arXiv subjects

David Dahmen

Publications and source records attributed to David Dahmen.

25 records · Page 2Linked to original sources

arXiv2020

Capacity of the covariance perceptron

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.

cond-mat.dis-nn↗

arXiv2019

Statistical field theory for neural networks

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.

cond-mat.dis-nn↗

arXiv2016

Integration of continuous-time dynamics in a spiking neural network simulator

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.

q-bio.NC↗

arXiv2016

Functional methods for disordered neural networks

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.

cond-mat.dis-nn↗

arXiv2016

Distributions of covariances as a window into the operational regime of neuronal networks

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.

cond-mat.dis-nn↗

arXiv2016

Hybrid scheme for modeling local field potentials from point-neuron networks

Due to rapid advances in multielectrode recording technology, the local field potential (LFP) has again become a popular measure of neuronal activity in both basic research and clinical applications. Proper understanding of the LFP requires detailed mathematical modeling incorporating the anatomical and electrophysiological features of neurons near the recording electrode, as well as synaptic inputs from the entire network. Here we propose a hybrid modeling scheme combining the efficiency of commonly used simplified point-neuron network models with the biophysical principles underlying LFP generation by real neurons. The scheme can be used with an arbitrary number of point-neuron network populations. The LFP predictions rely on populations of network-equivalent, anatomically reconstructed multicompartment neuron models with layer-specific synaptic connectivity. The present scheme allows for a full separation of the network dynamics simulation and LFP generation. For illustration, we apply the scheme to a full-scale cortical network model for a $\sim$1 mm$^2$ patch of primary visual cortex and predict laminar LFPs for different network states, assess the relative LFP contribution from different laminar populations, and investigate the role of synaptic input correlations and neuron density on the LFP. The generic nature of the hybrid scheme and its publicly available implementation in \texttt{hybridLFPy} form the basis for LFP predictions from other point-neuron network models, as well as extensions of the current application to larger circuitry and additional biological detail.

q-bio.NC↗

arXiv2015

Correlated fluctuations in strongly-coupled binary networks beyond equilibrium

Randomly coupled Ising spins constitute the classical model of collective phenomena in disordered systems, with applications covering ferromagnetism, combinatorial optimization, protein folding, stock market dynamics, and social dynamics. The phase diagram of these systems is obtained in the thermodynamic limit by averaging over the quenched randomness of the couplings. However, many applications require the statistics of activity for a single realization of the possibly asymmetric couplings in finite-sized networks. Examples include reconstruction of couplings from the observed dynamics, learning in the central nervous system by correlation-sensitive synaptic plasticity, and representation of probability distributions for sampling-based inference. The systematic cumulant expansion for kinetic binary (Ising) threshold units with strong, random and asymmetric couplings presented here goes beyond mean-field theory and is applicable outside thermodynamic equilibrium; a system of approximate non-linear equations predicts average activities and pairwise covariances in quantitative agreement with full simulations down to hundreds of units. The linearized theory yields an expansion of the correlation- and response functions in collective eigenmodes, leads to an efficient algorithm solving the inverse problem, and shows that correlations are invariant under scaling of the interaction strengths.

cond-mat.dis-nn↗