Moving bumps in theta neuron networks

We consider large networks of theta neurons on a ring, synaptically coupled with an asymmetric kernel. Such networks support stable "bumps" of activity, which move along the ring if the coupling kernel is asymmetric. We investigate the effects of the kernel asymmetry on the existence, stability and speed of these moving bumps using continuum equations formally describing infinite networks. Depending on the level of heterogeneity within the network we find complex sequences of bifurcations as the amount of asymmetry is varied, in strong contrast to the behaviour of a classical neural field model.Comment: To appear in Chao

Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons

We study the macroscopic dynamics of large networks of excitable type 1 neurons composed of two populations interacting with disparate but symmetric intra- and inter-population coupling strengths. This nonuniform coupling scheme facilitates symmetric equilibria, where both populations display identical firing activity, characterized by either quiescent or spiking behavior, or asymmetric equilibria, where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior. Oscillations in the firing rate are possible if neurons emit pulses with non-zero width but are otherwise quenched. Here, we explore how collective oscillations emerge for two statistically identical neuron populations in the limit of an infinite number of neurons. A detailed analysis reveals how collective oscillations are born and destroyed in various bifurcation scenarios and how they are organized around higher codimension bifurcation points. Since both symmetric and asymmetric equilibria display bistable behavior, a large configuration space with steady and oscillatory behavior is available. Switching between configurations of neural activity is relevant in functional processes such as working memory and the onset of collective oscillations in motor control

Effect of Cauchy noise on a network of quadratic integrate-and-fire neurons with non-Cauchy heterogeneities

We analyze the dynamics of large networks of pulse-coupled quadratic integrate-and-fire neurons driven by Cauchy noise and non-Cauchy heterogeneous inputs. Two types of heterogeneities defined by families of $q$-Gaussian and flat distributions are considered. Both families are parametrized by an integer $n$, so that as $n$ increases, the first family tends to a normal distribution, and the second tends to a uniform distribution. For both families, exact systems of mean-field equations are derived and their bifurcation analysis is carried out. We show that noise and heterogeneity can have qualitatively different effects on the collective dynamics of neurons.Comment: 9 pages, 5 figure

Dynamics on networks: the role of local dynamics and global networks on the emergence of hypersynchronous neural activity.

Published onlineJournal ArticleResearch Support, Non-U.S. Gov'tGraph theory has evolved into a useful tool for studying complex brain networks inferred from a variety of measures of neural activity, including fMRI, DTI, MEG and EEG. In the study of neurological disorders, recent work has discovered differences in the structure of graphs inferred from patient and control cohorts. However, most of these studies pursue a purely observational approach; identifying correlations between properties of graphs and the cohort which they describe, without consideration of the underlying mechanisms. To move beyond this necessitates the development of computational modeling approaches to appropriately interpret network interactions and the alterations in brain dynamics they permit, which in the field of complexity sciences is known as dynamics on networks. In this study we describe the development and application of this framework using modular networks of Kuramoto oscillators. We use this framework to understand functional networks inferred from resting state EEG recordings of a cohort of 35 adults with heterogeneous idiopathic generalized epilepsies and 40 healthy adult controls. Taking emergent synchrony across the global network as a proxy for seizures, our study finds that the critical strength of coupling required to synchronize the global network is significantly decreased for the epilepsy cohort for functional networks inferred from both theta (3-6 Hz) and low-alpha (6-9 Hz) bands. We further identify left frontal regions as a potential driver of seizure activity within these networks. We also explore the ability of our method to identify individuals with epilepsy, observing up to 80% predictive power through use of receiver operating characteristic analysis. Collectively these findings demonstrate that a computer model based analysis of routine clinical EEG provides significant additional information beyond standard clinical interpretation, which should ultimately enable a more appropriate mechanistic stratification of people with epilepsy leading to improved diagnostics and therapeutics.Funding was from Epilepsy Research UK (http://www.epilepsyresearch.org.uk) via grant number A1007 and the Medical Research Council (http://www.mrc.ac.uk) via grants (MR/K013998/1 and G0701310)

Paradoxical phase response of gamma rhythms facilitates their entrainment in heterogeneous networks

The synchronization of different $\gamma$-rhythms arising in different brain areas has been implicated in various cognitive functions. Here, we focus on the effect of the ubiquitous neuronal heterogeneity on the synchronization of PING (pyramidal-interneuronal network gamma) and ING (interneuronal network gamma) rhythms. The synchronization properties of rhythms depends on the response of their collective phase to external input. We therefore determined the macroscopic phase-response curve for finite-amplitude perturbations (fmPRC), using numerical simulation of all-to-all coupled networks of integrate-and-fire (IF) neurons exhibiting either PING or ING rhythms. We show that the intrinsic neuronal heterogeneity can qualitatively modify the fmPRC. While the phase-response curve for the individual IF-neurons is strictly positive (type I), the fmPRC can be biphasic and exhibit both signs (type II). Thus, for PING rhythms, an external excitation to the excitatory cells can, in fact, delay the collective oscillation of the network, even though the same excitation would lead to an advance when applied to uncoupled neurons. This paradoxical delay arises when the external excitation modifies the internal dynamics of the network by causing additional spikes of inhibitory neurons, whose delaying within-network inhibition outweighs the immediate advance caused by the external excitation. These results explain how intrinsic heterogeneity allows the PING rhythm to become synchronized with a periodic forcing or another PING rhythm for a wider range in the mismatch of their frequencies. We demonstrate a similar mechanism for the synchronization of ING rhythms. Our results identify a potential function of neuronal heterogeneity in the synchronization of coupled $\gamma$-rhythms, which may play a role in neural information transfer via communication through coherence.Comment: 24 pages, 7 Figs, 3 Supp Fig

Synchrony-induced modes of oscillation of a neural field model

We investigate the modes of oscillation of heterogeneous ring-networks of quadratic integrate-and-fire (QIF) neurons with non-local, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogous to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network’s oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model (QIF-NFM) describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially-homogeneous state, and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead, is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially-inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states, and are maintained away from onset