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The infinite width limit of random neural networks is known to result in Neural Networks as Gaussian Process (NNGP) (Lee et al. [2018]), characterized by task-independent kernels. It is widely accepted that larger network widths contribute to improved generalization (Park et al. [2019]). However, this work challenges this notion by investigating the narrow width limit of the Bayesian Parallel Branching Graph Neural Network (BPB-GNN), an architecture that resembles residual networks. We demonstrate that when the width of a BPB-GNN is significantly smaller compared to the number of training examples, each branch exhibits more robust learning due to a symmetry breaking of branches in kernel renormalization. Surprisingly, the performance of a BPB-GNN in the narrow width limit is generally superior or comparable to that achieved in the wide width limit in bias-limited scenarios. Furthermore, the readout norms of each branch in the narrow width limit are mostly independent of the architectural hyperparameters but generally reflective of the nature of the data. Our results characterize a newly defined narrow-width regime for parallel branching networks in general.

Continual learning (CL) enables animals to learn new tasks without erasing prior knowledge. CL in artificial neural networks (NNs) is challenging due to catastrophic forgetting, where new learning degrades performance on older tasks. While various techniques exist to mitigate forgetting, theoretical insights into when and why CL fails in NNs are lacking. Here, we present a statistical-mechanics theory of CL in deep, wide NNs, which characterizes the network's input-output mapping as it learns a sequence of tasks. It gives rise to order parameters (OPs) that capture how task relations and network architecture influence forgetting and knowledge transfer, as verified by numerical evaluations. We found that the input and rule similarity between tasks have different effects on CL performance. In addition, the theory predicts that increasing the network depth can effectively reduce overlap between tasks, thereby lowering forgetting. For networks with task-specific readouts, the theory identifies a phase transition where CL performance shifts dramatically as tasks become less similar, as measured by the OPs. Sufficiently low similarity leads to catastrophic anterograde interference, where the network retains old tasks perfectly but completely fails to generalize new learning. Our results delineate important factors affecting CL performance and suggest strategies for mitigating forgetting., Comment: 26 pages, 8 figures

Neural networks posses the crucial ability to generate meaningful representations of task-dependent features. Indeed, with appropriate scaling, supervised learning in neural networks can result in strong, task-dependent feature learning. However, the nature of the emergent representations, which we call the `coding scheme', is still unclear. To understand the emergent coding scheme, we investigate fully-connected, wide neural networks learning classification tasks using the Bayesian framework where learning shapes the posterior distribution of the network weights. Consistent with previous findings, our analysis of the feature learning regime (also known as `non-lazy', `rich', or `mean-field' regime) shows that the networks acquire strong, data-dependent features. Surprisingly, the nature of the internal representations depends crucially on the neuronal nonlinearity. In linear networks, an analog coding scheme of the task emerges. Despite the strong representations, the mean predictor is identical to the lazy case. In nonlinear networks, spontaneous symmetry breaking leads to either redundant or sparse coding schemes. Our findings highlight how network properties such as scaling of weights and neuronal nonlinearity can profoundly influence the emergent representations.

Despite the remarkable empirical performance of Transformers, their theoretical understanding remains elusive. Here, we consider a deep multi-head self-attention network, that is closely related to Transformers yet analytically tractable. We develop a statistical mechanics theory of Bayesian learning in this model, deriving exact equations for the network's predictor statistics under the finite-width thermodynamic limit, i.e., $N,P\rightarrow\infty$, $P/N=\mathcal{O}(1)$, where $N$ is the network width and $P$ is the number of training examples. Our theory shows that the predictor statistics are expressed as a sum of independent kernels, each one pairing different 'attention paths', defined as information pathways through different attention heads across layers. The kernels are weighted according to a 'task-relevant kernel combination' mechanism that aligns the total kernel with the task labels. As a consequence, this interplay between attention paths enhances generalization performance. Experiments confirm our findings on both synthetic and real-world sequence classification tasks. Finally, our theory explicitly relates the kernel combination mechanism to properties of the learned weights, allowing for a qualitative transfer of its insights to models trained via gradient descent. As an illustration, we demonstrate an efficient size reduction of the network, by pruning those attention heads that are deemed less relevant by our theory.

Recently, growth in our understanding of the computations performed in both biological and artificial neural networks has largely been driven by either low-level mechanistic studies or global normative approaches. However, concrete methodologies for bridging the gap between these levels of abstraction remain elusive. In this work, we investigate the internal mechanisms of neural networks through the lens of neural population geometry, aiming to provide understanding at an intermediate level of abstraction, as a way to bridge that gap. Utilizing manifold capacity theory (MCT) from statistical physics and manifold alignment analysis (MAA) from high-dimensional statistics, we probe the underlying organization of task-dependent manifolds in deep neural networks and macaque neural recordings. Specifically, we quantitatively characterize how different learning objectives lead to differences in the organizational strategies of these models and demonstrate how these geometric analyses are connected to the decodability of task-relevant information. These analyses present a strong direction for bridging mechanistic and normative theories in neural networks through neural population geometry, potentially opening up many future research avenues in both machine learning and neuroscience., Comment: To appear in the proceedings of the Conference on Parsimony and Learning (CPAL) 2024

Artificial neural networks have revolutionized machine learning in recent years, but a complete theoretical framework for their learning process is still lacking. Substantial progress has been made for infinitely wide networks. In this regime, two disparate theoretical frameworks have been used, in which the network's output is described using kernels: one framework is based on the Neural Tangent Kernel (NTK) which assumes linearized gradient descent dynamics, while the Neural Network Gaussian Process (NNGP) kernel assumes a Bayesian framework. However, the relation between these two frameworks has remained elusive. This work unifies these two distinct theories using a Markov proximal learning model for learning dynamics in an ensemble of randomly initialized infinitely wide deep networks. We derive an exact analytical expression for the network input-output function during and after learning, and introduce a new time-dependent Neural Dynamical Kernel (NDK) from which both NTK and NNGP kernels can be derived. We identify two learning phases characterized by different time scales: gradient-driven and diffusive learning. In the initial gradient-driven learning phase, the dynamics is dominated by deterministic gradient descent, and is described by the NTK theory. This phase is followed by the diffusive learning stage, during which the network parameters sample the solution space, ultimately approaching the equilibrium distribution corresponding to NNGP. Combined with numerical evaluations on synthetic and benchmark datasets, we provide novel insights into the different roles of initialization, regularization, and network depth, as well as phenomena such as early stopping and representational drift. This work closes the gap between the NTK and NNGP theories, providing a comprehensive framework for understanding the learning process of deep neural networks in the infinite width limit.

Recently proposed Gated Linear Networks present a tractable nonlinear network architecture, and exhibit interesting capabilities such as learning with local error signals and reduced forgetting in sequential learning. In this work, we introduce a novel gating architecture, named Globally Gated Deep Linear Networks (GGDLNs) where gating units are shared among all processing units in each layer, thereby decoupling the architectures of the nonlinear but unlearned gatings and the learned linear processing motifs. We derive exact equations for the generalization properties in these networks in the finite-width thermodynamic limit, defined by $P,N\rightarrow\infty, P/N\sim O(1)$, where P and N are the training sample size and the network width respectively. We find that the statistics of the network predictor can be expressed in terms of kernels that undergo shape renormalization through a data-dependent matrix compared to the GP kernels. Our theory accurately captures the behavior of finite width GGDLNs trained with gradient descent dynamics. We show that kernel shape renormalization gives rise to rich generalization properties w.r.t. network width, depth and L2 regularization amplitude. Interestingly, networks with sufficient gating units behave similarly to standard ReLU networks. Although gatings in the model do not participate in supervised learning, we show the utility of unsupervised learning of the gating parameters. Additionally, our theory allows the evaluation of the network's ability for learning multiple tasks by incorporating task-relevant information into the gating units. In summary, our work is the first exact theoretical solution of learning in a family of nonlinear networks with finite width. The rich and diverse behavior of the GGDLNs suggests that they are helpful analytically tractable models of learning single and multiple tasks, in finite-width nonlinear deep networks.

A central question in computational neuroscience is how structure determines function in neural networks. The emerging high-quality large-scale connectomic datasets raise the question of what general functional principles can be gleaned from structural information such as the distribution of excitatory/inhibitory synapse types and the distribution of synaptic weights. Motivated by this question, we developed a statistical mechanical theory of learning in neural networks that incorporates structural information as constraints. We derived an analytical solution for the memory capacity of the perceptron, a basic feedforward model of supervised learning, with constraint on the distribution of its weights. Our theory predicts that the reduction in capacity due to the constrained weight-distribution is related to the Wasserstein distance between the imposed distribution and that of the standard normal distribution. To test the theoretical predictions, we use optimal transport theory and information geometry to develop an SGD-based algorithm to find weights that simultaneously learn the input-output task and satisfy the distribution constraint. We show that training in our algorithm can be interpreted as geodesic flows in the Wasserstein space of probability distributions. We further developed a statistical mechanical theory for teacher-student perceptron rule learning and ask for the best way for the student to incorporate prior knowledge of the rule. Our theory shows that it is beneficial for the learner to adopt different prior weight distributions during learning, and shows that distribution-constrained learning outperforms unconstrained and sign-constrained learning. Our theory and algorithm provide novel strategies for incorporating prior knowledge about weights into learning, and reveal a powerful connection between structure and function in neural networks., Comment: 38 pages, 13 figures. Updated introduction part and fixed several typos

When neural circuits learn to perform a task, it is often the case that there are many sets of synaptic connections that are consistent with the task. However, only a small number of possible solutions are robust to noise in the input and are capable of generalizing their performance of the task to new inputs. Finding such good solutions is an important goal of learning systems in general and neuronal circuits in particular. For systems operating with static inputs and outputs, a well known approach to the problem is the large margin methods such as Support Vector Machines (SVM). By maximizing the distance of the data vectors from the decision surface, these solutions enjoy increased robustness to noise and enhanced generalization abilities. Furthermore, the use of the kernel method enables SVMs to perform classification tasks that require nonlinear decision surfaces. However, for dynamical systems with event based outputs, such as spiking neural networks and other continuous time threshold crossing systems, this optimality criterion is inapplicable due to the strong temporal correlations in their input and output. We introduce a novel extension of the static SVMs - The Temporal Support Vector Machine (T-SVM). The T-SVM finds a solution that maximizes a new construct - the dynamical margin. We show that T-SVM and its kernel extensions generate robust synaptic weight vectors in spiking neurons and enable their learning of tasks that require nonlinear spatial integration of synaptic inputs. We propose T-SVM with nonlinear kernels as a new model of the computational role of the nonlinearities and extensive morphologies of neuronal dendritic trees.

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous works introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs as well as the octonion and previously-proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain., Comment: 32 pages, 9 figures

A neural population responding to multiple appearances of a single object defines a manifold in the neural response space. The ability to classify such manifolds is of interest, as object recognition and other computational tasks require a response that is insensitive to variability within a manifold. Linear classification of object manifolds was previously studied for max-margin classifiers. Soft-margin classifiers are a larger class of algorithms and provide an additional regularization parameter used in applications to optimize performance outside the training set by balancing between making fewer training errors and learning more robust classifiers. Here we develop a mean-field theory describing the behavior of soft-margin classifiers applied to object manifolds. Analyzing manifolds with increasing complexity, from points through spheres to general manifolds, a mean-field theory describes the expected value of the linear classifier's norm, as well as the distribution of fields and slack variables. By analyzing the robustness of the learned classification to noise, we can predict the probability of classification errors and their dependence on regularization, demonstrating a finite optimal choice. The theory describes a previously unknown phase transition, corresponding to the disappearance of a non-trivial solution, thus providing a soft version of the well-known classification capacity of max-margin classifiers.

Networks of strongly-coupled neurons with random connectivity exhibit chaotic, asynchronous fluctuations. In previous work, we showed that when endowed with an additional low-rank connectivity consisting of the outer product of orthogonal vectors, these networks generate large-scale coherent fluctuations. Although a striking phenomenon, that result depended on a fine-tuned choice of low-rank structure. Here we extend that work by generalizing the theory of excitation-inhibition balance to networks with arbitrary low-rank structure and show that low-dimensional variability emerges intrinsically through what we call incomplete recurrent alignment. We say that a low-rank connectivity structure exhibits incomplete alignment if its row-space is not contained in its column-space. In the setting of incomplete alignment, recurrent connectivity can be decomposed into a subspace-recurrent component and an effective-feedforward component. We show that high-dimensional, microscopic fluctuations are propagated via the effective-feedforward component to a low-dimensional subspace where they are dynamically balanced by macroscopic fluctuations. We present biologically plausible examples from excitation-inhibition networks and networks with heterogeneous degree distributions. Finally, we define the alignment matrix as the overlap between left and right-singular vectors of the structured connectivity, and show that the singular values of the alignment matrix determine the amplitude of macroscopic variability, while its singular vectors determine the structure. Our work shows how macroscopic fluctuations can emerge generically in strongly-coupled networks with low-rank structure. Furthermore, by generalizing excitation-inhibition balance to arbitrary low-rank structure our work may find relevance in any setting with strongly interacting units, whether in biological, social, or technological networks.

The success of deep learning in many real-world tasks has triggered an intense effort to understand the power and limitations of deep learning in the training and generalization of complex tasks, so far with limited progress. In this work, we study the statistical mechanics of learning in Deep Linear Neural Networks (DLNNs) in which the input-output function of an individual unit is linear. Despite the linearity of the units, learning in DLNNs is nonlinear, hence studying its properties reveals some of the features of nonlinear Deep Neural Networks (DNNs). Importantly, we solve exactly the network properties following supervised learning using an equilibrium Gibbs distribution in the weight space. To do this, we introduce the Back-Propagating Kernel Renormalization (BPKR), which allows for the incremental integration of the network weights starting from the network output layer and progressing backward until the first layer's weights are integrated out. This procedure allows us to evaluate important network properties, such as its generalization error, the role of network width and depth, the impact of the size of the training set, and the effects of weight regularization and learning stochasticity. BPKR does not assume specific statistics of the input or the task's output. Furthermore, by performing partial integration of the layers, the BPKR allows us to compute the properties of the neural representations across the different hidden layers. We have proposed an extension of the BPKR to nonlinear DNNs with ReLU. Surprisingly, our numerical simulations reveal that despite the nonlinearity, the predictions of our theory are largely shared by ReLU networks in a wide regime of parameters. Our work is the first exact statistical mechanical study of learning in a family of DNNs, and the first successful theory of learning through successive integration of DoFs in the learned weight space., Comment: main text 31 pages, 15 figures

Many sensory pathways in the brain rely on sparsely active populations of neurons downstream from the input stimuli. The biological reason for the occurrence of expanded structure in the brain is unclear, but may be because expansion can increase the expressive power of a neural network. In this work, we show that expanding a neural network can improve its generalization performance even in cases in which the expanded structure is pruned after the learning period. To study this setting we use a teacher-student framework where a perceptron teacher network generates labels which are corrupted with small amounts of noise. We then train a student network that is structurally matched to the teacher and can achieve optimal accuracy if given the teacher's synaptic weights. We find that sparse expansion of the input of a student perceptron network both increases its capacity and improves the generalization performance of the network when learning a noisy rule from a teacher perceptron when these expansions are pruned after learning. We find similar behavior when the expanded units are stochastic and uncorrelated with the input and analyze this network in the mean field limit. We show by solving the mean field equations that the generalization error of the stochastic expanded student network continues to drop as the size of the network increases. The improvement in generalization performance occurs despite the increased complexity of the student network relative to the teacher it is trying to learn. We show that this effect is closely related to the addition of slack variables in artificial neural networks and suggest possible implications for artificial and biological neural networks., Comment: 13+10 pages, 13 figures

A recent line of works studied wide deep neural networks (DNNs) by approximating them as Gaussian Processes (GPs). A DNN trained with gradient flow was shown to map to a GP governed by the Neural Tangent Kernel (NTK), whereas earlier works showed that a DNN with an i.i.d. prior over its weights maps to the so-called Neural Network Gaussian Process (NNGP). Here we consider a DNN training protocol, involving noise, weight decay and finite width, whose outcome corresponds to a certain non-Gaussian stochastic process. An analytical framework is then introduced to analyze this non-Gaussian process, whose deviation from a GP is controlled by the finite width. Our contribution is three-fold: (i) In the infinite width limit, we establish a correspondence between DNNs trained with noisy gradients and the NNGP, not the NTK. (ii) We provide a general analytical form for the finite width correction (FWC) for DNNs with arbitrary activation functions and depth and use it to predict the outputs of empirical finite networks with high accuracy. Analyzing the FWC behavior as a function of $n$, the training set size, we find that it is negligible for both the very small $n$ regime, and, surprisingly, for the large $n$ regime (where the GP error scales as $O(1/n)$). (iii) We flesh out algebraically how these FWCs can improve the performance of finite convolutional neural networks (CNNs) relative to their GP counterparts on image classification tasks., Comment: 8 pages + appendix, 7 figures overall

Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object recognition and discrimination requires classifying the manifolds in a manner that is insensitive to variability within a manifold. How neuronal systems give rise to invariant object classification and recognition is a fundamental problem in brain theory as well as in machine learning. Here we study the ability of a readout network to classify objects from their perceptual manifold representations. We develop a statistical mechanical theory for the linear classification of manifolds with arbitrary geometry revealing a remarkable relation to the mathematics of conic decomposition. Novel geometrical measures of manifold radius and manifold dimension are introduced which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including L2 ellipsoids prototypical of strictly convex manifolds, L1 balls representing polytopes consisting of finite sample points, and orientation manifolds which arise from neurons tuned to respond to a continuous angle variable, such as object orientation. The effects of label sparsity on the classification capacity of manifolds are elucidated, revealing a scaling relation between label sparsity and manifold radius. Theoretical predictions are corroborated by numerical simulations using recently developed algorithms to compute maximum margin solutions for manifold dichotomies. Our theory and its extensions provide a powerful and rich framework for applying statistical mechanics of linear classification to data arising from neuronal responses to object stimuli, as well as to artificial deep networks trained for object recognition tasks., Comment: 24 pages, 12 figures, Supplementary Materials

We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation methods rely upon sampling large numbers of training examples from these manifolds; instead, we propose an iterative algorithm called M_{CP} based upon a cutting-plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of M_{CP} are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet dataset. Our results indicate that M_{CP} is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin methods using data augmentation methods.

Neurons and networks in the cerebral cortex must operate reliably despite multiple sources of noise. To evaluate the impact of both input and output noise, we determine the robustness of single-neuron stimulus selective responses, as well as the robustness of attractor states of networks of neurons performing memory tasks. We find that robustness to output noise requires synaptic connections to be in a balanced regime in which excitation and inhibition are strong and largely cancel each other. We evaluate the conditions required for this regime to exist and determine the properties of networks operating within it. A plausible synaptic plasticity rule for learning that balances weight configurations is presented. Our theory predicts an optimal ratio of the number of excitatory and inhibitory synapses for maximizing the encoding capacity of balanced networks for a given statistics of afferent activations. Previous work has shown that balanced networks amplify spatio-temporal variability and account for observed asynchronous irregular states. Here we present a novel type of balanced network that amplifies small changes in the impinging signals, and emerges automatically from learning to perform neuronal and network functions robustly., Comment: Article and supplementary information

Objects are represented in sensory systems by continuous manifolds due to sensitivity of neuronal responses to changes in physical features such as location, orientation, and intensity. What makes certain sensory representations better suited for invariant decoding of objects by downstream networks? We present a theory that characterizes the ability of a linear readout network, the perceptron, to classify objects from variable neural responses. We show how the readout perceptron capacity depends on the dimensionality, size, and shape of the object manifolds in its input neural representation., Comment: 5 pages, 3 figures, accepted in Physical Review E as Rapid Communication on 14th May. 2016

Firing patterns in the central nervous system often exhibit strong temporal irregularity and heterogeneity in their time averaged response properties. Previous studies suggested that these properties are outcome of an intrinsic chaotic dynamics. Indeed, simplified rate-based large neuronal networks with random synaptic connections are known to exhibit sharp transition from fixed point to chaotic dynamics when the synaptic gain is increased. However, the existence of a similar transition in neuronal circuit models with more realistic architectures and firing dynamics has not been established. In this work we investigate rate based dynamics of neuronal circuits composed of several subpopulations and random connectivity. Nonzero connections are either positive-for excitatory neurons, or negative for inhibitory ones, while single neuron output is strictly positive; in line with known constraints in many biological systems. Using Dynamic Mean Field Theory, we find the phase diagram depicting the regimes of stable fixed point, unstable dynamic and chaotic rate fluctuations. We characterize the properties of systems near the chaotic transition and show that dilute excitatory-inhibitory architectures exhibit the same onset to chaos as a network with Gaussian connectivity. Interestingly, the critical properties near transition depend on the shape of the single- neuron input-output transfer function near firing threshold. Finally, we investigate network models with spiking dynamics. When synaptic time constants are slow relative to the mean inverse firing rates, the network undergoes a sharp transition from fast spiking fluctuations and static firing rates to a state with slow chaotic rate fluctuations. When the synaptic time constants are finite, the transition becomes smooth and obeys scaling properties, similar to crossover phenomena in statistical mechanics, Comment: 28 Pages, 12 Figures, 5 Appendices

We construct and analyze a rate-based neural network model in which self-interacting units represent clusters of neurons with strong local connectivity and random inter-unit connections reflect long-range interactions. When sufficiently strong, the self-interactions make the individual units bistable. Simulation results, mean-field calculations and stability analysis reveal the different dynamic regimes of this network and identify the locations in parameter space of its phase transitions. We identify an interesting dynamical regime exhibiting transient but long-lived chaotic activity that combines features of chaotic and multiple fixed-point attractors.

We study the computational capacity of a model neuron, the Tempotron, which classifies sequences of spikes by linear-threshold operations. We use statistical mechanics and extreme value theory to derive the capacity of the system in random classification tasks. In contrast to its static analog, the Perceptron, the Tempotron's solutions space consists of a large number of small clusters of weight vectors. The capacity of the system per synapse is finite in the large size limit and weakly diverges with the stimulus duration relative to the membrane and synaptic time constants., Comment: 4 page, 4 figures, Accepted to Physical Review Letters on 19th Oct. 2010

Trial-to-trial variability is an essential feature of neural responses, but its source is a subject of active debate. Response variability (Mast and Victor, 1991; Arieli et al., 1995 & 1996; Anderson et al., 2000 & 2001; Kenet et al., 2003; Petersen et al., 2003a & b; Fiser, Chiu and Weliky, 2004; MacLean et al., 2005; Yuste et al., 2005; Vincent et al., 2007) is often treated as random noise, generated either by other brain areas, or by stochastic processes within the circuitry being studied. We call such sources of variability external to stress the independence of this form of noise from activity driven by the stimulus. Variability can also be generated internally by the same network dynamics that generates responses to a stimulus. How can we distinguish between external and internal sources of response variability? Here we show that internal sources of variability interact nonlinearly with stimulus-induced activity, and this interaction yields a suppression of noise in the evoked state. This provides a theoretical basis and potential mechanism for the experimental observation that, in many brain areas, stimuli cause significant suppression of neuronal variability (Werner and Mountcastle, 1963; Fortier, Smith and Kalaska, 1993; Anderson et al., 2000; Friedrich and Laurent, 2004; Churchland et al., 2006; Finn, Priebe and Ferster, 2007; Mitchell, Sundberg and Reynolds, 2007; Churchland et al., 2009). The combined theoretical and experimental results suggest that internally generated activity is a significant contributor to response variability in neural circuits.

Neuronal activity arises from an interaction between ongoing firing generated spontaneously by neural circuits and responses driven by external stimuli. Using mean-field analysis, we ask how a neural network that intrinsically generates chaotic patterns of activity can remain sensitive to extrinsic input. We find that inputs not only drive network responses, they also actively suppress ongoing activity, ultimately leading to a phase transition in which chaos is completely eliminated. The critical input intensity at the phase transition is a non-monotonic function of stimulus frequency, revealing a "resonant" frequency at which the input is most effective at suppressing chaos even though the power spectrum of the spontaneous activity peaks at zero and falls exponentially. A prediction of our analysis is that the variance of neural responses should be most strongly suppressed at frequencies matching the range over which many sensory systems operate., Comment: 12 pages, 3 figures

We consider a threshold-crossing spiking process as a simple model for the activity within a population of neurons. Assuming that these neurons are driven by a common fluctuating input with Gaussian statistics, we evaluate the cross-correlation of spike trains in pairs of model neurons with different thresholds. This correlation function tends to be asymmetric in time, indicating a preference for the neuron with the lower threshold to fire before the one with the higher threshold, even if their inputs are identical. The relationship between these results and spike statistics in other models of neural activity are explored. In particular, we compare our model with an integrate-and-fire model in which the membrane voltage resets following each spike. The qualitative properties of spike cross correlations, emerging from the threshold-crossing model, are similar to those of bursting events in the integrate-and-fire model. This is particularly true for generalized integrate-and-fire models in which spikes tend to occur in bursts as observed, for example, in retinal ganglion cells driven by a rapidly fluctuating visual stimulus. The threshold crossing model thus provides a simple, analytically tractable description of event onsets in these neurons., Comment: In press in Neural Computation. 43 pages including 5 appendices and 9 figures

We study the ability of linear recurrent networks obeying discrete time dynamics to store long temporal sequences that are retrievable from the instantaneous state of the network. We calculate this temporal memory capacity for both distributed shift register and random orthogonal connectivity matrices. We show that the memory capacity of these networks scales with system size., Comment: 4 pages, 4 figures, to be published in Phys. Rev. Lett

We studied the mutual information between a stimulus and a large system consisting of stochastic, statistically independent elements that respond to a stimulus. The Mutual Information (MI) of the system saturates exponentially with system size. A theory of the rate of saturation of the MI is developed. We show that this rate is controlled by a distance function between the response probabilities induced by different stimuli. This function, which we term the {\it Confusion Distance} between two probabilities, is related to the Renyi $\alpha$-Information., Comment: 11 pages, 3 figures, accepted to PRL

Using methods of Statistical Physics, we investigate the generalization performance of support vector machines (SVMs), which have been recently introduced as a general alternative to neural networks. For nonlinear classification rules, the generalization error saturates on a plateau, when the number of examples is too small to properly estimate the coefficients of the nonlinear part. When trained on simple rules, we find that SVMs overfit only weakly. The performance of SVMs is strongly enhanced, when the distribution of the inputs has a gap in feature space., Comment: REVTeX, 4 pages, 2 figures, accepted by Phys. Rev. Lett (typos corrected)