Contributions of Theoretical Modeling to the Understanding of Neural Map Development

A tonically active master neuron modulates mutually exclusive motor states at two timescales

Continuity of behaviors requires animals to make smooth transitions between mutually exclusive behavioral states. Neural principles that govern these transitions are not well understood. Caenorhabditis elegans spontaneously switch between two opposite motor states, forward and backward movement, a phenomenon thought to reflect the reciprocal inhibition between interneurons AVB and AVA. Here, we report that spontaneous locomotion and their corresponding motor circuits are not separately controlled. AVA and AVB are neither functionally equivalent nor strictly reciprocally inhibitory. AVA, but not AVB, maintains a depolarized membrane potential. While AVA phasically inhibits the forward promoting interneuron AVB at a fast timescale, it maintains a tonic, extrasynaptic excitation on AVB over the longer timescale. We propose that AVA, with tonic and phasic activity of opposite polarities on different timescales, acts as a master neuron to break the symmetry between the underlying forward and backward motor circuits. This master neuron model offers a parsimonious solution for sustained locomotion consisted of mutually exclusive motor states.

Emergence of radial orientation selectivity: Effect of cell density changes and eccentricity in a layered network

We establish a simple mechanism by which radially oriented simple cells can emerge in the primary visual cortex. In 1986, R. Linsker. proposed a means by which radially symmetric, spatial opponent cells can evolve, driven entirely by noise, from structure in the initial synaptic connectivity distribution. We provide an analytical derivation of Linsker's results, and further show that radial eigenfunctions can be expressed as a weighted sum of degenerate Cartesian eigenfunctions, and vice-versa. These results are extended to allow for radially dependent cell density, from which we show that, despite a circularly symmetric synaptic connectivity distribution, radially biased orientation selectivity emerges in the third layer when cell density in the first layer, or equivalently, synaptic radius, changes with eccentricity; i.e., distance to the center of the lamina. This provides a potential mechanism for the emergence of radial orientation in the primary visual cortex before eye opening and the onset of structured visual input after birth.

Development and Reorganization of Orientation Representation in the Cat Visual Cortex: Experience-Dependent Synaptic Rewiring in Early Life

To date, numerous mathematical models have been proposed on the basis of some types of Hebbian synaptic learning to account for the activity-dependent development of orientation maps as well as neuronal orientation selectivity. These models successfully reproduced orientation map-like spatial patterns. Nevertheless, we still have questions: (1) How does synaptic rewiring occur in the visual cortex during the formation of orderly orientation maps in early life? (2) How does visual experience contribute to the maturation of orientation selectivity of visual cortical neurons and reorganize orientation maps? (3) How does the sensitive period for orientation plasticity end? In this study, we performed animal experiments and mathematical modeling to understand the mechanisms underlying synaptic rewiring for experience-dependent formation and reorganization of orientation maps. At first, we visualized orientation maps from the intrinsic signal optical imaging in area 17 of kittens reared under single-orientation exposure through cylindrical-lens-fitted goggles. The experiments revealed that the degree of expansion of cortical domains representing the experienced orientation depends on the age at which the single-orientation exposure starts. As a result, we obtained the sensitive period profile for orientation plasticity. Next, we refined our previously proposed mathematical model for the activity-dependent self-organization of thalamo-cortical inputs on the assumption that rewiring is caused by the competitive interactions among transient synaptic contacts on the same dendritic spine. Although various kinds of molecules have been reported to be involved in such interactions, we attempt to build a mathematical model to describe synaptic rewiring focusing on brain-derived neurotrophic factor (BDNF) and its related molecules. Performing computer simulations based on the refined model, we successfully reproduced orientation maps reorganized in kittens reared under single-orientation exposure as well as normal visual experience. We also reproduced the experimentally obtained sensitive period profile for orientation plasticity. The excellent agreement between experimental observations and theoretical reproductions suggests that the BDNF-induced competitive interaction among synaptic contacts from different axons on the same spine is an important factor for the experience-dependent formation and reorganization of orientation selectivity and orientation maps.

Origins of Functional Organization in the Visual Cortex

How are the complex maps for orientation selectivity (OS) created in the primary visual cortex (V1)? Rodents and rabbits have a random distribution of OS preferences across V1 while in cats, ferrets, and all primates cells with similar OS preferences cluster together into relatively wide cortical columns. Given other clear similarities in the organization of the visual pathways, why is it that maps coding OS preferences are so radically different? Prominent models have been created of cortical OS mapping that incorporate Hebbian plasticity, intracortical interactions, and the properties of growing axons. However, these models suggest that the maps arise primarily through intracortical interactions. Here we focus on several other features of the visual system and brain that may influence V1 structure. These are: eye divergence, the total number of cells in V1, the thalamocortical networks, the topography of the retina and phylogeny. We outline the evidence for and against these factors contributing to map formation. One promising theory is that the central-to-peripheral ratio (CP ratio) of retinal cell density can be used to predict whether or not a species has pinwheel maps. Animals with high CP ratios (>7) have orientation columns while those with low CP ratios (<4) have random OS maps. The CP ratio is related to the total number of cells in cortex, which also appears to be a reasonable contributing factor. However, while these factors correlate with map structure to some extent, there is a gray area where certain species do not fit elegantly into the theory. A problem with the existing literature is that OS maps have been investigated in only a small number of mammals, from a small fraction of the mammalian phylogenetic tree. We suggest four species (agouti, fruit bat, sheep, and wallaby) that have a range of interesting characteristics, which sit at intermediate locations between primates and rodents, that make them good targets for filling in the missing gaps in the literature. We make predictions about the map structures of these species based on the organization of their brains and visual systems and, in doing so, set possible paths for future research.

Neurod1 Is Essential for the Primary Tonotopic Organization and Related Auditory Information Processing in the Midbrain

Hearing depends on extracting frequency, intensity, and temporal properties from sound to generate an auditory map for acoustical signal processing. How physiology intersects with molecular specification to fine tune the developing properties of the auditory system that enable these aspects remains unclear. We made a novel conditional deletion model that eliminates the transcription factor NEUROD1 exclusively in the ear. These mice (both sexes) develop a truncated frequency range with no neuroanatomically recognizable mapping of spiral ganglion neurons onto distinct locations in the cochlea nor a cochleotopic map presenting topographically discrete projections to the cochlear nuclei. The disorganized primary cochleotopic map alters tuning properties of the inferior colliculus units, which display abnormal frequency, intensity, and temporal sound coding. At the behavioral level, animals show alterations in the acoustic startle response, consistent with altered neuroanatomical and physiological properties. We demonstrate that absence of the primary afferent topology during embryonic development leads to dysfunctional tonotopy of the auditory system. Such effects have never been investigated in other sensory systems because of the lack of comparable single gene mutation models.SIGNIFICANCE STATEMENT All sensory systems form a topographical map of neuronal projections from peripheral sensory organs to the brain. Neuronal projections in the auditory pathway are cochleotopically organized, providing a tonotopic map of sound frequencies. Primary sensory maps typically arise by molecular cues, requiring physiological refinements. Past work has demonstrated physiologic plasticity in many senses without ever molecularly undoing the specific mapping of an entire primary sensory projection. We genetically manipulated primary auditory neurons to generate a scrambled cochleotopic projection. Eliminating tonotopic representation to auditory nuclei demonstrates the inability of physiological processes to restore a tonotopic presentation of sound in the midbrain. Our data provide the first insights into the limits of physiology-mediated brainstem plasticity during the development of the auditory system.

A stochastic framework to model axon interactions within growing neuronal populations

The confined and crowded environment of developing brains imposes spatial constraints on neuronal cells that have evolved individual and collective strategies to optimize their growth. These include organizing neurons into populations extending their axons to common target territories. How individual axons interact with each other within such populations to optimize innervation is currently unclear and difficult to analyze experimentally in vivo. Here, we developed a stochastic model of 3D axon growth that takes into account spatial environmental constraints, physical interactions between neighboring axons, and branch formation. This general, predictive and robust model, when fed with parameters estimated on real neurons from the Drosophila brain, enabled the study of the mechanistic principles underlying the growth of axonal populations. First, it provided a novel explanation for the diversity of growth and branching patterns observed in vivo within populations of genetically identical neurons. Second, it uncovered that axon branching could be a strategy optimizing the overall growth of axons competing with others in contexts of high axonal density. The flexibility of this framework will make it possible to investigate the rules underlying axon growth and regeneration in the context of various neuronal populations.

Understanding how neuronal cells establish complex circuits with specific functions within a developing brain is a major current challenge. Over the last past years, enormous progress has been done to precisely resolve brain anatomy and to dissect the mechanisms controlling the establishment of precise neuronal networks. However, due to the extreme complexity of the brain, it is still experimentally difficult to investigate in vivo how neurons interact with each other and with their physical environments to innervate target territories during development. Here, we have developed a framework that integrates a dynamic 3D mathematical model of single axonal growth with parameters estimated from neurons grown in vivo and simulations of entire populations of growing axons. The emergent properties of our model enable the study of the mechanistic principles underlying the growth of axonal population in developing brains. Specifically, our results highlight the impact of mechanical interactions on both individual and collective axon growth, and uncover how branching regulate this process.

Theoretical Models of Neural Development

Constructing a functioning nervous system requires the precise orchestration of a vast array of mechanical, molecular, and neural-activity-dependent cues. Theoretical models can play a vital role in helping to frame quantitative issues, reveal mathematical commonalities between apparently diverse systems, identify what is and what is not possible in principle, and test the abilities of specific mechanisms to explain the data. This review focuses on the progress that has been made over the last decade in our theoretical understanding of neural development.

Emergence of spontaneous assembly activity in developing neural networks without afferent input

Spontaneous activity is a fundamental characteristic of the developing nervous system. Intriguingly, it often takes the form of multiple structured assemblies of neurons. Such assemblies can form even in the absence of afferent input, for instance in the zebrafish optic tectum after bilateral enucleation early in life. While the development of neural assemblies based on structured afferent input has been theoretically well-studied, it is less clear how they could arise in systems without afferent input. Here we show that a recurrent network of binary threshold neurons with initially random weights can form neural assemblies based on a simple Hebbian learning rule. Over development the network becomes increasingly modular while being driven by initially unstructured spontaneous activity, leading to the emergence of neural assemblies. Surprisingly, the set of neurons making up each assembly then continues to evolve, despite the number of assemblies remaining roughly constant. In the mature network assembly activity builds over several timesteps before the activation of the full assembly, as recently observed in calcium-imaging experiments. Our results show that Hebbian learning is sufficient to explain the emergence of highly structured patterns of neural activity in the absence of structured input.

Mechanisms of Winner-Take-All and Group Selection in Neuronal Spiking Networks

A major function of central nervous systems is to discriminate different categories or types of sensory input. Neuronal networks accomplish such tasks by learning different sensory maps at several stages of neural hierarchy, such that different neurons fire selectively to reflect different internal or external patterns and states. The exact mechanisms of such map formation processes in the brain are not completely understood. Here we study the mechanism by which a simple recurrent/reentrant neuronal network accomplish group selection and discrimination to different inputs in order to generate sensory maps. We describe the conditions and mechanism of transition from a rhythmic epileptic state (in which all neurons fire synchronized and indiscriminately to any input) to a winner-take-all state in which only a subset of neurons fire for a specific input. We prove an analytic condition under which a stable bump solution and a winner-take-all state can emerge from the local recurrent excitation-inhibition interactions in a three-layer spiking network with distinct excitatory and inhibitory populations, and demonstrate the importance of surround inhibitory connection topology on the stability of dynamic patterns in spiking neural network.

Rapid Long-Range Disynaptic Inhibition Explains the Formation of Cortical Orientation Maps

Competitive interactions are believed to underlie many types of cortical processing, ranging from memory formation, attention and development of cortical functional organization (e.g., development of orientation maps in primary visual cortex). In the latter case, the competitive interactions happen along the cortical surface, with local populations of neurons reinforcing each other, while competing with those displaced more distally. This specific configuration of lateral interactions is however in stark contrast with the known properties of the anatomical substrate, i.e., excitatory connections (mediating reinforcement) having longer reach than inhibitory ones (mediating competition). No satisfactory biologically plausible resolution of this conflict between anatomical measures, and assumed cortical function has been proposed. Recently a specific pattern of delays between different types of neurons in cat cortex has been discovered, where direct mono-synaptic excitation has approximately the same delay, as the combined delays of the disynaptic inhibitory interactions between excitatory neurons (i.e., the sum of delays from excitatory to inhibitory and from inhibitory to excitatory neurons). Here we show that this specific pattern of delays represents a biologically plausible explanation for how short-range inhibition can support competitive interactions that underlie the development of orientation maps in primary visual cortex. We demonstrate this statement analytically under simplifying conditions, and subsequently show using network simulations that development of orientation maps is preserved when long-range excitation, direct inhibitory to inhibitory interactions, and moderate inequality in the delays between excitatory and inhibitory pathways is added.

Novel Models of Visual Topographic Map Alignment in the Superior Colliculus

The establishment of precise neuronal connectivity during development is critical for sensing the external environment and informing appropriate behavioral responses. In the visual system, many connections are organized topographically, which preserves the spatial order of the visual scene. The superior colliculus (SC) is a midbrain nucleus that integrates visual inputs from the retina and primary visual cortex (V1) to regulate goal-directed eye movements. In the SC, topographically organized inputs from the retina and V1 must be aligned to facilitate integration. Previously, we showed that retinal input instructs the alignment of V1 inputs in the SC in a manner dependent on spontaneous neuronal activity; however, the mechanism of activity-dependent instruction remains unclear. To begin to address this gap, we developed two novel computational models of visual map alignment in the SC that incorporate distinct activity-dependent components. First, a Correlational Model assumes that V1 inputs achieve alignment with established retinal inputs through simple correlative firing mechanisms. A second Integrational Model assumes that V1 inputs contribute to the firing of SC neurons during alignment. Both models accurately replicate in vivo findings in wild type, transgenic and combination mutant mouse models, suggesting either activity-dependent mechanism is plausible. In silico experiments reveal distinct behaviors in response to weakening retinal drive, providing insight into the nature of the system governing map alignment depending on the activity-dependent strategy utilized. Overall, we describe novel computational frameworks of visual map alignment that accurately model many aspects of the in vivo process and propose experiments to test them.

The temporal-spatial dynamics of feature maps during monocular deprivation revealed by chronic imaging and self-organization model simulation

Experiments on the adult visual cortex of cats, ferrets and monkeys have revealed organized spatial relationships between multiple feature maps which can also be reproduced by the Kohonen and elastic net self-organization models. However, attempts to apply these models to simulate the temporal kinetics of monocular deprivation (MD) during the critical period, and their effects on the spatial arrangement of feature maps, have led to conflicting results. In this study, we performed MD and chronic imaging in the ferret visual cortex during the critical period of ocular dominance (OD) plasticity. We also used the Kohonen model to simulate the effects of MD on OD and orientation map development. Both the experiments and simulations demonstrated two general parameter-insensitive findings. Specifically, our first finding demonstrated that the OD index shift resulting from MD, and its subsequent recovery during binocular vision (BV), were both nonlinear, with a significantly stronger shift occurring during the initial period. Meanwhile, spatial reorganization of feature maps led to globally unchanged but locally shifted map patterns. In detail, we found that the periodicity of OD and orientation maps remained unchanged during, and after, deprivation. Relationships between OD and orientation maps remained similar but were significantly weakened due to OD border shifts. These results indicate that orthogonal gradient relationships between maps may be preset and are only mildly modifiable during the critical period. The Kohonen model was able to reproduce these experimental results, hence its role is further extended to the description of cortical feature map dynamics during development.

Sensory experience modifies feature map relationships in visual cortex

The extent to which brain structure is influenced by sensory input during development is a critical but controversial question. A paradigmatic system for studying this is the mammalian visual cortex. Maps of orientation preference (OP) and ocular dominance (OD) in the primary visual cortex of ferrets, cats and monkeys can be individually changed by altered visual input. However, the spatial relationship between OP and OD maps has appeared immutable. Using a computational model we predicted that biasing the visual input to orthogonal orientation in the two eyes should cause a shift of OP pinwheels towards the border of OD columns. We then confirmed this prediction by rearing cats wearing orthogonally oriented cylindrical lenses over each eye. Thus, the spatial relationship between OP and OD maps can be modified by visual experience, revealing a previously unknown degree of brain plasticity in response to sensory input.

DOI:http://dx.doi.org/10.7554/eLife.13911.001

The structure of the brain results from a combination of nature (genes) and nurture (environment). The brain’s ability to adapt to changes in the environment is known as plasticity, and the young brain is especially plastic. An animal’s sensory experiences in early life help to determine how its brain will process sensory input as an adult. One of the best sensory systems in which to study this process is the visual system.

Within the visual system, some brain cells respond only to input from the left eye and others only to input from the right eye. Cells that respond to input from the same eye are arranged to form columns. Within each column, some cells respond only to lines with a particular orientation. Cells with different preferred orientations are grouped together in patterns that resemble pinwheels. The relative positions of the pinwheels and eye-specific columns within the brain tissue belonging to the visual system have so far been robust to changes in visual experience during development, suggesting that they are determined by an animal’s genes.

However, Cloherty, Hughes et al. have now tested the unexpected predictions of a computer model. The model suggested that rearing animals so that they saw mostly vertical lines through one eye, and mostly horizontal lines through the other, would cause a form of plasticity that had never been observed before. Specifically, it would change the relative positions of the pinwheels and eye-specific columns within the visual parts of the brain. This prediction turned out to be correct. Young cats that wore special lenses – which slightly distorted what they saw but did not obviously affect their behavior – showed the predicted changes in brain structure.

The results confirm that this aspect of brain structure is partly determined by nurture, as opposed to being entirely specified by nature. A key future challenge is to identify the chemical signaling that enables sensory input to have these effects on brain structure. It might then be possible to use drugs to restore normal brain activity in cases where abnormal sensory input has altered the brain, for example in the condition known as amblyopia (or “lazy eye”).

DOI:http://dx.doi.org/10.7554/eLife.13911.002

Random Wiring, Ganglion Cell Mosaics, and the Functional Architecture of the Visual Cortex

The architecture of iso-orientation domains in the primary visual cortex (V1) of placental carnivores and primates apparently follows species invariant quantitative laws. Dynamical optimization models assuming that neurons coordinate their stimulus preferences throughout cortical circuits linking millions of cells specifically predict these invariants. This might indicate that V1's intrinsic connectome and its functional architecture adhere to a single optimization principle with high precision and robustness. To validate this hypothesis, it is critical to closely examine the quantitative predictions of alternative candidate theories. Random feedforward wiring within the retino-cortical pathway represents a conceptually appealing alternative to dynamical circuit optimization because random dimension-expanding projections are believed to generically exhibit computationally favorable properties for stimulus representations. Here, we ask whether the quantitative invariants of V1 architecture can be explained as a generic emergent property of random wiring. We generalize and examine the stochastic wiring model proposed by Ringach and coworkers, in which iso-orientation domains in the visual cortex arise through random feedforward connections between semi-regular mosaics of retinal ganglion cells (RGCs) and visual cortical neurons. We derive closed-form expressions for cortical receptive fields and domain layouts predicted by the model for perfectly hexagonal RGC mosaics. Including spatial disorder in the RGC positions considerably changes the domain layout properties as a function of disorder parameters such as position scatter and its correlations across the retina. However, independent of parameter choice, we find that the model predictions substantially deviate from the layout laws of iso-orientation domains observed experimentally. Considering random wiring with the currently most realistic model of RGC mosaic layouts, a pairwise interacting point process, the predicted layouts remain distinct from experimental observations and resemble Gaussian random fields. We conclude that V1 layout invariants are specific quantitative signatures of visual cortical optimization, which cannot be explained by generic random feedforward-wiring models.

Origins of feature selectivities and maps in the mammalian primary visual cortex

A common feature of the mammalian striate cortex is the arrangement of 'orientation domains' containing neurons preferring similar stimulus orientations. They are arranged as spokes of a pinwheel that converge at singularities known as 'pinwheel centers'. We propose that a cortical network of feedforward and intracortical lateral connections elaborates a full set of optimum orientations from geniculate inputs that show a bias to stimulus orientation and form a set of two or a small number of 'Cartesian' coordinates. Because each geniculate afferent carries signals only from one eye and its receptive field (RF) is either ON or OFF center, the network constructs also ocular dominance columns and a quasi-segregation of ON and OFF responses across the horizontal extent of the striate cortex.

Conjugating time and frequency: hemispheric specialization, acoustic uncertainty, and the mustached bat

A prominent hypothesis of hemispheric specialization for human speech and music states that the left and right auditory cortices (ACs) are respectively specialized for precise calculation of two canonically-conjugate variables: time and frequency. This spectral-temporal asymmetry does not account for sex, brain-volume, or handedness, and is in opposition to closed-system hypotheses that restrict this asymmetry to humans. Mustached bats have smaller brains, but greater ethological pressures to develop such a spectral-temporal asymmetry, than humans. Using the Heisenberg-Gabor Limit (i.e., the mathematical basis of the spectral-temporal asymmetry) to frame mustached bat literature, we show that recent findings in bat AC (1) support the notion that hemispheric specialization for speech and music is based on hemispheric differences in temporal and spectral resolution, (2) discredit closed-system, handedness, and brain-volume theories, (3) underscore the importance of sex differences, and (4) provide new avenues for phonological research.

Self-organization of control circuits for invariant fiber projections

Assuming that patterns in memory are represented as two-dimensional arrays of local features, just as they are in primary visual cortices, pattern recognition can take the form of elastic graph matching (Lades et al., 1993 ). Neural implementation of this may be based on preorganized fiber projections that can be activated rapidly with the help of control units (Wolfrum, Wolff, Lücke, & von der Malsburg, 2008 ). Each control unit governs a set of projection fibers that form part of a coherent mapping. We describe a mathematical model for the ontogenesis of the underlying connectivity based on a principle of network self-organization as described by the Häussler system (Häussler & von der Malsburg, 1983 ), modified to be sensitive to pattern similarity and to support formation of multiple mappings, each under the command of a control unit. The process takes the form of a soft-winner-take-all, where units compete for the representation of maps. We show simulations for invariant point-to-point and feature-to-feature mappings.

The mapping of eccentricity and meridional angle onto orthogonal axes in the primary visual cortex: an activity-dependent developmental model

Primate vision research has shown that in the retinotopic map of the primary visual cortex, eccentricity and meridional angle are mapped onto two orthogonal axes: whereas the eccentricity is mapped onto the nasotemporal axis, the meridional angle is mapped onto the dorsoventral axis. Theoretically such a map has been approximated by a complex log map. Neural models with correlational learning have explained the development of other visual maps like orientation maps and ocular-dominance maps. In this paper it is demonstrated that activity based mechanisms can drive a self-organizing map (SOM) into such a configuration that dilations and rotations of a particular image (in this case a rectangular bar) are mapped onto orthogonal axes. We further demonstrate using the Laterally Interconnected Synergetically Self Organizing Map (LISSOM) model, with an appropriate boundary and realistic initial conditions, that a retinotopic map which maps eccentricity and meridional angle to the horizontal and vertical axes respectively can be developed. This developed map bears a strong resemblance to the complex log map. We also simulated lesion studies which indicate that the lateral excitatory connections play a crucial role in development of the retinotopic map.

Multimap formation in visual cortex

An extrastriate visual area such as V2 or V4 contains neurons selective for a multitude of complex shapes, all sharing a common topographic organization. Simultaneously developing multiple interdigitated maps--hereafter a "multimap"--is challenging in that neurons must compete to generate a diversity of response types locally, while cooperating with their dispersed same-type neighbors to achieve uniform visual field coverage for their response type at all orientations, scales, etc. Previously proposed map development schemes have relied on smooth spatial interaction functions to establish both topography and columnar organization, but by locally homogenizing cells' response properties, local smoothing mechanisms effectively rule out multimap formation. We found in computer simulations that the key requirements for multimap development are that neurons are enabled for plasticity only within highly active regions of cortex designated "learning eligibility regions" (LERs), but within an LER, each cell's learning rate is determined only by its activity level with no dependence on location. We show that a hybrid developmental rule that combines spatial and activity-dependent learning criteria in this way successfully produces multimaps when the input stream contains multiple distinct feature types, or in the degenerate case of a single feature type, produces a V1-like map with "salt-and-pepper" structure. Our results support the hypothesis that cortical maps containing a fine mixture of different response types, whether in monkey extrastriate cortex, mouse V1 or elsewhere in the cortex, rather than signaling a breakdown of map formation mechanisms at the fine scale, are a product of a generic cortical developmental scheme designed to map cells with a diversity of response properties across a shared topographic space.

Quantitative assessment of computational models for retinotopic map formation

Molecular and activity-based cues acting together are thought to guide retinal axons to their terminal sites in vertebrate optic tectum or superior colliculus (SC) to form an ordered map of connections. The details of mechanisms involved, and the degree to which they might interact, are still not well understood. We have developed a framework within which existing computational models can be assessed in an unbiased and quantitative manner against a set of experimental data curated from the mouse retinocollicular system. Our framework facilitates comparison between models, testing new models against known phenotypes and simulating new phenotypes in existing models. We have used this framework to assess four representative models that combine Eph/ephrin gradients and/or activity-based mechanisms and competition. Two of the models were updated from their original form to fit into our framework. The models were tested against five different phenotypes: wild type, Isl2-EphA3(ki/ki), Isl2-EphA3(ki/+), ephrin-A2,A3,A5 triple knock-out (TKO), and Math5(-/-) (Atoh7). Two models successfully reproduced the extent of the Math5(-/-) anteromedial projection, but only one of those could account for the collapse point in Isl2-EphA3(ki/+). The models needed a weak anteroposterior gradient in the SC to reproduce the residual order in the ephrin-A2,A3,A5 TKO phenotype, suggesting either an incomplete knock-out or the presence of another guidance molecule. Our article demonstrates the importance of testing retinotopic models against as full a range of phenotypes as possible, and we have made available MATLAB software, we wrote to facilitate this process.

Computational and mathematical methods for morphogenetic gradient analysis, boundary formation and axonal targeting

Morphogenesis and axonal targeting are key processes during development that depend on complex interactions at molecular, cellular and tissue level. Mathematical modeling is essential to bridge this multi-scale gap in order to understand how the emergence of large structures is controlled at molecular level by interactions between various signaling pathways. We summarize mathematical modeling and computational methods for time evolution and precision of morphogenetic gradient formation. We discuss tissue patterning and the formation of borders between regions labeled by different morphogens. Finally, we review models and algorithms that reveal the interplay between morphogenetic gradients and patterned activity for axonal pathfinding and the generation of the retinotopic map in the visual system.

Chemoaffinity in topographic mapping revisited--is it more about fiber-fiber than fiber-target interactions?

Axonal projections between two populations of neurons, which preserve neighborhood relationships, are called topographic. They are ubiquitous in the brain. The development of the retinotectal projection, mapping the retinal output onto the roof of the midbrain, has been studied for decades as a model system. The rigid precision of normal retinotopic mapping has prompted the chemoaffinity hypothesis, positing axonal targeting to be based on fixed biochemical affinities between fibers and targets. In addition, however, abundant evidence has been gathered mainly in the 1970s and 80s that the mapping can adjust to variegated targets with stunning flexibility demonstrating the extraordinary robustness of the guidance process. The identification of ephrins and Eph-receptors as the underlying molecular cues has mostly been interpreted as supporting the fiber-target chemoaffinity hypothesis, while the evidence on mapping robustness has largely been neglected. By having a fresh look on the old data, we expound that they indicate, in addition to fiber-target chemoaffinity, the existence of a second autonomous guidance influence, which we call fiber-fiber chemoaffinity. Classical in vitro observations suggest both influences be composed of opposing monofunctional guidance activities. Based on the molecular evidence, we propose that those might be ephrin/Eph forward and reverse signaling, not only in fiber-target but also in fiber-fiber interactions. In fact, computational models based on this assumption can reconcile the seemingly conflicting findings on rigid and flexible topographic mapping. Supporting the suggested parsimonious and powerful mechanism, they contribute to an understanding of the evolutionary success of robust topographic mass wiring of axons.

A computational model unifies apparently contradictory findings concerning phantom pain

Amputation often leads to painful phantom sensations, whose pathogenesis is still unclear. Supported by experimental findings, an explanatory model has been proposed that identifies maladaptive reorganization of the primary somatosensory cortex (S1) as a cause of phantom pain. However, it was recently found that BOLD activity during voluntary movements of the phantom positively correlates with phantom pain rating, giving rise to a model of persistent representation. In the present study, we develop a physiologically realistic, computational model to resolve the conflicting findings. Simulations yielded that both the amount of reorganization and the level of cortical activity during phantom movements were enhanced in a scenario with strong phantom pain as compared to a scenario with weak phantom pain. These results suggest that phantom pain, maladaptive reorganization, and persistent representation may all be caused by the same underlying mechanism, which is driven by an abnormally enhanced spontaneous activity of deafferented nociceptive channels.

Mechanisms for stable, robust, and adaptive development of orientation maps in the primary visual cortex

Development of orientation maps in ferret and cat primary visual cortex (V1) has been shown to be stable, in that the earliest measurable maps are similar in form to the eventual adult map, robust, in that similar maps develop in both dark rearing and in a variety of normal visual environments, and yet adaptive, in that the final map pattern reflects the statistics of the specific visual environment. How can these three properties be reconciled? Using mechanistic models of the development of neural connectivity in V1, we show for the first time that realistic stable, robust, and adaptive map development can be achieved by including two low-level mechanisms originally motivated from single-neuron results. Specifically, contrast-gain control in the retinal ganglion cells and the lateral geniculate nucleus reduces variation in the presynaptic drive due to differences in input patterns, while homeostatic plasticity of V1 neuron excitability reduces the postsynaptic variability in firing rates. Together these two mechanisms, thought to be applicable across sensory systems in general, lead to biological maps that develop stably and robustly, yet adapt to the visual environment. The modeling results suggest that topographic map stability is a natural outcome of low-level processes of adaptation and normalization. The resulting model is more realistic, simpler, and far more robust, and is thus a good starting point for future studies of cortical map development.

Sparse coding can predict primary visual cortex receptive field changes induced by abnormal visual input

Receptive fields acquired through unsupervised learning of sparse representations of natural scenes have similar properties to primary visual cortex (V1) simple cell receptive fields. However, what drives in vivo development of receptive fields remains controversial. The strongest evidence for the importance of sensory experience in visual development comes from receptive field changes in animals reared with abnormal visual input. However, most sparse coding accounts have considered only normal visual input and the development of monocular receptive fields. Here, we applied three sparse coding models to binocular receptive field development across six abnormal rearing conditions. In every condition, the changes in receptive field properties previously observed experimentally were matched to a similar and highly faithful degree by all the models, suggesting that early sensory development can indeed be understood in terms of an impetus towards sparsity. As previously predicted in the literature, we found that asymmetries in inter-ocular correlation across orientations lead to orientation-specific binocular receptive fields. Finally we used our models to design a novel stimulus that, if present during rearing, is predicted by the sparsity principle to lead robustly to radically abnormal receptive fields.

The responses of neurons in the primary visual cortex (V1), a region of the brain involved in encoding visual input, are modified by the visual experience of the animal during development. For example, most neurons in animals reared viewing stripes of a particular orientation only respond to the orientation that the animal experienced. The responses of V1 cells in normal animals are similar to responses that simple optimisation algorithms can learn when trained on images. However, whether the similarity between these algorithms and V1 responses is merely coincidental has been unclear. Here, we used the results of a number of experiments where animals were reared with modified visual experience to test the explanatory power of three related optimisation algorithms. We did this by filtering the images for the algorithms in ways that mimicked the visual experience of the animals. This allowed us to show that the changes in V1 responses in experiment were consistent with the algorithms. This is evidence that the precepts of the algorithms, notably sparsity, can be used to understand the development of V1 responses. Further, we used our model to propose a novel rearing condition which we expect to have a dramatic effect on development.

Coordinated optimization of visual cortical maps (I) symmetry-based analysis

In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of orientation columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about a hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference. From basic symmetry assumptions we obtain a comprehensive phenomenological classification of possible inter-map coupling energies and examine representative examples. We show that each individual coupling energy leads to a different class of OP solutions with different correlations among the maps such that inferences about the optimization principle from map layout appear viable. We systematically assess whether quantitative laws resembling experimental observations can result from the coordinated optimization of orientation columns with other feature maps.

Functional maps within a single neuron

The presence and plasticity of dendritic ion channels are well established. However, the literature is divided on what specific roles these dendritic ion channels play in neuronal information processing, and there is no consensus on why neuronal dendrites should express diverse ion channels with different expression profiles. In this review, we present a case for viewing dendritic information processing through the lens of the sensory map literature, where functional gradients within neurons are considered as maps on the neuronal topograph. Under such a framework, drawing analogies from the sensory map literature, we postulate that the formation of intraneuronal functional maps is driven by the twin objectives of efficiently encoding inputs that impinge along different dendritic locations and of retaining homeostasis in the face of changes that are required in the coding process. In arriving at this postulate, we relate intraneuronal map physiology to the vast literature on sensory maps and argue that such a metaphorical association provides a fresh conceptual framework for analyzing and understanding single-neuron information encoding. We also describe instances where the metaphor presents specific directions for research on intraneuronal maps, derived from analogous pursuits in the sensory map literature. We suggest that this perspective offers a thesis for why neurons should express and alter ion channels in their dendrites and provides a framework under which active dendrites could be related to neural coding, learning theory, and homeostasis.

Reappraisal of the somatosensory homunculus and its discontinuities

Neuroscience folklore has it that somatotopy in human primary somatosensory cortex (SI) has two significant discontinuities: the hands and face map onto adjacent regions in SI, as do the feet and genitalia. It has been proposed that these conjunctions in SI result from coincident sources of stimulation in the fetal position, where the hands frequently touch the face, and the feet the genitalia. Computer modeling using a Hebbian variant of the self-organizing Kohonen net is consistent with this proposal. However, recent work reveals that the genital representation in SI for cutaneous sensations (as opposed to tumescence) is continuous with that of the lower trunk and thigh. This result, in conjunction with reports of separate face innervation and its earlier onset of sensory function, compared to that of the rest of the body, allows a reappraisal of homuncular organization. It is proposed that the somatosensory homunculus comprises two distinct somatotopic regions: the face representation and that of the rest of the body. Principles of self-organization do not account satisfactorily for the overall homuncular map. These results may serve to alert computational modelers that intrinsic developmental factors can override simple rules of plasticity.

Influence fields: a quantitative framework for representation and analysis of active dendrites

Neuronal dendrites express numerous voltage-gated ion channels (VGICs), typically with spatial gradients in their densities and properties. Dendritic VGICs, their gradients, and their plasticity endow neurons with information processing capabilities that are higher than those of neurons with passive dendrites. Despite this, frameworks that incorporate dendritic VGICs and their plasticity into neurophysiological and learning theory models have been far and few. Here, we develop a generalized quantitative framework to analyze the extent of influence of a spatially localized VGIC conductance on different physiological properties along the entire stretch of a neuron. Employing this framework, we show that the extent of influence of a VGIC conductance is largely independent of the conductance magnitude but is heavily dependent on the specific physiological property and background conductances. Morphologically, our analyses demonstrate that the influences of different VGIC conductances located on an oblique dendrite are confined within that oblique dendrite, thus providing further credence to the postulate that dendritic branches act as independent computational units. Furthermore, distinguishing between active and passive propagation of signals within a neuron, we demonstrate that the influence of a VGIC conductance is spatially confined only when propagation is active. Finally, we reconstruct functional gradients from VGIC conductance gradients using influence fields and demonstrate that the cumulative contribution of VGIC conductances in adjacent compartments plays a critical role in determining physiological properties at a given location. We suggest that our framework provides a quantitative basis for unraveling the roles of dendritic VGICs and their plasticity in neural coding, learning, and homeostasis.

Deletion of Ten-m3 induces the formation of eye dominance domains in mouse visual cortex

The visual system is characterized by precise retinotopic mapping of each eye, together with exquisitely matched binocular projections. In many species, the inputs that represent the eyes are segregated into ocular dominance columns in primary visual cortex (V1), whereas in rodents, this does not occur. Ten-m3, a member of the Ten-m/Odz/Teneurin family, regulates axonal guidance in the retinogeniculate pathway. Significantly, ipsilateral projections are expanded in the dorsal lateral geniculate nucleus and are not aligned with contralateral projections in Ten-m3 knockout (KO) mice. Here, we demonstrate the impact of altered retinogeniculate mapping on the organization and function of V1. Transneuronal tracing and c-fos immunohistochemistry demonstrate that the subcortical expansion of ipsilateral input is conveyed to V1 in Ten-m3 KOs: Ipsilateral inputs are widely distributed across V1 and are interdigitated with contralateral inputs into eye dominance domains. Segregation is confirmed by optical imaging of intrinsic signals. Single-unit recording shows ipsilateral, and contralateral inputs are mismatched at the level of single V1 neurons, and binocular stimulation leads to functional suppression of these cells. These findings indicate that the medial expansion of the binocular zone together with an interocular mismatch is sufficient to induce novel structural features, such as eye dominance domains in rodent visual cortex.

Building a mechanistic model of the development and function of the primary visual cortex

Researchers have used a very wide range of different experimental and theoretical approaches to help understand mammalian visual systems. These approaches tend to have quite different assumptions, strengths, and weaknesses. Computational models of the visual cortex, in particular, have typically implemented either a proposed circuit for part of the visual cortex of the adult, assuming a very specific wiring pattern based on findings from adults, or else attempted to explain the long-term development of a visual cortex region from an initially undifferentiated starting point. Previous models of adult V1 have been able to account for many of the measured properties of V1 neurons, while not explaining how these properties arise or why neurons have those properties in particular. Previous developmental models have been able to reproduce the overall organization of specific feature maps in V1, such as orientation maps, but are generally formulated at an abstract level that does not allow testing with real images or analysis of detailed neural properties relevant for visual function. In this review of results from a large set of new, integrative models developed from shared principles and a set of shared software components, I show how these models now represent a single, consistent explanation for a wide body of experimental evidence, and form a compact hypothesis for much of the development and behavior of neurons in the visual cortex. The models are the first developmental models with wiring consistent with V1, the first to have realistic behavior with respect to visual contrast, and the first to include all of the demonstrated visual feature dimensions. The goal is to have a comprehensive explanation for why V1 is wired as it is in the adult, and how that circuitry leads to the observed behavior of the neurons during visual tasks.

Coverage, continuity, and visual cortical architecture

BACKGROUND

The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far.

RESULTS

We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all the previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise.

CONCLUSIONS

Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.

Generalized spin models for coupled cortical feature maps obtained by coarse graining correlation based synaptic learning rules

We derive generalized spin models for the development of feedforward cortical architecture from a Hebbian synaptic learning rule in a two layer neural network with nonlinear weight constraints. Our model takes into account the effects of lateral interactions in visual cortex combining local excitation and long range effective inhibition. Our approach allows the principled derivation of developmental rules for low-dimensional feature maps, starting from high-dimensional synaptic learning rules. We incorporate the effects of smooth nonlinear constraints on net synaptic weight projected from units in the thalamic layer (the fan-out) and on the net synaptic weight received by units in the cortical layer (the fan-in). These constraints naturally couple together multiple feature maps such as orientation preference and retinotopic organization. We give a detailed illustration of the method applied to the development of the orientation preference map as a special case, in addition to deriving a model for joint pattern formation in cortical maps of orientation preference, retinotopic location, and receptive field width. We show that the combination of Hebbian learning and center-surround cortical interaction naturally leads to an orientation map development model that is closely related to the XY magnetic lattice model from statistical physics. The results presented here provide justification for phenomenological models studied in Cowan and Friedman (Advances in neural information processing systems 3, 1991), Thomas and Cowan (Phys Rev Lett 92(18):e188101, 2004) and provide a developmental model realizing the synaptic weight constraints previously assumed in Thomas and Cowan (Math Med Biol 23(2):119-138, 2006).

Self-Organization of Topographic Bilinear Networks for Invariant Recognition

We present a model for the emergence of ordered fiber projections that may serve as a basis for invariant recognition. After invariance transformations are self-organized, so-called control units competitively activate fiber projections for different transformation parameters. The model builds on a well-known ontogenetic mechanism, activity-based development of retinotopy, and it employs activity blobs of varying position and size to install different transformations. We provide a detailed analysis for the case of 1D input and output fields for schematic input patterns that shows how the model is able to develop specific mappings. We discuss results that show that the proposed learning scheme is stable for complex, biologically more realistic input patterns. Finally, we show that the model generalizes to 2D neuronal fields driven by simulated retinal waves.

Genetic dissection of EphA receptor signaling dynamics during retinotopic mapping

Retinal ganglion cells (RGCs) project axons from their cell bodies in the eye to targets in the superior colliculus of the midbrain. The wiring of these axons to their synaptic targets creates an ordered representation, or "map," of retinal space within the brain. Many lines of experiments have demonstrated that the development of this map requires complementary gradients of EphA receptor tyrosine kinases and their ephrin-A ligands, yet basic features of EphA signaling during mapping remain to be resolved. These include the individual roles played by the multiple EphA receptors that make up the retinal EphA gradient. We have developed a set of ratiometric "relative signaling" (RS) rules that quantitatively predict how the composite low-nasal-to-high-temporal EphA gradient is translated into topographic order among RGCs. A key feature of these rules is that the component receptors of the gradient--in the mouse, EphA4, EphA5, and EphA6--must be functionally equivalent and interchangeable. To test this aspect of the model, we generated compound mutant mice in which the periodicity, slope, and receptor composition of the gradient are systematically altered with respect to the levels of EphA4, EphA5, and a closely related receptor, EphA3, that we ectopically express. Analysis of the retinotopic maps of these new mouse mutants establishes the general utility of the RS rules for predicting retinocollicular topography, and demonstrates that individual EphA gene products are approximately equivalent with respect to axon guidance and target selection.

The role of long-range connectivity for the characterization of the functional-anatomical organization of the cortex

This review focuses on the role of long-range connectivity as one element of brain structure that is of key importance for the functional–anatomical organization of the cortex. In this context, we discuss the putative guiding principles for mapping brain function and structure onto the cortical surface. Such mappings reveal a high degree of functional–anatomical segregation. Given that brain regions frequently maintain characteristic connectivity profiles and the functional repertoire of a cortical area is closely related to its anatomical connections, long-range connectivity may be used to define segregated cortical areas. This methodology is called connectivity-based parcellation. Within this framework, we investigate different techniques to estimate connectivity profiles with emphasis given to non-invasive methods based on diffusion magnetic resonance imaging (dMRI) and diffusion tractography. Cortical parcellation is then defined based on similarity between diffusion tractograms, and different clustering approaches are discussed. We conclude that the use of non-invasively acquired connectivity estimates to characterize the functional–anatomical organization of the brain is a valid, relevant, and necessary endeavor. Current and future developments in dMRI technology, tractography algorithms, and models of the similarity structure hold great potential for a substantial improvement and enrichment of the results of the technique.

Computational modeling of neuronal map development: insights into disease

The study of the formation of neuronal maps in the brain has greatly increased our understanding of how the brain develops and, in some cases, regenerates. Computational modeling of neuronal map development has been invaluable in integrating complex biological phenomena and synthesizing them into quantitative and predictive frameworks. These models allow us to investigate how neuronal map development is perturbed under conditions of altered development, disease and regeneration. In this article, we use examples of activity-dependent and activity-independent models of retinotopic map formation to illustrate how they can aid our understanding of developmental and acquired disease processes. We note that fully extending these models to specific clinically relevant problems is a largely unexplored domain and suggest future work in this direction. We argue that this type of modeling will be necessary in furthering our understanding of the pathophysiology of neurological diseases and in developing treatments for them. Furthermore, we discuss how the nature of computational and theoretical approaches uniquely places them to bridge the gap between the bench and the clinic.

Intrinsic activity in the fly brain gates visual information during behavioral choices

The small insect brain is often described as an input/output system that executes reflex-like behaviors. It can also initiate neural activity and behaviors intrinsically, seen as spontaneous behaviors, different arousal states and sleep. However, less is known about how intrinsic activity in neural circuits affects sensory information processing in the insect brain and variability in behavior. Here, by simultaneously monitoring Drosophila's behavioral choices and brain activity in a flight simulator system, we identify intrinsic activity that is associated with the act of selecting between visual stimuli. We recorded neural output (multiunit action potentials and local field potentials) in the left and right optic lobes of a tethered flying Drosophila, while its attempts to follow visual motion (yaw torque) were measured by a torque meter. We show that when facing competing motion stimuli on its left and right, Drosophila typically generate large torque responses that flip from side to side. The delayed onset (0.1–1 s) and spontaneous switch-like dynamics of these responses, and the fact that the flies sometimes oppose the stimuli by flying straight, make this behavior different from the classic steering reflexes. Drosophila, thus, seem to choose one stimulus at a time and attempt to rotate toward its direction. With this behavior, the neural output of the optic lobes alternates; being augmented on the side chosen for body rotation and suppressed on the opposite side, even though the visual input to the fly eyes stays the same. Thus, the flow of information from the fly eyes is gated intrinsically. Such modulation can be noise-induced or intentional; with one possibility being that the fly brain highlights chosen information while ignoring the irrelevant, similar to what we know to occur in higher animals.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks V: self-organization schemes and weight dependence

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity on a (much) slower time scale. This paper examines the effect of STDP in a recurrently connected network stimulated by external pools of input spike trains, where both input and recurrent synapses are plastic. Our previously developed theoretical framework is extended to incorporate weight-dependent STDP and dendritic delays. The weight dynamics is determined by an interplay between the neuronal activation mechanisms, the input spike-time correlations, and the learning parameters. For the case of two external input pools, the resulting learning scheme can exhibit a symmetry breaking of the input connections such that two neuronal groups emerge, each specialized to one input pool only. In addition, we show how the recurrent connections within each neuronal group can be strengthened by STDP at the expense of those between the two groups. This neuronal self-organization can be seen as a basic dynamical ingredient for the emergence of neuronal maps induced by activity-dependent plasticity.

Gaussian process methods for estimating cortical maps

A striking feature of cortical organization is that the encoding of many stimulus features, for example orientation or direction selectivity, is arranged into topographic maps. Functional imaging methods such as optical imaging of intrinsic signals, voltage sensitive dye imaging or functional magnetic resonance imaging are important tools for studying the structure of cortical maps. As functional imaging measurements are usually noisy, statistical processing of the data is necessary to extract maps from the imaging data. We here present a probabilistic model of functional imaging data based on Gaussian processes. In comparison to conventional approaches, our model yields superior estimates of cortical maps from smaller amounts of data. In addition, we obtain quantitative uncertainty estimates, i.e. error bars on properties of the estimated map. We use our probabilistic model to study the coding properties of the map and the role of noise-correlations by decoding the stimulus from single trials of an imaging experiment.

Towards reproducible descriptions of neuronal network models

Progress in science depends on the effective exchange of ideas among scientists. New ideas can be assessed and criticized in a meaningful manner only if they are formulated precisely. This applies to simulation studies as well as to experiments and theories. But after more than 50 years of neuronal network simulations, we still lack a clear and common understanding of the role of computational models in neuroscience as well as established practices for describing network models in publications. This hinders the critical evaluation of network models as well as their re-use.

We analyze here 14 research papers proposing neuronal network models of different complexity and find widely varying approaches to model descriptions, with regard to both the means of description and the ordering and placement of material. We further observe great variation in the graphical representation of networks and the notation used in equations. Based on our observations, we propose a good model description practice, composed of guidelines for the organization of publications, a checklist for model descriptions, templates for tables presenting model structure, and guidelines for diagrams of networks. The main purpose of this good practice is to trigger a debate about the communication of neuronal network models in a manner comprehensible to humans, as opposed to machine-readable model description languages.

We believe that the good model description practice proposed here, together with a number of other recent initiatives on data-, model-, and software-sharing, may lead to a deeper and more fruitful exchange of ideas among computational neuroscientists in years to come. We further hope that work on standardized ways of describing—and thinking about—complex neuronal networks will lead the scientific community to a clearer understanding of high-level concepts in network dynamics, and will thus lead to deeper insights into the function of the brain.

Scientists make precise, testable statements about their observations and models of nature. Other scientists can then evaluate these statements and attempt to reproduce or extend them. Results that cannot be reproduced will be duly criticized to arrive at better interpretations of experimental results or better models. Over time, this discourse develops our joint scientific knowledge. A crucial condition for this process is that scientists can describe their own models in a manner that is precise and comprehensible to others. We analyze in this paper how well models of neuronal networks are described in the scientific literature and conclude that the wide variety of manners in which network models are described makes it difficult to communicate models successfully. We propose a good model description practice to improve the communication of neuronal network models.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. II. Input selectivity--symmetry breaking

Spike-timing-dependent plasticity (STDP) is believed to structure neuronal networks by slowly changing the strengths (or weights) of the synaptic connections between neurons depending upon their spiking activity, which in turn modifies the neuronal firing dynamics. In this paper, we investigate the change in synaptic weights induced by STDP in a recurrently connected network in which the input weights are plastic but the recurrent weights are fixed. The inputs are divided into two pools with identical constant firing rates and equal within-pool spike-time correlations, but with no between-pool correlations. Our analysis uses the Poisson neuron model in order to predict the evolution of the input synaptic weights and focuses on the asymptotic weight distribution that emerges due to STDP. The learning dynamics induces a symmetry breaking for the individual neurons, namely for sufficiently strong within-pool spike-time correlation each neuron specializes to one of the input pools. We show that the presence of fixed excitatory recurrent connections between neurons induces a group symmetry-breaking effect, in which neurons tend to specialize to the same input pool. Consequently STDP generates a functional structure on the input connections of the network.