A kinetic theory approach to capturing interneuronal correlation: the feed-forward case

Liu, Chin Yueh; Nykamp, Duane Q.

doi:10.1007/s10827-008-0116-4

Cited by 9 publications

(8 citation statements)

References 60 publications

(88 reference statements)

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“…Therefore, it seems that for these models, structural (synaptic weights, mean synaptic input) and dynamical (mean firing rate, correlations) statistics are related to each other in a hierarchical manner, as already observed in a simpler setting (Liu and Nykamp 2009). We therefore do not conclude that connectivity is completely "decoupled" from correlation, but rather that this detail of description is irrelevant at a large-scale level of observation.…”

Section: Discussionmentioning

confidence: 57%

Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons

et al. 2011

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The relationship between the dynamics of neural networks and their patterns of connectivity is far from clear, despite its importance for understanding functional properties. Here, we have studied sparsely-connected networks of conductance-based integrate-and-fire (IF) neurons with balanced excitatory and inhibitory connections and with finite axonal propagation speed. We focused on the genesis of states with highly irregular spiking activity and synchronous firing patterns at low rates, called slow Synchronous Irregular (SI) states. In such balanced networks, we examined the "macroscopic" properties of the spiking activity, such as ensemble correlations and mean firing rates, for different intracortical connectivity profiles ranging from randomly connected networks to networks with Gaussian-distributed local connectivity. We systematically computed the distance-dependent correlations at the extracellular (spiking) and intracellular (membrane potential) levels between randomly assigned pairs of neurons. The main finding is that such properties, when they are averaged at a macroscopic scale, are invariant with respect to the different connectivity patterns, provided the excitatory-inhibitory balance is the same. In particular, the same correlation structure holds for different connectivity profiles. In addition, we examined the response of such networks to external input, and found that the correlation landscape can be modulated by the mean level of synchrony imposed by the external drive. This modulation was found again to be independent of the external connectivity profile. We conclude that first and second-order "mean-field" statistics of such networks do not depend on the details of the connectivity at a microscopic scale. This study is an encouraging step toward a mean-field description of topological neuronal networks.

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Section: Discussionmentioning

confidence: 57%

Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons

et al. 2011

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“…Let ρ(v, t) be the probability density function for the state of the I&F population receiving only excitatory inputs [57]. The population density approach to modeling a variety of noisy I&F neurons in many contexts has attracted much attention as an analytic [20,51,9,10,40,24,52,37,35,43,11,38] and time-saving computational tool [32,31,3]. Interacting populations of leaky I&F neurons have been simulated with these methods [9].…”

Section: Application To Neural Oscillators Coupled To a Population Ofmentioning

confidence: 99%

Analysis of Recurrent Networks of Pulse-Coupled Noisy Neural Oscillators

Ly¹,

Ermentrout²

2010

SIAM J. Appl. Dyn. Syst.

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Synchronization of neural oscillators has been well studied by both theorists and experimentalists. However, realistic details are often disregarded for tractability. Here, we consider a recurrent network of pulse-coupled neural oscillators since synaptic communication is often mediated by spikes. Neurons receive many stochastic inputs that have effects depending on the state of the neuron; thus, we incorporate phase-dependent (multiplicative) noise. Previous analysis of neural oscillators with additive noise (not necessarily weak) cannot be directly applied here because this results in analytically intractable equations that possibly have singularities. However, assuming weak coupling and weak noise, we accurately and analytically characterize various phenomena by a linear stability analysis around an asymptotic steady-state density approximation. Depending on the phase resetting curve, the system can undergo a supercritical Andronov-Hopf bifurcation as the recurrent coupling strength of the oscillator is increased, leading to synchronous and oscillatory population activity. The analysis is extended to include recurrent input through synapses with kinetics-it generally stabilizes the incoherent state. Moreover, input via synapses can uncover a bifurcation that does not exist with instantaneous recurrent input. The analysis is further generalized to recurrent input that is a smooth function of the population firing rate. The results are applied to a closed-loop system of neural oscillators that receive feedback mediated by a noisy population of excitable integrate-and-fire neurons. Our results extend the power of perturbation methods for dealing with equations that, a priori, appear intractable.

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“…Several theoretical studies have analysed the effects of correlations on neuron response [28], [29] and the transmission of correlations [30]–[34], also through several layers [35]. However, the description of the interaction of recurrent connectivity, correlations and neuron dynamics in a self-consistent theory has not been presented yet.…”

Section: Introductionmentioning

confidence: 99%

How Structure Determines Correlations in Neuronal Networks

et al. 2011

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Networks are becoming a ubiquitous metaphor for the understanding of complex biological systems, spanning the range between molecular signalling pathways, neural networks in the brain, and interacting species in a food web. In many models, we face an intricate interplay between the topology of the network and the dynamics of the system, which is generally very hard to disentangle. A dynamical feature that has been subject of intense research in various fields are correlations between the noisy activity of nodes in a network. We consider a class of systems, where discrete signals are sent along the links of the network. Such systems are of particular relevance in neuroscience, because they provide models for networks of neurons that use action potentials for communication. We study correlations in dynamic networks with arbitrary topology, assuming linear pulse coupling. With our novel approach, we are able to understand in detail how specific structural motifs affect pairwise correlations. Based on a power series decomposition of the covariance matrix, we describe the conditions under which very indirect interactions will have a pronounced effect on correlations and population dynamics. In random networks, we find that indirect interactions may lead to a broad distribution of activation levels with low average but highly variable correlations. This phenomenon is even more pronounced in networks with distance dependent connectivity. In contrast, networks with highly connected hubs or patchy connections often exhibit strong average correlations. Our results are particularly relevant in view of new experimental techniques that enable the parallel recording of spiking activity from a large number of neurons, an appropriate interpretation of which is hampered by the currently limited understanding of structure-dynamics relations in complex networks.

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A kinetic theory approach to capturing interneuronal correlation: the feed-forward case

Cited by 9 publications

References 60 publications

Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons

Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons

Analysis of Recurrent Networks of Pulse-Coupled Noisy Neural Oscillators

How Structure Determines Correlations in Neuronal Networks

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