arXiv subjects

Fred Wolf

Publications and source records attributed to Fred Wolf.

35 records · Page 2Linked to original sources

arXiv2011

Spike Onset Dynamics and Response Speed in Neuronal Populations

Recent studies of cortical neurons driven by fluctuating currents revealed cutoff frequencies for action potential encoding of several hundred Hz. Theoretical studies of biophysical neuron models have predicted a much lower cutoff frequency of the order of average firing rate or the inverse membrane time constant. The biophysical origin of the observed high cutoff frequencies is thus not well understood. Here we introduce a neuron model with dynamical action potential generation, in which the linear response can be analytically calculated for uncorrelated synaptic noise. We find that the cutoff frequencies increase to very large values when the time scale of action potential initiation becomes short.

q-bio.NC↗

arXiv2011

Reservoirs of Stability: Flux Tubes in the Dynamics of Cortical Circuits

Triggering a single additional spike in a cerebral cortical neuron was recently demonstrated to cause a cascade of extra spikes in the network that is likely to rapidly decorrelate the network's microstate. The mechanisms involved in this extreme sensitivity of cortical networks are currently not well understood. Here, we show in a minimal model of cortical circuit dynamics that exponential state separation after single spike and even single synapse perturbations coexists with dynamical stability to infinitesimal state perturbations. We propose a unifying picture of exponentially separating flux tubes enclosing unique stable trajectories composing the networks' state spaces.

q-bio.NC↗

arXiv2010

Dynamical Entropy Production in Spiking Neuron Networks in the Balanced State

We demonstrate deterministic extensive chaos in the dynamics of large sparse networks of theta neurons in the balanced state. The analysis is based on numerically exact calculations of the full spectrum of Lyapunov exponents, the entropy production rate and the attractor dimension. Extensive chaos is found in inhibitory networks and becomes more intense when an excitatory population is included. We find a strikingly high rate of entropy production that would limit information representation in cortical spike patterns to the immediate stimulus response.

cond-mat.dis-nn↗

arXiv2009

Correlations and Synchrony in Threshold Neuron Models

We study how threshold model neurons transfer temporal and interneuronal input correlations to correlations of spikes. We find that the low common input regime is governed by firing rate dependent spike correlations which are sensitive to the detailed structure of input correlation functions. In the high common input regime the spike correlations are insensitive to the firing rate and exhibit a universal peak shape independent of input correlations. Rate heterogeneous pairs driven by common inputs in general exhibit asymmetric spike correlations. All predictions are confirmed in in vitro experiments with cortical neurons driven by synthesized fluctuating input currents.

q-bio.NC↗

arXiv2009

Pinwheel stabilization by ocular dominance segregation

We present an analytical approach for studying the coupled development of ocular dominance and orientation preference columns. Using this approach we demonstrate that ocular dominance segregation can induce the stabilization and even the production of pinwheels by their crystallization in two types of periodic lattices. Pinwheel crystallization depends on the overall dominance of one eye over the other, a condition that is fulfilled during early cortical development. Increasing the strength of inter-map coupling induces a transition from pinwheel-free stripe solutions to intermediate and high pinwheel density states.

q-bio.NC↗

arXiv2008

The simplest problem in the collective dynamics of neural networks: Is synchrony stable?

For spiking neural networks we consider the stability problem of global synchrony, arguably the simplest non-trivial collective dynamics in such networks. We find that even this simplest dynamical problem -- local stability of synchrony -- is non-trivial to solve and requires novel methods for its solution. The dynamics in the vicinity of the synchronous state is determined by a multitude of linear operators, in contrast to a single stability matrix in conventional linear stability theory. This unusual property qualitatively depends on network topology and may be neglected for globally coupled homogeneous networks. For generic networks, however, the number of operators increases exponentially with the size of the network. We present methods to treat this multi-operator problem exactly. First, based on the Gershgorin and Perron-Frobenius theorems, we derive bounds on the eigenvalues that provide important information about the synchronization process but are not sufficient to establish the asymptotic stability or instability of the synchronous state. We then present a complete analysis of asymptotic stability for topologically strongly connected networks using simple graph-theoretical considerations. For inhibitory interactions between dissipative (leaky) oscillatory neurons the synchronous state is stable, independent of the parameters and the network connectivity. These results indicate that pulse-like interactions play a profound role in network dynamical systems, and in particular in the dynamics of biological synchronization, unless the coupling is homogeneous and all-to-all.

math.DS↗

arXiv2008

Inter-areal coordination of columnar architectures during visual cortical development

The occurrence of a critical period of plasticity in the visual cortex has long been established, yet its function in normal development is not fully understood. Here we show that as the late phase of the critical period unfolds, different areas of cat visual cortex develop in a coordinated manner. Orientation columns in areas V1 and V2 become matched in size in regions that are mutually connected. The same age trend is found for such regions in the left and right brain hemisphere. Our results indicate that a function of critical period plasticity is to progressively coordinate the functional architectures of different cortical areas - even across hemispheres.

q-bio.NC↗

arXiv2008

Pinwheel stability, pattern selection and the geometry of visual space

It has been proposed that the dynamical stability of topological defects in the visual cortex reflects the Euclidean symmetry of the visual world. We analyze defect stability and pattern selection in a generalized Swift-Hohenberg model of visual cortical development symmetric under the Euclidean group E(2). Euclidean symmetry strongly influences the geometry and multistability of model solutions but does not directly impact on defect stability.

q-bio.NC↗

arXiv2008

Self-organization and the selection of pinwheel density in visual cortical development

Self-organization of neural circuitry is an appealing framework for understanding cortical development, yet its applicability remains unconfirmed. Models for the self-organization of neural circuits have been proposed, but experimentally testable predictions of these models have been less clear. The visual cortex contains a large number of topological point defects, called pinwheels, which are detectable in experiments and therefore in principle well suited for testing predictions of self-organization empirically. Here, we analytically calculate the density of pinwheels predicted by a pattern formation model of visual cortical development. An important factor controlling the density of pinwheels in this model appears to be the presence of non-local long-range interactions, a property which distinguishes cortical circuits from many nonliving systems in which self-organization has been studied. We show that in the limit where the range of these interactions is infinite, the average pinwheel density converges to $π$. Moreover, an average pinwheel density close to this value is robustly selected even for intermediate interaction ranges, a regime arguably covering interaction-ranges in a wide range of different species. In conclusion, our paper provides the first direct theoretical demonstration and analysis of pinwheel density selection in models of cortical self-organization and suggests to quantitatively probe this type of prediction in future high-precision experiments.

q-bio.NC↗

arXiv2005

Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.

q-bio.NC↗

arXiv2004

Long Chaotic Transients in Complex Networks

We show that long chaotic transients dominate the dynamics of randomly diluted networks of pulse-coupled oscillators. This contrasts with the rapid convergence towards limit cycle attractors found in networks of globally coupled units. The lengths of the transients strongly depend on the network connectivity and varies by several orders of magnitude, with maximum transient lengths at intermediate connectivities. The dynamics of the transient exhibits a novel form of robust synchronization. An approximation to the largest Lyapunov exponent characterizing the chaotic nature of the transient dynamics is calculated analytically.

cond-mat.dis-nn↗

arXiv2003

The Intrinsic Time Scale of Transient Neuronal Responses

In a generic neuron model, we present the linear response theory for the firing rate in response to both time dependent input currents and noise amplitudes. In both cases the signal transmission is strongly attenuated for frequencies above the stationary firing rate. For high frequencies both the mean input and the noise transmission function decay as ω^{-2}, independent of model details. Our results indicate that previously suggested mechanisms for near instantaneous transmission of information are not consistent with the spike generation mechanism of real neurons.

physics.bio-ph↗

arXiv2003

Breaking Synchrony by Heterogeneity in Complex Networks

For networks of pulse-coupled oscillators with complex connectivity, we demonstrate that in the presence of coupling heterogeneity precisely timed periodic firing patterns replace the state of global synchrony that exists in homogenous networks only. With increasing disorder, these patterns persist until they reach a critical temporal extent that is of the order of the interaction delay. For stronger disorder these patterns cease to exist and only asynchronous, aperiodic states are observed. We derive self-consistency equations to predict the precise temporal structure of a pattern from the network heterogeneity. Moreover, we show how to design heterogenous coupling architectures to create an arbitrary prescribed pattern.

cond-mat.dis-nn↗

arXiv2003

Topological Speed Limits to Network Synchronization

We study collective synchronization of pulse-coupled oscillators interacting on asymmetric random networks. We demonstrate that random matrix theory can be used to accurately predict the speed of synchronization in such networks in dependence on the dynamical and network parameters. Furthermore, we show that the speed of synchronization is limited by the network connectivity and stays finite, even if the coupling strength becomes infinite. In addition, our results indicate that synchrony is robust under structural perturbations of the network dynamics.

cond-mat.dis-nn↗

arXiv2003

Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators

For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multi-operator problem analytically and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying \textit{regular} dynamics, coexists with a balanced state that exhibits \textit{irregular} dynamics such that external signals may switch the network between qualitatively distinct states.

cond-mat.dis-nn↗

arXiv2002

Prevalence of unstable attractors in networks of pulse-coupled oscillators

We present and analyze the first example of a dynamical system that naturally exhibits attracting periodic orbits that are \textit{unstable}. These unstable attractors occur in networks of pulse-coupled oscillators where they prevail for large networks and a wide range of parameters. They are enclosed by basins of attraction of other attractors but are remote from their own basin volume such that arbitrarily small noise leads to a switching among attractors.

cond-mat.dis-nn↗

arXiv2002

Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which \textit{unstable attractors} arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.

cond-mat.dis-nn↗