SOM-Model for the Development of Oriented ... - Semantic Scholar

SOM-Model for the Development of Oriented Receptive Fields and Orientation Maps from Non-Oriented ON-center OFF-center Inputs D. Brockmann1, H.-U. Bauer1, M. Riesenhuber2, and T. Geisel1 1

Max-Planck-Institut fur Stromungsforschung, Postfach 28 53, 37018 Gottingen, Germany 2 Department of Brain and Cognitive Sciences and Center for Biological and Computational Learning, Massachusetts Institute of Technology, E25-221, Cambridge, MA 02139, USA

Abstract. We investigate the development of oriented receptive elds and an orientation map in a SOM-model which is driven by rotationally symmetric stimuli in ON-center and OFF-center input layers. To this end we use the high-dimensional variant of the SOM-algorithm which allows to develop the internal structure of receptive elds as well as the layout of a neural map. We calculate a state diagram for this model, identify parameter regimes in which the rotational symmetry of the stimuli is broken, corroborate the analytical results by simulations, and investigate an extended version of the model which includes ocular dominance development.

1 Introduction Many aspects of the development of individual receptive elds, and of topographic neural maps have been explained by activity-driven self-organization processes. Assumptions about underlying mechanisms can be cast in mathematical form in such a general fashion that many phenomena, even in dierent sensory modalities, can be explained within the same modeling framework. One such general framework is Kohonen's Self-Organizing Map (SOM [4]) which has successfully accounted for various aspects of visual, auditory and somatosensory maps. The development of oriented receptive elds and orientation maps in the visual cortex has been hypothesized to result from a competition of correlated ON-center and OFF-center inputs to the map neurons which could be driven by prenatal spontaneous retinal activity [5, 6]. Is is an interesting question whether or not SOM-models can also exhibit this pattern formation behavior. If the do, consistency with other modeling frameworks is maintained, and we come closer to a uni ed view on map formation. If they do not, the reason for the discrepancy to the other models must be identi ed; possible conclusions about underlying mechanisms can be drawn. To support this hypothesis in the SOM-framework we need:

a) a \high-dimensional" SOM-model which describes receptive elds as a distribution of synaptic weights over some input layer (as opposed to receptive elds as points in some feature space) b) a break of rotational symmetry from the unoriented stimuli driving the system to the oriented receptive elds forming the map. We present here mathematical and numerical results for such a SOM-model with ON-center and OFF-center inputs. The mathematical results were obtained using a recently described analysis technique for high-dimensional SOMs. In the nal section of the paper we also describe results for an extended version of the model which includes the development of ocular dominance columns. A more detailed description of this work has been submitted elsewhere [9].

2 Analysis of Orientation Map Model Neurons in a SOM are characterized by positions r in a map lattice A, and receptive elds wr in a map input space V . The input space is assumed to consist of one (or several) layer(s) of input channels. A stimulus v consists of a distribution of activity over these input channels, normalized to a constant sum. The stimulus is mapped onto that neuron s 2 A, the receptive eld ws of which matches v best, e.g. s = arg maxr2A fwr
vg. Presenting a random sequence of stimuli and performing adaptation steps,


wr = h(r ? s) (v ? wr) ;

(1)

the internal shape of individual receptive elds as well as the map layout ? self-organize simultaneously. The neighborhood function h(r ? s) =
exp ?kr ? sk2 =22 ensures that neighboring neurons align their receptive elds. We consider a projection geometry analogous to that proposed in [5, 6]. Layers of ON-center and OFF-center cells serve as a map input space and project to the map layer (Fig. 1). We assume our stimuli to consist of an activity peak in one layer, complemented by an activity annulus in the other layer. Stimuli are represented as dierence-of-Gaussians (DOG), the positive part of which resembles the acitvity in one input-layer (e.g. ON or OFF), the negative part in the other. Stimulus parameters are the relative widths of the gaussians comprising the DOG, and the relative amplitude k of the annulus-shaped negative part of the DOG. In the simulations, the center positions and polarity (e.g peak-activity in the ON- or OFF-layer) are chosen at random. The receptive elds which result after the self-organization process belong to one out of three qualitatively dierent possible states. The states can be distinguished by the way the set of all stimuli is tesselated, i.e. distributed among the map neurons. System B: Each neuron responds to both an ON- and an OFF-stimulus, each located at the same retinal position. This tesselation yields neurons with orientation insensitive receptive elds. System S : As in system B, each neuron responds to stimuli of both polarities, but now displaced one step

ON-Layer

OFF-Layer

Cortical Layer

Fig. 1. Architecture of the SOM model of orientation map development. along one retinal coordinate. The displacement breaks isotropy. It causes the receptive elds to exhibit internal ON- and OFF-center structure, with orientation speci city. System O: Here each neuron responds to two retinally neighboring stimuli of identical polarity. Although this tesselation induces an orientation speci city, it also breaks the symmetry between ON-center and OFF-center inputs to each neuron. Neurons segregate into ON-center and OFF-center dominated populations, analogous to an ocular dominance map. For each of these tesselations we can evaluate the distortion measure X X X Ev = h(r ? r0 ) (v ? v0 )2 ; (2) ;

r r0

v

2 r v 2 r 0

0

where r denotes all stimuli which are mapped to neuron r. Each term in Ev consists of the mean squared dierence between stimuli within the same, or between neighboring r 's, weighted by the neighborhood function h(r ? r0 ). Assuming that the SOM-algorithm leads to a minimization of Ev , the nal state of the receptive eld vectors and the map can be determined by comparing the values of EvB , EvS and EvO (For a motivation and a more detailed description of this method see [8]). Fig. 2a shows a state diagram in the parameter-plane which is calculated in this way. All three possible map states are predicted to occur in some region of parameter space. To corroborate the mathematical analysis above and to actually obtain orientation maps we also investigated the model numerically. We ran simulations with 16  16 neuron maps, at various parameters. Classifying the resulting receptive

Fig. 2. Analytical (a) and numerical (b) phase diagrams for the SOM-orientation map model. The parameters  and k denote the neighborhood width of the SOM-algorithm, and the annulus amplitude of the stimuli, respectively. The + symbol denotes the non-oriented state B, ? denotes the oriented state S , and 4 the (non-biological) state O.

elds with regard to the states B; S ; O, we obtained the state diagram depicted in Fig. 2b. The correspondence to the mathematically obtained diagram (Fig. 2a) is strikingly good, underlining the value of the distortion measure method for the analysis of high-dimensional SOMs. Fig. 3 shows exemplary receptive elds of neurons in a segment of the map. The receptive elds show a multi-lobed structure and are clearly oriented.

Fig. 3. Sample receptive elds of an orientation map with parameters chosen to yield map state S .

3 Orientation and Ocular Dominance Finally, we complement the two ON-center and OFF-center input cell layers for one eye by two further ON-center and OFF-center cell layers for the other eye. The repertoire of possible patterns in this extended model should go beyond merely oriented receptive elds in an orientation map, it should also include monocular receptive elds and ocular dominance maps, and combinations of the two types of patterns. Stimuli in the extended model consist of activity distributions in all four input layers. While the dierence in shape of the activity distributions between ON-center and OFF-center layers is the same as before, the partial stimuli are assumed to be of identical shape in the corresponding layers for either eye, but attenuated by a factor of c, 0  c  1, in one of the eyes (analogous to the assumptions underlying a recently analyzed SOM-based model for ocular dominance formation [3, 2]. The analysis technique introduced above can be applied to this more complicated case as well, considering the dierent tesselation possibilities for four stimuli per neuron. The parameter space in this extended model is rather large, which renders simulating the SOM for every set of parameters computationally unfeasible. Evaluation of the distortion measure for dierent map-layouts, however, enabled us to obtain phase-diagrams analogous to those depicted in g. 2., and lead us directly to a parameter-regime in which to expect the combined development of orientation and ocular dominance. Simulating in this promising parameter region indeed produced maps with monocular, oriented receptive elds. Fig. 4 shows one such combined map in a plot that displays the boundaries of the isoocularity domains superimposed on the orientation map. Determining the transition lines between isoocularity regions in the simulated map, and computing the intersection angles with the isoorientation lines at these locations, one nds that isoorientation lines intersect the boundaries between isoocularity regions preferably at large angles, consistent with experimental observations [1, 7]. This eect cannot be accounted for in linear models.

Acknowledgement This work has been supported by the Deutsche Forschungsgemeinschaft through SFB 185 \Nichtlineare Dynamik", TP E6.

References 1. Bartfeld, E., Grinvald A.: Relationship between orientation preference pinwheels, cytochrome oxidase blobs and ocular-dominance columns in primate striate cortex. Proc. Nat. Acad. Sci. USA 89 (1992) 11905{11909 2. Bauer, H.-U., Brockmann, D., Geisel, T.: Analysis of ocular dominance pattern formation in a high-dimensional self-organizing-map-model. Network 8 (1997) 17{ 33

Fig. 4. Combined ocular dominance and orientation map. Isoocularity domain boundaries are superimposed on the orientation map as black lines. 3. Goodhill, G. J.: Topography and ocular dominance: a model exploring positive correlations. Biol. Cyb. 69 (1993) 109{118 4. Kohonen, T.: The Self-Organizing Map, Springer Berlin (1995) 5. Miller, K. D.: A model for the development of simple-cell receptive elds and the ordered arrangement of orientation columns through activity dependent competition between On- and O-center inputs. J. Neurosci. 14 (1994) 409{441 6. Miyashita, M., Tanaka, S.: A mathematical model for the self-organization of orientation columns in visual cortex. NeuroRep. 3 (1992) 69{72 7. Obermayer, K., and Blasdel, G. G.: Geometry of orientation and ocular dominance columns in monkey striate cortex. J. Neurosci. 13 (1993) 4114{4129 8. Riesenhuber, M., Bauer, H.-U., Geisel, T.: Analyzing phase transitions in highdimensional self-organizing maps. Biol. Cyb. 75 (1996) 397{407 9. Riesenhuber M., Bauer H.-U., Brockmann D., and T. Geisel: Breaking rotational symmetry in a self-organizing map-model for orientation map development. submitted to Neur. Comp. (1997)

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