Stability versus neuronal specialization for STDP: long-tail weight distributions solve the dilemma

Spike-timing-dependent plasticity (STDP) modifies the weight (or strength) of synaptic connections between neurons and is considered to be crucial for generating network structure. It has been observed in physiology that, in addition to spike timing, the weight update also depends on the current value of the weight. The functional implications of this feature are still largely unclear. Additive STDP gives rise to strong competition among synapses, but due to the absence of weight dependence, it requires hard boundaries to secure the stability of weight dynamics. Multiplicative STDP with linear weight dependence for depression ensures stability, but it lacks sufficiently strong competition required to obtain a clear synaptic specialization. A solution to this stability-versus-function dilemma can be found with an intermediate parametrization between additive and multiplicative STDP. Here we propose a novel solution to the dilemma, named log-STDP, whose key feature is a sublinear weight dependence for depression. Due to its specific weight dependence, this new model can produce significantly broad weight distributions with no hard upper bound, similar to those recently observed in experiments. Log-STDP induces graded competition between synapses, such that synapses receiving stronger input correlations are pushed further in the tail of (very) large weights. Strong weights are functionally important to enhance the neuronal response to synchronous spike volleys. Depending on the input configuration, multiple groups of correlated synaptic inputs exhibit either winner-share-all or winner-take-all behavior. When the configuration of input correlations changes, individual synapses quickly and robustly readapt to represent the new configuration. We also demonstrate the advantages of log-STDP for generating a stable structure of strong weights in a recurrently connected network. These properties of log-STDP are compared with those of previous models. Through long-tail weight distributions, log-STDP achieves both stable dynamics for and robust competition of synapses, which are crucial for spike-based information processing.     

Depression-biased reverse plasticity rule is required for stable learning at top-down connections

Top-down synapses are ubiquitous throughout neocortex and play a central role in cognition, yet little is known about their development and specificity. During sensory experience, lower neocortical areas are activated before higher ones, causing top-down synapses to experience a preponderance of post-synaptic activity preceding pre-synaptic activity. This timing pattern is the opposite of that experienced by bottom-up synapses, which suggests that different versions of spike-timing dependent synaptic plasticity (STDP) rules may be required at top-down synapses. We consider a two-layer neural network model and investigate which STDP rules can lead to a distribution of top-down synaptic weights that is stable, diverse and avoids strong loops. We introduce a temporally reversed rule (rSTDP) where top-down synapses are potentiated if post-synaptic activity precedes pre-synaptic activity. Combining analytical work and integrate-and-fire simulations, we show that only depression-biased rSTDP (and not classical STDP) produces stable and diverse top-down weights. The conclusions did not change upon addition of homeostatic mechanisms, multiplicative STDP rules or weak external input to the top neurons. Our prediction for rSTDP at top-down synapses, which are distally located, is supported by recent neurophysiological evidence showing the existence of temporally reversed STDP in synapses that are distal to the post-synaptic cell body.     

RNA conformation in rod-shaped viruses: a possible molecular model for RNA in narcissus mosaic virus.

A model for visual adaptation to spatial grating is developed based on the assumption that inhibitory synapses within the visual system may be temporarily modified as a function of recent usage. Specifically, it is hypothesized that inhibitory synaptic weights are altered as a function of the correlation between recent presynaptic and postsynaptic activity. When such modifiable synapses are incorporated into a simple neural network model having the spatial filtering properties of the human visual system, two coupled equations are obtained which may be solved analytically. The model accounts for experimental data on adaptation to sinusoidal gratings, square wave gratings, single bars, and tilted gratings. The relationship of the model to single and multiple channel models of the human visual system is discussed.     

A synaptic model for spatial frequency adaptation

A model for visual adaptation to spatial grating is developed based on the assumption that inhibitory synapses within the visual system may be temporarily modified as a function of recent usage. Specifically, it is hypothesized that inhibitory synaptic weights are altered as a function of the correlation between recent presynaptic and postsynaptic activity. When such modifiable synapses are incorporated into a simple neural network model having the spatial filtering properties of the human visual system, two coupled equations are obtained which may be solved analytically. The model accounts for experimental data on adaptation to sinusoidal gratings, square wave gratings, single bars, and tilted gratings. The relationship of the model to single and multiple channel models of the human visual system is discussed.     

An improved test for detecting multiplicative homeostatic synaptic scaling

Homeostatic scaling of synaptic strengths is essential for maintenance of network "gain", but also poses a risk of losing the distinctions among relative synaptic weights, which are possibly cellular correlates of memory storage. Multiplicative scaling of all synapses has been proposed as a mechanism that would preserve the relative weights among them, because they would all be proportionately adjusted. It is crucial for this hypothesis that all synapses be affected identically, but whether or not this actually occurs is difficult to determine directly. Mathematical tests for multiplicative synaptic scaling are presently carried out on distributions of miniature synaptic current amplitudes, but the accuracy of the test procedure has not been fully validated. We now show that the existence of an amplitude threshold for empirical detection of miniature synaptic currents limits the use of the most common method for detecting multiplicative changes. Our new method circumvents the problem by discarding the potentially distorting subthreshold values after computational scaling. This new method should be useful in assessing the underlying neurophysiological nature of a homeostatic synaptic scaling transformation, and therefore in evaluating its functional significance.     

Self-organization in Balanced State Networks by STDP and Homeostatic Plasticity

Structural inhomogeneities in synaptic efficacies have a strong impact on population response dynamics of cortical networks and are believed to play an important role in their functioning. However, little is known about how such inhomogeneities could evolve by means of synaptic plasticity. Here we present an adaptive model of a balanced neuronal network that combines two different types of plasticity, STDP and synaptic scaling. The plasticity rules yield both long-tailed distributions of synaptic weights and firing rates. Simultaneously, a highly connected subnetwork of driver neurons with strong synapses emerges. Coincident spiking activity of several driver cells can evoke population bursts and driver cells have similar dynamical properties as leader neurons found experimentally. Our model allows us to observe the delicate interplay between structural and dynamical properties of the emergent inhomogeneities. It is simple, robust to parameter changes and able to explain a multitude of different experimental findings in one basic network.     

The Formation of Multi-synaptic Connections by the Interaction of Synaptic and Structural Plasticity and Their Functional Consequences

Cortical connectivity emerges from the permanent interaction between neuronal activity and synaptic as well as structural plasticity. An important experimentally observed feature of this connectivity is the distribution of the number of synapses from one neuron to another, which has been measured in several cortical layers. All of these distributions are bimodal with one peak at zero and a second one at a small number (3–8) of synapses.In this study, using a probabilistic model of structural plasticity, which depends on the synaptic weights, we explore how these distributions can emerge and which functional consequences they have.We find that bimodal distributions arise generically from the interaction of structural plasticity with synaptic plasticity rules that fulfill the following biological realistic constraints: First, the synaptic weights have to grow with the postsynaptic activity. Second, this growth curve and/or the input-output relation of the postsynaptic neuron have to change sub-linearly (negative curvature). As most neurons show such input-output-relations, these constraints can be fulfilled by many biological reasonable systems.Given such a system, we show that the different activities, which can explain the layer-specific distributions, correspond to experimentally observed activities.Considering these activities as working point of the system and varying the pre- or postsynaptic stimulation reveals a hysteresis in the number of synapses. As a consequence of this, the connectivity between two neurons can be controlled by activity but is also safeguarded against overly fast changes.These results indicate that the complex dynamics between activity and plasticity will, already between a pair of neurons, induce a variety of possible stable synaptic distributions, which could support memory mechanisms.     

Gating of action potential propagation by an axonal A-like potassium conductance in the hippocampus: a new type of non-synaptic plasticity.

Synaptic plasticity is usually considered as the main form of activity-dependent functional plasticity in the mammalian brain. Short- and long-term regulation of synaptic transmission have been shown in various types of excitatory synapses including cortical and hippocampal synapses. In this review, we discuss a novel form of non-synaptic plasticity that involves voltage-gated K+ conductances in CA3 pyramidal cell axons. With experimental and theoretical arguments, we show that axons cannot only be considered as a simple structure that transmit reliably the action potential (AP) from the cell body to the nerve terminals. The axon is also able to express conduction failures in specific axonal pathways. We discuss possible physiological conditions in which these axonal plasticity may occur and its incidence on hippocampal network properties.     

The Influence of Synaptic Weight Distribution on Neuronal Population Dynamics

The manner in which different distributions of synaptic weights onto cortical neurons shape their spiking activity remains open. To characterize a homogeneous neuronal population, we use the master equation for generalized leaky integrate-and-fire neurons with shot-noise synapses. We develop fast semi-analytic numerical methods to solve this equation for either current or conductance synapses, with and without synaptic depression. We show that its solutions match simulations of equivalent neuronal networks better than those of the Fokker-Planck equation and we compute bounds on the network response to non-instantaneous synapses. We apply these methods to study different synaptic weight distributions in feed-forward networks. We characterize the synaptic amplitude distributions using a set of measures, called tail weight numbers, designed to quantify the preponderance of very strong synapses. Even if synaptic amplitude distributions are equated for both the total current and average synaptic weight, distributions with sparse but strong synapses produce higher responses for small inputs, leading to a larger operating range. Furthermore, despite their small number, such synapses enable the network to respond faster and with more stability in the face of external fluctuations.     

Optimal information storage and the distribution of synaptic weights: perceptron versus Purkinje cell

It is widely believed that synaptic modifications underlie learning and memory. However, few studies have examined what can be deduced about the learning process from the distribution of synaptic weights. We analyze the perceptron, a prototypical feedforward neural network, and obtain the optimal synaptic weight distribution for a perceptron with excitatory synapses. It contains more than 50% silent synapses, and this fraction increases with storage reliability: silent synapses are therefore a necessary byproduct of optimizing learning and reliability. Exploiting the classical analogy between the perceptron and the cerebellar Purkinje cell, we fitted the optimal weight distribution to that measured for granule cell-Purkinje cell synapses. The two distributions agreed well, suggesting that the Purkinje cell can learn up to 5 kilobytes of information, in the form of 40,000 input-output associations.     

Spike-timing dependent synaptic plasticity: a phenomenological framework

 In this paper a phenomenological model of spike-timing dependent synaptic plasticity (STDP) is developed that is based on a Volterra series-like expansion. Synaptic weight changes as a function of the relative timing of pre- and postsynaptic spikes are described by integral kernels that can easily be inferred from experimental data. The resulting weight dynamics can be stated in terms of statistical properties of pre- and postsynaptic spike trains. Generalizations to neurons that fire two different types of action potentials, such as cerebellar Purkinje cells where synaptic plasticity depends on correlations in two distinct presynaptic fibers, are discussed. We show that synaptic plasticity, together with strictly local bounds for the weights, can result in synaptic competition that is required for any form of pattern formation. This is illustrated by a concrete example where a single neuron equipped with STDP can selectively strengthen those synapses with presynaptic neurons that reliably deliver precisely timed spikes at the expense of other synapses which transmit spikes with a broad temporal distribution. Such a mechanism may be of vital importance for any neuronal system where information is coded in the timing of individual action potentials. Received: 23 January 2002 / Accepted: 28 March 2002 Correspondence to: W.M. Kistler (e-mail: kistler@anat.fgg.eur.nl Fax: +31 10 408 5459)     

Astrocytes in endocannabinoid signalling

Astrocytes are emerging as integral functional components of synapses, responding to synaptically released neurotransmitters and regulating synaptic transmission and plasticity. Thus, they functionally interact with neurons establishing tripartite synapses: a functional concept that refers to the existence of communication between astrocytes and neurons and its crucial role in synaptic function. Here, we discuss recent evidence showing that astrocytes are involved in the endocannabinoid (ECB) system, responding to exogenous cannabinoids as well as ECBs through activation of type 1 cannabinoid receptors, which increase intracellular calcium and stimulate the release of glutamate that modulates synaptic transmission and plasticity. We also discuss the consequences of ECB signalling in tripartite synapses on the astrocyte-mediated regulation of synaptic function, which reveal novel properties of synaptic regulation by ECBs, such as the spatially controlled dual effect on synaptic strength and the lateral potentiation of synaptic efficacy. Finally, we discuss the potential implications of ECB signalling for astrocytes in brain pathology and animal behaviour.     

Going Beyond a Mean-field Model for the Learning Cortex: Second-Order Statistics

Mean-field models of the cortex have been used successfully to interpret the origin of features on the electroencephalogram under situations such as sleep, anesthesia, and seizures. In a mean-field scheme, dynamic changes in synaptic weights can be considered through fluctuation-based Hebbian learning rules. However, because such implementations deal with population-averaged properties, they are not well suited to memory and learning applications where individual synaptic weights can be important. We demonstrate that, through an extended system of equations, the mean-field models can be developed further to look at higher-order statistics, in particular, the distribution of synaptic weights within a cortical column. This allows us to make some general conclusions on memory through a mean-field scheme. Specifically, we expect large changes in the standard deviation of the distribution of synaptic weights when fluctuation in the mean soma potentials are large, such as during the transitions between the “up” and “down” states of slow-wave sleep. Moreover, a cortex that has low structure in its neuronal connections is most likely to decrease its standard deviation in the weights of excitatory to excitatory synapses, relative to the square of the mean, whereas a cortex with strongly patterned connections is most likely to increase this measure. This suggests that fluctuations are used to condense the coding of strong (presumably useful) memories into fewer, but dynamic, neuron connections, while at the same time removing weaker (less useful) memories.     

Both NMDA and non-NMDA subtypes of glutamate receptors are concentrated at synapses on cerebral cortical neurons in culture.

N-methyl-D-aspartate (NMDA) and non-NMDA receptors play a key role in synaptic transmission and plasticity in the vertebrate central nervous system. Previous studies have suggested that although both receptor types are present at synapses, the NMDA receptors may be relatively uniformly distributed. We have combined iontophoretic mapping of NMDA and non-NMDA receptors with immunohistochemical localization of synaptic vesicles along dendrites of single neocortical neurons to determine the relationship between NMDA and non-NMDA receptor distribution and the location of synapses. We find that when corrections for glutamate diffusion are made, NMDA responses are concentrated at focal "hot spots" that coincide with non-NMDA hot spots and that there is an excellent correlation between these hot spots and synapses.     

Fine-tuning and the stability of recurrent neural networks

A central criticism of standard theoretical approaches to constructing stable, recurrent model networks is that the synaptic connection weights need to be finely-tuned. This criticism is severe because proposed rules for learning these weights have been shown to have various limitations to their biological plausibility. Hence it is unlikely that such rules are used to continuously fine-tune the network in vivo. We describe a learning rule that is able to tune synaptic weights in a biologically plausible manner. We demonstrate and test this rule in the context of the oculomotor integrator, showing that only known neural signals are needed to tune the weights. We demonstrate that the rule appropriately accounts for a wide variety of experimental results, and is robust under several kinds of perturbation. Furthermore, we show that the rule is able to achieve stability as good as or better than that provided by the linearly optimal weights often used in recurrent models of the integrator. Finally, we discuss how this rule can be generalized to tune a wide variety of recurrent attractor networks, such as those found in head direction and path integration systems, suggesting that it may be used to tune a wide variety of stable neural systems.     

N-Cadherin Relocalizes from the Periphery to the Center of the Synapse after Transient Synaptic Stimulation in Hippocampal Neurons

N-cadherin is a cell adhesion molecule which is enriched at synapses. Binding of N-cadherin molecules to each other across the synaptic cleft has been postulated to stabilize adhesion between the presynaptic bouton and the postsynaptic terminal. N-cadherin is also required for activity-induced changes at synapses, including hippocampal long term potentiation and activity-induced spine expansion and stabilization. We hypothesized that these activity-dependent changes might involve changes in N-cadherin localization within synapses. To determine whether synaptic activity changes the localization of N-cadherin, we used structured illumination microscopy, a super-resolution approach which overcomes the conventional resolution limits of light microscopy, to visualize the localization of N-cadherin within synapses of hippocampal neurons. We found that synaptic N-cadherin exhibits a spectrum of localization patterns, ranging from puncta at the periphery of the synapse adjacent to the active zone to an even distribution along the synaptic cleft. Furthermore, the N-cadherin localization pattern within synapses changes during KCl depolarization and after transient synaptic stimulation. During KCl depolarization, N-cadherin relocalizes away from the central region of the synaptic cleft to the periphery of the synapse. In contrast, after transient synaptic stimulation with KCl followed by a period of rest in normal media, fewer synapses have N-cadherin present as puncta at the periphery and more synapses have N-cadherin present more centrally and uniformly along the synapse compared to unstimulated cells. This indicates that transient synaptic stimulation modulates N-cadherin localization within the synapse. These results bring new information to the structural organization and activity-induced changes occurring at synapses, and suggest that N-cadherin relocalization may contribute to activity dependent changes at synapses.     

Mechanisms of long-term plasticity of hippocampal GABAergic synapses

Long-term potentiation and depression of synaptic transmission have been considered as cellular mechanisms of memory in studies conducted in recent decades. These studies were predominantly focused on mechanisms underlying plasticity at excitatory synapses. Nevertheless, normal central nervous system functioning requires maintenance of a balance between inhibition and excitation, suggesting existence of similar modulation of glutamatergic and GABAergic synapses. Here we review the involvement of G-protein-coupled receptors in the generation of long-term changes in synaptic transmission of inhibitory synapses. We considered the role of endocannabinoid and glutamate systems, GABA_B and opioid receptors in the induction of long-term potentiation and long-term depression in inhibitory synapses. The preand postsynaptic effects of activation of these receptors are also discussed.     

Extracellular glutamate diffusion determines the occupancy of glutamate receptors at CA1 synapses in the hippocampus

Following exocytosis at excitatory synapses in the brain, glutamate binds to several subtypes of postsynaptic receptors. The degree of occupancy of AMPA and NMDA receptors at hippocampal synapses is, however, not known. One approach to estimate receptor occupancy is to examine quantal amplitude fluctuations of postsynaptic signals in hippocampal neurons studied in vitro. The results of such experiments suggest that NMDA receptors at CA1 synapses are activated not only by glutamate released from the immediately apposed presynaptic terminals, but also by glutamate spillover from neighbouring terminals. Numerical simulations point to the extracellular diffusion coefficient as a critical parameter that determines the extent of activation of receptors positioned at different distances from the release site. We have shown that raising the viscosity of the extracellular medium can modulate the diffusion coefficient, providing an experimental tool to investigate the role of diffusion in activation of synaptic and extrasynaptic receptors. Whether intersynaptic cross-talk mediated by NMDA receptors occurs in vivo remains to be determined. The theoretical and experimental approaches described here also promise to shed light on the roles of metabotropic and kainate receptors, which often occur in an extrasynaptic distribution, and are therefore positioned to sense glutamate escaping from the synaptic cleft.     

Asymmetry of glia near central synapses favors presynaptically directed glutamate escape

Recent findings demonstrate that synaptically released excitatory neurotransmitter glutamate activates receptors outside the immediate synaptic cleft and that the extent of such extrasynaptic actions is regulated by the high affinity glutamate uptake. The bulk of glutamate transporter systems are evenly distributed in the synaptic neuropil, and it is generally assumed that glutamate escaping the cleft affects pre- and postsynaptic receptors to a similar degree. To test whether this is indeed the case, we use quantitative electron microscopy and establish the stochastic pattern of glial occurrence in the three-dimensional (3D) vicinity of two common types of excitatory central synapses, stratum radiatum synapses in hippocampus and parallel fiber synapses in cerebellum. We find that the occurrence of glia postsynaptically is strikingly higher (3-4-fold) than presynaptically, in both types of synapses. To address the functional consequences of this asymmetry, we simulate diffusion and transport of synaptically released glutamate in these two brain areas using a detailed 3D compartmental model of the extracellular space with glutamate transporters arranged unevenly, in accordance with the obtained experimental data. The results predict that glutamate escaping the synaptic cleft is 2-4 times more likely to activate presynaptic compared to postsynaptic receptors. Simulations also show that postsynaptic neuronal transporters (EAAT4 type) at dendritic spines of cerebellar Purkinje cells exaggerate this asymmetry further. Our data suggest that the perisynaptic environment of these common central synapses favors fast presynaptic feedback in the information flow while preserving the specificity of the postsynaptic input.