Filtering of Spatial Bias and Noise Inputs by Spatially Structured Neural Networks

Neural population densities shape network correlations

The way sensory microcircuits manage cellular response correlations is a crucial question in understanding how such systems integrate external stimuli and encode information. Most sensory systems exhibit heterogeneities in terms of population sizes and features, which all impact their dynamics. This work addresses how correlations between the dynamics of neural ensembles depend on the relative size or density of excitatory and inhibitory populations. To do so, we study an apparently symmetric system of coupled stochastic differential equations that model the evolution of the populations' activities. Excitatory and inhibitory populations are connected by reciprocal recurrent connections, and both receive different stimuli exhibiting a certain level of correlation with each other. A stability analysis is performed, which reveals an intrinsic asymmetry in the distribution of the fixed points with respect to the threshold of the nonlinearities. Based on this, we show how the cross correlation between the population responses depends on the density of the inhibitory population, and that a specific ratio between both population sizes leads to a state of zero correlation. We show that this so-called asynchronous state subsists, despite the presence of stimulus correlation, and most importantly, that it occurs only in asymmetrical systems where one population outnumbers the other. Using linear approximations, we derive analytical expressions for the root of the cross-correlation function and study how the asynchronous state is impacted by the model's parameters. This work suggests a possible explanation for why inhibitory cells outnumber excitatory cells in the visual system.

Stochastic resonance on weakly paced scale-free networks

We study the impact of additive Gaussian noise and weak periodic forcing on the dynamics of a scale-free network of bistable overdamped oscillators. The periodic forcing is introduced to a single oscillator and therefore acts as a pacemaker trying to impose its rhythm on the whole ensemble. We show that an intermediate intensity of temporally and spatially uncorrelated noise is able to optimally assist the pacemaker in achieving this goal, thus providing evidence for stochastic resonance on weakly paced scale-free networks. Because of the inherent degree inhomogeneity of individual oscillators forming the scale-free network, the placement of the pacemaker within the network is thereby crucial. As two extremes, we consider separately the introduction of the pacemaker to the oscillator with the highest degree and to one of the oscillators having the lowest degree. In both cases the coupling strength plays a crucial role, since it determines to what extent the whole network will follow the pacemaker on the expense of a weaker correlation between the pacemaker and the units that are directly linked with the paced oscillator. Higher coupling strengths facilitate the global outreach of the pacemaker, but require higher noise intensities for the optimal response. In contrast, lower coupling strengths and comparatively low noise intensities localize the optimal response to immediate neighbors of the paced oscillator. If the pacemaker is introduced to the main hub, the transition between the locally and globally optimal responses is characterized by a double resonance that postulates the existence of an optimal coupling strength for the transmission of weak rhythmic activity across scale-free networks. We corroborate the importance of the inhomogeneous structure of scale-free networks by additionally considering regular networks of oscillators with different degrees of coupling.

Driving neural oscillations with correlated spatial input and topographic feedback

We consider how oscillatory activity in networks of excitable systems depends on spatial correlations of random inputs and the spatial range of feedback coupling. Analysis of a neural field model with topographic delayed recurrent feedback reveals how oscillations in certain frequency bands, including the gamma band, are enhanced by increases in the input correlation length. Further, the enhancement is maximal when this length exceeds the feedback coupling range. Suppression of oscillatory power occurs concomitantly in other bands. These effects depend solely on the ratio of input and feedback length scales. The precise positions of these bands are determined by the synaptic constants and the delays. The results agree with numerical simulations of the model and of a network of stochastic spiking neurons, and are expected for any noise-driven excitable element networks.

Stochastic resonance on excitable small-world networks via a pacemaker

We show that the correlation between the frequency of subthreshold pacemaker activity and the response of an excitable array is resonantly dependent on the intensity of additive spatiotemporal noise. Thereby, the effect of the underlying network, defining the interactions among excitable units, largely depends on the coupling strength. Only for intermediate coupling strengths is the small world property able to enhance the stochastic resonance, whereas for smaller and larger couplings the impact of the transition from diffusive to random networks is less profound. Thus, the optimal interplay between a localized source of weak rhythmic activity and the response of the whole array demands a delicate balance between the strength of excitation transfer and the effectiveness of the network structure to support it.