Intrinsic stability of temporally shifted spike-timing dependent plasticity

Spike-timing dependent plasticity (STDP), a widespread synaptic modification mechanism, is sensitive to correlations between presynaptic spike trains and it generates competition among synapses. However, STDP has an inherent instability because strong synapses are more likely to be strengthened than weak ones, causing them to grow in strength until some biophysical limit is reached. Through simulations and analytic calculations, we show that a small temporal shift in the STDP window that causes synchronous, or nearly synchronous, pre- and postsynaptic action potentials to induce long-term depression can stabilize synaptic strengths. Shifted STDP also stabilizes the postsynaptic firing rate and can implement both Hebbian and anti-Hebbian forms of competitive synaptic plasticity. Interestingly, the overall level of inhibition determines whether plasticity is Hebbian or anti-Hebbian. Even a random symmetric jitter of a few milliseconds in the STDP window can stabilize synaptic strengths while retaining these features. The same results hold for a shifted version of the more recent "triplet" model of STDP. Our results indicate that the detailed shape of the STDP window function near the transition from depression to potentiation is of the utmost importance in determining the consequences of STDP, suggesting that this region warrants further experimental study.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. I. Input selectivity–strengthening correlated input pathways

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. In this paper, we extend previous studies of input selectivity induced by (STDP) for single neurons to the biologically interesting case of a neuronal network with fixed recurrent connections and plastic connections from external pools of input neurons. We use a theoretical framework based on the Poisson neuron model to analytically describe the network dynamics (firing rates and spike-time correlations) and thus the evolution of the synaptic weights. This framework incorporates the time course of the post-synaptic potentials and synaptic delays. Our analysis focuses on the asymptotic states of a network stimulated by two homogeneous pools of “steady” inputs, namely Poisson spike trains which have fixed firing rates and spike-time correlations. The (STDP) model extends rate-based learning in that it can implement, at the same time, both a stabilization of the individual neuron firing rates and a slower weight specialization depending on the input spike-time correlations. When one input pathway has stronger within-pool correlations, the resulting synaptic dynamics induced by (STDP) are shown to be similar to those arising in the case of a purely feed-forward network: the weights from the more correlated inputs are potentiated at the expense of the remaining input connections.     

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.     

The Effect of STDP Temporal Kernel Structure on the Learning Dynamics of Single Excitatory and Inhibitory Synapses

Spike-Timing Dependent Plasticity (STDP) is characterized by a wide range of temporal kernels. However, much of the theoretical work has focused on a specific kernel – the “temporally asymmetric Hebbian” learning rules. Previous studies linked excitatory STDP to positive feedback that can account for the emergence of response selectivity. Inhibitory plasticity was associated with negative feedback that can balance the excitatory and inhibitory inputs. Here we study the possible computational role of the temporal structure of the STDP. We represent the STDP as a superposition of two processes: potentiation and depression. This allows us to model a wide range of experimentally observed STDP kernels, from Hebbian to anti-Hebbian, by varying a single parameter. We investigate STDP dynamics of a single excitatory or inhibitory synapse in purely feed-forward architecture. We derive a mean-field-Fokker-Planck dynamics for the synaptic weight and analyze the effect of STDP structure on the fixed points of the mean field dynamics. We find a phase transition along the Hebbian to anti-Hebbian parameter from a phase that is characterized by a unimodal distribution of the synaptic weight, in which the STDP dynamics is governed by negative feedback, to a phase with positive feedback characterized by a bimodal distribution. The critical point of this transition depends on general properties of the STDP dynamics and not on the fine details. Namely, the dynamics is affected by the pre-post correlations only via a single number that quantifies its overlap with the STDP kernel. We find that by manipulating the STDP temporal kernel, negative feedback can be induced in excitatory synapses and positive feedback in inhibitory. Moreover, there is an exact symmetry between inhibitory and excitatory plasticity, i.e., for every STDP rule of inhibitory synapse there exists an STDP rule for excitatory synapse, such that their dynamics is identical.     

Beyond spike timing: the role of nonlinear plasticity and unreliable synapses

 Spike-timing-dependent plasticity (STDP) strengthens synapses that are activated immediately before a postsynaptic spike, and weakens those that are activated after a spike. To prevent an uncontrolled growth of the synaptic strengths, weakening must dominate strengthening for uncorrelated spike times. However, this weight-normalization property would preclude Hebbian potentiation when the pre- and postsynaptic neurons are strongly active without specific spike-time correlations. We show that nonlinear STDP as inherent in the data of Markram et al. [(1997) Science 275:213–215] can preserve the benefits of both weight normalization and Hebbian plasticity, and hence can account for learning based on spike-time correlations and on mean firing rates. As examples we consider the moving-threshold property of the Bienenstock–Cooper–Munro rule, the development of direction-selective simple cells by changing short-term synaptic depression, and the joint adaptation of axonal and dendritic delays. Without threshold nonlinearity at low frequencies, the development of direction selectivity does not stabilize in a natural stimulation environment. Without synaptic unreliability there is no causal development of axonal and dendritic delays. Received: 22 April 2002 / Accepted: 23 May 2002 Acknowledgements. This study was supported by the Swiss National Science Foundation (grant 3152-065234.01) and the Silva-Casa foundation. The author thanks Stefano Fusi, Henry Markram, and Misha Tsodyks for helpful discussions, Nissim Buchs and Martin Schneider for their simulations, and Jan Reutimann for proof reading. Correspondence to: e-mail: wsenn@cns.unibe.ch, Tel.: +41-31-6318721, Fax: 41-31-6314611     

Maturation of GABAergic Inhibition Promotes Strengthening of Temporally Coherent Inputs among Convergent Pathways

Spike-timing-dependent plasticity (STDP), a form of Hebbian plasticity, is inherently stabilizing. Whether and how GABAergic inhibition influences STDP is not well understood. Using a model neuron driven by converging inputs modifiable by STDP, we determined that a sufficient level of inhibition was critical to ensure that temporal coherence (correlation among presynaptic spike times) of synaptic inputs, rather than initial strength or number of inputs within a pathway, controlled postsynaptic spike timing. Inhibition exerted this effect by preferentially reducing synaptic efficacy, the ability of inputs to evoke postsynaptic action potentials, of the less coherent inputs. In visual cortical slices, inhibition potently reduced synaptic efficacy at ages during but not before the critical period of ocular dominance (OD) plasticity. Whole-cell recordings revealed that the amplitude of unitary IPSCs from parvalbumin positive (Pv+) interneurons to pyramidal neurons increased during the critical period, while the synaptic decay time-constant decreased. In addition, intrinsic properties of Pv+ interneurons matured, resulting in an increase in instantaneous firing rate. Our results suggest that maturation of inhibition in visual cortex ensures that the temporally coherent inputs (e.g. those from the open eye during monocular deprivation) control postsynaptic spike times of binocular neurons, a prerequisite for Hebbian mechanisms to induce OD plasticity.     

Spectral analysis of input spike trains by spike-timing-dependent plasticity

Spike-timing-dependent plasticity (STDP) has been observed in many brain areas such as sensory cortices, where it is hypothesized to structure synaptic connections between neurons. Previous studies have demonstrated how STDP can capture spiking information at short timescales using specific input configurations, such as coincident spiking, spike patterns and oscillatory spike trains. However, the corresponding computation in the case of arbitrary input signals is still unclear. This paper provides an overarching picture of the algorithm inherent to STDP, tying together many previous results for commonly used models of pairwise STDP. For a single neuron with plastic excitatory synapses, we show how STDP performs a spectral analysis on the temporal cross-correlograms between its afferent spike trains. The postsynaptic responses and STDP learning window determine kernel functions that specify how the neuron "sees" the input correlations. We thus denote this unsupervised learning scheme as 'kernel spectral component analysis' (kSCA). In particular, the whole input correlation structure must be considered since all plastic synapses compete with each other. We find that kSCA is enhanced when weight-dependent STDP induces gradual synaptic competition. For a spiking neuron with a "linear" response and pairwise STDP alone, we find that kSCA resembles principal component analysis (PCA). However, plain STDP does not isolate correlation sources in general, e.g., when they are mixed among the input spike trains. In other words, it does not perform independent component analysis (ICA). Tuning the neuron to a single correlation source can be achieved when STDP is paired with a homeostatic mechanism that reinforces the competition between synaptic inputs. Our results suggest that neuronal networks equipped with STDP can process signals encoded in the transient spiking activity at the timescales of tens of milliseconds for usual STDP.     

Spike timing-dependent plasticity of neural circuits

Recent findings of spike timing-dependent plasticity (STDP) have stimulated much interest among experimentalists and theorists. Beyond the traditional correlation-based Hebbian plasticity, STDP opens up new avenues for understanding information coding and circuit plasticity that depend on the precise timing of neuronal spikes. Here we summarize experimental characterization of STDP at various synapses, the underlying cellular mechanisms, and the associated changes in neuronal excitability and dendritic integration. We also describe STDP in the context of complex spike patterns and its dependence on the dendritic location of the synapse. Finally, we discuss timing-dependent modification of neuronal receptive fields and human visual perception and the computational significance of STDP as a synaptic learning rule.     

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.     

Bayesian Computation Emerges in Generic Cortical Microcircuits through Spike-Timing-Dependent Plasticity

The principles by which networks of neurons compute, and how spike-timing dependent plasticity (STDP) of synaptic weights generates and maintains their computational function, are unknown. Preceding work has shown that soft winner-take-all (WTA) circuits, where pyramidal neurons inhibit each other via interneurons, are a common motif of cortical microcircuits. We show through theoretical analysis and computer simulations that Bayesian computation is induced in these network motifs through STDP in combination with activity-dependent changes in the excitability of neurons. The fundamental components of this emergent Bayesian computation are priors that result from adaptation of neuronal excitability and implicit generative models for hidden causes that are created in the synaptic weights through STDP. In fact, a surprising result is that STDP is able to approximate a powerful principle for fitting such implicit generative models to high-dimensional spike inputs: Expectation Maximization. Our results suggest that the experimentally observed spontaneous activity and trial-to-trial variability of cortical neurons are essential features of their information processing capability, since their functional role is to represent probability distributions rather than static neural codes. Furthermore it suggests networks of Bayesian computation modules as a new model for distributed information processing in the cortex.     

Emergence of Connectivity Motifs in Networks of Model Neurons with Short- and Long-Term Plastic Synapses

Recent experimental data from the rodent cerebral cortex and olfactory bulb indicate that specific connectivity motifs are correlated with short-term dynamics of excitatory synaptic transmission. It was observed that neurons with short-term facilitating synapses form predominantly reciprocal pairwise connections, while neurons with short-term depressing synapses form predominantly unidirectional pairwise connections. The cause of these structural differences in excitatory synaptic microcircuits is unknown. We show that these connectivity motifs emerge in networks of model neurons, from the interactions between short-term synaptic dynamics (SD) and long-term spike-timing dependent plasticity (STDP). While the impact of STDP on SD was shown in simultaneous neuronal pair recordings in vitro, the mutual interactions between STDP and SD in large networks are still the subject of intense research. Our approach combines an SD phenomenological model with an STDP model that faithfully captures long-term plasticity dependence on both spike times and frequency. As a proof of concept, we first simulate and analyze recurrent networks of spiking neurons with random initial connection efficacies and where synapses are either all short-term facilitating or all depressing. For identical external inputs to the network, and as a direct consequence of internally generated activity, we find that networks with depressing synapses evolve unidirectional connectivity motifs, while networks with facilitating synapses evolve reciprocal connectivity motifs. We then show that the same results hold for heterogeneous networks, including both facilitating and depressing synapses. This does not contradict a recent theory that proposes that motifs are shaped by external inputs, but rather complements it by examining the role of both the external inputs and the internally generated network activity. Our study highlights the conditions under which SD-STDP might explain the correlation between facilitation and reciprocal connectivity motifs, as well as between depression and unidirectional motifs.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks IV

In neuronal networks, the changes of synaptic strength (or weight) performed by spike-timing-dependent plasticity (STDP) are hypothesized to give rise to functional network structure. This article investigates how this phenomenon occurs for the excitatory recurrent connections of a network with fixed input weights that is stimulated by external spike trains. We develop a theoretical framework based on the Poisson neuron model to analyze the interplay between the neuronal activity (firing rates and the spike-time correlations) and the learning dynamics, when the network is stimulated by correlated pools of homogeneous Poisson spike trains. STDP can lead to both a stabilization of all the neuron firing rates (homeostatic equilibrium) and a robust weight specialization. The pattern of specialization for the recurrent weights is determined by a relationship between the input firing-rate and correlation structures, the network topology, the STDP parameters and the synaptic response properties. We find conditions for feed-forward pathways or areas with strengthened self-feedback to emerge in an initially homogeneous recurrent network.     

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)     

Coactivation of pre- and postsynaptic signaling mechanisms determines cell-specific spike-timing-dependent plasticity

Synapses may undergo long-term increases or decreases in synaptic strength dependent on critical differences in the timing between pre-and postsynaptic activity. Such spike-timing-dependent plasticity (STDP) follows rules that govern how patterns of neural activity induce changes in synaptic strength. Synaptic plasticity in the dorsal cochlear nucleus (DCN) follows Hebbian and anti-Hebbian patterns in a cell-specific manner. Here we show that these opposing responses to synaptic activity result from differential expression of two signaling pathways. Ca2+/calmodulin-dependent protein kinase II (CaMKII) signaling underlies Hebbian postsynaptic LTP in principal cells. By contrast, in interneurons, a temporally precise anti-Hebbian synaptic spike-timing rule results from the combined effects of postsynaptic CaMKII-dependent LTP and endocannabinoid-dependent presynaptic LTD. Cell specificity in the circuit arises from selective targeting of presynaptic CB1 receptors in different axonal terminals. Hence, pre- and postsynaptic sites of expression determine both the sign and timing requirements of long-term plasticity in interneurons.     

Cortical development and remapping through spike timing-dependent plasticity.

Long-term modification of synaptic efficacy can depend on the timing of pre- and postsynaptic action potentials. In model studies, such spike timing-dependent plasticity (STDP) introduces the desirable features of competition among synapses and regulation of postsynaptic firing characteristics. STDP strengthens synapses that receive correlated input, which can lead to the formation of stimulus-selective columns and the development, refinement, and maintenance of selectivity maps in network models. The temporal asymmetry of STDP suppresses strong destabilizing self-excitatory loops and allows a group of neurons that become selective early in development to direct other neurons to become similarly selective. STDP, acting alone without further hypothetical global constraints or additional forms of plasticity, can also reproduce the remapping seen in adult cortex following afferent lesions.     

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.     

Computational consequences of experimentally derived spike-time and weight dependent plasticity rules

We present two weight- and spike-time dependent synaptic plasticity rules consistent with the physiological data of Bi and Poo (J Neurosci 18:10464–10472, 1998). One rule assumes synaptic saturation, while the other is scale free. We extend previous analyses of the asymptotic consequences of weight-dependent STDP to the case of strongly correlated pre- and post-synaptic spiking, more closely resembling associative learning. We further provide a general formula for the contribution of any number of spikes to synaptic drift. Asymptotic weights are shown to principally depend on the correlation and rate of pre- and post-synaptic activity, decreasing with increasing rate under correlated activity, and increasing with rate under uncorrelated activity. Spike train statistics reveal a quantitative effect only in the pre-asymptotic regime, and we provide a new interpretation of the relation between BCM and STDP data.     

Pre- and postsynaptically expressed spike-timing-dependent plasticity contribute differentially to neuronal learning

A plethora of experimental studies have shown that long-term synaptic plasticity can be expressed pre- or postsynaptically depending on a range of factors such as developmental stage, synapse type, and activity patterns. The functional consequences of this diversity are not clear, although it is understood that whereas postsynaptic expression of plasticity predominantly affects synaptic response amplitude, presynaptic expression alters both synaptic response amplitude and short-term dynamics. In most models of neuronal learning, long-term synaptic plasticity is implemented as changes in connective weights. The consideration of long-term plasticity as a fixed change in amplitude corresponds more closely to post- than to presynaptic expression, which means theoretical outcomes based on this choice of implementation may have a postsynaptic bias. To explore the functional implications of the diversity of expression of long-term synaptic plasticity, we adapted a model of long-term plasticity, more specifically spike-timing-dependent plasticity (STDP), such that it was expressed either independently pre- or postsynaptically, or in a mixture of both ways. We compared pair-based standard STDP models and a biologically tuned triplet STDP model, and investigated the outcomes in a minimal setting, using two different learning schemes: in the first, inputs were triggered at different latencies, and in the second a subset of inputs were temporally correlated. We found that presynaptic changes adjusted the speed of learning, while postsynaptic expression was more efficient at regulating spike timing and frequency. When combining both expression loci, postsynaptic changes amplified the response range, while presynaptic plasticity allowed control over postsynaptic firing rates, potentially providing a form of activity homeostasis. Our findings highlight how the seemingly innocuous choice of implementing synaptic plasticity by single weight modification may unwittingly introduce a postsynaptic bias in modelling outcomes. We conclude that pre- and postsynaptically expressed plasticity are not interchangeable, but enable complimentary functions.     

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.     

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.