An integrative dynamical perspective for graph theory and the analysis of complex networks.

Built upon the shoulders of graph theory, the field of complex networks has become a central tool for studying real systems across various fields of research. Represented as graphs, different systems can be studied using the same analysis methods, which allows for their comparison. Here, we challenge the widespread idea that graph theory is a universal analysis tool, uniformly applicable to any kind of network data. Instead, we show that many classical graph metrics-including degree, clustering coefficient, and geodesic distance-arise from a common hidden propagation model: the discrete cascade. From this perspective, graph metrics are no longer regarded as combinatorial measures of the graph but as spatiotemporal properties of the network dynamics unfolded at different temporal scales. Once graph theory is seen as a model-based (and not a purely data-driven) analysis tool, we can freely or intentionally replace the discrete cascade by other canonical propagation models and define new network metrics. This opens the opportunity to design-explicitly and transparently-dedicated analyses for different types of real networks by choosing a propagation model that matches their individual constraints. In this way, we take stand that network topology cannot always be abstracted independently from network dynamics but shall be jointly studied, which is key for the interpretability of the analyses. The model-based perspective here proposed serves to integrate into a common context both the classical graph analysis and the more recent network metrics defined in the literature which were, directly or indirectly, inspired by propagation phenomena on networks.

Whole-brain analyses indicate the impairment of posterior integration and thalamo-frontotemporal broadcasting in disorders of consciousness.

The study of the brain's dynamical activity is opening a window to help the clinical assessment of patients with disorders of consciousness. For example, glucose uptake and the dysfunctional spread of naturalistic and synthetic stimuli has proven useful to characterize hampered consciousness. However, understanding of the mechanisms behind loss of consciousness following brain injury is still missing. Here, we study the propagation of endogenous and in-silico exogenous perturbations in patients with disorders of consciousness, based upon directed and causal interactions estimated from resting-state fMRI data, fitted to a linear model of activity propagation. We found that patients with disorders of consciousness suffer decreased capacity for neural propagation and responsiveness to events, and that this can be related to severe reduction of glucose metabolism as measured with [18 F]FDG-PET. In particular, we show that loss of consciousness is related to the malfunctioning of two neural circuits: the posterior cortical regions failing to convey information, in conjunction with reduced broadcasting of information from subcortical, temporal, parietal and frontal regions. These results shed light on the mechanisms behind disorders of consciousness, triangulating network function with basic measures of brain integrity and behavior.

The covariance perceptron: A new paradigm for classification and processing of time series in recurrent neuronal networks.

Learning in neuronal networks has developed in many directions, in particular to reproduce cognitive tasks like image recognition and speech processing. Implementations have been inspired by stereotypical neuronal responses like tuning curves in the visual system, where, for example, ON/OFF cells fire or not depending on the contrast in their receptive fields. Classical models of neuronal networks therefore map a set of input signals to a set of activity levels in the output of the network. Each category of inputs is thereby predominantly characterized by its mean. In the case of time series, fluctuations around this mean constitute noise in this view. For this paradigm, the high variability exhibited by the cortical activity may thus imply limitations or constraints, which have been discussed for many years. For example, the need for averaging neuronal activity over long periods or large groups of cells to assess a robust mean and to diminish the effect of noise correlations. To reconcile robust computations with variable neuronal activity, we here propose a conceptual change of perspective by employing variability of activity as the basis for stimulus-related information to be learned by neurons, rather than merely being the noise that corrupts the mean signal. In this new paradigm both afferent and recurrent weights in a network are tuned to shape the input-output mapping for covariances, the second-order statistics of the fluctuating activity. When including time lags, covariance patterns define a natural metric for time series that capture their propagating nature. We develop the theory for classification of time series based on their spatio-temporal covariances, which reflect dynamical properties. We demonstrate that recurrent connectivity is able to transform information contained in the temporal structure of the signal into spatial covariances. Finally, we use the MNIST database to show how the covariance perceptron can capture specific second-order statistical patterns generated by moving digits.

Propagation of BOLD Activity Reveals Task-dependent Directed Interactions Across Human Visual Cortex.

It has recently been shown that large-scale propagation of blood-oxygen-level-dependent (BOLD) activity is constrained by anatomical connections and reflects transitions between behavioral states. It remains to be seen, however, if the propagation of BOLD activity can also relate to the brain's anatomical structure at a more local scale. Here, we hypothesized that BOLD propagation reflects structured neuronal activity across early visual field maps. To explore this hypothesis, we characterize the propagation of BOLD activity across V1, V2, and V3 using a modeling approach that aims to disentangle the contributions of local activity and directed interactions in shaping BOLD propagation. It does so by estimating the effective connectivity (EC) and the excitability of a noise-diffusion network to reproduce the spatiotemporal covariance structure of the data. We apply our approach to 7T fMRI recordings acquired during resting state (RS) and visual field mapping (VFM). Our results reveal different EC interactions and changes in cortical excitability in RS and VFM, and point to a reconfiguration of feedforward and feedback interactions across the visual system. We conclude that the propagation of BOLD activity has functional relevance, as it reveals directed interactions and changes in cortical excitability in a task-dependent manner.

Beyond the disconnectivity hypothesis of schizophrenia.

To go beyond the disconnectivity hypothesis of schizophrenia, directed (effective) connectivity was measured between 94 brain regions, to provide evidence on the source of the changes in schizophrenia and a mechanistic model. Effective connectivity (EC) was measured in 180 participants with schizophrenia and 208 controls. For the significantly different effective connectivities in schizophrenia, on average the forward (stronger) effective connectivities were smaller, whereas the backward connectivities tended to be larger. Further, higher EC in schizophrenia was found from the precuneus and posterior cingulate cortex (PCC) to areas such as the parahippocampal, hippocampal, temporal, fusiform, and occipital cortices. These are backward effective connectivities and were positively correlated with the positive symptoms of schizophrenia. Lower effective connectivities were found from temporal and other regions and were negatively correlated with the symptoms, especially the negative and general symptoms. Further, a signal variance parameter was increased for areas that included the parahippocampal gyrus and hippocampus, consistent with the hypothesis that hippocampal overactivity is involved in schizophrenia. This investigation goes beyond the disconnectivity hypothesis by drawing attention to differences in schizophrenia between backprojections and forward connections, with the backward connections from the precuneus and PCC implicated in memory stronger in schizophrenia.

Network analysis of whole-brain fMRI dynamics: A new framework based on dynamic communicability.

Neuroimaging techniques such as MRI have been widely used to explore the associations between brain areas. Structural connectivity (SC) captures the anatomical pathways across the brain and functional connectivity (FC) measures the correlation between the activity of brain regions. These connectivity measures have been much studied using network theory in order to uncover the distributed organization of brain structures, in particular FC for task-specific brain communication. However, the application of network theory to study FC matrices is often "static" despite the dynamic nature of time series obtained from fMRI. The present study aims to overcome this limitation by introducing a network-oriented analysis applied to whole-brain effective connectivity (EC) useful to interpret the brain dynamics. Technically, we tune a multivariate Ornstein-Uhlenbeck (MOU) process to reproduce the statistics of the whole-brain resting-state fMRI signals, which provides estimates for MOU-EC as well as input properties (similar to local excitabilities). The network analysis is then based on the Green function (or network impulse response) that describes the interactions between nodes across time for the estimated dynamics. This model-based approach provides time-dependent graph-like descriptor, named communicability, that characterize the roles that either nodes or connections play in the propagation of activity within the network. They can be used at both global and local levels, and also enables the comparison of estimates from real data with surrogates (e.g. random network or ring lattice). In contrast to classical graph approaches to study SC or FC, our framework stresses the importance of taking the temporal aspect of fMRI signals into account. Our results show a merging of functional communities over time, moving from segregated to global integration of the network activity. Our formalism sets a solid ground for the analysis and interpretation of fMRI data, including task-evoked activity.

Distinct modes of functional connectivity induced by movie-watching.

A fundamental question in systems neuroscience is how endogenous neuronal activity self-organizes during particular brain states. Recent neuroimaging studies have demonstrated systematic relationships between resting-state and task-induced functional connectivity (FC). In particular, continuous task studies, such as movie watching, speak to alterations in coupling among cortical regions and enhanced fluctuations in FC compared to the resting-state. This suggests that FC may reflect systematic and large-scale reorganization of functionally integrated responses while subjects are watching movies. In this study, we characterized fluctuations in FC during resting-state and movie-watching conditions. We found that the FC patterns induced systematically by movie-watching can be explained with a single principal component. These condition-specific FC fluctuations overlapped with inter-subject synchronization patterns in occipital and temporal brain regions. However, unlike inter-subject synchronization, condition-specific FC patterns were characterized by increased correlations within frontal brain regions and reduced correlations between frontal-parietal brain regions. We investigated these condition-specific functional variations as a shorter time scale, using time-resolved FC. The time-resolved FC showed condition-specificity over time; notably when subjects watched both the same and different movies. To explain self-organisation of global FC through the alterations in local dynamics, we used a large-scale computational model. We found that condition-specific reorganization of FC could be explained by local changes that engendered changes in FC among higher-order association regions, mainly in frontal and parietal cortices.

Imaging Connectomics and the Understanding of Brain Diseases.

Neuroimaging-based personalized medicine is emerging to characterize brain disorders and their evolution at the patient level. In this chapter, we present the most classic methods used to infer large-scale brain connectivity based on functional MRI. We adopt a modeling perspective where every connectivity measure is linked to a specific model that allows to interpret the connectivity estimate. This perspective allows to analyze the quality of retrieved connectivity profiles in terms of modeling error and estimation error. In the first part of the chapter, we present undirected functional connectivity (Pearson's correlation and MI) and effective connectivity (partial correlation), as well as directed effective connectivity (VAR, MOU, Granger causality, DCM). In addition, some of these measures correspond to fully connected graphs (Pearson's correlation) while others to sparse ones (MOU, DCM), where the sparsity can come from the integration of functional and structural data. In the second part, we claim that machine learning tools are better suited than null-hypothesis testing to link the estimated connectomes with diagnosis and prognosis of neuropsychiatric diseases. Finally, we propose that linear models and features selection are preferable to more complex and nonlinear tools (when prediction performance is on a par) for building interpretable algorithms to predict clinical variables.

Stereotypical modulations in dynamic functional connectivity explained by changes in BOLD variance.

Spontaneous activity measured in human subject under the absence of any task exhibits complex patterns of correlation that largely correspond to large-scale functional topographies obtained with a wide variety of cognitive and perceptual tasks. These "resting state networks" (RSNs) fluctuate over time, forming and dissolving on the scale of seconds to minutes. While these fluctuations, most prominently those of the default mode network, have been linked to cognitive function, it remains unclear whether they result from random noise or whether they index a nonstationary process which could be described as state switching. In this study, we use a sliding windows-approach to relate temporal dynamics of RSNs to global modulations in correlation and BOLD variance. We compare empirical data, phase-randomized surrogate data, and data simulated with a stationary model. We find that RSN time courses exhibit a large amount of coactivation in all three cases, and that the modulations in their activity are closely linked to global dynamics of the underlying BOLD signal. We find that many properties of the observed fluctuations in FC and BOLD, including their ranges and their correlations amongst each other, are explained by fluctuations around the average FC structure. However, we also report some interesting characteristics that clearly support nonstationary features in the data. In particular, we find that the brain spends more time in the troughs of modulations than can be expected from stationary dynamics.

Extracting orthogonal subject- and condition-specific signatures from fMRI data using whole-brain effective connectivity.

The study of brain communication based on fMRI data is often limited because such measurements are a mixture of session-to-session variability with subject- and condition-related information. Disentangling these contributions is crucial for real-life applications, in particular when only a few recording sessions are available. The present study aims to define a reliable standard for the extraction of multiple signatures from fMRI data, while verifying that they do not mix information about the different modalities (e.g., subjects and conditions such as tasks performed by them). In particular, condition-specific signatures should not be contaminated by subject-related information, since they aim to generalize over subjects. Practically, signatures correspond to subnetworks of directed interactions between brain regions (typically 100 covering the whole brain) supporting the subject and condition identification for single fMRI sessions. The key for robust prediction is using effective connectivity instead of functional connectivity. Our method demonstrates excellent generalization capabilities for subject identification in two datasets, using only a few sessions per subject as reference. Using another dataset with resting state and movie viewing, we show that the two signatures related to subjects and tasks correspond to distinct subnetworks, which are thus topologically orthogonal. Our results set solid foundations for applications tailored to individual subjects, such as clinical diagnostic.

Effective connectivity inferred from fMRI transition dynamics during movie viewing points to a balanced reconfiguration of cortical interactions.

Our behavior entails a flexible and context-sensitive interplay between brain areas to integrate information according to goal-directed requirements. However, the neural mechanisms governing the entrainment of functionally specialized brain areas remain poorly understood. In particular, the question arises whether observed changes in the regional activity for different cognitive conditions are explained by modifications of the inputs to the brain or its connectivity? We observe that transitions of fMRI activity between areas convey information about the tasks performed by 19 subjects, watching a movie versus a black screen (rest). We use a model-based framework that explains this spatiotemporal functional connectivity pattern by the local variability for 66 cortical regions and the network effective connectivity between them. We find that, among the estimated model parameters, movie viewing affects to a larger extent the local activity, which we interpret as extrinsic changes related to the increased stimulus load. However, detailed changes in the effective connectivity preserve a balance in the propagating activity and select specific pathways such that high-level brain regions integrate visual and auditory information, in particular boosting the communication between the two brain hemispheres. These findings speak to a dynamic coordination underlying the functional integration in the brain.

Task-related effective connectivity reveals that the cortical rich club gates cortex-wide communication.

Higher cognition may require the globally coordinated integration of specialized brain regions into functional networks. A collection of structural cortical hubs-referred to as the rich club-has been hypothesized to support task-specific functional integration. In the present paper, we use a whole-cortex model to estimate directed interactions between 68 cortical regions from functional magnetic resonance imaging activity for four different tasks (reflecting different cognitive domains) and resting state. We analyze the state-dependent input and output effective connectivity (EC) of the structural rich club and relate these to whole-cortex dynamics and network reconfigurations. We find that the cortical rich club exhibits an increase in outgoing EC during task performance as compared with rest while incoming connectivity remains constant. Increased outgoing connectivity targets a sparse set of peripheral regions with specific regions strongly overlapping between tasks. At the same time, community detection analyses reveal massive reorganizations of interactions among peripheral regions, including those serving as target of increased rich club output. This suggests that while peripheral regions may play a role in several tasks, their concrete interplay might nonetheless be task-specific. Furthermore, we observe that whole-cortex dynamics are faster during task as compared with rest. The decoupling effects usually accompanying faster dynamics appear to be counteracted by the increased rich club outgoing EC. Together our findings speak to a gating mechanism of the rich club that supports fast-paced information exchange among relevant peripheral regions in a task-specific and goal-directed fashion, while constantly listening to the whole network.

Analysis of fMRI data using noise-diffusion network models: a new covariance-coding perspective.

Since the middle of the 1990s, studies of resting-state fMRI/BOLD data have explored the correlation patterns of activity across the whole brain, which is referred to as functional connectivity (FC). Among the many methods that have been developed to interpret FC, a recently proposed model-based approach describes the propagation of fluctuating BOLD activity within the recurrently connected brain network by inferring the effective connectivity (EC). In this model, EC quantifies the strengths of directional interactions between brain regions, viewed from the proxy of BOLD activity. In addition, the tuning procedure for the model provides estimates for the local variability (input variances) to explain how the observed FC is generated. Generalizing, the network dynamics can be studied in the context of an input-output mapping-determined by EC-for the second-order statistics of fluctuating nodal activities. The present paper focuses on the following detection paradigm: observing output covariances, how discriminative is the (estimated) network model with respect to various input covariance patterns? An application with the model fitted to experimental fMRI data-movie viewing versus resting state-illustrates that changes in local variability and changes in brain coordination go hand in hand.

Resting state networks in empirical and simulated dynamic functional connectivity.

It is well-established that patterns of functional connectivity (FC) - measures of correlated activity between pairs of voxels or regions observed in the human brain using neuroimaging - are robustly expressed in spontaneous activity during rest. These patterns are not static, but exhibit complex spatio-temporal dynamics. Over the last years, a multitude of methods have been proposed to reveal these dynamics on the level of the whole brain. One finding is that the brain transitions through different FC configurations over time, and substantial effort has been put into characterizing these configurations. However, the dynamics governing these transitions are more elusive, specifically, the contribution of stationary vs. non-stationary dynamics is an active field of inquiry. In this study, we use a whole-brain approach, considering FC dynamics between 66 ROIs covering the entire cortex. We combine an innovative dimensionality reduction technique, tensor decomposition, with a mean field model which possesses stationary dynamics. It has been shown to explain resting state FC averaged over time and multiple subjects, however, this average FC summarizes the spatial distribution of correlations while hiding their temporal dynamics. First, we apply tensor decomposition to resting state scans from 24 healthy controls in order to characterize spatio-temporal dynamics present in the data. We simultaneously utilize temporal and spatial information by creating tensors that are subsequently decomposed into sets of brain regions ("communities") that share similar temporal dynamics, and their associated time courses. The tensors contain pairwise FC computed inside of overlapping sliding windows. Communities are discovered by clustering features pooled from all subjects, thereby ensuring that they generalize. We find that, on the group level, the data give rise to four distinct communities that resemble known resting state networks (RSNs): default mode network, visual network, control networks, and somatomotor network. Second, we simulate data with our stationary mean field model whose nodes are connected according to results from DTI and fiber tracking. In this model, all spatio-temporal structure is due to noisy fluctuations around the average FC. We analyze the simulated data in the same way as the empirical data in order to determine whether stationary dynamics can explain the emergence of distinct FC patterns (RSNs) which have their own time courses. We find that this is the case for all four networks using the spatio-temporal information revealed by tensor decomposition if nodes in the simulation are connected according to model-based effective connectivity. Furthermore, we find that these results require only a small part of the FC values, namely the highest values that occur across time and ROI pair. Our findings show that stationary dynamics can account for the emergence of RSNs. We provide an innovative method that does not make strong assumptions about the underlying data and is generally applicable to resting state or task data from different subject populations.

Estimation of Directed Effective Connectivity from fMRI Functional Connectivity Hints at Asymmetries of Cortical Connectome.

The brain exhibits complex spatio-temporal patterns of activity. This phenomenon is governed by an interplay between the internal neural dynamics of cortical areas and their connectivity. Uncovering this complex relationship has raised much interest, both for theory and the interpretation of experimental data (e.g., fMRI recordings) using dynamical models. Here we focus on the so-called inverse problem: the inference of network parameters in a cortical model to reproduce empirically observed activity. Although it has received a lot of interest, recovering directed connectivity for large networks has been rather unsuccessful so far. The present study specifically addresses this point for a noise-diffusion network model. We develop a Lyapunov optimization that iteratively tunes the network connectivity in order to reproduce second-order moments of the node activity, or functional connectivity. We show theoretically and numerically that the use of covariances with both zero and non-zero time shifts is the key to infer directed connectivity. The first main theoretical finding is that an accurate estimation of the underlying network connectivity requires that the time shift for covariances is matched with the time constant of the dynamical system. In addition to the network connectivity, we also adjust the intrinsic noise received by each network node. The framework is applied to experimental fMRI data recorded for subjects at rest. Diffusion-weighted MRI data provide an estimate of anatomical connections, which is incorporated to constrain the cortical model. The empirical covariance structure is reproduced faithfully, especially its temporal component (i.e., time-shifted covariances) in addition to the spatial component that is usually the focus of studies. We find that the cortical interactions, referred to as effective connectivity, in the tuned model are not reciprocal. In particular, hubs are either receptors or feeders: they do not exhibit both strong incoming and outgoing connections. Our results sets a quantitative ground to explore the propagation of activity in the cortex.

Delay selection by spike-timing-dependent plasticity in recurrent networks of spiking neurons receiving oscillatory inputs.

Learning rules, such as spike-timing-dependent plasticity (STDP), change the structure of networks of neurons based on the firing activity. A network level understanding of these mechanisms can help infer how the brain learns patterns and processes information. Previous studies have shown that STDP selectively potentiates feed-forward connections that have specific axonal delays, and that this underlies behavioral functions such as sound localization in the auditory brainstem of the barn owl. In this study, we investigate how STDP leads to the selective potentiation of recurrent connections with different axonal and dendritic delays during oscillatory activity. We develop analytical models of learning with additive STDP in recurrent networks driven by oscillatory inputs, and support the results using simulations with leaky integrate-and-fire neurons. Our results show selective potentiation of connections with specific axonal delays, which depended on the input frequency. In addition, we demonstrate how this can lead to a network becoming selective in the amplitude of its oscillatory response to this frequency. We extend this model of axonal delay selection within a single recurrent network in two ways. First, we show the selective potentiation of connections with a range of both axonal and dendritic delays. Second, we show axonal delay selection between multiple groups receiving out-of-phase, oscillatory inputs. We discuss the application of these models to the formation and activation of neuronal ensembles or cell assemblies in the cortex, and also to missing fundamental pitch perception in the auditory brainstem.

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.

Entropy production of multivariate Ornstein-Uhlenbeck processes correlates with consciousness levels in the human brain.

Consciousness is supported by complex patterns of brain activity which are indicative of irreversible nonequilibrium dynamics. While the framework of stochastic thermodynamics has facilitated the understanding of physical systems of this kind, its application to infer the level of consciousness from empirical data remains elusive. We faced this challenge by calculating entropy production in a multivariate Ornstein-Uhlenbeck process fitted to Functional magnetic resonance imaging brain activity recordings. To test this approach, we focused on the transition from wakefulness to deep sleep, revealing a monotonous relationship between entropy production and the level of consciousness. Our results constitute robust signatures of consciousness while also advancing our understanding of the link between consciousness and complexity from the fundamental perspective of statistical physics.

Frequency selectivity emerging from spike-timing-dependent plasticity.

Periodic neuronal activity has been observed in various areas of the brain, from lower sensory to higher cortical levels. Specific frequency components contained in this periodic activity can be identified by a neuronal circuit that behaves as a bandpass filter with given preferred frequency, or best modulation frequency (BMF). For BMFs typically ranging from 10 to 200 Hz, a plausible and minimal configuration consists of a single neuron with adjusted excitatory and inhibitory synaptic connections. The emergence, however, of such a neuronal circuitry is still unclear. In this letter, we demonstrate how spike-timing-dependent plasticity (STDP) can give rise to frequency-dependent learning, thus leading to an input selectivity that enables frequency identification. We use an in-depth mathematical analysis of the learning dynamics in a population of plastic inhibitory connections. These provide inhomogeneous postsynaptic responses that depend on their dendritic location. We find that synaptic delays play a crucial role in organizing the weight specialization induced by STDP. Under suitable conditions on the synaptic delays and postsynaptic potentials (PSPs), the BMF of a neuron after learning can match the training frequency. In particular, proximal (distal) synapses with shorter (longer) dendritic delay and somatically measured PSP time constants respond better to higher (lower) frequencies. As a result, the neuron will respond maximally to any stimulating frequency (in a given range) with which it has been trained in an unsupervised manner. The model predicts that synapses responding to a given BMF form clusters on dendritic branches.

STDP allows fast rate-modulated coding with Poisson-like spike trains.

Spike timing-dependent plasticity (STDP) has been shown to enable single neurons to detect repeatedly presented spatiotemporal spike patterns. This holds even when such patterns are embedded in equally dense random spiking activity, that is, in the absence of external reference times such as a stimulus onset. Here we demonstrate, both analytically and numerically, that STDP can also learn repeating rate-modulated patterns, which have received more experimental evidence, for example, through post-stimulus time histograms (PSTHs). Each input spike train is generated from a rate function using a stochastic sampling mechanism, chosen to be an inhomogeneous Poisson process here. Learning is feasible provided significant covarying rate modulations occur within the typical timescale of STDP (~10-20 ms) for sufficiently many inputs (~100 among 1000 in our simulations), a condition that is met by many experimental PSTHs. Repeated pattern presentations induce spike-time correlations that are captured by STDP. Despite imprecise input spike times and even variable spike counts, a single trained neuron robustly detects the pattern just a few milliseconds after its presentation. Therefore, temporal imprecision and Poisson-like firing variability are not an obstacle to fast temporal coding. STDP provides an appealing mechanism to learn such rate patterns, which, beyond sensory processing, may also be involved in many cognitive tasks.