Ординатура / Офтальмология / Английские материалы / Computational Maps in the Visual Cortex_Miikkulainen_2005

Fig. 17.3. Example Topographica screenshot. In this example session with Topographica, the user is studying the behavior of an orientation map in the primary visual cortex, using a model similar to the one depicted in Figure 17.2. The window at the bottom labeled “Orientation 1” shows the self-organized orientation map and the orientation selectivity in V1. The five windows labeled “Activity” show a sample visual image along with the responses of the retinal ganglion cells and V1 (labeled “Primary”; both the initial and the settled responses are shown). The input patterns were generated using the “Test pattern parameters” dialog at left. The window labeled “Weights 1” (lower right) shows the strengths of the connections to one neuron in V1. This neuron has afferent receptive fields in the ganglion cells and lateral receptive fields within V1. The afferent weights for 8 × 8 and 4 × 4 samplings of the V1 neurons are shown in the two “Weights Array” windows at right; most neurons are selective for Gabor-like patches of oriented lines. The inhibitory lateral connections for an 8 × 8 sampling of neurons are shown in the “Weights Array 3” window at lower left; neurons tend to receive connections from their immediate neighbors and from distant neurons of the same orientation. Topographica is designed to make this type of large-scale analysis of topographic maps practical, in addition to providing effective tools for constructing the models and their training and testing environments.

to see that many of the ideas interact. Models like LISSOM bring together several research areas and facilitate gaining a deep understanding about perceptual phenomena. Many of these ideas can be tested immediately, although some of them depend on large-scale models and high-performance computing. Importantly, at the current

408 17 Future Work: Computational Directions

rate of technological progress, computers should be powerful enough to simulate the visual cortex at realistic detail within a decade. An appropriate goal for computational neuroscience is to produce models that can make use of that power. Such confluence is likely to lead to a fundamental understanding of perception and higher brain function, and result in novel algorithms for pattern recognition and artificial vision.

18

Conclusion

In the beginning of this book, three computational hypotheses about the development, structure, and function of the visual cortex were proposed:

1.Self-organization, plasticity and perceptual phenomena in the visual cortex are mediated by a single computational process based on recurrent lateral interactions between neurons and cooperatively adapting afferent and lateral connections.

2.A functioning sensory system can be constructed from a specification of a rough initial structure, internal training pattern generators, and self-organizing algorithm that learns from both internal and environmental inputs.

3.Perceptual grouping can be established through synchronized activity between neuronal groups, mediated by self-organized lateral connections.

In order to verify these hypotheses, a unified computational theory and a concrete simulation model called LISSOM was developed based on the known biological and psychological constraints, and a number of simulated experiments were performed with it. The results strongly support the hypotheses, matching biological and psychological data, and suggesting specific biological and psychophysical experiments for further verification. In this chapter, the main contributions of each main chapter are summarized, concluding with future prospects for computational understanding of the visual cortex.

18.1 Contributions

In Chapter 4, an algorithm called LISSOM that combines the self-organization of afferents and lateral connections was developed. The afferent weights of LISSOM develop like in other self-organizing algorithms and form nonlinear approximating surfaces for input distributions. The self-organizing lateral connections, an original contribution of the model, learn correlations in activity between the neurons, dynamically modulating the map response, which allows modeling several new phenomena. In addition, an LGN component in the model makes it possible to self-organize from

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moving natural inputs. Such a design is primarily biologically motivated, with the goal of establishing a computational interpretation for several experimental observations.

In Chapter 5, LISSOM was used to demonstrate how the observed patterns of orientation, ocular dominance, direction selectivity, and their combinations can arise from input-driven self-organization, together with patchy lateral connectivity. The maps were analyzed using the same techniques as those for experimental data, and shown to agree qualitatively and quantitatively. The model developed shaped receptive fields and maps with structures similar to those in the primary visual cortex. The patterns of lateral connections follow the organization of the map, matching the experimental data known to date. The model makes several predictions about the lateral connection patterns and interactions of the different features, much of which have not been studied experimentally nor computationally before.

Chapter 6 extended the input-driven self-organization to understanding cortical plasticity. The objective was to demonstrate that the self-organized network is in a dynamic equilibrium with the inputs and reorganizes like the cortex when the inputs are altered. No single model so far had accounted for both plasticity and development of the primary visual cortex: LISSOM constitutes such a unified model. Phenomena such as compensation for blind spots and dynamic receptive fields were shown to result from the rapid reorganization of afferent and lateral connection weights. Similar reorganization occurred after lesions in the cortical network. Based on the computational simulations, the model suggested techniques to hasten recovery following stroke and cortical surgery.

Chapter 7 showed how the same self-organizing processes can account for functional phenomena in the adult. The tilt aftereffect was studied in detail, and shown to result from increased inhibition during adaptation. The results from the model match human performance very well. The direct aftereffect with small angles was shown to take place as predicted, and the model suggested a novel explanation for the indirect effect: It arises indirectly as a result of weight normalization. The study demonstrated how a computational model can be used in lieu of a biological system to gain insight into the biological process.

In Chapter 8, the HLISSOM extension of LISSOM outward to subcortical and higher level visual areas was introduced. HLISSOM is the first model to show how genetic and environmental influences can interact in multiple cortical areas. This level of detail is crucial for validating the model on experimental data, and for making specific predictions for future experiments in biology and psychology. Also crucial is that the results of the self-organizing process depend on the stream of input patterns seen during development, not on the initial connection weight values. This result was demonstrated experimentally on the HLISSOM model.

In Chapter 9, V1 neurons in HLISSOM were shown to develop biologically realistic, multi-lobed receptive fields and patterned intracortical connections through unsupervised learning of internally generated and visually evoked activity. These neurons organized into biologically realistic topographic maps, matching those found at birth and in older animals. Postnatal experience gradually modified the orientation map into a precise match to the distribution of orientations present in the en-

vironment. This smooth transition has been measured in animals, but had not been demonstrated computationally before.

In Chapter 10, prenatal internally generated activity in HLISSOM was shown to result in a newborn visual system that prefers facelike patterns, and also detects faces in real images. This hypothesis follows naturally from experimental studies in early vision, but had not been previously proposed and tested. Further, HLISSOM was used to show how postnatal learning with real faces can explain how newborns learn to prefer their mothers, and why the preference for schematic faces disappears over time. The model suggests that the psychological studies claiming that newborns learn face outlines, and the studies claiming that responses to faces in the periphery decrease over the first month of age, should probably be reinterpreted. Instead, newborns may learn all parts of the face, and only responses to specific schematic stimuli may decline.

The LISSOM model was then expanded inward to PGLISSOM in Chapter 11, by replacing the firing-rate model of the neuron with a spiking model, and opening the cortical column to include an excitatory and an inhibitory component. The map still self-organizes as before, but it now can also implement perceptual grouping functions. PGLISSOM is the first model where these two process have been brought together, showing that they can coexist and both be due to adapting lateral interaction. If inhibition is strong enough, it will drive the self-organization of the whole system, and allow excitation to implement the grouping function in the time domain.

Conditions for synchronization in the model were studied in detail in Chapter 12. Synchronization can be robustly controlled in a network of spiking neurons, by adjusting the PSP decay rate and connection range. Since decay may be easier to regulate then delay, PGLISSOM suggests that it may be the mechanism used in biological systems to modulate synchronization. Such a network can be robust against noise, provided there is strong excitation and the refractory period is long enough. Thus, the model demonstrates that synchronization may be possible even in the noisy natural environment of the neuron, which has long been an open question.

Chapter 13 presented a series of experiments demonstrating how PGLISSOM can account for perceptual grouping phenomena. Contour integration performance in the model matches human performance, and contour segmentation is achieved simultaneously in the same network. The model predicts that differences in input distribution cause the network to develop different structure and functionality, as seen in the different areas of the vision system. The network also performs contour completion, thereby accounting for a class of illusory contours as well.

In Chapter 14, the representations of visual input formed in the LISSOM map were analyzed computationally. The self-organized inhibitory lateral connections decorrelate the activation on the map, resulting in a sparse, redundancy-reduced code that retains the original information well. Such coding allows representing more information with fixed resources, but it also provides an advantage for information processing: The patterns are more easily separable and generalizable, making further visual processing such as pattern recognition easier.

In order to make future research with larger models possible, in Chapter 15 a set of scaling equations was derived, showing how quantitatively equivalent maps can

412 18 Conclusion

be developed over a wide range of simulation sizes. These equations are systematically utilized in a map growing method called GLISSOM, allowing the entire V1 be simulated at the column level with existing desktop computers.

Together, the results demonstrate a comprehensive approach to understanding the development and function of the visual cortex. They suggest that a simple but powerful set of self-organizing principles can account for a wide range of experimental results from animals and infants.

Perhaps the most important potential contribution of the LISSOM project, however, is to serve as a foundation and catalyst to further research in the area. As reviewed in Chapters 16 and 17, many future projects are possible based on the results presented in this book, some immediately, others in the near future. It is equally important to provide proper tools; the Topographica simulator reviewed in Section 17.4 is intended to serve that role, making it easy to initiate and carry out new research in computational modeling of the visual cortex.

18.2 Conclusion

The research reviewed in this book demonstrates how computational models can play a crucial role in understanding biological phenomena. In order to make a theory computational, it must be specified precisely and completely. It is then possible to test the theory as if it was the real system, in effect running simulated experiments that would be difficult to set up in biology. The results can be observed and analyzed exactly and completely, allowing insights that would otherwise not be possible. These insights must eventually be verified experimentally, but the experiments can be chosen more carefully if they are based on a solid computational theory. As our understanding of brain structures and mechanisms becomes more sophisticated, such computational models are likely to become an increasingly important part of neuroscience research.

LISSOM has already led to several insights and proposed experiments at the level of computations in cortical maps. It provides a framework for understanding the synergy of nature and nurture in development, the dynamic nature of a continuously adapting visual system, and the low-level automatic mechanisms of binding and segmentation. In the future, the same principles can be applied to understanding higher visual functions as well, and how the visual system is maintained, and how it can be repaired in case of damage.

The research in visual cortex is at an exciting stage. For the first time, we have the technology to look into the brain in enough detail to constrain computational models, and the computing power to build large models that help understand perceptual behavior. LISSOM and the Topographica software tool that accompanies it are intended to serve as a platform on which future research can be based, eventually aiming at complete understanding of computational maps in the cortex.

Appendices

A

LISSOM Simulation Specifications

Appendices A–F give the specifications and parameters for the models and computational experiments in this book. This appendix details the basic LISSOM model, starting with a generalized version of the activation equation that serves as a reference for how the activation parameters are used in practice. Later appendices describe the HLISSOM and PGLISSOM extensions, as well as the reduced LISSOM and SOM abstractions of self-organizing maps, and the experiments on sparse coding and pattern recognition. Appendix G then describes how the various map visualizations were calculated.

The specifications listed in these appendices can be used to reproduce the LISSOM simulation results on general-purpose simulation platforms, such as the Topographica simulator for cortical maps (Section 17.4). The executables and source code for this simulator are freely available at http://topographica.org; the site also contains implementations of a few LISSOM models as examples, including demos and animations illustrating how they work, specific instructions on how to run them, and how to modify them for other purposes.

A.1 Generalized Activation Equation

For clarity, the LISSOM activation equations in Chapter 4 were presented in their most concrete form, showing how activations are computed for a single retina, one pair of ON and OFF channels, and V1. This section generalizes those equations into a single equation applicable to all of the LISSOM simulations in this book, with an arbitrary number of input and LGN regions. Although more abstract, this unified version is concise and easily extensible to additional input dimensions and cortical areas in future work. This form is also the one implemented in the Topographica simulator, which makes it easy determine the simulator parameters from the values listed in this appendix.

In the general case, the activation of unit (i, j) in a LISSOM map at time t is

where the index ρ indicates a particular receptive field (RF; afferent, lateral, or feedback), Xkl (t − 1) is the activation of unit (k, l) in that receptive field, and wkl,ij is the weight from that unit to unit (i, j). The sign of scaling factor γρ is positive for afferent and lateral excitatory connections, and negative for lateral inhibitory connections. This equation can also be extended to HLISSOM by including afferent normalization as in Equation 8.1, and to PGLISSOM by including spiking as in Equations 11.3–11.6.

As an example of how this equation is used, the combined orientation, ocular dominance, and direction simulation in Section 5.6.3 consisted of two eyes, 16 LGN regions, and V1. In V1, each neuron has 16 afferent RFs (four types of lag for both ON and OFF channels for the two eyes), and two lateral RFs (excitatory and inhibitory). Thus, ρ iterates over 18 RFs while the sum of the contributions from each RF is accumulated. A sigmoid is then applied to this entire sum to determine the actual response of the neuron. Other simulations have fewer RFs, but otherwise operate through the same steps.

For the first settling iteration, the lateral contributions are zero, because all units are initialized to zero at each input presentation. Thus, Equation A.1 reduces to the initial activation equation 4.6 for the first settling step. This equation also applies to LGN units: They have only had one RF so far in this book (i.e., a single eye), but multiple RFs can be included, e.g. if color inputs are to be processed (Section 17.2.1).

A.2 Default Parameters

All of the LISSOM simulations in this book were based on the same set of default parameters, with small modifications to these defaults as necessary to study different phenomena. The default model corresponds approximately to a 5 mm×5 mm area of macaque V1; the V1 size was chosen to match the estimated number of columns in such an area and the other parameters were set to simulate it realistically. This model was introduced in Section 4.5 and used to organize an orientation map in Section 5.3. This section describes the default parameter values in detail, and later sections in this appendix show how each simulation differed from the defaults.

Although all of the parameters are listed here for completeness, most of these can be left unchanged or calculated from known values. Most of the rest can be set systematically and without an extensive search for the correct values. Each simulation has relatively few free parameters in practice, which makes it straightforward to use the model to simulate new phenomena.

Because a large number of closely related simulations will be covered by this appendix, the default parameter values are listed in a format that makes it convenient to calculate new values when some of the defaults are changed. That is, instead of listing numeric values, most parameters are shown as formulas derived from the

Table A.1. Parameters for the LISSOM reference simulation. These values define a reference simulation that serves as a basis for calculating the parameters for other simulations, as specified in Table A.2. The subscript “o” in each name stands for “original”, as in the scaling equations in Section 15.2. These parameters have the same value in every simulation.

scaling equations in Section 15.2. These formulas differ slightly from the ones in Section 15.2 because they have been extended to support networks with an LGN, to add scaling for other additional parameters, and to make it easier to change the retina and cortex sizes.

The scaling equations require that one particular network size is used as a reference from which parameters for other sizes can be calculated; Table A.1 lists these reference values. Based on these values, the default LISSOM parameters are presented in Tables A.2 and A.3. The parameters in Table A.2 are constant for any particular simulation, and the parameters in Table A.3 vary systematically throughout each simulation. Figure A.1 illustrates how the parameters for retina, LGN and V1 sizes map to each other.

Sections A.3–A.9 explain how these three tables were used to compute the parameters for each different simulation (sometimes overriding the defaults). As an example, the parameters for the default simulation, i.e. the orientation map study presented in Section 5.3 can be computed by following Table A.2 line by line. The numerical value for each parameter is calculated by filling in the constants from Table A.1 and the previous lines. For instance, parameter Nd can be calculated as 142, Ld as 24, rA as 6.5, and so on.

Most of the parameters in Table A.2 are temporary variables used only in later entries in this table and in Table A.3. They are introduced to make the notation easier to follow, and are not part of the LISSOM model itself. For the actual LISSOM parameters, the tables list the equation or section where each parameter is used. Once the LISSOM parameter values are obtained, the temporary values can be discarded.