ofComparative Neurology, 374, 204–222.

Mulvanny, P. (1979). Discrimination of line orientation by vi-sual nuclei. In A. Granda, & J. Maxwell (Eds.), Neural


mechanisms of behavior in the pigeon (pp. 199–222).New York: Plenum.

Nakayama, K. (1985). Biological image motion processing: Areview. Vision Research, 25, 625–660.

Ngo, T., Davies, D., Egedi, G., & Tömböl, T. (1994). Phase-olus lectin anterograde tracing study of the tectorotundalprojections in the domestic chick. Journal of Anatomy, 184,129–136.

Nguyen, A., Spetch, M., Crowder, N., Winship, I., Hurd, P.,& Wylie, D. (2004). A dissociation of motion and spatial-pattern vision in the avian telencephalon: Implications forthe evolution of visual streams. Journal of Neuroscience, 26,4962–4970.

Pakkenberg, B., Pelvig, D., Marner, L., Bundgaard, M. J.,Gundersen, H. J., Nyengaard, J. R., et al. (2003). Aging andthe human neocortex. Experimental Gerontology, 38(1–2),95–99.

Prevost, F., Lepore, F., & Guillemot, J. P. (2007). Spatio-temporal receptive field properties of cells in the rat’s supe-rior colliculus. Brain Research, 1142, 80–81.

Revzin, A. (1970). Some characteristics of wide-field units in thebrain of the pigeon. Brain Research, 2, 264–276.

Schmidt, A., & Bischof, H. J. (2001). Neurons with complex re-ceptive fields in the stratum griseum centrale of the zebrafinch (taeniopygia guffata castanotis gould) optic tectum.Journal of Comparative Physiology. A, 187(11), 913–924.

Sun, H., & Frost, B. (1998). Computation of different optical vari-ables of looming objects in pigeon nucleus rotundus neurons.Nature Neuroscience, 1, 296–303.

Sun, H., Zhao, J., Southall, T., & Xu, B. (2002). Contextual influ-ences on the directional responses of tectal cells in pigeons.Visual Neuroscience, 19, 133–144.

Troje, N., & Frost, B. (1998). The physiological fine structureof motion sensitive neurons in the pigeon’s tectum opticum.Society for Neuroscience Abstract, 24, 642.9.

Wang, Y., & Frost, B. (1990). Functional organization of the nu-cleus rotundus of pigeons. Society for Neuroscience Abstract,16, 1314.

Wang, Y., Jing, S., & Frost, B. (1993). Visual processing in pigeonnucleus rotundus: Luminance, color, motion, and loomingsubdivisions. Visual Neuroscience, 10, 21–30.

Watanabe, S. (1991). Effects of ectostriatal lesions on naturalconcept, pseudoconcept, and artificial pattern discriminationin pigeons. Visual Neuroscience, 6, 497–506.

Wörgötter, F., & Koch, C. (1991). A detailed model of theprimary visual pathway in the cat: Comparison of afferentexcitatory and intracortical inhibitory connection schemesfor orientation selectivity. Journal of Neuroscience, 11(7),1959–1979.

Wu, L., Niu, Y., & Wang, S. (2005). Tectal neurons signal im-pending collision of looming objects in the pigeon. EuropeanJournal of Neuroscience, 22, 2325–2331.

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