A principle for the formation of the spatial structure of cortical feature maps

Cortical reorganization consistent with spike timing–but not correlation-dependent plasticity

The receptive fields of neurons in primary visual cortex that are inactivated by retinal damage are known to 'shift' to nondamaged retinal locations, seemingly due to the plasticity of intracortical connections. We have observed in cats that these shifts occur in a pattern that is highly convergent, even among receptive fields that are separated by large distances before inactivation. Here we show, using a computational model of primary visual cortex, that the observed convergent shifts are inconsistent with the common assumption that the underlying intracortical connection plasticity is dependent on the temporal correlation of pre- and postsynaptic action potentials. The shifts are, however, consistent with the hypothesis that this plasticity is dependent on the temporal order of pre- and postsynaptic action potentials. This convergent reorganization seems to require increased neuronal gain, revealing a mechanism that networks may use to selectively facilitate the didactic transfer of neuronal response properties.

Contribution of lateral interactions in V1 to organization of response properties

We propose a model of self-organization of synaptic connections in V1, emphasizing lateral interactions. Subject to Hebbian learning with decay, evolution of synaptic strengths proceeds to a stable state in which all synapses are either saturated, or have minimum pre/post-synaptic coincidence. The most stable configuration gives rise to anatomically realistic "local maps", each of macro-columnar size, and each organized as Mobius projections of retinotopic space. A tiling of V1, constructed of approximately mirror-image reflections of each local map by its neighbors is formed, accounting for orientation-preference singularities, linear zones, and saddle points-with each map linked by connections between sites of common orientation preference. Ocular dominance columns are partly explained as a special case of the same process. The occurrence of direction preference fractures always in odd numbers around singularities is a specific feature explained by the Mobius configuration of the local map. Effects of stimulus velocity, orientation relative to direction of motion, and extension, upon orientation preference, which are not accounted for by spatial filtering, are explained by interactions between the classic receptive field and global V1.

The Coordinated Mapping of Visual Space and Response Features in Visual Cortex

Whether general principles can explain the layouts of cortical maps remains unresolved. In primary visual cortex of ferret, the relationships between the maps of visual space and response features are predicted by a "dimension-reduction" model. The representation of visual space is anisotropic, with the elevation and azimuth axes having different magnification. This anisotropy is reflected in the orientation, ocular dominance, and spatial frequency domains, which are elongated such that their directions of rapid change, or high-gradient axes, are orthogonal to the high-gradient axis of the visual map. The feature maps are also strongly interdependent-their high-gradient regions avoid one another and intersect orthogonally where essential, so that overlap is minimized. Our results demonstrate a clear influence of the visual map on each feature map. In turn, the local representation of visual space is smooth, as predicted when many features are mapped within a cortical area.

Mirror Symmetric Topographic Maps Can Arise from Activity-Dependent Synaptic Changes

Multiple adjacent, roughly mirror-image topographic maps are commonly observed in the sensory neocortex of many species. The cortical regions occupied by these maps are generally believed to be determined initially by genetically controlled chemical markers during development, with thalamocortical afferent activity subsequently exerting a progressively increasing influence over time. Here we use a computational model to show that adjacent topographic maps with mirror-image symmetry can arise from activity-dependent synaptic changes whenever the distribution radius of afferents sufficiently exceeds that of horizontal intracortical interactions. Which map edges become adjacent is strongly influenced by the probability distribution of input stimuli during map formation. Our results suggest that activity-dependent synaptic changes may play a role in influencing how adjacent maps become oriented following the initial establishment of cortical areas via genetically determined chemical markers. Further, the model unexpectedly predicts the occasional occurrence of adjacent maps with a different rotational symmetry. We speculate that such atypically oriented maps, in the context of otherwise normally interconnected cortical regions, might contribute to abnormal cortical information processing in some neuro developmental disorders.

A Computational Model for the Development of Multiple Maps in Primary Visual Cortex

Primary visual cortex contains multiple maps of features of the visual scene, including visual field position, orientation, direction, ocular dominance and spatial frequency. The complex relationships between these maps provide clues to the strategies the cortex uses for representing and processing information. Here we simulate the combined development of all these map systems using a computational model, the elastic net. We show that this model robustly produces combined maps of these four variables that bear a close resemblance to experimental maps. In addition we show that the experimentally observed effects of monocular deprivation and single-orientation rearing can be reproduced in this model, and we make some testable predictions. These results provide strong support for the hypothesis that cortical representations attempt to optimize a trade-off between coverage and continuity.

Theoretical consideration of olfactory axon projection with an activity-dependent neural network model

Olfactory sensory neurons (OSNs) that express a given odorant receptor (OR) target their axons onto a few specific glomeruli in the olfactory bulb. Although the odorant receptor plays an indispensable role in olfactory axon targeting, the mechanisms underlying this guidance are largely unknown. In particular, there is much controversy regarding the involvement of activity-dependent mechanism in the targeting process. In this study, we developed an activity-dependent self-organization model of the glomerular layer in the olfactory bulb and simulated the targeting of olfactory axons onto the layer. Our model successfully constructed discrete glomeruli that received olfactory axons expressing a common odorant receptor through odorant-evoked neural activities. Furthermore, our model explained the perplexing experimental results that have been reported in olfactory axon targeting. For example, dispersion of olfactory axons in knockout mice for the odorant receptor gene was reasonably reproduced in the simulation. The segregated projection of the axons that express the same odorant receptor transcribed from the different alleles was also successfully simulated if the genetically modified allele was assumed to express a smaller amount of the receptor protein. The activity-dependent model even explained the inconsistent effects of disruption of the activity-evoking ion channel on axons expressing different odorant receptors, although some of these results were regarded as evidence for activity-independency of the olfactory targeting. Taken together, the activity-dependent targeting of olfactory axons seems to be a simple probable mechanism that can provide a unified explanation of glomerular formation.

Hierarchical self-organization of minicolumnar receptive fields

We study self-organization of receptive fields (RFs) of cortical minicolumns. Input driven self-organization is induced by Hebbian synaptic plasticity of afferent fibers to model minicolumns based on spiking neurons and background oscillations. If input in the form of spike patterns is presented during learning, the RFs of minicolumns hierarchically specialize to increasingly small groups of similar RFs in a series of nested group subdivisions. In a number of experiments we show that the system finds clusters of similar spike patterns, that it is capable of evenly cover the input space if the input is continuously distributed, and that it extracts basic features from input consisting of superpositions of spike patterns. With a continuous version of the bars test we, furthermore, demonstrate the system's ability to evenly cover the space of extracted basic input features. The hierarchical nature and its flexibility with respect to input distinguishes the presented type of self-organization from others including similar but non-hierarchical self-organization as discussed in [Lucke J., & von der Malsburg, C. (2004). Rapid processing and unsupervised learning in a model of the cortical macrocolumn. Neural Computation 16, 501-533]. The capabilities of the presented system match crucial properties of the plasticity of cortical RFs and we suggest it as a model for their hierarchical formation.

Pattern formation and cortical maps

The response selectivities of neurons in adult primary sensory cortices depend on intricate patterns of synaptic connections; these selectivities are arranged over cortex in equally rich fashion. Characterising these patterns, and particularly the activity-dependence (and independence) of their developmental trajectories, has been a major task for experimental and theoretical neuroscience. Here, we describe and analyse a paradigmatic algorithm for activity-dependent development of the refinement and generation of neuronal selectivities, and relate it to some of the wealth of suggestions in the literature.

Intrinsic connections in tree shrew V1 imply a global to local mapping

The local-global map hypothesis states that locally organized response properties--such as orientation preference--result from visuotopically organized local maps of non-retinotopic response properties. In the tree shrew, the lateral extent of horizontal patchy connections is as much as 80-100% of V1 and is consistent with the length summation property. We argue that neural signals can be transmitted across the entire extent of V1 and this allows the formation of maps at the local scale that are visuotopically organized. We describe mechanisms relevant to the formation of local maps and report modeling results showing the same patterns of horizontal connectivity, and relationships to orientation preference, seen in vivo. The structure of the connectivity that emerges in the simulations reveals a 'hub and spoke' organization. Singularities form the centers of local maps, and linear zones and saddle-points arise as smooth border transitions between maps. These findings are used to present the case for the local-global map hypothesis for tree shrew V1.

Symmetry considerations and development of pinwheels in visual maps

Neurons in the visual cortex respond best to rod-like stimuli of given orientation. While the preferred orientation varies continuously across most of the cortex, there are prominent pinwheel centers around which all orientations are present. Oriented segments abound in natural images and tend to be collinear; neurons are also more likely to be connected if their preferred orientations are aligned to their topographic separation. These are indications of a reduced symmetry requiring joint rotations of both orientation preference and the underlying topography. We verify that this requirement extends to cortical maps of monkey and cat by direct statistical analysis. Furthermore, analytical arguments and numerical studies indicate that pinwheels are generically stable in evolving field models that couple orientation and topography.

Shaping of receptive fields in the visual cortex during retinal maturation

We present a computational study of the formation of simple-cell receptive field patterns in the primary visual cortex. Based on the observation that the spatial frequency of the retinal filter increases postnatally, our results explain differences in the time course of the development of orientation selectivity in binocularly deprived and normally reared kittens. Development after eye-opening in normal animals is modelled by training with natural images, whereas in the case of binocular deprivation noise-like stimulation continues. Further, it is shown that different orientation selectivities are obtained for network models trained with natural images in contrast to random phase images of identical second order statistics. The latter finding suggests that higher-order statistics of the inputs influences development of primary visual cortex. Finally, we search for quantities that identify possible signatures of natural image statistics in order to specify the amount of constructiveness that visual experience has on the formation of receptive fields.

Spatial coding of position and orientation in primary visual cortex

We examined the spatial distribution of population activity in primary visual cortex (V1) of tree shrews with optical imaging and electrophysiology. A line stimulus, thinner than the average V1 receptive field, evoked a broad strip of neural activity of nearly constant size for all stimulus locations tested within the central 10 degrees of visual space. Stimuli in adjacent positions activated highly overlapping populations of neurons; nevertheless, small changes in stimulus position produced orderly changes in the location of the peak of the population response. Statistically significant shifts in the population response were found for stimulus displacements an order of magnitude smaller than receptive field width, down to the limit of optical imaging resolution. Based on the pattern of population activity, we conclude that the map of visual space in V1 is orderly at a fine scale and has uniform coverage of position and orientation without local relationships in the mapping of these features.

Self-organization in the basal ganglia with modulation of reinforcement signals

Self-organization is one of fundamental brain computations for forming efficient representations of information. Experimental support for this idea has been largely limited to the developmental and reorganizational formation of neural circuits in the sensory cortices. We now propose that self-organization may also play an important role in short-term synaptic changes in reward-driven voluntary behaviors. It has recently been shown that many neurons in the basal ganglia change their sensory responses flexibly in relation to rewards. Our computational model proposes that the rapid changes in striatal projection neurons depend on the subtle balance between the Hebb-type mechanisms of excitation and inhibition, which are modulated by reinforcement signals. Simulations based on the model are shown to produce various types of neural activity similar to those found in experiments.

A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images

The classical receptive fields of simple cells in the visual cortex have been shown to emerge from the statistical properties of natural images by forcing the cell responses to be maximally sparse, i.e. significantly activated only rarely. Here, we show that this single principle of sparseness can also lead to emergence of topography (columnar organization) and complex cell properties as well. These are obtained by maximizing the sparsenesses of locally pooled energies, which correspond to complex cell outputs. Thus, we obtain a highly parsimonious model of how these properties of the visual cortex are adapted to the characteristics of the natural input.

Intracortical origin of visual maps

Previous experiments indicate that the shape of maps of preferred orientation in the primary visual cortex does not depend on visual experience. We propose a network model that demonstrates that the orientation and direction selectivity of individual units and the structure of the corresponding angle maps could emerge from local recurrent connections. Our model reproduces the structure of preferred orientation and direction maps, and explains the origin of their interrelationship. The model also provides an explanation for the correlation between position shifts of receptive fields and changes of preferred orientations of single neurons across the surface of the cortex.

Self-organization of topographic mixture networks using attentional feedback

This article proposes a neural network model of supervised learning that employs biologically motivated constraints of using local, on-line, constructive learning. The model possesses two novel learning mechanisms. The first is a network for learning topographic mixtures. The network's internal category nodes are the mixture components, which learn to encode smooth distributions in the input space by taking advantage of topography in the input feature maps. The second mechanism is an attentional biasing feedback circuit. When the network makes an incorrect output prediction, this feedback circuit modulates the learning rates of the category nodes, by amounts based on the sharpness of their tuning, in order to improve the network's prediction accuracy. The network is evaluated on several standard classification benchmarks and shown to perform well in comparison to other classifiers.

A neural model of how horizontal and interlaminar connections of visual cortex develop into adult circuits that carry out perceptual grouping and learning

A neural model suggests how horizontal and interlaminar connections in visual cortical areas V1 and V2 develop within a laminar cortical architecture and give rise to adult visual percepts. The model suggests how mechanisms that control cortical development in the infant lead to properties of adult cortical anatomy, neurophysiology and visual perception. The model clarifies how excitatory and inhibitory connections can develop stably by maintaining a balance between excitation and inhibition. The growth of long-range excitatory horizontal connections between layer 2/3 pyramidal cells is balanced against that of short-range disynaptic interneuronal connections. The growth of excitatory on-center connections from layer 6-to-4 is balanced against that of inhibitory interneuronal off-surround connections. These balanced connections interact via intracortical and intercortical feedback to realize properties of perceptual grouping, attention and perceptual learning in the adult, and help to explain the observed variability in the number and temporal distribution of spikes emitted by cortical neurons. The model replicates cortical point spread functions and psychophysical data on the strength of real and illusory contours. The on-center, off-surround layer 6-to-4 circuit enables top-down attentional signals from area V2 to modulate, or attentionally prime, layer 4 cells in area V1 without fully activating them. This modulatory circuit also enables adult perceptual learning within cortical area V1 and V2 to proceed in a stable way.

Modular organization of frequency integration in primary auditory cortex

Two fundamental aspects of frequency analysis shape the functional organization of primary auditory cortex. For one, the decomposition of complex sounds into different frequency components is reflected in the tonotopic organization of auditory cortical fields. Second, recent findings suggest that this decomposition is carried out in parallel for a wide range of frequency resolutions by neurons with frequency receptive fields of different sizes (bandwidths). A systematic representation of the range of frequency resolution and, equivalently, spectral integration shapes the functional organization of the iso-frequency domain. Distinct subregions, or "modules," along the iso-frequency domain can be demonstrated with various measures of spectral integration, including pure-tone tuning curves, noise masking, and electrical cochlear stimulation. This modularity in the representation of spectral integration is expressed by intrinsic cortical connections. This organization has implications for our understanding of psychophysical spectral integration measures such as the critical band and general cortical coding strategies.

Visual cortex maps are optimized for uniform coverage

Cat visual cortex contains a topographic map of visual space, plus superimposed, spatially periodic maps of ocular dominance, spatial frequency and orientation. It is hypothesized that the layout of these maps is determined by two constraints: continuity or smooth mapping of stimulus properties across the cortical surface, and coverage uniformity or uniform representation of combinations of map features over visual space. Here we use a quantitative measure of coverage uniformity (c') to test the hypothesis that cortical maps are optimized for coverage. When we perturbed the spatial relationships between ocular dominance, spatial frequency and orientation maps obtained in single regions of cortex, we found that cortical maps are at a local minimum for c'. This suggests that coverage optimization is an important organizing principle governing cortical map development.

Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex

Revealing the layout of cortical maps is important both for understanding the processes involved in their development and for uncovering the mechanisms underlying neural computation. The typical organization of orientation maps in the cat visual cortex is radial; complete orientation cycles are mapped around orientation singularities. In contrast, long linear zones of orientation representation have been detected in the primary visual cortex of the tree shrew. In this study, we searched for the existence of long linear sequences and wide linear zones within orientation preference maps of the cat visual cortex. Optical imaging based on intrinsic signals was used. Long linear sequences and wide linear zones of preferred orientation were occasionally detected along the border between areas 17 and 18, as well as within area 18. Adjacent zones of distinct radial and linear organizations were observed across area 18 of a single hemisphere. However, radial and linear organizations were not necessarily segregated; long (7.5 mm) linear sequences of preferred orientation were found embedded within a typical pinwheel-like organization of orientation. We conclude that, although the radial organization is dominant, perfectly linear organization may develop and perform the processing related to orientation in the cat visual cortex.

Can Hebbian volume learning explain discontinuities in cortical maps?

It has recently been shown that orientation and retinotopic position, both of which are mapped in primary visual cortex, can show correlated jumps (Das & Gilbert, 1997). This is not consistent with maps generated by Kohonen's algorithm (Kohonen, 1982), where changes in mapped variables tend to be anticorrelated. We show that it is possible to obtain correlated jumps by introducing a Hebbian component (Hebb, 1949) into Kohonen's algorithm. This correspondents to a volume learning mechanism where synaptic facilitation depends not only on the spread of a signal from a maximally active neuron but also requires postsynaptic activity at a synapse. The maps generated by this algorithm show discontinuities across which both orientation and retinotopic position change rapidly, but these regions, which include the orientation singularities, are also aligned with the edges of ocular dominance columns, and this is not a realistic feature of cortical maps. We conclude that cortical maps are better modeled by standard, non-Hebbian volume learning, perhaps coupled with some other mechanism (e.g., that of Ernst, Pawelzik, Tsodyks, & Sejnowski, 1999) to produce receptive field shifts.

Theory of ocular dominance pattern formation

We investigate a general and analytically tractable model for the activity-dependent formation of neuronal connectivity patterns. Previous models are contained as limiting cases. As an important example we analyze the formation of ocular dominance patterns in the visual cortex. A linear stability analysis reveals that the model undergoes a Turing-type instability as a function of interaction range and receptive field size. The phase transitions is of second order. After the linear instability the patterns may reorganize which we analyze in terms of a potential for the dynamics. Our analysis demonstrates that the experimentally observed dependency of ocular dominance patterns on interocular correlations of visual experience during development can emerge according to two generic scenarios: either the system is driven through the phase transition during development thereby selecting and stabilizing the first unstable mode or a primary pattern reorganizes towards larger wavelength according their lower energy. Experimentally observing the time course of ocular dominance pattern formation will decide which scenario is realized in the brain.

Breaking Rotational Symmetry in a Self&hyphenOrganizing Map Model for Orientation Map Development

We analyze the pattern formation behavior of a high&hyphendimensional self&hyphenorganizing map (SOM) model for the competitive projection of ON&hyphencenter&hyphentype and OFF&hyphencenter&hyphentype inputs to a common map layer. We mathematically show, and numerically confirm, that even isotropic stimuli can drive the development of oriented receptive fields and an orientation map in this model. This result provides an important missing link in the spectrum of pattern formation behaviors observed in SOM models. Extending the model by including further layers for binocular inputs, we also investigate the combined development of orientation and ocular dominance maps. A parameter region for combined patterns exists; corresponding maps show a preference for perpendicular intersection angles between iso&hyphenorientation lines and ocularity domain boundaries, consistent with experimental observations.

A neural network model for the development of simple and complex cell receptive fields within cortical maps of orientation and ocular dominance

Prenatal development of the primary visual cortex leads to simple cells with spatially distinct and oriented ON and OFF subregions. These simple cells are organized into spatial maps of orientation and ocular dominance that exhibit singularities, fractures, and linear zones. On a finer spatial scale, simple cells occur that are sensitive to similar orientations but opposite contrast polarities, and exhibit both even-symmetric and odd-symmetric receptive fields. Pooling of outputs from oppositely polarized simple cells leads to complex cells that respond to both contrast polarities. A neural network model is described which simulates how simple and complex cells self-organize starting from unsegregated and unoriented geniculocortical inputs during prenatal development. Neighboring simple cells that are sensitive to opposite contrast polarities develop from a combination of spatially short-range inhibition and high-gain recurrent habituative excitation between cells that obey membrane equations. Habituation, or depression, of synapses controls reset of cell activations both through enhanced ON responses and OFF antagonistic rebounds. Orientation and ocular dominance maps form when high-gain medium-range recurrent excitation and long-range inhibition interact with the short-range mechanisms. The resulting structure clarifies how simple and complex cells contribute to perceptual processes such as texture segregation and perceptual grouping.

Influences on the global structure of cortical maps

Cortical maps often contain global spatial structure: however, theoretical accounts for their development have generally concentrated on reproducing only local structure. We show that the elastic net model of cortical map formation can closely approximate the global structure of the ocular dominance column map observed in macaque primary visual cortex. A key component is the assumption of spatially non-uniform and anisotropic correlations in the retina. This work shows how genetic and epigenetic effects could combine to establish characteristic global structure in cortical maps.

Singularities in primate orientation maps

We report the results of an analysis of orientation maps in primate striate cortex with focus on singularities and their distribution. Data were obtained from squirrel monkeys and macaque monkeys of different ages. We find the approximately 80% of singularities that are nearest neighbors have the opposite sign and that the spatial distribution of singularities differs significantly from a random distribution of points. We do not find evidence for consistent geometric patterns that singularities may form across the cortex. Except for a different overall alignment of orientation bands and different periods of repetition, maps obtained from different animals and different ages are found similar with respect to the measures used. Orientation maps are then compared with two different pattern models that are currently discussed in the literature: bandpass-filtered white noise, which accounts very well for the overall map structure, and the field analogy model, which specifies the orientation map by the location of singularities and their properties. The bandpass-filtered noise approach to orientation patterns correctly predicts the sign correlations between singularities and accounts for the deviations in the spatial distribution of singularities away from a random dot pattern. The field analogy model can account for the structure of certain local patches of the orientation map but not for the whole map. Neither of the models is completely satisfactory, and the structure of the orientation map remains to be fully explained.

Order and disorder in auditory cortical maps

Recent physiological experiments suggest that several basic receptive field properties of neurons show non-uniform spatial distributions in the primary auditory cortex of cats and primates. The spatial distribution patterns of some of these receptive field parameters are suggestive of a parallel coding scheme for processing sound information onto several superimposed cortical 'maps'. The representations of these parameters in the auditory cortex are compatible with general features of self-organizing mapping algorithms, as the spatial representations exhibit global parameter gradients with overlaid functional patchiness. Recent studies have also revealed that within a 'representational' map, the degree of local coherence varies over a wide range and that the map can contain a substantial degree of disorder in its parametric representation of sounds. Although the causes and consequences of this representational disorder are not known, it may reflect yet unresolved organizational principles in the auditory cortex.

Models of orientation and ocular dominance columns in the visual cortex: a critical comparison

Orientation and ocular dominance maps in the primary visual cortex of mammals are among the most thoroughly investigated of the patterns in the cerebral cortex. A considerable amount of work has been dedicated to unraveling both their detailed structure and the neural mechanisms that underlie their formation and development. Many schemes have been proposed, some of which are in competition. Some models focus on development of receptive fields while others focus on the structure of cortical maps, i.e., the arrangement of receptive field properties across the cortex. Each model used different means to determine its success at reproducing experimental map patterns, often relying principally on visual comparison. Experimental data are becoming available that allow a more careful evaluation of models. In this contribution more than 10 of the most prominent models of cortical map formation and structure are critically evaluated and compared with the most recent experimental findings from macaque striate cortex. Comparisons are based on properties of the predicted or measured cortical map patterns. We introduce several new measures for comparing experimental and model map data that reveal important differences between models. We expect that the use of these measures will improve current models by helping determine parameters to match model maps to experimental data now becoming available from a variety of species. Our study reveals that (1) despite apparent differences, many models are based on similar principles and consequently make similar predictions, (2) several models produce orientation map patterns that are not consistent with the experimental data from macaques, regardless of the plausibility of the models' suggested physiological implementations, and (3) no models have yet fully accounted for both the local and the global relationships between orientation and ocular dominance map patterns.

Development of oriented ocular dominance bands as a consequence of areal geometry

It has been hypothesized that the different appearance of ocular dominance bands in the cat and the monkey is a consequence of the different mapping geometries in these species (LeVay et al. 1985; Anderson et al. 1988). Here I investigate the impact of areal geometries on the preferred direction of ocular dominance bands in two adaptive map formation models, the self-organizing feature map and the elastic net algorithm. In the case of the self-organizing feature map, the occurrence of instabilities that correspond to ocular dominance bands can be analytically investigated. The instabilities automatically yield stripes of correct orientation. These analytic results are complemented by simulations. In the case of the elastic net algorithm, simulations reveal two different parameter regimes of the algorithm, only one of which leads to stripes of correct orientation. The results suggest that neighborhood preservation in visual maps is enforced in the backward direction, such that neighboring cells in the cortex have neighboring receptive fields, and not vice versa.

A Type of Duality between Self-Organizing Maps and Minimal Wiring

I show here that two interpretations of neural maps are closely related. The first, due to Kohonen, sees these maps as forming by an adaptive process in response to stimuli. The second—the minimal wiring or dimension-reduction perspective—interprets the maps as the solution of a minimization problem, where the goal is to keep the “wiring” between neurons with similar receptive fields as short as possible. Recent work by Luttrell provides a bridging concept, by showing that Kohonen's algorithm can be regarded as an approximation to gradient descent on a certain functional. I show how this functional can be generalized in a way that allows it to be interpreted as a measure of wirelength.

Modeling neural mechanisms of vertebrate habituation: locus specificity and pattern discrimination

A critical problem in neurobiology is to explain how the central nervous system coordinates pattern discrimination and locus specificity in learning. This problem is investigated in anuran amphibians who demonstrate both locus specificity and pattern discrimination in visual habituation. A neural mechanism is proposed whereby neural circuitry for pattern discrimination is shared by a spatial memory system. Such learning processes are argued to occur in the medial pallium (MP), the anuran's homolog of mammalian hippocampus. Necessary mapping from the shared network to spatial memory is set up by a mechanism that forms topographical connections, with desired orientation determined by activity gradient in presynaptic and postsynaptic layers. The model of MP is tested on both locus and stimulus specific habituation, which involve short-term as well as long-term synaptic plasticity. Successful modeling yields a set of predictions concerning MP organization and learning properties.

Design Principles of Columnar Organization in Visual Cortex

Visual space is represented by cortical cells in an orderly manner. Only little variation in the cell behavior is found with changing depth below the cortical surface, that is, all cells in a column with axis perpendicular to the cortical plane have approximately the same properties (Hubel and Wiesel 1962, 1963, 1968). Therefore, the multiple features of the visual space (e.g., position in visual space, preferred orientation, and orientation tuning strength) are mapped on a two-dimensional space, the cortical plane. Such a dimension reduction leads to complex maps (Durbin and Mitchison 1990) that so far have evaded an intuitive understanding. Analyzing optical imaging data (Blasdel 1992a, b; Blasdel and Salama 1986; Grinvald et al. 1986) using a theoretical approach we will show that the most salient features of these maps can be understood from a few basic design principles: local correlation, modularity, isotropy, and homogeneity. These principles can be defined in a mathematically exact sense in the Fourier domain by a rather simple annulus-like spectral structure. Many of the models that have been developed to explain the mapping of the preferred orientations (Cooper et al. 1979; Legendy 1978; Linsker 1986a, b; Miller 1992; Nass and Cooper 1975; Obermayer et al. 1990, 1992; Soodak 1987; Swindale 1982, 1985, 1992; von der Malsburg 1973; von der Malsburg and Cowan 1982) are quite successful in generating maps that are close to experimental maps. We suggest that this success is due to these principles, which are common properties of the models and of biological maps.

Formation of field discontinuities and islands in visual cortical maps

The representations of visual hemifields in the extrastriate areas of various species exhibit field discontinuities and islands. We propose that these violations of retinotopy are a developmental consequence of the elongated shape of the respective cortical areas. To substantiate this claim, we investigated a model of activity-driven map formation. In agreement with observations, this model yields maps with field discontinuities if the cortical areas exceed a threshold elongation. Moreover, within the same model island representations in the periphery and the area centralis can also be understood. A multistability of the solutions in the model gives a very simple explanation for the observed interindividual variability of maps in cats. The model leads to a prediction of the radial dependence of the areal magnification factor near field discontinuities, which could be accessible for a high precision mapping experiment.

Models of activity-dependent neural development

What makes a useful model of neural development? One important contribution of modeling is to demonstrate that proposed biological mechanisms can be sufficient to account for experimental results. The Von der Malsburg model is a classic example. But such demonstrations alone do not provide tools to experimentally distinguish one mechanism from another. To draw such distinctions, the connection between measurable biological quantities and developmental outcomes must be established. Perhaps the most important task for the future of developmental modeling is to deepen the connection between theory and experiment. Experimentally, this requires detailed and difficult measurements or experimental perturbations of the correlations among inputs and the intracortical connectivity existing during development. Simultaneous measurement of the maps of spatial phase and orientation of mature simple cells will provide important information for the understanding of orientation column development. Theoretically, the number of open problems is enormous. How will inclusion of additional plasticity mechanisms, such as sprouting and retraction of synapses or plasticity of intracortical connections, alter the analytical understanding thus far achieved? What precisely determines the width of orientation columns in the model presented here? Can the relationship between ocular dominance and orientation columns be understood from developmental rules in a testable way? The existing framework may be extended to a three-dimensional cortex and to more complex models of intracortical connectivity. It may also be applied to other developmental phenomena including the development of lamination in the LGN (Shatz and Stryker, 1988; Hahm et al., 1991), the formation of visual maps in experimentally altered auditory cortex (Roe et al., 1990, 1992), and the mapping of visual and auditory maps in the optic tectum (Knudsen and Brainard, 1991; Brainard and Knudsen, 1993). For each system the goal is to develop testable predictions as to the patterns of activity and connectivity that could or could not lead to the results observed given a proposed mechanism of plasticity. Incorporation of deeper levels of biophysical realism will extend, deepen, and perhaps fundamentally alter the framework presented here. An important goal for the future will be to understand the computational and functional significance of developmental rules. Activity-dependent, competitive mechanisms of synaptic plasticity appear to play an important role in many processes of late neural development, where an initially rough connectivity pattern refines to a precise, mature pattern. A prominent example is the formation of ocular dominance columns in the visual cortex of many mammals. These processes may be modeled at several levels.(ABSTRACT TRUNCATED AT 400 WORDS)

The predictive brain: temporal coincidence and temporal order in synaptic learning mechanisms

Some forms of synaptic plasticity depend on the temporal coincidence of presynaptic activity and postsynaptic response. This requirement is consistent with the Hebbian, or correlational, type of learning rule used in many neural network models. Recent evidence suggests that synaptic plasticity may depend in part on the production of a membrane permeant-diffusible signal so that spatial volume may also be involved in correlational learning rules. This latter form of synaptic change has been called volume learning. In both Hebbian and volume learning rules, interaction among synaptic inputs depends on the degree of coincidence of the inputs and is otherwise insensitive to their exact temporal order. Conditioning experiments and psychophysical studies have shown, however, that most animals are highly sensitive to the temporal order of the sensory inputs. Although these experiments assay the behavior of the entire animal or perceptual system, they raise the possibility that nervous systems may be sensitive to temporally ordered events at many spatial and temporal scales. We suggest here the existence of a new class of learning rule, called a predictive Hebbian learning rule, that is sensitive to the temporal ordering of synaptic inputs. We show how this predictive learning rule could act at single synaptic connections and through diffuse neuromodulatory systems.

Topography and ocular dominance: a model exploring positive correlations

The map from eye to brain in vertebrates is topographic, i.e., neighbouring points in the eye map to neighbouring points in the brain. In addition, when two eyes innervate the same target structure, the two sets of fibres segregate to form ocular dominance stripes. Experimental evidence from the frog and goldfish suggests that these two phenomena may be subserved by the same mechanisms. We present a computational model that addresses the formation of both topography and ocular dominance. The model is based on a form of competitive learning with subtractive enforcement of a weight normalization rule. Inputs to the model are distributed patterns of activity presented simultaneously in both eyes. An important aspect of this model is that ocular dominance segregation can occur when the two eyes are positively correlated, whereas previous models have tended to assume zero or negative correlations between the eyes. This allows investigation of the dependence of the pattern of stripes on the degree of correlation between the eyes: we find that increasing correlation leads to narrower stripes. Experiments are suggested to test this prediction.