Presynaptic and postsynaptic competition in models for the development of neuromuscular connections

Single-neuron analysis of axon arbors reveals distinct presynaptic organizations between feedforward and feedback projections

The morphology and spatial distribution of axon arbors and boutons are crucial for neuron presynaptic functions. However, the principles governing their whole-brain organization at the single-neuron level remain unclear. We developed a machine-learning method to separate axon arbors from passing axons in single-neuron reconstruction from fluorescence micro-optical sectioning tomography imaging data and obtained 62,374 axon arbors that displayed distinct morphology, spatial patterns, and scaling laws dependent on neuron types and targeted brain areas. Focusing on the axon arbors in the thalamus and cortex, we revealed the segregated spatial distributions and distinct morphology but shared topographic gradients between feedforward and feedback projections. Furthermore, we uncovered an association between arbor complexity and microglia density. Finally, we found that the boutons on terminal arbors show branch-specific clustering with a log-normal distribution that again differed between feedforward and feedback terminal arbors. Together, our study revealed distinct presynaptic structural organizations underlying diverse functional innervation of single projection neurons.

Synaptic self-organization of spatio-temporal pattern selectivity

Spiking model neurons can be set up to respond selectively to specific spatio-temporal spike patterns by optimization of their input weights. It is unknown, however, if existing synaptic plasticity mechanisms can achieve this temporal mode of neuronal coding and computation. Here it is shown that changes of synaptic efficacies which tend to balance excitatory and inhibitory synaptic inputs can make neurons sensitive to particular input spike patterns. Simulations demonstrate that a combination of Hebbian mechanisms, hetero-synaptic plasticity and synaptic scaling is sufficient for self-organizing sensitivity for spatio-temporal spike patterns that repeat in the input. In networks inclusion of hetero-synaptic plasticity that depends on the pre-synaptic neurons leads to specialization and faithful representation of pattern sequences by a group of target neurons. Pattern detection is robust against a range of distortions and noise. The proposed combination of Hebbian mechanisms, hetero-synaptic plasticity and synaptic scaling is found to protect the memories for specific patterns from being overwritten by ongoing learning during extended periods when the patterns are not present. This suggests a novel explanation for the long term robustness of memory traces despite ongoing activity with substantial synaptic plasticity. Taken together, our results promote the plausibility of precise temporal coding in the brain.

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.

The impact of encoding-decoding schemes and weight normalization in spiking neural networks

Spike-timing Dependent Plasticity (STDP) is a learning mechanism that can capture causal relationships between events. STDP is considered a foundational element of memory and learning in biological neural networks. Previous research efforts endeavored to understand the functionality of STDP's learning window in spiking neural networks (SNNs). In this study, we investigate the interaction among different encoding/decoding schemes, STDP learning windows and normalization rules for the SNN classifier, trained and tested on MNIST, NIST and ETH80-Contour datasets. The results show that when no normalization rules are applied, classical STDP typically achieves the best performance. Additionally, first-spike decoding classifiers require much less decoding time than a spike count decoding classifier. Thirdly, when no normalization rule is applied, the classifier accuracy decreases as the encoding duration increases from 10ms to 34ms using count decoding scheme. Finally, normalization of output weights is shown to improve the performance of a first-spike decoding classifier, which reveals the importance of weight normalization to SNN.

Simple modification of Oja rule limits L1-norm of weight vector and leads to sparse connectivity

This letter describes a simple modification of the Oja learning rule, which asymptotically constrains the L1-norm of an input weight vector instead of the L2-norm as in the original rule. This constraining is local as opposed to commonly used instant normalizations, which require the knowledge of all input weights of a neuron to update each one of them individually. The proposed rule converges to a weight vector that is sparser (has more zero weights) than the vector learned by the original Oja rule with or without the zero bound, which could explain the developmental synaptic pruning.

Dual constraints on synapse formation and regression in schizophrenia: neuregulin, neuroligin, dysbindin, DISC1, MuSK and agrin

During adolescence there is a loss of approximately 30% of the synapses formed in the cortex during childhood. Comprehensive studies of the visual cortex show that this loss of synapses does not occur as a consequence of less appropriate projections being eliminated in favour of more appropriate ones. Rather it seems that synapses with low efficacy for transmission are eliminated in favour of those with higher efficacy. The loss of low-efficacy synapses is known, on theoretical grounds, to enhance the function of neural networks, but large synapse losses lead to failure of network function. In the dorsolateral prefrontal cortex (DLPC) of those suffering from schizophrenia the number of synapses is relatively very low, approximately 60% lower than that observed in normal childhood. It is not known if this is due to an additional loss over that during normal adolescence or whether it results from a failure to form a normal complement of synapses during childhood. The first study of synapse loss in the mammalian nervous system was made on the neuromuscular junction at Sydney University in 1974. Since then this junction has provided principal insights into the molecular basis of synapse formation and regression, so providing a paradigm for investigations of these phenomena in the DLPC. For example the molecules muscle-specific receptor tyrosine kinase (MuSK), agrin and neuregulin have been identified and their critical roles in the formation and maintenance of synapses elucidated. Loss of function of MuSK or agrin leads to failure of neuromuscular synapse formation as well as a loss of approximately 30% of excitatory synapses in the cortex. Similar synapse loss occurs on failure of neuregulin in vitro and of neuroligin in vivo. It is suggested that three important questions need to be answered: first, over what development period are the synapse numbers in DLPC of subjects with schizophrenia lower than normal; second, what are the relative importance of MuSK/agrin, neuregulin/ErB and neurexin/neuroligin in synapse formation and regression in the DLPC; and third, to what extent have these molecules gone awry in schizophrenia.

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.

Mathematical analysis of competition between sensory ganglion cells for neurotrophic factor in the skin

A model is presented of competition between sensory axons for trophic molecules (e.g. a neurotrophin such as NGF), produced in a region of skin small enough to permit their free diffusion throughout it; e.g., a touch dome, or a vibrissal follicle hair sinus. The variables specified are the number of high affinity trophic factor receptors per axon terminal and the concentration of trophic factor in the extracellular space. Previous models of this class predicted the loss of all the axons innervating the region except the one requiring least trophic factor for its maintenance, even with high rates of trophic factor production. In the present model, we have imposed upper limits to axonal growth, thereby introducing new equilibria, and we show by a global analysis using LaSalle's theorem, and also by local analysis, that several axons can then coexist if the rate of production of trophic molecules is sufficiently high.

Neuronal cell death, nerve growth factor and neurotrophic models: 50 years on

Viktor Hamburger has just died at the age of 100. It is 50 years since he and Rita Levi-Montalcini laid the foundations for the study of naturally occurring cell death and of neurotrophic factors in the nervous system. In a period of less than 10 years, from 1949 to 1958, Hamburger and Levi-Montalcini made the following seminal discoveries: that neuron cell death occurs in dorsal root ganglia, sympathetic ganglia and the cervical column of motoneurons; that the predictions arising from this observation, namely that survival is dependent on the supply of a trophic factor, could be substantiated by studying the effects of a sarcoma on the proliferation of ganglionic processes both in vivo and in vitro; and that the proliferation of these processes could be used as an assay system to isolate the factor. This work provides a short review mostly of the early history of this subject in the context of the Hamburger/Levi-Montalcini paradigm. This acts as an introduction to a consideration of models that have been proposed to account for how the different sources of growth factors provide for the survival of neurons during development. It is suggested that what has been called the 'social-control' model provides the most parsimonious quantitative description of the contribution of trophic factors to neuronal survival, a concept for which we are in debt to Viktor Hamburger and Rita Levi-Montalcini.

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.

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).

Competition for neurotrophic factor in the development of nerve connections

The development of nerve connections is thought to involve competition among axons for survival promoting factors, or neurotrophins, which are released by the cells that are innervated by the axons. Although the notion of competition is widely used within neurobiology, there is little understanding of the nature of the competitive process and the underlying mechanisms. We present a new theoretical model to analyse competition in the development of nerve connections. According to the model, the precise manner in which neurotrophins regulate the growth of axons, in particular the growth of the amount of neurotrophin receptor, determines what patterns of target innervation can develop. The regulation of neurotrophin receptors is also involved in the degeneration and regeneration of connections. Competition in our model can be influenced by factors dependent on and independent of neuronal electrical activity. Our results point to the need to measure directly the specific form of the regulation by neurotrophins of their receptors.

Synapse formation molecules in muscle and autonomic ganglia: the dual constraint hypothesis

In 1970 it was thought that if the motor-nerve supply to a muscle was interrupted and then allowed to regenerate into the muscle, motor-synaptic terminals most often formed presynaptic specializations at random positions over the surface of the constituent muscle fibres, so that the original spatial pattern of synapses was not restored. However, in the early 1970s a systematic series of experiments were carried out showing that if injury to muscles was avoided then either reinnervation or cross-reinnervation reconstituted the pattern of synapses on the muscle fibres according to an analysis using the combined techniques of electrophysiology, electronmicroscopy and histology on the muscles. It was thus shown that motor-synaptic terminals are uniquely restored to their original synaptic positions. This led to the concept of the synaptic site, defined as that region on a muscle fibre that contains molecules for triggering synaptic terminal formation. However, nerves in developing muscles were found to form connections at random positions on the surface of the very short muscle cells, indicating that these molecules are not generated by the muscle but imprinted by the nerves themselves; growth in length of the cells on either side of the imprint creates the mature synaptic site in the approximate middle of the muscle fibres. This process is accompanied at first by the differentiation of an excess number of terminals at the synaptic site, and then the elimination of all but one of the terminals. In the succeeding 25 years, identification of the synaptic site molecules has been a major task of molecular neurobiology. This review presents an historical account of the developments this century of the idea that synaptic-site formation molecules exist in muscle. The properties that these molecules must possess if they are to guide the differentiation and elimination of synaptic terminals is considered in the context of a quantitative model of this process termed the dual-constraint hypothesis. It is suggested that the molecules agrin, ARIA, MuSK and S-laminin have suitable properties according to the dual-constraint hypothesis to subserve this purpose. The extent to which there is evidence for similar molecules at neuronal synapses such as those in autonomic ganglia is also considered.

Marr's theory of the neocortex as a self-organizing neural network

Marr's proposal for the functioning of the neocortex (Marr, 1970) is the least known of his various theories for specific neural circuitries. He suggested that the neocortex learns by self-organization to extract the structure from the patterns of activity incident upon it. He proposed a feedforward neural network in which the connections to the output cells (identified with the pyramidal cells of the neocortex) are modified by a mechanism of competitive learning. It was intended that each output cell comes to be selective for the input patterns from a different class and is able to respond to new patterns from the same class that have not been seen before. The learning rule that Marr proposed was underspecified, but a logical extension of the basic idea results in a synaptic learning rule in which the total amount of synaptic strength of the connections from each input ("presynaptic") cell is kept at a constant level. In contrast, conventional competitive learning involves rules of the "postsynaptic" type. The network learns by exploiting the structure that Marr assumed to exist within the ensemble of input patterns. For this case, analysis is possible that extends that carried out by Marr, which was restricted to the binary classification task. This analysis is presented here, together with results from computer simulations of different types of competitive learning mechanisms. The presynaptic mechanism is best known in the computational neuroscience literature. In neural network applications, it may be a more suitable mechanism of competitive learning than those normally considered.

Competitive exclusion between axons dependent on a single trophic substance: a mathematical analysis

A mathematical model is presented of competition between axons for a trophic substance, such as is believed to occur particularly during development. The model is biologically realistic. The growth-stimulating activity of the trophic molecules is assumed to result from their binding to high-affinity receptors on neurons and their axons, but the model also incorporates uptake by nonneuronal cells possessing only lower affinity receptors. Plausible and fairly general assumptions are made concerning the kinetics of binding and internalization and the effects on axonal growth. The model takes into account the possibility that trophic factor production may be regulated by the afferent axons or autoregulated. The variables specified are the "axonal vigor" of each axon, representing the ability of each axon to take up trophic molecules, and the concentration of trophic molecules in the extracellular space of the axonal target region. Of the several parameters introduced, the most important turns out to be the "zero vigor-growth parameter," which is defined as the concentration of trophic molecules that gives zero growth of the vigor of a given axon. By means of a Lyapunov function, it is shown that the system will approach asymptotically to a stable equilibrium characterized by the survival of only the axon whose zero-growth parameter is lowest. Or, if several axons share the same lowest zero-growth parameter, these will all survive. The model may be particularly relevant to the elimination of polyneuronal innervation from developing muscle fibers and from autonomic ganglion cells.