The computation of feature selective responses in the primary visual cortex

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(1)THE COMPUTATION OF FEATURE SELECTIVE RESPONSES IN THE PRIMARY VISUAL CORTEX. vorgelegt von Dipl. Phys. Peter Wiesing aus Berlin ¨ t IV - Elektrotechnik und Informatik Von der Fakulta ¨ t Berlin der Technischen Universita zur Erlangung des akademischen Grades Doktor der Naturwissenschaften - Dr. rer. nat. genehmigte Dissertation. Promotionsausschuss: Vorsitzender: Prof. Dr. O. Hellwich Berichter: Prof. Dr. K. Obermayer Berichter: Prof. M. Sur, PhD (MIT, Cambridge) Tag der wissenschaftlichen Aussprache: 15.12.2005. Berlin 2006 D 83.

(2) Abstract. We investigate the processing of visual input in the early visual system of mammals. Cells in the primary visual cortex (V1) often appear to compute a weighted sum of the light intensity distribution of the visual stimuli that fall on their receptive fields. A linear model of these cells has the advantage of simplicity and captures a number of basic aspects of cell function. On the other hand, it fails to account for important observed response nonlinearities which appear to have an intra-cortical origin. Much experimental and theoretical work has been published on this subject. However, the mechanisms that give rise to the integration of a cortical cell’s input to its final response are still controversial and are still part of intense study and debate. In this manuscript, we illustrate the use of combined theoretical approaches and experimental measurements for understanding neuronal and circuit dynamics. Firstly, we use an abstract correlation-based learning (CBL) model to study the role of lateral interaction on the emergence of ocular dominance (OD) and topographic (TP) maps. In detail, this mechanism resembles the effect of a variable degree of intra-cortical competition in which neurons excite each other at short distances and inhibit each other at larger distances and is implemented through a nonlinear response function followed by divisive normalization mechanism. Furthermore, we explore the effect of this quantity at different locations of the OD map, which in many species dissolves in the periphery of the visual cortex. We investigate to what extent these OD pattern changes can be related - either as a cause or an effect - to changes in the cortical magnification factor as well as to changes in the relative strength of the projection from the left and the right eye. We find an inverse proportionality of the cortical magnification factor and cortical receptive field size, similar to what had been measured in macaque primary visual cortex. While cortical magnification neither affects the emergence nor the stability of OD bands and of localized receptive fields as a function of cortical competition, the difference between the overall right-eye and left-eye projections do. The larger the difference is, the higher the degree of cortical competition or the strength of input correlations must be in order for localized receptive fields. ii.

(3) and topographic maps to form. Within a similar model framework we investigate, in addition, the isolated role of powerlaw nonlinear response function (without a divisive normalization mechanism) on stable OD and TP map representation. We show that there is a critical degree of nonlinearity required for the emergence of localized receptive fields and ocular dominance. However, the pure nonlinear response function without normalization is not sufficient to develop stable topography, rather all cortical cells will tend to receive inputs from the same LGN cells. We show that ocular dominance emerges beyond a critical value of nonlinearity, i.e. one eye dominates, but in contrast to the case where divisive normalization is considered, typical alternating ocular dominance stripes do not form. Therefore we conclude that a certain degree of an intra-cortical inhibitory mechanism is important for the emergence of stable (biologically realistic) feature maps. A pure nonlinearity in the response function of neurons can not account for these effects. Secondly, we present electrophysiological measurements of the excitatory and inhibitory synaptic conductances in neurons at different orientation (OR) map positions - ranging from pinwheels to orientation domains - and anatomical measurements of the inputs to these neurons in the cat adult primary visual cortex. The local arrangement of cells with different response properties vary with spatial location in the OR map and influence information processing in local neuronal circuits. We find that visually evoked excitatory and inhibitory synaptic conductances are balanced exquisitely in cortical neurons and thus keep the spike response sharply tuned at all map locations. This balance derives from the anatomically measured spatially isotropic local connectivity of both excitatory and inhibitory cells. This pattern of anatomical connections can be combined with the strength of synaptic drive to explain the physiological responses of neurons. In a second step we investigate, using a combination of mathematical analysis and computer simulations, what constraints these experimental data imposes on the cortical ”operating regime” (feed-forward, dominated inhibitory, dominated excitatory, intermediate recurrent or highly recurrent ”marginal phase”). In a simple ”superposition” model (describing the interplay of the afferent and the excitatory recurrent drive), we show that the experimental findings can only be explained if a sharply tuned feed-forward input is processed by a cortical network of intermediate recurrency. Using a geometrical firing rate. iii.

(4) network model we calculate the tuning curves of the total excitatory and inhibitory conductances and the output spike-rates as a function of a large range of model parameters. We find, that the experimental data are best explained, when the recurrent inputs dominate the feed-forward input, when the orientation tuning of recurrent excitation and inhibition is balanced and co-varies with OR map locations. For all other parameter regimes including the highly recurrent marginal phase, no parameter setting was found which is equally consistent with the data. The results observed by the firing rate network model are finally confirmed for a biologically more realistic Hodgkin-Huxley (HH) type network model, which can additionally account for the orientation tuning of the membrane potential.. iv.

(5) Previously Published Work. The content of chapter 2 is submitted to the journal ”Biological Cybernetics”. Chapter 4 is published in its entirety in Mari˜ no et al. [2005]. The physiological measurements have been accomplished by Jorge Marino and James Schummers and the anatomical measurements by David C. Lyon under the supervision of Mriganka Sur at Massachusetts Institute of Technology, Massachusetts, USA. Chapter 5 is in preparation for the submission to the journal ”Neuron”. The simulations of the Hodgkin-Huxley type network model have been conducted by Oliver Beck at Berlin University of Technology, Berlin, Germany.. v.

(6) Acknowledgements. This work was supervised by Prof. Klaus Obermayer. Prof. Obermayer was one of the persons who initially directed me towards the exciting problems and insights of theoretical neuroscience, but also towards a highly demanding and challenging approach of solving research problems. He has set up an excellent lab with enormous work and attention and has always been an exemplary head of his group. I am exceedingly thankful for his supervision and for providing me the freedom to organize the International Neuroscience Summit (INS 2002) together with Lars Schwabe and Gregor Wenning. The content of this thesis is also a product of the close cooperation with Prof. Mriganka Sur, the second supervisor of this work. I am highly indebted to Prof. Sur to have been a member of his excellent research environment and have had the chance to enjoy outstanding experimental research in neuroscience. Abha Sur and Prof. Sur have always shown me great hospitality and I have many fond memories of time spent in their company. The work presented in this thesis was funded by the German Research Foundation (DFG) and by the Wellcome Trust. Special thanks is addressed to Christine Waite. Her generous hospitality and her cooperativeness have made my numerous visits at the Sur Laboratory in Cambridge highly enjoyable. I am greatly indebted to all my colleagues at the Neural Information Processing Group and at the Sur Laboratory. In particular, I would like to thank Oliver Beck, David Lyon, Jorge Mari˜ no, Lars Schwabe and James Schummers for their assistance and support, as well as interesting discussions. It was great fun to work with them together in a close and fruitful collaboration. I would like express special thanks to Oliver Beck, who supported me throughout my entire research time in a very close friendship.. vi.

(7) Oliver Beck, James Schummers, Lars Schwabe and Josh Young read all or parts of the manuscript and gave me valuable feedback. I would like to accord my mother, my brother, Anneli and Peter Schehka my greatest gratitude for their invaluable love and support.. vii.

(8) Contents 1 Introduction. 1. 1.1. Mechanisms of orientation selectivity . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. Questions addressed in this manuscript . . . . . . . . . . . . . . . . . . . . .. 4. 1.3. Plan of the manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2 Lateral competition in OD and TP map development. 9. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.2. The computational model . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.2.1. Model architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.2.2. The input patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 2.2.3. The input-output mapping . . . . . . . . . . . . . . . . . . . . . . .. 13. 2.2.4. The learning dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. Mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 2.3.1. Stationary states of the learning rule . . . . . . . . . . . . . . . . . .. 15. 2.3.2. Stability of stationary states. . . . . . . . . . . . . . . . . . . . . . .. 17. 2.4. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 2.3. 3 Nonlinearities in OD and TP map development. 26. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 3.2. The computational model . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3.2.1. Model architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. Mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30. 3.3.1. Stationary states of the learning rule . . . . . . . . . . . . . . . . . .. 30. 3.3.2. Stability of stationary states. 31. 3.3. . . . . . . . . . . . . . . . . . . . . . .. viii.

(9) 3.4. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 3.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 4 Orientation selectivity in V1: experimental results. 39. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 4.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. 4.2.1. Synaptic conductances at different map locations . . . . . . . . . . .. 40. 4.2.2. Anatomical inputs to different map locations . . . . . . . . . . . . .. 46. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 4.3.1. The role of inhibition in orientation selectivity . . . . . . . . . . . .. 48. 4.3.2. Invariant tuning with balanced excitation and inhibition . . . . . . .. 49. 4.3. 5 Orientation selectivity in V1: modeling studies. 51. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53. 5.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 5.2.1. The superposition model . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 5.2.2. The mean-field network model . . . . . . . . . . . . . . . . . . . . .. 62. 5.2.3. The Hodgkin-Huxley network model . . . . . . . . . . . . . . . . . .. 68. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76. 5.3.1. The role of sharp feed-forward input . . . . . . . . . . . . . . . . . .. 76. 5.3.2. Feed-forward versus recurrent network behavior . . . . . . . . . . . .. 77. 5.3.3. The marginal phase . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 5.3.4. Limitations of the study . . . . . . . . . . . . . . . . . . . . . . . . .. 80. 5.3. A Chapter 2: derivations. 83. A.1 Stationary states of the learning rule (Gaussian weights) . . . . . . . . . . .. 84. A.2 Stationary states of the learning rule (flat weights) . . . . . . . . . . . . . .. 86. A.3 Stability of fixed points (general considerations) . . . . . . . . . . . . . . . .. 87. A.4 Stability of fixed points (flat weights) . . . . . . . . . . . . . . . . . . . . . .. 89. A.5 CBL and SOM as limit cases . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. A.6 Used Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93. B Chapter 3: derivations. 94. B.1 Stationary states of the learning rule (Gaussian weights in topograph order). 94. B.2 Stationary states of the learning rule (Gaussian weights) . . . . . . . . . . .. 95. ix.

(10) B.3 Stationary states of the learning rule (flat weights) . . . . . . . . . . . . . .. 98. B.4 Stability of fixed points (general considerations) . . . . . . . . . . . . . . . .. 99. B.5 Stability of fixed points (flat weights) . . . . . . . . . . . . . . . . . . . . . .. 101. B.6 Used Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102. C Chapter 4: methods. 103. C.1 Animals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103. C.2 Optical imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103. C.3 Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 104. C.4 Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106. D Chapter 5: methods. 107. D.1 Orientation selectivity index (OSI) . . . . . . . . . . . . . . . . . . . . . . .. 107. D.2 Construction of the model orientation maps . . . . . . . . . . . . . . . . . .. 107. D.3 Calculating the Bayesian posterior . . . . . . . . . . . . . . . . . . . . . . .. 111. D.4 The superposition model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112. D.5 The mean-field network model. . . . . . . . . . . . . . . . . . . . . . . . . .. 113. D.6 The Hodgkin-Huxley network model . . . . . . . . . . . . . . . . . . . . . .. 115. x.

(11) Chapter 1. Introduction The representation of the visual world in the cerebral cortex has been the target of intense research which has provided fundamental insights into the anatomical and physiological foundations of the early visual cortical areas which have greatly increased our understanding of the initial steps of visual information processing. Particular attention has been paid to the early visual pathway from the retina via the lateral geniculate nucleus (LGN) to the primary visual cortex (V1) because, relative to higher processing levels of the visual cortex, the operations performed in these structures appear easier to understand. Even for V1 however, research has not provided a final answer to its circuitry, function and action yet. Neurons in V1 receive most of their input from the LGN and are further connected into circuits by excitatory and inhibitory synapses that show a variety of amplitudes, time courses and time-dependent changes in synaptic strength. This connectivity pattern provides a first suggestion of how the cortex processes visual information. An important question is: are cortical response properties determined principally by thalamocortical inputs or by a feedback process that encompasses the entire cortical circuit? Furthermore, what is the best way to study the circuits underlying response properties? And how can we determine which of these broad categories of inputs - thalamic, intra-cortical excitatory, intra-cortical inhibitory, or some combination of all three - are integrated to produce a neuron’s response? It has been thought for decades that orientation tuning - as the most prominent response property in V1 - must be important to how the cortex works and why it provides such a specific circuitry.. 1.

(12) CHAPTER 1. INTRODUCTION. 1.1. 2. Mechanisms of orientation selectivity. Hubel and Wiesel [1962] offered the first model to explain the emergence of orientation selectivity. They proposed that simple cells in cat striate cortex are orientation selective because they receive segregated ON and OFF-input from appropriately elongated areas of the LGN retinotopic map. According to their hypothesis, the axis of elongation of the afferent projection determines orientation preference while the aspect ratio of the receptive-field determines the specificity of the response. Pharmacological manipulations in ferret visual cortex provided the first experimental support for the role of spatial arrangement of LGN inputs in generating orientation selectivity (Chapman et al. [1991]). It has also been shown that the probability of finding a monosynaptic connection between an LGN cell and a cortical cell is strongly related to the overlap between their receptive fields (Reid and Alonso [1995], Alonso and Swadlow [2005]). Furthermore, the alignment of the geniculate inputs appears to be sufficient for the generation of orientation tuning in a simple cell (Ferster [1996], Chung and Ferster [1998]). These researchers reasoned that during either cooling (Ferster [1996]) or shocking (Chung and Ferster [1998]) the visual input to a simple cell will come solely from the thalamic afferents. Both studies concluded that orientation tuning was almost unchanged during cortical deactivation, which let them conclude that intra-cortical dynamics are of little or no importance for orientation tuning. In summary, there is considerable empirical evidence that orientation selectivity is conferred to V1 neurons by specifically aligned feed-forward thalamic input (Chapman et al. [1991], Reid and Alonso [1995]). It is not clear, however, whether this input is sufficient to account for the sharpness of tuning (Gardner et al. [1999]). Such sharpening may be accomplished by intra-cortical mechanisms. The role of inhibition in the sharpening of orientation tuning is uncertain. Thalamocortical synapses are purely excitatory (Freund et al. [1989], Callaway [1998]), therefore, if inhibition plays an important role, it must come from the intra-cortical network. It is known that within each cortical layer horizontal excitatory and inhibitory connections built the intra-cortical circuitry (Callaway [1998]). Sillito’s experiments with bicuculline suggest that lateral inhibition is necessary for orientation tuning (Sillito [1975]). However, experiments by Nelson et al. [1994] where inhibition was blocked intracellularly indicate that inhibitory input to a single neuron is not necessary for the neuron to exhibit orientation.

(13) CHAPTER 1. INTRODUCTION. 3. tuning. On the other hand, an important role for intra-cortical inhibition on the computation of orientation selectivity has been suggested by pharmacological experiments (Sato et al. [1996], Crook et al. [1998]). Albrecht and Geisler [1991], Heeger [1992], Carandini et al. [1997] proposed normalization models, in which divisive intra-cortical inhibition can account for contrast invariant orientation tuning (Sclar and Freeman [1982], Skottun et al. [1987]). In these models, the feed-forward geniculate input is divided or normalized just prior to threshold by an intra-cortical inhibitory input. A combination of the inhibition and excitation then yields a sigmoidal, saturating function of contrast. However, recent results suggested that suppression might not come from intra-cortical interactions. Carandini et al. [2002] for example have proposed an alternative biophysical foundation for the control of cortical responsiveness, namely, thalamocortical synaptic depression. There are several models that argue for both recurrent cortical excitation and inhibition as crucial elements of orientation selectivity (Somers et al. [1995], Ben-Yishai et al. [1995], McLaughlin et al. [2000]). In particular, these models explain cortical orientation selectivity in terms of broadly tuned inhibition and more narrowly tuned excitation. Ben-Yishai et al. [1995] offered an analytical model from which they made several qualitative and quantitative predictions. One of their theoretical results is that if recurrent feedback is strong enough one will observe a ”marginal phase” state, in which V1 behaves like a set of attractors for orientation (see also Tsodyks et al. [1999], Adorjan et al. [1999], Ernst et al. [2001] for similar studies). The concept is that the tuning of very weakly orientation-tuned feedforward signals can be massively sharpened by considerably strong recurrent excitatory feedback. In such a network, the neurons will respond to any visual signal by relaxing into a state of activity governed by the pattern of intra-cortical feedback. Such a model of strong recurrency can, for example, resolve the issue of contrast invariant orientation tuning (Ben-Yishai et al. [1995]). Finally, attempts have been made to estimate the relative contributions of afferent and intra-cortical mechanisms contributing to orientation tuning (Gardner et al. [1999]). These researchers used a simple cell model consisting of a linear filter followed by a static nonlinearity. They found that the empirically observed tuning curves were more sharply tuned for orientation than the curves predicted by the model. This observation led them conclude that cortical factors must play an important role in sharpening the orientation tuning of simple cells. They suggested that recurrent cortical excitation could amplify input signals (Douglas et al. [1995]) and could therefore provide the neuronal basis for the expansive.

(14) CHAPTER 1. INTRODUCTION. 4. nonlinearity. Miller et al. [2001], Miller and Troyer [2002], Troyer et al. [2002] used such an expansive nonlinearity of cortical response to demonstrate that it can transform contrastinvariant voltage responses into contrast-invariant spike responses. These studies suggested the key to an expansive nonlinearity lies in the noisiness of the membrane potential. Noise may smooth the average relation between membrane potential and spike rate throughout the brain (Anderson et al. [2000]).. 1.2. Questions addressed in this manuscript. In conclusion, it seems likely that afferent thalamocortical input provides a significant part of the selectivity of some cortical cells. In addition however, intra-cortical mechanisms appear to modulate cortical responses and thus are essential for the integration of thalamic visual information. In this manuscript we investigate in the processing of visual information using a combination of theoretical approaches and experimental measurements in two different feature selective phenomena: (i) long term changes, i.e. the development and the plasticity of ocular dominance (OD) and topographic (TP) maps and (ii) the representation of orientation selectivity. In the first part of the manuscript (chapter 2 and 3) we examine - by means of abstract computational models - the role of lateral recurrency in the development of feature selective receptive field structures and the underlying feature maps. Many different models have accounted for the emergence of feature map formation, including ocular dominance, topography and preferred orientation maps (von der Malsburg [1973], Miller et al. [1989], Obermayer et al. [1990], Piepenbrock and Obermayer [2000], see Erwin et al. [1995], Swindale [2004] for a review) and looked, for example, at the spatial relationship of multiple emerging maps (Hubener et al. [1997], Swindale [2004], Yu et al. [2005]). More precisely, there have been two major modeling frameworks to explain the activity driven development of the response properties of cortical neurons, and their spatial distribution across the brain’s surface, the so-called correlation-based learning (CBL) (von der Malsburg [1973], Linsker [1986], Yuille et al. [1989], Miller et al. [1989], Tanaka [1991], Miller [1994]) and self-organizing map (SOM) (Takeuchi and Amari [1979], Kohonen.

(15) CHAPTER 1. INTRODUCTION. 5. [1982], Swindale [1982], Durbin and Mitchison [1990], Obermayer et al. [1990, 1992], Goodhill and Willshaw [1990], Bauer [1995], Swindale [2004]) models. Both approaches are based on firing rates and use Hebbian learning as the rule of change, but differ significantly in the way lateral cortical interactions are taken into account. CBL models treat the influence of intra-cortical interactions by the application of a linear filter to the total afferent inputs. As a result of this, receptive field profiles are determined by the average values and by the spatial correlations of second order of the afferent neural activity patterns and not through the intra-cortical dynamics. Self-organizing map approaches, on the other hand, include a winner-takes-all operation before the linear filter is applied, i.e. only a small localized region in the cortex is active for a given afferent input at a given time located at the region which receives the strongest total input for the current afferent pattern. This leads to feature extraction and feature mapping instead. Most likely, cortical interactions are neither linear nor do they implement a winnertakes-all operation, and one may argue that the assumptions underlying the correlationbased learning and self-organizing map approaches are inadequate - even on a conceptual level. This issue raises the following questions: • Which role plays cortico-cortical dynamics during the development of the thalamocortical synaptic input connectivity? • Is there a critical degree of cortical competition which is necessary for the emergence of ordinary feature maps? • If so, does this critical degree change within V1 and could those changes, for example, be a potential reason for the dissolving OD maps of typical alternating stripes in the periphery of macaque’s primary visual cortex (Anderson et al. [1988], Horton and Hocking [1996a], Lund et al. [2003])? • Furthermore, if intra-cortical dynamics are of general importance in the developing/plastic brain, what are the individual roles played by lateral excitation and lateral inhibition? In order to investigate these questions we use a computational model for TP maps and OD columns where the winner-takes-all step is replaced by a ”soft” competition for neural activity in the target layer (Piepenbrock and Obermayer [1999, 2000]). This approach.

(16) CHAPTER 1. INTRODUCTION. 6. considers a variable degree of cortical competition and shows in the limit of low competition CBL model behaviour and for high values of competition SOM model behaviour. Piepenbrock and Obermayer [1999, 2000] showed, that the degree of cortical competition is indeed an important quantity, because its value determines whether topographic maps and ocular dominance patterns are stable or not. We therefore estimate the role and a potential critical degree of lateral competition, i.e. lateral recurrency, which is necessary for feature map development. Within this framework, we then examine the role of dissolving OD maps in the periphery of macaque’s visual cortex. Finally, we investigate in more detail in the ”soft” competition mechanism, to examine the isolated role of intra-cortical excitatory and inhibitory influence in the pattern formation process. In the second part of the manuscript (chapter 4 and 5) we investigate in the computation of orientation selectivity. Little work has attempted to incorporate information about the structure of the orientation map, in order to observe the origins of orientation selectivity (McLaughlin et al. [2000], Shelley et al. [2002], Kang et al. [2003]). The orientation map, however, creates a fundamental local inhomogeneity in the nature of the recurrent inputs of V1 neurons. The orientation map is characterized by pinwheels, around which all orientations are represented radially, arranged periodically between orientation domains, across which the preferred orientation changes smoothly (Bonhoeffer and Grinvald [1991], Blasdel [1992]). Short range projections in V1 are spatially isotropic and independent of orientation map locations (Malach et al. [1993], Bosking et al. [1997], Kisvarday et al. [1997], Yousef et al. [2001]). If local recurrent connections play an important role in identifying the response properties of neurons, one would therefore expect differences in the orientation tuning of neurons in different map locations. But however, Maldonado et al. [1997], Schummers et al. [2002] showed that neurons in V1 are sharply tuned, regardless of their OR map locations. This issue raise the following questions: • Which processes are responsible for the observation of Maldonado et al. [1997], Schummers et al. [2002], namely that neurons near pinwheels are sharply tuned for orientation and show no differences to the tuning in orientation domains? • Can a specific integration of different inputs to a neuron - thalamic, intra-cortical excitatory, intra-cortical inhibitory - account for this finding? In order to answer these questions we investigate the response properties of neurons located.

(17) CHAPTER 1. INTRODUCTION. 7. at different locations of the orientation map which reflect the differences in local inputs at these map locations. By recording the activity of neurons at different map locations, one can potentially gain additional insight into the integration of inputs and the shaping of response tuning. For the first time, information about the neuron’s spike output is combined with information about its direct inputs and the behavior of its membrane potential. Using this experimental data, we can finally estimate the possible excitatory/inhibitory operating point of the visual cortex, i.e. we can define the role of excitatory and inhibitory recurrency in the computation of feature selective responses. In conclusion, each part of this manuscript investigates a different aspect of the computation of feature selective responses. For a more thorough understanding of the computation processes in V1 both parts could be integrated to yield a developmental model of orientation selectivity whose emerging circuitry accounts also for the cortical integration and dynamics observed in the second part of the thesis (chapter 4 and 5). Future studies need to deal with model classes such as mean-field approaches (as used in chapter 5), where the recurrent cortical circuitry can be more realistically incorporated and feature selectivity as well as developmental aspects could - at least numerically - be taken into account.. 1.3. Plan of the manuscript. In chapter 2, we study (mainly) analytically by means of the ”soft” competition model the role of lateral competition on the emergence of OD and TP maps (Piepenbrock and Obermayer [1999, 2000]). This approach describes the effect of a variable degree of intra-cortical competition in which neurons excite each other at short distances and inhibit each other at larger distances. In order to consider different positions in the visual cortex, i.e. to incorporate changes in the OD patterns, we consider a variable degree of cortical magnification as well as a variable degree of the relative strength of the projection from the left and the right eye. We examine in a linear stability analysis under which parameter conditions OD and TP maps emerge. We can therefore estimate, which influence lateral competition, i.e. which degree of lateral recurrency, might be necessary for the onset of OD and TP maps at different OD map locations, i.e. for different values of cortical magnification and differences in inter-eye arborization..

(18) CHAPTER 1. INTRODUCTION. 8. In chapter 3, we investigate in more detail the ”soft” competitive mechanism, which was used in chapter 2. This cortical mechanism is mathematically realized by a so-called ”softmax” functionality, which considers a power-law nonlinear transfer of the linear afferent input, followed by an divisive normalization. In this chapter we examine if a pure powerlaw nonlinear transfer function - either caused by excitatory recurrent effects (Douglas et al. [1995]) or by the noisiness of the membrane potential (Anderson et al. [2000], Troyer et al. [2002]) or both - is sufficient to explain OD and TP pattern formation. Using a similar model architecture as in chapter 2, this analysis examines the role of lateral excitation in pattern formation in the absence of an inhibitory mechanism, such as the divisive normalization. By performing also a linear stability analysis we compare the results with results of chapter 2. In both chapters analytical results are also quantified by means of numerical simulations. In chapter 4, we present electrophysiological measurements of the excitatory and inhibitory synaptic conductances in neurons at different OR map positions - ranging from pinwheels to orientation domains - and anatomical measurements of the inputs to these neurons in the cat adult primary visual cortex. In chapter 5, we provide a systematic investigation of the implications of the experimental data introduced in chapter 4 using a combination of mathematical analysis and computer simulations of network models of different complexity. We test various network parameters which modulate the degree of recurrency, like the tuning and the strength of the feed-forward drive and the level of recurrent synaptic input into a cortical model cell. These model parameters define different recurrent network regimes i.e. feed-forward, dominated inhibitory, dominated excitatory, intermediate recurrent or highly recurrent ”marginal phase”, which we relate to the experimental data via a Bayesian posterior approach. In the appendix A - D, we provide detailed analytical derivations and methods of the previous chapters 2 - 5..

(19) Chapter 2. Lateral competition in OD and TP map development Hebbian learning and lateral competition: The influence of cortical magnification and unequal arbor functions on topography and ocular dominance maps In many species, which show a segregation of left eye and right eye afferents in the primary visual cortex, the spatial pattern of ocular dominance maps change with eccentricity (Anderson et al. [1988], Horton and Hocking [1996a], Lund et al. [2003]). Here we investigate, to what extent these changes can be related to - either as a cause or an effect - to changes in the cortical magnification factor as well as to changes in the relative strength of the projection from the left and the right eye (Horton and Hocking [1996b]). Using a Hebbian model of activity driven neural development which includes nonlinear lateral interactions within the cortical layers (Piepenbrock and Obermayer [1999, 2000], Dayan [2000, 2003]), we therefore investigate the influence of these quantities on the final spatial pattern of topographic maps and ocular dominance bands. We find an inverse proportionality of the cortical magnification factor and the cortical receptive field size, similar to what had been measured in macaque primary visual cortex (Connolly and Van Essen [1984]). Then we investigate, how spatial patterns depend on these quantities for different strengths of the intra-cortical competition. While cortical magnification neither affect the emergence nor the stability of OD bands and of localized receptive fields as a function of cortical 9.

(20) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 10. competition, the difference between the overall right-eye and left-eye projections do. The larger the difference is, the higher the degree of cortical competition or the strength of input correlations must be in order for localized receptive fields and topographic maps to form. We quantitatively characterize these dependencies by means of analytical calculations and numerical simulations.. 2.1. Introduction. Topographic maps and ocular dominance columns in the primary visual cortex emerge early in development and there is an ongoing debate whether intrinsic factors or neural activity dominate this process (Sur and Leamey [2001], Katz and Crowley [2002]). Later in cortical development - during so-called critical periods - neural activity has a crucial influence on cortical organization. During adulthood, changes in neural activity lead to longterm changes in cortical organization, a phenomenon called activity-dependent plasticity. The simplicity of the observed spatial patterns (continuous maps, stripes and patches), their common appearance when two neural layers innervate a third target layer, and their sensitivity to neural activity made the ”topographic map - ocular dominance” system a playground also for theoreticians, who use this system to illustrate hypotheses about pattern formation, self-organization, and the role of neural activity in the brain. Piepenbrock and Obermayer [1999, 2000] have investigated a computational model for topographic maps and ocular dominance columns, using a ”soft” competition for neural activity in the target layer - implemented by a so-called ”softmax” function. The results obtained with the ”soft” competition model showed that there is an important connection between the competitiveness of cortical interactions and cortical map structure. These findings raise the hypothesis that also in real neural systems the strength of cortical competition, mediated by the strength of recurrent excitation and lateral inhibition, must be properly adjusted and must match with the structure of the afferent activity patterns. If this would not be the case, topographic maps and ocular dominance columns would either not emerge in early development or they would destabilize during a critical period or whenever cortical circuits become ”plastic”. A second, related hypothesis states, that region-specific differences in the primary visual areas and differences between species are related to differences in the strength of cortical competition. Before these hypotheses can be addressed, however, several other important factors must.

(21) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 11. be taken into account which are likely to affect the pattern forming process. Two of these factors which are likely to be important are the local cortical magnification factor and a potentially unequal total density of afferents from the two eyes. The cortical magnification factor, defined as the number of linear millimeters in the cortex allotted to a given angle of visual space, changes by two order of magnitude between the fovea and the periphery and is different for different visual areas and different species (Waessle et al. [1990]). Also, the density of afferents from the two eyes changes from the binocular fovea to a monocular periphery whose spatial extend varies between species. Experiments show that while the topographic mapping is preserved with eccentricity the ocular dominance pattern changes. In the macaque monkey, for example, it changes from regular stripes to irregular stripes and patches to a monocular area in the periphery (Horton and Hocking [1996b]). Ocular dominance patterns also change between visual areas and between species. In this chapter we, therefore, investigate the role of the cortical magnification factor and the consequences of an unequal strength of projections from the two eyes on the properties and the stability of a topographic map and an ocular dominance pattern. Our model is a modification of the approach by Piepenbrock and Obermayer [1999, 2000] and Dayan [2000, 2003]. Its architecture consists of two input layers and one output layer with a quasi-continuous number of simple connectionistic neurons. Artificial punctiform input with a variable degree of correlation between both eyes is propagated from two input to one output layer. Cortical interaction is also modelled by a softmax function followed by a linear spatial filter. The different strength of projections from the input layers and different cortical magnification factors are incorporated by appropriate constraints on the connection pattern (”arbor function”). The chapter is structured as follows. In section 2.2 we describe the model architecture and the activity dependent learning rule. In section 2.3 we calculate the stationary states of the learning dynamics and their stability under perturbations. In section 2.4 we compare and augment the analytical results by numerical simulations. In section 2.5 we discuss the results regarding their biological significance. Finally, the appendix A provides the analytical derivations..

(22) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 12. Figure 2.1: Model architecture. The model consists of two input layers (left, L, and right, R, eye) and one output layer. Input and output layers are of length l and Mf ·l, where Mf denotes the local cortical magnification factor. Gaussian-shaped activity patterns L/R Pi , presented simultaneously to both input layers at corresponding locations i and with an amplitude difference γ, propagate through connection with a total strength L/R L/R of Six · Aix to the output layer and elicit output activities Ox . Intra-cortical interactions are described by a softmax function (see subsection ”Input-output mapping”) followed by the application of a linear interaction function Ixx0 .. 2.2 2.2.1. The computational model Model architecture. The model consists of two input layers, one for each eye (L for ”left eye”, R for ”right eye”), and one output layer, which represents primary visual cortex (see figure 2.1.) Input and output layers consist of a quasi-continuous set of simple connectionist neurons, which are laid out in a single spatial dimension. We assume a length of l for the input layers. Input and output layers are fully connected, and the strengths of the connections are represented by synaptic weights S. A variable cortical magnification factor Mf is implemented by choosing the length of the output layer to be Mf · l. Constraints on the connectivity pattern between layers are implemented by an arbor function A (cf. Miller [1994]), whose components are multiplied with the synaptic weights. The arbor function is used to impose constraints on the connectivity pattern by setting some of the components of the arbor function to zero and it can be used to bias the connection pattern towards one or the other eye as we do it in the following. Intra-cortical interactions are described by a softmax nonlinearity followed by the application of a linear interaction function I as specified below. The particular form of the intra-cortical constraints was.

(23) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 13. chosen in order (i) to include nonlinear interactions (via the softmax-function), (ii) to add a constraint which implements spatial correlations in the output layer and generates smooth cortical maps (interaction function), (iii) to allow for an interpolation between CBL and SOM models (see appendix A.5 for details), and (iv) to allow for a mathematical analysis. Note, however, that although the constraints are chosen such that they consistently implement important aspects of the integration of afferent inputs, they have not yet been derived from an underlying microscopic model.. 2.2.2. The input patterns. In the following we use localized stimuli with a Gaussian shape, ignoring the difference between ON-center OFF-center cells. The input activity functions are given by: L/R Pi. = =. ½ ¾ 1 1 ± γz 2 1 √ exp − (i − ξ) 2 2πσp 2σp2 1 1 (1 ± γz)G (i − ξ, σP ) = (1 ± γz)Pi . 2 2. (2.1). These functions describe the activity of the neurons of the input layers at the positions i which are caused by a Gaussian-shaped stimulus of size σp at position ξ presented to both layers simultaneously, but with different total strength 1±γz, z = ±1. Patterns are selected with equal probability at every learning step, i.e. P (z = ±1) = 0.5 and P (ξ) = 1/l = const. The activity patterns described by eq. 2.1 were chosen (i) because they are localized, (ii) because they allow to tune the degree of correlation between both eyes using the model parameter γ, and (iii) because they are simple enough to make a mathematical analysis possible. It is clear that real activity patterns are substantially more complicated than a single Gaussian bump, however, their complexity may not be adequate given the abstract model under investigations (but see Piepenbrock and Obermayer [2001] for a study using natural images as input).. 2.2.3. The input-output mapping. For every input presentation, the activity of a neuron in the output layer is calculated using a three-step procedure. In the first step, a ”total afferent input” Oxl at the position x of the output layer,. Z Oxl. =. ¡ ¢ L L R R R di AL , ix Six Pi + Aix Six Pi. (2.2).

(24) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 14. is calculated as the sum over all input activities PiL,R , weighted by the synaptic strengths L,R Six and the arbor function AL,R ix . Here we choose a flat arbor function: L/R. Aix. = (1 ± c) · A0 ,. (2.3). where c introduces an ocular bias in the strength of connections and A0 is a constant value. Miller et al. [1989], Dayan [2000] and Dayan [2003] introduced localized arbor functions, in order to consider the effect of an initial topography and to limit the extension of cortical activation. Here we have chosen a constant function, because we assume that the whole output layer represents a small region in cortex defined by the spatial extension of its intracortical lateral interactions, and because we focus on the role of unequal projections from the different eyes rather than the role of initial topography. In the second step, a nonlinear cortical activation function is applied to the total afferent input Oxl . Here we have chosen a modified softmax function (Dayan [2000, 2003]) Oxc = R. (Oxl )β , dx0 (Oxl 0 )β. (2.4). where the power function with exponent β can be loosely interpreted as a single population transfer function where the denominator implements a divisive normalization. Both effects can emerge, for example, as a result of recurrent excitation (Douglas et al. [1995]) and lateral inhibition (Heeger [1992], Carandini et al. [1997]). The exponent β determines therefore the degree competition between cortical cells, and - as already mentioned above - it allows −→ −→∞) competition to interpolate between the limit cases of weak (β − CBL 1) and strong (β− SOM (see appendix A.5 for details). Finally, a Gaussian-shaped linear filter Ixx0 , ¡ ¢ Ixx0 = G x − x0 , σI ,. (2.5). of width σI , the so-called interaction function, is applied which then leads to the activations Ox ,. Z Ox =. dx0 Ixx0 Oxc 0 ,. (2.6). of the cortical neurons at positions x. This filter implements the effect of spatially local interactions during learning. It is necessary for the emergence of spatially smooth cortical.

(25) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 15. maps and may be mediated, e.g., by diffusive substances as suggested by Miller et al. [1989], Miller [1994].. 2.2.4. The learning dynamics. The afferent synaptic weights of the cortical cells are trained according to a Hebbian learning rule with multiplicative normalization, which includes the constraint of a constant sum of synaptic weights for each output neuron (see Miller and MacKay [1994] for a detailed analL/R. ysis of normalization constraints). The cortical interaction Ixx0 and arbor functions Aix. are fixed during the adaptation process, because they implement the models constraints. The learning rule is given by: L/R Six. →. L/R Six. µD E L/R +ε Ox Pi. where ε is the learning step size and τx =. R. ¶ ξ,z. −. L/R τx Six. ,. (2.7). diA0 h(1 + cγz)Ox Pi iξ,z (cf. A.35) is chosen to. fulfill the normalization constraint: Z. 2.3 2.3.1. £. ¤ L R R AL ix Six + Aix Six di = 1 , ∀x. (2.8). Mathematical analysis Stationary states of the learning rule L/R. The stationary states of the learning rule, eq. (2.7), for the synaptic strengths Six fulfill the condition. L/R ∆Six. must. = 0 from which we obtain D. L/R. O(x)Pi. E ξ,z. L/R. = τx Six .. (2.9). The condition for stationarity cannot be solved explicitly for the synaptic strengths without further assumptions. Motivated by the results of previous studies (Piepenbrock and Obermayer [1999, 2000], Dayan [2000, 2003]) we investigate two cases: (i) Stationary states which are characterized by constant weights, L/R. Six. = ω = const.,. (2.10).

(26) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 16. Figure 2.2: Size of σS of the ”anatomical” receptive field as a function of the strength β of the competition parameter plotted using eq. (2.12) for σI = σP . and (ii) stationary states which are characterized by a perfect topographic map and Gaussianshaped profiles of synaptic strengths (localized receptive fields), L/R. Six. = ωG (x − Mf i, Mf σS ) .. (2.11). Mf is the local cortical magnification factor and σs is the width of the profile of the afferent connection strengths. In the appendix A.1 and A.2 it is shown that constant weights with ω = 1/(2A0 l) are always a stationary state of the learning dynamics, i.e. they fulfill eq. (2.9). A topographic map with Gaussian-shaped receptive field profiles, however, is stationary only if the width σs of the profile is given by: σS2 =. β+1 2 β σP + σ2, 2 β−1 Mf (β − 1) I. (2.12). and if the lateral competition is strong enough (β > 1) and ω = 1/(2A0 ). Both stationary states are independent of the parameter c which describes the ocular bias in the arbor function, eq. (2.3). Figure 2.2 shows the relationship between σS and β for the ”topographic” stationary state for σI = σp . The receptive field size diverges for β → 1, and the state with Gaussian receptive field profiles is no longer stationary for smaller values of β..

(27) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 17. Equation (2.12) implements an inverse magnification rule, i.e. the receptive field size is proportional to the inverse of the magnification factor. This is a relationship which is commonly observed for topographic maps in the primary visual cortex (Essen et al. [1983], Connolly and Van Essen [1984]) and is in agreement with previous modelling studies for −→ the β − SOM ∞ limit case (Obermayer et al. [1990]).. 2.3.2. Stability of stationary states. In the following, we will assess the stability of the stationary states eqs. (2.10,2.11,2.12) of the learning rule, eq. (2.7), by a linear stability analysis, i.e. by evaluating the learning dynamics for small perturbations. ∗ . The Taylor expansion of eq. (2.7) with respect Let us consider a stationary state Six L/R. to small perturbations δSix. ∗ leads to: away from the point of stationarity Six. · 1 − γ2 dx0 di0 (1 − c) Oixi0 x0 1 − (γc)2 ¸ Z 00 − (1 − c)2A0 di Oi00 xi0 x0 δSiR0 x0 · Z κ 1 + γ 2 − 2γ 2 c 0 0 +εβ dx di (1 + c) Oii0 xx0 2 1 − (γc)2 ¸ Z L − (1 + c)2A0 di00 Oi00 xi0 x0 δSiL0 x0 − εκδSix. L ∆δSix = εβ. κ 2. Z. · 1 − γ2 dx0 di0 (1 + c) Oixi0 x0 1 − (γc)2 ¸ Z 00 − (1 + c)2A0 di Oi00 xi0 x0 δSiL0 x0 · Z 1 + γ 2 (1 + 2c) κ 0 0 +εβ dx di (1 − c) Oixi0 x0 2 1 − (γc)2 ¸ Z R − (1 − c)2A0 di00 Oi00 xi0 x0 δSiR0 x0 − εκδSix. R ∆δSix = εβ. with Oixi0 x0. κ 2. (2.13). Z. µ ¡ ¢ = G x − x0 , σI −. 1 l Mf. ¶. ³ ´ √ G i − i0 , 2σP. (2.14). (2.15). and κ = A0 /(l Mf ) (see appendix A.3 and A.4). Equations (2.13) and (2.14) can be decoupled, if the weight vectors are transformed into the joint eigenbasis of the integral operators.

(28) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 18. Z L/R. L/R. dx0 di0 Oixi0 x0 δSi0 x0 = E 1 δSix Z Z L/R L/R dx0 di0 di00 Oi00 xi0 x0 δSi0 x0 = E 2 δSix , L/R. where E 1 and E 2 are the corresponding eigenvalues and δSix. (2.16) (2.17). are the corresponding eigen-. vectors. Inserting eqs. (2.16, 2.17) into eqs. (2.13, 2.14) leads to · ¸ 1 + γ 2 − 2γ 2 c 1 2 κ 2 L (1 + c) E − (1 + c)2A E − δSix 0 2 1 − (γc)2 β ¸ · κ 1 − γ2 1 2 R +εβ (1 − c) E − (1 − c)2A0 E δSix 2 1 − (γc)2. L ∆δSix = εβ. · ¸ κ 1 − γ2 1 2 L (1 + c) E − (1 + c)2A E δSix 0 2 1 − (γc)2 · ¸ 1 + γ 2 (1 + 2c) 1 κ 2 R 2 (1 − c) δSix +εβ E − (1 − c)2A0 E − 2 1 − (γc)2 β. R ∆δSix = εβ. (2.18). (2.19). The differential equation system can be decoupled (using any symbolic eigenvalue solver) which leads to (1). ∆δSix. (2). ∆δSix. ¡ ¢ (1) = εκ βE 1 − β2A0 E 2 − 1 δSix ¡ ¢ (2) εκ (1 − c2 )βγ 2 E 1 + γ 2 c2 − 1 δSix = 2 2 1−γ c. (1). (2.20) (2.21). (2). L + δS R and δS L R with δSix = δSix ix ix = (1 + c)/(1 − c)δSix − δSix . The corresponding. eigenvectors in the basis of left and right eye’s weights are Ã (1). ~eS. =. 1 1. !. Ã (2). , ~eS. =. 1+c 1−c. −1. ! .. (2.22). Mode (1) corresponds to the total sum of synaptic weights with both eyes contributing equally. For c = 0, mode (2) corresponds to the difference between the strengths of the weights from the different eyes and describes the phenomenon of ocular dominance. For c 6= 0, this mode is still related to between eye segregation, although the mode is no longer symmetric with respect to the eyes..

(29) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 19. The sign of the coefficients on the right side of eqs. (2.20) and (2.21) determine the stability of the stationary state. If both coefficients are negative, small perturbations are damped and the stationary state is stable, otherwise for values greater zero the stationary state is unstable and small perturbations are amplified. Using the convolution theorem and eq. (A.34) the condition of instability for mode (1) is given by: ³ ´ (1) ˆ1 ˆ2 εκ β E − β2A E − 1 δ Sˆki ,kx > 0 0 ki ,kx ki ,kx µ ¶ 1 eqs.(2.15,2.16,2.17) ⇔ β Gkx Gki − Gk δk l Mf i x µ ¶ 1 −β2A0 2πGkx δki − δk δk − 1 > 0 l Mf x i with Gki =. (2.23). √ √ √ √ 2π/( 2σP )G(ki , 1/( 2σP )) and Gkx = 2π/σI G(kI , 1/σI ). In the case of. ki = kx = 0 the positive delta term becomes maximal. However, for l ≥ 1 and Mf ≥ 1 the negative delta terms dominate even for ki = kx = 0. The delta terms getting minimal if kx 6= 0 and ki 6= 0. For β → 0 the coefficient is negative, and mode (1) decays. If β increases, the maximum value of the coefficient increases and becomes positive for βcrit. > 1. (2.24). and the eigenmode which becomes unstable first is given by kx = ki = 0+ . That means that the smallest mode of kx and ki are responsible for the stability. Then the Gaussian functions getting maximal but δkx = δki = 0. The condition of instability for mode (2) is given by: ³ ´ εκ (2) 2 2 ˆ1 2 2 (1 − c )βγ E + γ c − 1 δ Sˆix > 0 ki ,kx 1 − γ 2 c2 µ ¶ 1 eqs.(2.15,2.16) 2 2 Gk δk + γ 2 c2 − 1 > 0 ⇔ βγ (1 − c ) Gkx Gki − l Mf i x. (2.25). For β → 0 the coefficient is negative, and mode (2) decays. If β increases, the maximum value of the coefficient increases and becomes positive for βcrit. >. 1 − (γc)2 . γ 2 (1 − c2 ). (2.26).

(30) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 20. Figure 2.3: Critical value βcrit. of the competition parameter as a function of the difference c of arborization for the onset of ocular dominance (mode (2)) and the onset of localized receptive fields (mode (1)). Values are shown for two different values of the input correlation γ. The eigenmode which becomes unstable first are given by kx = 0+ and ki = 0, since ki does not affect negative terms as in mode (1) and the positive term becomes maximal for ki = 0. Figure 2.3 shows the critical value βcrit. of the competition parameter as a function of the difference c in arborization for different values of γ and for the two modes (1) and (2) (cf. eqs. (2.24, 2.26)). Because γ ≤ 1 the critical value of β for mode (1) is always smaller than the critical value for mode (2). If β is increased from small to large values, localized receptive fields and a topographic map always form before ocular dominance segregation occurs. The critical value for mode (2) was derived by considering the stationary state with flat weights, so that the corresponding instability for the ”topographic state” may occur at a different value of β. Figure 2.2, however, shows that the receptive field size σS is up to three times larger than the size σP and the size σI of the input pattern and the cortical interaction, at least for sufficiently correlated input pattern. Hence the ”flat weight” approximation still holds to some extend and the critical value of mode (2) should be close to the critical value given by eq. (2.26), see figure 2.6. The critical values of both modes are independent of the magnification factor Mf , but they depend on the difference in arborization c. Since the difference in arborization is likely to vary across cortex, this dependency may explain some of the differences in the OD.

(31) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 21. patterns seen for varying eccentricity. The dependence on c is much stronger for the OD than for the mode (1), as can be seen in figure 2.3.. 2.4. Numerical simulations. We explored the properties of our model for a one dimensional network with N = 100 neurons for each input layer. Periodic boundary conditions were chosen to avoid edge effects. The output layer consisted of M = Mf · 100 cells taking into account the effect of cortical magnification. The number of neurons was chosen such that (i) the numerical results are close to the analytical results which hold for the continuous case (infinite number of neurons), but (ii) computation time was still reasonable. Gaussian stimuli with variance σP = 0.075/Mf (with l=1) were applied to random positions ξ (uniform probability), and each input layer got the stronger input pattern with the same probability. The stimulus was then propagated through the network (variance of the cortical interaction function was σI = 0.075), and after each stimulus presentation the synaptic weights were updated according to the Hebbian learning rule, eq. (2.7). The learning rate was set to ε = 0.002. This was found to be a good compromise between computation time and taking a proper on-line average over the input pattern. The synaptic weights were initialized as flat weights. Starting with a high value of the competition parameter (β = 17) the competition strength was increased after every 30000 stimulus presentations by a factor of 0.05. Values were chosen such that no remarkable variations in the results were found for smaller initial values of β, smaller stepsize, a higher number of pattern presentations per value of β, and for reversing the annealing schedule (high β −→ low β). + R + S L of the strengths of the For each value of β we calculated the sum Six = Six ix. ”left” and ”right” synaptic weights for every cortical location x. In order to determine the size σS of the receptive field in the stationary state, a Gaussian function was fitted to the weight distribution for every x (least square fit of variance and mean) and the value of σS was averaged over all x. Ocular dominance segregation was assessed by calculating − L − S R of the strengths of the ”left” and ”right” synaptic weights the difference Six = Six ix. for every cortical location x. The total amount of OD segregation was then quantified by − taking the average of the absolute value of Six for all i and x. + − Figure 2.4 shows plots of the sum Six and the difference Six of the connection strengths.

(32) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 22. TP. OD. output. competition strength. input. different arbors. + − Figure 2.4: Plots of the sum Six (left column, ”TP”) and the difference Six (three rightmost columns, ”OD”) of the connection strengths between the input and output layers for a one-dimensional model network. Every small rectangle corresponds to the connectivity matrix for one numerical simulation after 30000 learning iterations. Retinal and cortical locations are plotted along the vertical and horizontal axes. Left col+ umn (TP): Dark and bright indicates low and high values of Six . Parameters were: β = {1, 2.5, 3.0, 3.5, 4.0} (top to bottom) and γ = 0.6, σP = σI = 0.075, Mf = 1, c = 0. Right columns (OD): Dark and bright indicates negative and positive values − of Six . Parameters were: c = {0.2, 0.4, 0.6} (left to right).. for different model parameters using a grey value coded connectivity matrix. Every small rectangle in figure 2.4 corresponds to the connectivity matrix for one numerical simulation, with retinal and cortical locations plotted along the vertical and horizontal axes. The figure shows that the receptive field size changes from broad to narrow with increasing strength of competition (see left column ”TP”). Ocular dominance segregation occurs for higher values of β than the formation of localized receptive fields and its onset moves towards higher values of β for increasing difference c in arborization (see three right columns ”OD”). Note, that due to the unequal projection strength (quantified via the parameter c) ocular dominance as such is present for all values of β and that the value of OD(x) needs to be multilpied by 1 ± c to obtain the total segregation value. For values below the critical value of β, however, the projection is diffuse and no patterned projection, i.e. no OD stripes or patches, emerge. Figure 2.5 shows the average size of the receptive fields as a function of the strength β of the competition parameter for different values of the cortical magnification factor Mf . The size of the receptive field size is inverse proportional to the magnification factor (note.

(33) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 23. Figure 2.5: Average receptive field size as a function of the value β of the competition parameter for different values of the cortical magnification factor Mf . The critical values for mode (1) and (2) according to eqs. (2.24, 2.26) are indicated by the dashed vertical lines. Parameters: γ = 0.6, σP = 0.075/Mf , σI = 0.075 and c = 0. the log scale on the vertical axis) as expected from our calculations (eq. 2.12). Numerical simulations were done for c = 0 and γ = 0.6. According to the results of the stability analysis in section 2.3.2 the state with flat weights become unstable for β = 1 (dashed vertical line on the left). The onset of ocular dominance segregation according to eq. (2.26) is indicated by the dashed vertical line on the right. Figure 2.6 finally shows the average strength of OD segregation as a function of the value β of the competition parameter and for three different values of c. The critical values calculated from eq. (2.26) are indicated by the vertical lines. There is a reasonable agreement between the (approximative) analytical and the numerical results. The critical value of the competition parameter increases with larger values of c as expected from the analytical results.. 2.5. Summary. In this contribution, we have investigated a simple correlation based model of activity driven development and plasticity in the context of topographic maps and ocular dominance columns in primary visual cortex. We investigated the properties and the stability of two.

(34) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 24. 8. 1. 3. 5. 7. Figure 2.6: Average strength of ocular dominance segregation as a function of the value β of the competition parameter for different values of c. The vertical lines denote the analytical (2.26). ¢¯ The ocular dominance strength is given ¡ L according ¢ Pto¡ eq. P ¯¯Presults L R R ¯ by 1/M x i Six + Six . Parameters were: γ = 0.6, σP = σI = i Six − Six / 0.075 and Mf = 1. kinds of stationary states: A stationary state with flat weights (where any topography must be induced by constraints like an appropriate arbor function), and a stationary state which correspond to a topographic map with Gaussian receptive field profiles. In our analysis of the properties of fixed points of the learning dynamics (section 2.3) we found that the proportionality of receptive field size and eccentricity (as measured in macaque by van Essen and colleguages (Essen et al. [1983], Connolly and Van Essen [1984]) can be explained as a result of the change of the cortical magnification factor (inverse magnification rule). We also examined the influence of cortical magnification on the dependency of the final receptive field size on cortical competition, and we verified the inverse magnification rule for high cortical competition (as suggested in Obermayer et al. [1990]). Whereas differences in the projection strength between both eyes afferents do not change the final receptive field size they influence the onset of topographic maps and the emergence of ocular dominance. For a decreasing projection strength of the non-dominant eye a higher degree of cortical competition, i.e. a higher degree of intra-cortical interaction, is required for the onset of topographic maps and localized receptive fields. In the case of OD the degree of input correlation also influences the emergence of alternating monocular areas. For a decreasing density of arbors a higher degree of input correlation or a higher degree of cortical.

(35) CHAPTER 2. LATERAL COMPETITION IN OD AND TP MAP DEVELOPMENT 25. competition is required for the formation of ocular dominance maps. This dependency may explain the fact, that the well defined ocular dominance stripes in the primary visual cortex of the macaque tend to dissolve for higher eccentricities, giving rise to monocular areas for the highest degrees (Anderson et al. [1988], Horton and Hocking [1996a], Lund et al. [2003]). It would be interesting to experimentally measure the density of projections from the eye-specific layers of the LGN to area V1 in species like the macaque. If we assume that the degree of cortical competition is uniform across the entire primary visual cortex and the projection from LGN to V1 of the ipsilateral eye in the periphery is stronger than the projection of the contralateral eye, our analytical result provides a potential reason for the dissolving OD stripes in the macaque primary visual cortex. In addition, we confirmed the result of Piepenbrock and Obermayer [2000], Dayan [2003], namely that a certain level of cortical competition is necessary to explain stable ocular dominance and topographic map development. It remains to be answered which intracortical factors (lateral excitation and/or lateral inhibition) are finally important for this ordinary development. This issue is examined in the next chapter..

(36) Chapter 3. Nonlinearities in OD and TP map development Power-law nonlinearities and divisive normalization in Hebbian learning of topography and ocular dominance We study the potential role of power-law nonlinearities and divisive normalization in cortical response functions during the development of topographic and ocular dominance maps in the primary visual cortex. Previous work has concluded that such nonlinearities in cortical response functions (i) are able to explain experimentally observed response properties of cortical cells, such as contrast invariant orientation tuning, and (ii), as shown in the previous chapter, can explain stable feature map development with an additional divisive normalization mechanism. Here we use a Hebbian model of activity driven neural development which includes pure power-law nonlinear response functions to study their isolated role on feature map development. Furthermore, we compare this framework with the former softmax approach of the last chapter. As found for the softmax model, we show analytically that there is a critical degree of nonlinearity required for the emergence of localized receptive fields and ocular dominance. Unlike the softmax model, the pure nonlinear response function (without normalization) is not sufficient to develop stable topography, rather all cortical cells will tend to receive inputs from the same LGN cells. We show that ocular dominance emerges beyond a critical value of nonlinearity, i.e. one eye dominates, but in contrast to the softmax. 26.

(37) CHAPTER 3. NONLINEARITIES IN OD AND TP MAP DEVELOPMENT. 27. model, typical alternating ocular dominance stripes do not form. In summary, we conclude that pure nonlinearities in the response function explains localization and ocular dominance development, but within this framework a competitive (inhibitory) mechanism between cortical cells, such as divisive normalization, is necessary for the emergence of topography (of the localized receptive fields) and typical ocular dominance stripes. Therefore, since nonlinearities in the response function are thought to be an emergent property of excitatory recurrency, we show that both recurrent excitation and inhibition seem to play an important role in the emergence of stable feature maps. We quantify our analytical findings by numerical simulations.. 3.1. Introduction. Simple cells in the primary visual cortex (V1) often appear to compute a weighted sum of the light intensity distribution of the visual stimuli that fall on their receptive fields. A linear model of these cells captures a number of basic aspects of cell function. It fails to account for important response nonlinearities, such as the decrease in response gain and latency observed at high contrasts and the effects of masking by stimuli that fail to elicit responses when presented alone (Carandini et al. [1997]). To account for such nonlinear effects many phenomenological models of neuronal responses in the primary visual cortex have suggested that a cell’s firing rate should be given by its input raised to a power greater than one, i.e. cortical response function should be described by a power-law nonlinear function (Albrecht and Hamilton [1982], Emerson et al. [1989], Sclar et al. [1990], Murthy et al. [1998], Anzai et al. [1999], Gardner et al. [1999], Miller and Troyer [2002], Troyer et al. [2002]). Several studies demonstrated that such a power-law nonlinearity, for example, converts contrast-invariant input into contrast invariant output (Heeger et al. [1996], Miller et al. [2001], Miller and Troyer [2002], Troyer et al. [2002]) and its is assumed to be an emergent effect of the intra-cortical excitatory dynamics (Douglas et al. [1995]). On the other hand, several groups further proposed that various nonlinear contrast effects, such as the decrease in response gain for high contrast stimuli, could be explained with the addition of an cortico-cortical, divisive (inhibitory) normalization of cortical activity levels to the linear model (Albrecht and Geisler [1991], Heeger [1992], Carandini et al. [1997]). Little is known about the isolated role of such nonlinearities, either through power-law.

(38) CHAPTER 3. NONLINEARITIES IN OD AND TP MAP DEVELOPMENT. 28. nonlinearities or divisive normalization, on the development of cortical feature maps. The influence of theses factors could provide insights into the relevance of intra-cortical processing in the developing and/or plastic brain. In order to understand the role of (recurrent excitatory) nonlinearities and (recurrent inhibitory) normalization in cortical cell response we use here a pure power-law nonlinear response function and compare the results with the results of the softmax model. Our model architecture is similar to the architecture used in the previous chapter and consists of two input layers and one output layer with a variable degree of correlation between the input from both eyes. The input pattern is propagated through plastic synaptic weights from two input layers to one output layer. This linear geniculocortical input is applied to a nonlinear power-law function. This is followed by cortical interaction which is modelled by a linear spatial filter. The chapter is structured as follows: In section 3.2, the model architecture and its differences to the softmax approach are introduced. Section 3.3 presents am analytical treatment of the system and section 3.4 the numerical simulations results. In section 3.5, we summarize and discuss our results.. 3.2 3.2.1. The computational model Model architecture. As in the last chapter, the model consists of two LGN input layers, one for each eye and one output layer, which represents primary visual cortex (Fig. 3.1 a). Each layer has the length of l. Input and output layers are fully connected by synaptic weights S. The linear input of each cortical neuron is amplified by a power-law nonlinearity (Fig. 3.1 b, c). Local intra-cortical interactions are described by the application of a linear interaction function I of Gaussian shape. The input activity functions are given by a localized Gaussian stimuli of size σP at position i and with a certain positive correlation γ between eyes: L/R Pi. =. ( ) 1 1 ± γz (i − ξ)2 √ exp − 2 2πσp 2σp2. =. 1 1 (1 ± γz)G (i − ξ, σP ) = (1 ± γz)Pi . 2 2. (3.1).