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Graph neural networks (GNNs) have emerged as powerful tools for processing relational data in applications. However, GNNs suffer from the problem of oversmoothing, the property that the features of all nodes exponentially converge to the same vector over layers, prohibiting the design of deep GNNs. In this work we study oversmoothing in graph convolutional networks (GCNs) by using their Gaussian process (GP) equivalence in the limit of infinitely many hidden features. By generalizing methods from conventional deep neural networks (DNNs), we can describe the distribution of features at the output layer of deep GCNs in terms of a GP: as expected, we find that typical parameter choices from the literature lead to oversmoothing. The theory, however, allows us to identify a new, nonoversmoothing phase: if the initial weights of the network have sufficiently large variance, GCNs do not oversmooth, and node features remain informative even at large depth. We demonstrate the validity of this prediction in finite-size GCNs by training a linear classifier on their output. Moreover, using the linearization of the GCN GP, we generalize the concept of propagation depth of information from DNNs to GCNs. This propagation depth diverges at the transition between the oversmoothing and non-oversmoothing phase. We test the predictions of our approach and find good agreement with finite-size GCNs. Initializing GCNs near the transition to the non-oversmoothing phase, we obtain networks which are both deep and expressive.

A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width N . A large deviation approach, which is exact in the proportional limit for the number of data points $P = \alpha N \rightarrow \infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale., Comment: 31 pages, 7 figures, accepted at International Conference on Machine Learning 2024

The susceptible-infected-recovered (SIR) model and its variants form the foundation of our understanding of the spread of diseases. Here, each agent can be in one of three states (susceptible, infected, or recovered), and transitions between these states follow a stochastic process. The probability of an agent becoming infected depends on the number of its infected neighbors, hence all agents are correlated. A common mean-field theory of the same stochastic process however, assumes that the agents are statistically independent. This leads to a self-feedback effect in the approximation: when an agent infects its neighbors, this infection may subsequently travel back to the original agent at a later time, leading to a self-infection of the agent which is not present in the underlying stochastic process. We here compute the first order correction to the mean-field assumption, which takes fluctuations up to second order in the interaction strength into account. We find that it cancels the self-feedback effect, leading to smaller infection rates. In the SIR model and in the SIRS model, the correction significantly improves predictions. In particular, it captures how sparsity dampens the spread of the disease: this indicates that reducing the number of contacts is more effective than predicted by mean-field models.

It is frequently hypothesized that cortical networks operate close to a critical point. Advantages of criticality include rich dynamics well-suited for computation and critical slowing down, which may offer a mechanism for dynamic memory. However, mean-field approximations, while versatile and popular, inherently neglect the fluctuations responsible for such critical dynamics. Thus, a renormalized theory is necessary. We consider the Sompolinsky-Crisanti-Sommers model which displays a well studied chaotic as well as a magnetic transition. Based on the analogue of a quantum effective action, we derive self-consistency equations for the first two renormalized Greens functions. Their self-consistent solution reveals a coupling between the population level activity and single neuron heterogeneity. The quantitative theory explains the population autocorrelation function, the single-unit autocorrelation function with its multiple temporal scales, and cross correlations.

Recent advancements in measurement techniques have resulted in an increasing amount of data on neural activities recorded in parallel, revealing largely heterogeneous correlation patterns across neurons. Yet, the mechanistic origin of this heterogeneity is largely unknown because existing theoretical approaches linking structure and dynamics in neural circuits are restricted to population-averaged connectivity and activity. Here we present a systematic inclusion of heterogeneity in network connectivity to derive quantitative predictions for neuron-resolved covariances and their statistics in spiking neural networks. Our study shows that the heterogeneity in covariances is not a result of variability in single-neuron firing statistics but stems from the ubiquitously observed sparsity and variability of connections in brain networks. Linear-response theory maps these features to the effective connectivity between neurons, which in turn determines neuronal covariances. Beyond-mean-field tools reveal that synaptic heterogeneity modulates the variability of covariances and thus the complexity of neuronal coordination across many orders of magnitude.

Bayesian inference and kernel methods are well established in machine learning. The neural network Gaussian process in particular provides a concept to investigate neural networks in the limit of infinitely wide hidden layers by using kernel and inference methods. Here we build upon this limit and provide a field-theoretic formalism which covers the generalization properties of infinitely wide networks. We systematically compute generalization properties of linear, non-linear, and deep non-linear networks for kernel matrices with heterogeneous entries. In contrast to currently employed spectral methods we derive the generalization properties from the statistical properties of the input, elucidating the interplay of input dimensionality, size of the training data set, and variability of the data. We show that data variability leads to a non-Gaussian action reminiscent of a ($\varphi^3+\varphi^4$)-theory. Using our formalism on a synthetic task and on MNIST we obtain a homogeneous kernel matrix approximation for the learning curve as well as corrections due to data variability which allow the estimation of the generalization properties and exact results for the bounds of the learning curves in the case of infinitely many training data points., Comment: 56 pages, 8 figures

State-of-the-art neural networks require extreme computational power to train. It is therefore natural to wonder whether they are optimally trained. Here we apply a recent advancement in stochastic thermodynamics which allows bounding the speed at which one can go from the initial weight distribution to the final distribution of the fully trained network, based on the ratio of their Wasserstein-2 distance and the entropy production rate of the dynamical process connecting them. Considering both gradient-flow and Langevin training dynamics, we provide analytical expressions for these speed limits for linear and linearizable neural networks e.g. Neural Tangent Kernel (NTK). Remarkably, given some plausible scaling assumptions on the NTK spectra and spectral decomposition of the labels -- learning is optimal in a scaling sense. Our results are consistent with small-scale experiments with Convolutional Neural Networks (CNNs) and Fully Connected Neural networks (FCNs) on CIFAR-10, showing a short highly non-optimal regime followed by a longer optimal regime.

Residual networks have significantly better trainability and thus performance than feed-forward networks at large depth. Introducing skip connections facilitates signal propagation to deeper layers. In addition, previous works found that adding a scaling parameter for the residual branch further improves generalization performance. While they empirically identified a particularly beneficial range of values for this scaling parameter, the associated performance improvement and its universality across network hyperparameters yet need to be understood. For feed-forward networks, finite-size theories have led to important insights with regard to signal propagation and hyperparameter tuning. We here derive a systematic finite-size field theory for residual networks to study signal propagation and its dependence on the scaling for the residual branch. We derive analytical expressions for the response function, a measure for the network's sensitivity to inputs, and show that for deep networks the empirically found values for the scaling parameter lie within the range of maximal sensitivity. Furthermore, we obtain an analytical expression for the optimal scaling parameter that depends only weakly on other network hyperparameters, such as the weight variance, thereby explaining its universality across hyperparameters. Overall, this work provides a theoretical framework to study ResNets at finite size., Comment: 21 pages, 8 figures, under review

One challenge of physics is to explain how collective properties arise from microscopic interactions. Indeed, interactions form the building blocks of almost all physical theories and are described by polynomial terms in the action. The traditional approach is to derive these terms from elementary processes and then use the resulting model to make predictions for the entire system. But what if the underlying processes are unknown? Can we reverse the approach and learn the microscopic action by observing the entire system? We use invertible neural networks (INNs) to first learn the observed data distribution. By the choice of a suitable nonlinearity for the neuronal activation function, we are then able to compute the action from the weights of the trained model; a diagrammatic language expresses the change of the action from layer to layer. This process uncovers how the network hierarchically constructs interactions via nonlinear transformations of pairwise relations. We test this approach on simulated data sets of interacting theories. The network consistently reproduces a broad class of unimodal distributions; outside this class, it finds effective theories that approximate the data statistics up to the third cumulant. We explicitly show how network depth and data quantity jointly improve the agreement between the learned and the true model. This work shows how to leverage the power of machine learning to transparently extract microscopic models from data.

Many observables of brain dynamics appear to be optimized for computation. Which connectivity structures underlie this fine-tuning? We propose that many of these structures are naturally encoded in the space that more directly relates to network dynamics - the space of the connectivity eigenmodes. We develop a mathematical theory to impose eigenmode structures on connectivity, systematically characterizing their effect on network dynamics. We find the density of nearly-critical eigenvalues to be a particularly fundamental structure. It flexibly controls the power-law scaling of dynamical observables, in analogy with the system's spatial dimension in classical critical phenomena. This mechanism provides control over observables which are found to be fine-tuned in brain networks, but remained so far unexplained by traditionally studied structures, such as connectivity motifs. Specifically, the slope of the principal component spectrum of neural activity can be fine-tuned, as observed in primary visual cortex of mice. Furthermore, a novel transition between high and low dimensional activity allows for a wide and flexible tuning of dimensionality, as observed throughout cortex. The here discovered structures thus largely complement motifs. In fact, they are of a different, collective nature: they are not reflected by any local motif configuration. This result shows that many functionally relevant structures can remain hidden within the apparent randomness of highly heterogeneous cortical circuits. Our methods enable revealing these structures and investigate their effect on network dynamics.

Neuronal systems need to process temporal signals. We here show how higher-order temporal (co-)fluctuations can be employed to represent and process information. Concretely, we demonstrate that a simple biologically inspired feedforward neuronal model is able to extract information from up to the third order cumulant to perform time series classification. This model relies on a weighted linear summation of synaptic inputs followed by a nonlinear gain function. Training both - the synaptic weights and the nonlinear gain function - exposes how the non-linearity allows for the transfer of higher order correlations to the mean, which in turn enables the synergistic use of information encoded in multiple cumulants to maximize the classification accuracy. The approach is demonstrated both on a synthetic and on real world datasets of multivariate time series. Moreover, we show that the biologically inspired architecture makes better use of the number of trainable parameters as compared to a classical machine-learning scheme. Our findings emphasize the benefit of biological neuronal architectures, paired with dedicated learning algorithms, for the processing of information embedded in higher-order statistical cumulants of temporal (co-)fluctuations.

We analytically determine the number and distribution of fixed points in a canonical model of a chaotic neural network. This distribution reveals that fixed points and dynamics are confined to separate shells in phase space. Furthermore, the distribution enables us to determine the eigenvalue spectra of the Jacobian at the fixed points. Despite the radial separation of fixed points and dynamics, we find that nearby fixed points act as partially attracting landmarks for the dynamics., Comment: 37 pages, 11 figures. Main changes include: - additional analysis of the fixed points' role for the dynamics - replica calculation for the random determinant (removing a previous assumption) - extensive investigation of finite size effects - asymptotic results for the number of fixed points and their distribution at the edge of chaos and in the strongly chaotic limit

Neural networks can implement arbitrary functions. But, mechanistically, what are the tools at their disposal to construct the target? For classification tasks, the network must transform the data classes into a linearly separable representation in the final hidden layer. We show that a feed-forward architecture has one primary tool at hand to achieve this separability: progressive folding of the data manifold in unoccupied higher dimensions. The operation of folding provides a useful intuition in low-dimensions that generalizes to high ones. We argue that an alternative method based on shear, requiring very deep architectures, plays only a small role in real-world networks. The folding operation, however, is powerful as long as layers are wider than the data dimensionality, allowing efficient solutions by providing access to arbitrary regions in the distribution, such as data points of one class forming islands within the other classes. We argue that a link exists between the universal approximation property in ReLU networks and the fold-and-cut theorem (Demaine et al., 1998) dealing with physical paper folding. Based on the mechanistic insight, we predict that the progressive generation of separability is necessarily accompanied by neurons showing mixed selectivity and bimodal tuning curves. This is validated in a network trained on the poker hand task, showing the emergence of bimodal tuning curves during training. We hope that our intuitive picture of the data transformation in deep networks can help to provide interpretability, and discuss possible applications to the theory of convolutional networks, loss landscapes, and generalization. TL;DR: Shows that the internal processing of deep networks can be thought of as literal folding operations on the data distribution in the N-dimensional activation space. A link to a well-known theorem in origami theory is provided., Comment: preliminary version, comments are welcome

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the non-linearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification., Comment: Published in Physical Review Research Changes with respect to the previous version: - Added results with CIFAR-10 - Added sections to the supplementary: - Derivation of an analogous result to the depth scale of untrained deep networks. - Expanded discussion applicability of the Gaussian assumption when variables are weakly correlated. - Clarified main text in some areas. - Fixed typos

Understanding capabilities and limitations of different network architectures is of fundamental importance to machine learning. Bayesian inference on Gaussian processes has proven to be a viable approach for studying recurrent and deep networks in the limit of infinite layer width, $n\to\infty$. Here we present a unified and systematic derivation of the mean-field theory for both architectures that starts from first principles by employing established methods from statistical physics of disordered systems. The theory elucidates that while the mean-field equations are different with regard to their temporal structure, they yet yield identical Gaussian kernels when readouts are taken at a single time point or layer, respectively. Bayesian inference applied to classification then predicts identical performance and capabilities for the two architectures. Numerically, we find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks and the convergence speed depends non-trivially on the parameters of the weight prior as well as the depth or number of time steps, respectively. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in $1/n$ and we compute next-to-leading-order corrections which turn out to be architecture-specific. The formalism thus paves the way to investigate the fundamental differences between recurrent and deep architectures at finite widths $n$., Comment: Revision including next-to-leading-order corrections

Criticality is deeply related to optimal computational capacity. The lack of a renormalized theory of critical brain dynamics, however, so far limits insights into this form of biological information processing to mean-field results. These methods neglect a key feature of critical systems: the interaction between degrees of freedom across all length scales, which allows for complex nonlinear computation. We present a renormalized theory of a prototypical neural field theory, the stochastic Wilson-Cowan equation. We compute the flow of couplings, which parameterize interactions on increasing length scales. Despite similarities with the Kardar-Parisi-Zhang model, the theory is of a Gell-Mann-Low type, the archetypal form of a renormalizable quantum field theory. Here, nonlinear couplings vanish, flowing towards the Gaussian fixed point, but logarithmically slowly, thus remaining effective on most scales. We show this critical structure of interactions to implement a desirable trade-off between linearity, optimal for information storage, and nonlinearity, required for computation.

Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a non-trivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here, we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barab\'asi-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.

Cortical networks are strongly recurrent, and neurons have intrinsic temporal dynamics. This sets them apart from deep feed-forward networks. Despite the tremendous progress in the application of feed-forward networks and their theoretical understanding, it remains unclear how the interplay of recurrence and non-linearities in recurrent cortical networks contributes to their function. The purpose of this work is to present a solvable recurrent network model that links to feed forward networks. By perturbative methods we transform the time-continuous, recurrent dynamics into an effective feed-forward structure of linear and non-linear temporal kernels. The resulting analytical expressions allow us to build optimal time-series classifiers from random reservoir networks. Firstly, this allows us to optimize not only the readout vectors, but also the input projection, demonstrating a strong potential performance gain. Secondly, the analysis exposes how the second order stimulus statistics is a crucial element that interacts with the non-linearity of the dynamics and boosts performance.

We here unify the field theoretical approach to neuronal networks with large deviations theory. For a prototypical random recurrent network model with continuous-valued units, we show that the effective action is identical to the rate function and derive the latter using field theory. This rate function takes the form of a Kullback-Leibler divergence which enables data-driven inference of model parameters and calculation of fluctuations beyond mean-field theory. Lastly, we expose a regime with fluctuation-induced transitions between mean-field solutions., Comment: Extension to multiple populations

Due to the point-like nature of neuronal spiking, efficient neural network simulators often employ event-based simulation schemes for synapses. Yet many types of synaptic plasticity rely on the membrane potential of the postsynaptic cell as a third factor in addition to pre- and postsynaptic spike times. Synapses therefore require continuous information to update their strength which a priori necessitates a continuous update in a time-driven manner. The latter hinders scaling of simulations to realistic cortical network sizes and relevant time scales for learning. Here, we derive two efficient algorithms for archiving postsynaptic membrane potentials, both compatible with modern simulation engines based on event-based synapse updates. We theoretically contrast the two algorithms with a time-driven synapse update scheme to analyze advantages in terms of memory and computations. We further present a reference implementation in the spiking neural network simulator NEST for two prototypical voltage-based plasticity rules: the Clopath rule and the Urbanczik-Senn rule. For both rules, the two event-based algorithms significantly outperform the time-driven scheme. Depending on the amount of data to be stored for plasticity, which heavily differs between the rules, a strong performance increase can be achieved by compressing or sampling of information on membrane potentials. Our results on computational efficiency related to archiving of information provide guidelines for the design of learning rules in order to make them practically usable in large-scale networks., Comment: 45 pages, 13 figures, 7 tables

Wilson's original formulation of the renormalization group is perturbative in nature. We here present an alternative derivation of the infinitesimal momentum shell RG, akin to the Wegner and Houghton scheme, that is a priori exact. We show that the momentum-dependence of vertices is key to obtain a diagrammatic framework that has the same one-loop structure as the vertex expansion of the Wetterich equation. Momentum dependence leads to a delayed functional differential equation in the cutoff parameter. Approximations are then made at two points: truncation of the vertex expansion and approximating the functional form of the momentum dependence by a momentum-scale expansion. We exemplify the method on the scalar $\varphi^{4}$-theory, computing analytically the Wilson-Fisher fixed point, its anomalous dimension $\eta(d)$ and the critical exponent $\nu(d)$ non-perturbatively in $d\in[3,4]$ dimensions. The results are in reasonable agreement with the known values, despite the simplicity of the method.

Neurons in the brain communicate with spikes, which are discrete events in time and value. Functional network models often employ rate units that are continuously coupled by analog signals. Is there a qualitative difference implied by these two forms of signaling? We develop a unified mean-field theory for large random networks to show that first- and second-order statistics in rate and binary networks are in fact identical if rate neurons receive the right amount of noise. Their response to presented stimuli, however, can be radically different. We quantify these differences by studying how nearby state trajectories evolve over time, asking to what extent the dynamics is chaotic. Chaos in the two models is found to be qualitatively different. In binary networks we find a network-size-dependent transition to chaos and a chaotic submanifold whose dimensionality expands stereotypically with time, while rate networks with matched statistics are nonchaotic. Dimensionality expansion in chaotic binary networks aids classification in reservoir computing and optimal performance is reached within about a single activation per neuron; a fast mechanism for computation that we demonstrate also in spiking networks. A generalization of this mechanism extends to rate networks in their respective chaotic regimes., Comment: Final, shortened post-print version. (see v3 for extended material on microscopic and macroscopic chaos)

The classical perceptron is a simple neural network that performs a binary classification by a linear mapping between static inputs and outputs and application of a threshold. For small inputs, neural networks in a stationary state also perform an effectively linear input-output transformation, but of an entire time series. Choosing the temporal mean of the time series as the feature for classification, the linear transformation of the network with subsequent thresholding is equivalent to the classical perceptron. Here we show that choosing covariances of time series as the feature for classification maps the neural network to what we call a 'covariance perceptron'; a mapping between covariances that is bilinear in terms of weights. By extending Gardner's theory of connections to this bilinear problem, using a replica symmetric mean-field theory, we compute the pattern and information capacities of the covariance perceptron in the infinite-size limit. Closed-form expressions reveal superior pattern capacity in the binary classification task compared to the classical perceptron in the case of a high-dimensional input and low-dimensional output. For less convergent networks, the mean perceptron classifies a larger number of stimuli. However, since covariances span a much larger input and output space than means, the amount of stored information in the covariance perceptron exceeds the classical counterpart. For strongly convergent connectivity it is superior by a factor equal to the number of input neurons. Theoretical calculations are validated numerically for finite size systems using a gradient-based optimization of a soft-margin, as well as numerical solvers for the NP hard quadratically constrained quadratic programming problem, to which training can be mapped.

These notes attempt a self-contained introduction into statistical field theory applied to neural networks of rate units and binary spins. The presentation consists of three parts: First, the introduction of fundamental notions of probabilities, moments, cumulants, and their relation by the linked cluster theorem, of which Wick's theorem is the most important special case; followed by the diagrammatic formulation of perturbation theory, reviewed in the statistical setting. Second, dynamics described by stochastic differential equations in the Ito-formulation, treated in the Martin-Siggia-Rose-De Dominicis-Janssen path integral formalism. With concepts from disordered systems, we then study networks with random connectivity and derive their self-consistent dynamic mean-field theory, explaining the statistics of fluctuations and the emergence of different phases with regular and chaotic dynamics. Third, we introduce the effective action, vertex functions, and the loopwise expansion. These tools are illustrated by systematic derivations of self-consistency equations, going beyond the mean-field approximation. These methods are applied to the pairwise maximum entropy (Ising spin) model, including the recently-found diagrammatic derivation of the Thouless-Anderson-Palmer mean field theory.

Neural dynamics is often investigated with tools from bifurcation theory. However, many neuron models are stochastic, mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals. Noise changes the dynamics with respect to the deterministic model; in particular bifurcation theory cannot be applied. We formulate stochastic neuronal dynamics in the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism and present the fluctuation expansion of the effective action and the functional renormalization group (fRG) as two systematic ways to incorporate corrections to the mean dynamics and time-dependent statistics due to fluctuations in the presence of nonlinear neuronal gain. To formulate self-consistency equations, we derive a fundamental link between the effective action in the Onsager-Machlup(OM) formalism, which allows the study of phase transitions, and the MSRDJ effective action, which is computationally advantageous. These results in particular allow the derivation of an OM effective action for systems with non-Gaussian noise. This approach naturally leads to effective deterministic equations for the first moment of the stochastic system; they explain how nonlinearities and noise cooperate to produce memory effects. Moreover, the MSRDJ formulation yields an effective linear system that has identical power spectra and linear response. Starting from the better known loopwise approximation, we then discuss the use of the fRG as a method to obtain self-consistency beyond the mean. We present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the Blaizot, M\'endez and Wschebor approximation from the derivative expansion to the vertex expansion. The methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics., Comment: Equivalent to published version, including two minor typo fixes in Eq (6) and Fig 6. All conclusions unchanged. 7 figures

Spatiotemporal patterns such as traveling waves are frequently observed in recordings of neural activity. The mechanisms underlying the generation of such patterns are largely unknown. Previous studies have investigated the existence and uniqueness of different types of waves or bumps of activity using neural-field models, phenomenological coarse-grained descriptions of neural-network dynamics. But it remains unclear how these insights can be transferred to more biologically realistic networks of spiking neurons, where individual neurons fire irregularly. Here, we employ mean-field theory to reduce a microscopic model of leaky integrate-and-fire (LIF) neurons with distance-dependent connectivity to an effective neural-field model. In contrast to existing phenomenological descriptions, the dynamics in this neural-field model depends on the mean and the variance in the synaptic input, both determining the amplitude and the temporal structure of the resulting effective coupling kernel. For the neural-field model we employ liner stability analysis to derive conditions for the existence of spatial and temporal oscillations and wave trains, that is, temporally and spatially periodic traveling waves. We first prove that wave trains cannot occur in a single homogeneous population of neurons, irrespective of the form of distance dependence of the connection probability. Compatible with the architecture of cortical neural networks, wave trains emerge in two-population networks of excitatory and inhibitory neurons as a combination of delay-induced temporal oscillations and spatial oscillations due to distance-dependent connectivity profiles. Finally, we demonstrate quantitative agreement between predictions of the analytically tractable neural-field model and numerical simulations of both networks of nonlinear rate-based units and networks of LIF neurons., Comment: 36 pages, 8 figures, 4 tables

Neural networks with equal excitatory and inhibitory feedback show high computational performance. They operate close to a critical point characterized by the joint activation of large populations of neurons. Yet, in macaque motor cortex we observe very different dynamics with weak fluctuations on the population level. This suggests that motor cortex operates in a sub-optimal regime. Here we show the opposite: the large dispersion of correlations across neurons is a signature of a rich dynamical repertoire, hidden from macroscopic brain signals, but essential for high performance in such concepts as reservoir computing. Our findings suggest a refinement of the view on criticality in neural systems: network topology and heterogeneity endow the brain with two complementary substrates for critical dynamics of largely different complexities.

This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well known and composed of one-line irreducible diagrams only. For the expansions around an arbitrary, non-Gaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the Thouless-Anderson-Palmer mean-field theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the high-temperature expansion are special cases of the general formalism presented here., Comment: 37 pages, published version

Spiking neuronal networks are usually simulated with three main simulation schemes: the classical time-driven and event-driven schemes, and the more recent hybrid scheme. All three schemes evolve the state of a neuron through a series of checkpoints: equally spaced in the first scheme and determined neuron-wise by spike events in the latter two. The time-driven and the hybrid scheme determine whether the membrane potential of a neuron crosses a threshold at the end of of the time interval between consecutive checkpoints. Threshold crossing can, however, occur within the interval even if this test is negative. Spikes can therefore be missed. The present work derives, implements, and benchmarks a method for perfect retrospective spike detection. This method can be applied to neuron models with affine or linear subthreshold dynamics. The idea behind the method is to propagate the threshold with a time-inverted dynamics, testing whether the threshold crosses the neuron state to be evolved, rather than vice versa. Algebraically this translates into a set of inequalities necessary and sufficient for threshold crossing. This test is slower than the imperfect one, but faster than an alternative perfect tests based on bisection or root-finding methods. Comparison confirms earlier results that the imperfect test rarely misses spikes (less than a fraction $1/10^8$ of missed spikes) in biologically relevant settings. This study offers an alternative geometric point of view on neuronal dynamics., Comment: 9 figures, Preliminary results in proceedings of the Bernstein Conference 2016

Propagation of oscillatory signals through the cortex and coherence is shaped by the connectivity structure of neuronal circuits. This study systematically investigates the network and stimulus properties that shape network responses. The results show how input to a cortical column model of the primary visual cortex excites dynamical modes determined by the laminar pattern. Stimulating the inhibitory neurons in the upper layer reproduces experimentally observed resonances at gamma frequency whose origin can be traced back to two anatomical sub-circuits. We develop this result systematically: Initially, we highlight the effect of stimulus amplitude and filter properties of the neurons on their response to oscillatory stimuli. Subsequently, we analyze the amplification of oscillatory stimuli by the effective network structure. We demonstrate that the amplification of stimuli, as well as their visibility in different populations, can be explained by specific network patterns. Inspired by experimental results we ask whether the anatomical origin of oscillations can be inferred by applying oscillatory stimuli. We find that different network motifs can generate similar responses to oscillatory input, showing that resonances in the network response cannot, straightforwardly, be assigned to the motifs they emerge from. Applying the analysis to a spiking model of a cortical column, we characterize how the dynamic mode structure, which is induced by the laminar connectivity, processes external input. In particular, we show that a stimulus applied to specific populations typically elicits responses of several interacting modes. The resulting network response is therefore composed of a multitude of contributions and can therefore neither be assigned to a single mode nor do the observed resonances necessarily coincide with the intrinsic resonances of the circuit.

Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units. The unified simulation framework presented here supports the combination of the two for multi-scale modeling approaches, the quantitative validation of mean-field approaches by spiking network simulations, and an increase in reliability by usage of the same simulation code and the same network model specifications for both model classes. While most efficient spiking simulations rely on the communication of discrete events, rate models require time-continuous interactions between neurons. Exploiting the conceptual similarity to the inclusion of gap junctions in spiking network simulations, we arrive at a reference implementation of instantaneous and delayed interactions between rate-based models in a spiking network simulator. The separation of rate dynamics from the general connection and communication infrastructure ensures flexibility of the framework. We further demonstrate the broad applicability of the framework by considering various examples from the literature ranging from random networks to neural field models. The study provides the prerequisite for interactions between rate-based and spiking models in a joint simulation.

Population-wide oscillations are ubiquitously observed in mesoscopic signals of cortical activity. In these network states a global oscillatory cycle modulates the propensity of neurons to fire. Synchronous activation of neurons has been hypothesized to be a separate channel of signal processing information in the brain. A salient question is therefore if and how oscillations interact with spike synchrony and in how far these channels can be considered separate. Experiments indeed showed that correlated spiking co-modulates with the static firing rate and is also tightly locked to the phase of beta-oscillations. While the dependence of correlations on the mean rate is well understood in feed-forward networks, it remains unclear why and by which mechanisms correlations tightly lock to an oscillatory cycle. We here demonstrate that such correlated activation of pairs of neurons is qualitatively explained by periodically-driven random networks. We identify the mechanisms by which covariances depend on a driving periodic stimulus. Mean-field theory combined with linear response theory yields closed-form expressions for the cyclostationary mean activities and pairwise zero-time-lag covariances of binary recurrent random networks. Two distinct mechanisms cause time-dependent covariances: the modulation of the susceptibility of single neurons (via the external input and network feedback) and the time-varying variances of single unit activities. For some parameters, the effectively inhibitory recurrent feedback leads to resonant covariances even if mean activities show non-resonant behavior. Our analytical results open the question of time-modulated synchronous activity to a quantitative analysis., Comment: 57 pages, 12 figures, published version

Neural networks of the brain form one of the most complex systems we know. Many qualitative features of the emerging collective phenomena, such as correlated activity, stability, response to inputs, chaotic and regular behavior, can, however, be understood in simple models that are accessible to a treatment in statistical mechanics, or, more precisely, classical statistical field theory. This tutorial presents the fundamentals behind contemporary developments in the theory of neural networks of rate units that are based on methods from statistical mechanics of classical systems with a large number of interacting degrees of freedom. In particular we will focus on a relevant class of systems that have quenched (time independent) disorder. In neural networks, the main source of disorder arises from random synaptic couplings between neurons. These systems are in many respects similar to spin glasses. The tutorial therefore also explains the methods for these disordered systems as far as they are applied in neuroscience. The presentation consists of two parts. In the first part we introduce stochastic differential equations in the Martin - Siggia - Rose - De Dominicis - Janssen path integral formalism. In the second part we employ this language to derive the dynamic mean-field theory for deterministic random networks, the basis of the seminal work by Sompolinsky, Crisanti, Sommers 1988, as well as a recent extension to stochastic dynamics.

Pairwise maximum-entropy models have been used in recent neuroscientific literature to predict the activity of neuronal populations, given only the time-averaged correlations of the neuron activities. This paper provides evidence that the pairwise model, applied to experimental recordings, predicts a bimodal distribution for the population-averaged activity, and for some population sizes the second mode peaks at high activities, with 90% of the neuron population active within time-windows of few milliseconds. This bimodality has several undesirable consequences: 1. The presence of two modes is unrealistic in view of observed neuronal activity. 2. The prediction of a high-activity mode is unrealistic on neurobiological grounds. 3. Boltzmann learning becomes non-ergodic, hence the pairwise model found by this method is not the maximum entropy distribution; similarly, solving the inverse problem by common variants of mean-field approximations has the same problem. 4. The Glauber dynamics associated with the model is either unrealistically bistable, or does not reflect the distribution of the pairwise model. This bimodality is first demonstrated for an experimental dataset comprising 159 neuron activities recorded from the motor cortex of macaque monkey. Using a reduced maximum-entropy model, evidence is then provided that this bimodality affects typical neural recordings of population sizes of a couple of hundreds or more neurons. As a way to eliminate the bimodality and its ensuing problems, a modified pairwise model is presented, which -- most important -- has an associated pairwise Glauber dynamics. This model avoids bimodality thanks to a minimal asymmetric inhibition. It can be interpreted as a minimum-relative-entropy model with a particular prior, or as a maximum-entropy model with an additional constraint.

Massively parallel recordings of spiking activity in cortical networks show that covariances vary widely across pairs of neurons. Their low average is well understood, but an explanation for the wide distribution in relation to the static (quenched) disorder of the connectivity in recurrent random networks was so far elusive. We here derive a finite-size mean-field theory that reduces a disordered to a highly symmetric network with fluctuating auxiliary fields. The exposed analytical relation between the statistics of connections and the statistics of pairwise covariances shows that both, average and dispersion of the latter, diverge at a critical coupling. At this point, a network of nonlinear units transits from regular to chaotic dynamics. Applying these results to recordings from the mammalian brain suggests its operation close to this edge of criticality.