Competition for neurotrophic factors: mathematical analysis

Terminal Schwann cell and vacant site mediated synapse elimination at developing neuromuscular junctions

Synapses undergo transition from polyinnervation by multiple axons to single innervation a few weeks after birth. Synaptic activity of axons and interaxonal competition are thought to drive this developmental synapse elimination and tested as key parameters in quantitative models for further understanding. Recent studies of muscle synapses (endplates) show that there are also terminal Schwann cells (tSCs), glial cells associated with motor neurons and their functions, and vacant sites (or vacancies) devoid of tSCs and axons proposing tSCs as key effectors of synapse elimination. However, there is no quantitative model that considers roles of tSCs including vacancies. Here we develop a stochastic model of tSC and vacancy mediated synapse elimination. It employs their areas on individual endplates quantified by electron microscopy-based analyses assuming that vacancies form randomly and are taken over by adjacent axons or tSCs. The model reliably reproduced synapse elimination whereas equal or random probability models, similar to classical interaxonal competition models, did not. Furthermore, the model showed that synapse elimination is accelerated by enhanced synaptic activity of one axon and also by increased areas of vacancies and tSCs suggesting that the areas are important structural correlates of the rate of synapse elimination.

Network Plasticity as Bayesian Inference

General results from statistical learning theory suggest to understand not only brain computations, but also brain plasticity as probabilistic inference. But a model for that has been missing. We propose that inherently stochastic features of synaptic plasticity and spine motility enable cortical networks of neurons to carry out probabilistic inference by sampling from a posterior distribution of network configurations. This model provides a viable alternative to existing models that propose convergence of parameters to maximum likelihood values. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience, how cortical networks can generalize learned information so well to novel experiences, and how they can compensate continuously for unforeseen disturbances of the network. The resulting new theory of network plasticity explains from a functional perspective a number of experimental data on stochastic aspects of synaptic plasticity that previously appeared to be quite puzzling.

Development of feedforward receptive field structure of a simple cell and its contribution to the orientation selectivity: a modeling study

Recent experimental studies of hetero-synaptic interactions in various systems have shown the role of signaling in the plasticity, challenging the conventional understanding of Hebb's rule. It has also been found that activity plays a major role in plasticity, with neurotrophins acting as molecular signals translating activity into structural changes. Furthermore, role of synaptic efficacy in biasing the outcome of competition has also been revealed recently. Motivated by these experimental findings we present a model for the development of simple cell receptive field structure based on the competitive hetero-synaptic interactions for neurotrophins combined with cooperative hetero-synaptic interactions in the spatial domain. We find that with proper balance in competition and cooperation, the inputs from two populations (ON/OFF) of LGN cells segregate starting from the homogeneous state. We obtain segregated ON and OFF regions in simple cell receptive field. Our modeling study supports the experimental findings, suggesting the role of synaptic efficacy and the role of spatial signaling. We find that using this model we obtain simple cell RF, even for positively correlated activity of ON/OFF cells. We also compare different mechanism of finding the response of cortical cell and study their possible role in the sharpening of orientation selectivity. We find that degree of selectivity improvement in individual cells varies from case to case depending upon the structure of RF field and type of sharpening mechanism.

Competition in neurite outgrowth and the development of nerve connections

During the development of the nervous system, neurons form their characteristic morphologies and become assembled into synaptically connected networks. In both neuronal morphogenesis and the development of nerve connections, competition plays an important role. Although the notion of competition is commonly used in neurobiology, there is little understanding of the nature of the competitive process and the underlying molecular and cellular mechanisms. In this chapter, we review a model of competition between outgrowing neurites, as well as various models of competition that have been proposed for the refinement of connections that takes place in the development of the neuromuscular and visual systems. We describe in detail a model that links competition in the development of nerve connections with the underlying actions and biochemistry of neurotrophic factors.

Developmental robotics: manifesto and application

We argue that all embodied organisms, whether robots or animals, face the same challenge: of adapting to bodies, brains and environments that undergo constant and inevitable change. After highlighting the evidence for the universal role of a class of molecular factors called neurotrophic factors in the response of animals to this challenge, we suggest that implementing models of neurotrophic interactions on robots may confer on them the adaptability and robustness exhibited by animals. We briefly review a mathematical model of neurotrophic interactions and then discuss its application in a robotic context. Finally, we examine the potential, or otherwise, of our approach to developmental robotics.

An analysis of synaptic normalization in a general class of Hebbian models

In standard Hebbian models of developmental synaptic plasticity, synaptic normalization must be introduced in order to constrain synaptic growth and ensure the presence of activity-dependent, competitive dynamics. In such models, multiplicative normalization cannot segregate afferents whose patterns of electrical activity are positively correlated, while subtractive normalization can. It is now widely believed that multiplicative normalization cannot segregate positively correlated afferents in any Hebbian model. However, we recently provided a counterexample to this belief by demonstrating that our own neurotrophic model of synaptic plasticity, which can segregate positively correlated afferents, can be reformulated as a nonlinear Hebbian model with competition implemented through multiplicative normalization. We now perform an analysis of a general class of Hebbian models under general forms of synaptic normalization. In particular, we extract conditions on the forms of these rules that guarantee that such models possess a fixed-point structure permitting the segregation of all but perfectly correlated afferents. We find that the failure of multiplicative normalization to segregate positively correlated afferents in a standard Hebbian model is quite atypical.

Coupling an aVLSI neuromorphic vision chip to a neurotrophic model of synaptic plasticity: the development of topography

We couple a previously studied, biologically inspired neurotrophic model of activity-dependent competitive synaptic plasticity and neuronal development to a neuromorphic retina chip. Using this system, we examine the development and refinement of a topographic mapping between an array of afferent neurons (the retinal ganglion cells) and an array of target neurons. We find that the plasticity model can indeed drive topographic refinement in the presence of afferent activity patterns generated by a real-world device. We examine the resilience of the developing system to the presence of high levels of noise by adjusting the spontaneous firing rate of the silicon neurons.

Multiplicative synaptic normalization and a nonlinear Hebb rule underlie a neurotrophic model of competitive synaptic plasticity

Synaptic normalization is used to enforce competitive dynamics in many models of developmental synaptic plasticity. In linear and semilinear Hebbian models, multiplicative synaptic normalization fails to segregate afferents whose activity patterns are positively correlated. To achieve this, the biologically problematic device of subtractive synaptic normalization must be used instead. Our own model of competition for neurotrophic support, which can segregate positively correlated afferents, was developed in part in an attempt to overcome these problems by removing the need for synaptic normalization altogether. However, we now show that the dynamics of our model decompose into two decoupled subspaces, with competitive dynamics being implemented in one of them through a nonlinear Hebb rule and multiplicative synaptic normalization. This normalization is "emergent" rather than imposed. We argue that these observations permit biologically plausible forms of synaptic normalization to be viewed as abstract and general descriptions of the underlying biology in certain scaleless models of synaptic plasticity.

Development of nerve connections under the control of neurotrophic factors: parallels with consumer-resource systems in population biology

The development of connections between neurons and their target cells involves competition between axons for target-derived neurotrophic factors. Although the notion of competition is commonly used in neurobiology, the process is not well understood, and only a few formal models exist. In population biology, in contrast, the concept of competition is well developed and has been studied by means of many formal models of consumer-resource systems. Here we show that a recently formulated model of axonal competition can be rewritten as a general consumer-resource system. This allows neurobiological phenomena to be interpreted in population biological terms and, conversely, results from population biology to be applied to neurobiology. Using findings from population biology, we have studied two extensions of our axonal competition model. In the first extension, the spatial dimension of the target is explicitly taken into account. We show that distance between axons on their target mitigates competition and permits the coexistence of axons. The model can account for the fact that in many types of neurons a positive correlation exists between the size of the dendritic tree and the number of innervating axons surviving into adulthood. In the second extension, axons are allowed to respond to more than one neurotrophic factor. We show that this permits competitive exclusion among axons of one type, while at the same time there is coexistence with axons of another type innervating the same target. The model offers an explanation for the innervation pattern found on cerebellar Purkinje cells, where climbing fibres compete with each other until only a single one remains, which coexists with parallel fibre input to the same Purkinje cell.

A model of ocular dominance column development by competition for trophic factor: effects of excess trophic factor with monocular deprivation and effects of antagonist of trophic factor

Recent experimental evidence has implicated neurotrophic factors (NTs) in the competitive process believed to drive the development of ocular dominance (OD) columns. Application of excess amounts of particular NTs can prevent the segregation process, suggesting that they could be the substance for which geniculocortical afferents compete during development. We have previously presented a model that accounts for normal OD development as well as the prevention of that development with excess NT. The model uses a Hebbian learning rule in combination with competition for a limiting supply of cortical trophic factor to drive OD segregation, without any weight normalization procedures. Subsequent experimental evidence has further suggested that NTs may be causally involved in the competitive process. Application of NT antagonist can prevent OD columns by causing inputs from both eyes to be eliminated, suggesting that NTs may be the substance for which geniculocortical afferents compete. Also, excess NT can mitigate the shift to the open eye normally caused by monocular deprivation (MD). In this article, we show that the current model can account for these subsequent experiments. We show that deprivation of NT causes inputs from both eyes to decay and that excess NT can mitigate the shift to the open eye normally seen with MD. We then present predictions of the model concerning the effects of NT on the length of the critical period during which MD is effective. The model presents a novel mechanism for competition between neural populations inspired by particular biological evidence. It accounts for three specific experimental results, and provides several testable predictions.

Activity-driven synapse elimination leads paradoxically to domination by inactive neurons

In early postnatal life, multiple motor axons converge at individual neuromuscular junctions. However, during the first few weeks after birth, a competitive mechanism eliminates all the inputs but one. This phenomenon, known as synapse elimination, is thought to result from competition based on interaxonal differences in patterns or levels of activity (for review, see Lichtman,1995). Surprisingly, experimental data support two opposite views of the role of activity: that active axons have a competitive advantage (Ribchester and Taxt, 1983; Ridge and Betz, 1984; Balice-Gordon and Lichtman, 1994) and that inactive axons have a competitive advantage (Callaway et al., 1987, 1989). To understand this paradox, we have formulated a mathematical model of activity-mediated synapse elimination. We assume that the total amount of transmitter released, rather than the frequency of release, mediates synaptic competition. We further assume that the total synaptic area that a neuron can support is metabolically constrained by its activity level and size. This model resolves the paradox by showing that a competitive advantage of higher frequency axons early in development is overcome at later stages by greater synaptic efficacy of axons firing at a lower rate. This model both provides results consistent with experiments in which activity has been manipulated and an explanation for the origin of the size principle (Henneman, 1985).

A neurotrophic model of the development of the retinogeniculocortical pathway induced by spontaneous retinal waves

The development of the retinogeniculate pathway or the geniculocortical pathway, or both, occurs either before birth or before eye opening in many species. It is widely believed that spontaneous retinal activity could drive the segregation of afferents into eye-specific laminae or columns and the refinement of initially diffuse receptive fields and the emergence of orderly, retinotopic organization. We show that a recent computational model that generates a phenomenologically accurate representation of spontaneous retinal activity can indeed drive afferent segregation and, more particularly, topographic and receptive field refinement in the retinogeniculocortical system. We use a model of anatomical synaptic plasticity based on recent data suggesting that afferents might compete for limited amounts of retrograde neurotrophic factors (NTFs). We find that afferent segregation and receptive field formation are disrupted in the presence of exogenous NTFs. We thus predict that infusion of NTFs into the lateral geniculate nucleus should disrupt normal development and that the infusion of such factors into the striate cortex should disrupt receptive field refinement in addition to the well known disruption of ocular dominance column (ODC) formation. To demonstrate that the capacity of our model of plasticity to drive normal development is not restricted just to spontaneous retinal activity, we also use a coarse representation of visually evoked activity in some simulations. We find that such simulations can exhibit the formation of ODCs followed by their disappearance, reminiscent of the New World marmoset. A decrease in interocular correlations stabilizes these ODCs. Thus we predict that divergent strabismus should render marmoset ODCs stable into adulthood.

The role of presynaptic activity in monocular deprivation: comparison of homosynaptic and heterosynaptic mechanisms

Although investigations in computational neuroscience have been extensive, the opportunity (that has made such a marked difference in physical sciences) to test detailed and subtle quantitative consequences of a theory against experimental results is rare. In this paper, we outline a testable consequence of two contrasting theories of synaptic plasticity applied to the disconnection in visual cortex of the closed eye in monocular deprivation. This disconnection is sometimes thought to be the consequence of a process that stems from a competition of inputs for a limited resource such as neurotrophin. Such a process leads to what we call spatial competition, or heterosynaptic synaptic modification. A contrasting view-exemplified by the Bienenstock, Cooper, and Munro (BCM) theory-is that patterns of input activity compete in the temporal domain. This temporal competition is homosynaptic and does not require a conserved resource. The two mechanisms, homosynaptic and heterosynaptic, are the distinguishing characteristics of two general classes of learning rules we explore by using a realistic environment composed of natural scenes. These alternative views lead to opposite dependence on the level of presynaptic activity of the rate of disconnection of the closed eye in monocular deprivation. This strong and testable consequence sets the stage for a critical distinguishing experiment. This experiment has been done and supports the second view. These results have important implications for the processes of learning and memory storage in neocortex.

Competition for neurotrophic factors: ocular dominance columns

Activity-dependent competition between afferents in the primary visual cortex of many mammals is a quintessential feature of neuronal development. From both experimental and theoretical perspectives, understanding the mechanisms underlying competition is a significant challenge. Recent experimental work suggests that geniculocortical afferents might compete for retrograde neurotrophic factors. We show that a mathematically well-characterized model of retrograde neurotrophic interactions, in which the afferent uptake of neurotrophic factors is activity-dependent and in which the average level of uptake determines the complexity of the axonal arbors of afferents, permits the anatomical segregation of geniculocortical afferents into ocular dominance columns. The model induces segregation provided that the levels of neurotrophic factors available either by activity-independent release from cortical cells or by exogenous cortical infusion are not too high; otherwise segregation breaks down. We show that the model exhibits changes in ocular dominance column periodicity in response to changes in interocular image correlations and that the model predicts that changes in intraocular image correlations should also affect columnar periodicity.