Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions

In this manuscript we analyze the collective behavior of mean-field limits of large-scale, spatially extended stochastic neuronal networks with delays. Rigorously, the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro-differential McKean-Vlasov equation that remain impenetrable, leaving the stochastic collective dynamics of such networks poorly understood. In order to study these macroscopic dynamics, we analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics and sigmoidal interactions. In that case, we prove that the solution of the mean-field equation is Gaussian, hence characterized by its two first moments, and that these two quantities satisfy a set of coupled delayed integro-differential equations. These equations are similar to usual neural field equations, and incorporate noise levels as a parameter, allowing analysis of noise-induced transitions. We identify through bifurcation analysis several qualitative transitions due to noise in the mean-field limit. In particular, stabilization of spatially homogeneous solutions, synchronized oscillations, bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence in space of Brownian motion

The complexity of dynamics in small neural circuits

Mean-field theory is a powerful tool for studying large neural networks. However, when the system is composed of a few neurons, macroscopic differences between the mean-field approximation and the real behavior of the network can arise. Here we introduce a study of the dynamics of a small firing-rate network with excitatory and inhibitory populations, in terms of local and global bifurcations of the neural activity. Our approach is analytically tractable in many respects, and sheds new light on the finite-size effects of the system. In particular, we focus on the formation of multiple branching solutions of the neural equations through spontaneous symmetry-breaking, since this phenomenon increases considerably the complexity of the dynamical behavior of the network. For these reasons, branching points may reveal important mechanisms through which neurons interact and process information, which are not accounted for by the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of figures 8 and 9 fixed, results unchange

Conditions for wave trains in spiking neural networks

Spatiotemporal patterns such as traveling waves are frequently observed in recordings of neural activity. The mechanisms underlying the generation of such patterns are largely unknown. Previous studies have investigated the existence and uniqueness of different types of waves or bumps of activity using neural-field models, phenomenological coarse-grained descriptions of neural-network dynamics. But it remains unclear how these insights can be transferred to more biologically realistic networks of spiking neurons, where individual neurons fire irregularly. Here, we employ mean-field theory to reduce a microscopic model of leaky integrate-and-fire (LIF) neurons with distance-dependent connectivity to an effective neural-field model. In contrast to existing phenomenological descriptions, the dynamics in this neural-field model depends on the mean and the variance in the synaptic input, both determining the amplitude and the temporal structure of the resulting effective coupling kernel. For the neural-field model we employ liner stability analysis to derive conditions for the existence of spatial and temporal oscillations and wave trains, that is, temporally and spatially periodic traveling waves. We first prove that wave trains cannot occur in a single homogeneous population of neurons, irrespective of the form of distance dependence of the connection probability. Compatible with the architecture of cortical neural networks, wave trains emerge in two-population networks of excitatory and inhibitory neurons as a combination of delay-induced temporal oscillations and spatial oscillations due to distance-dependent connectivity profiles. Finally, we demonstrate quantitative agreement between predictions of the analytically tractable neural-field model and numerical simulations of both networks of nonlinear rate-based units and networks of LIF neurons.Comment: 36 pages, 8 figures, 4 table

Asymptotic Stability, Orbital Stability of Hopf-Bifurcating Periodic Solution of a Simple Three-Neuron Artificial Neural Network with Distributed Delay

A distributed delay model of a class of three-neuron network has been investigated. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. A closed interval for a parameter of the system is identified in which Hopf-bifurcating periodic solution occurs for each point of such interval. The orbital stability of such bifurcating periodic solution at the extreme points of the interval is ascertained. Lastly global bifurcation aspect of such periodic solutions is studied. The results are illustrated by numerical simulations

Dynamics in a Delayed Neural Network Model of Two Neurons with Inertial Coupling

A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results

On the role of arkypallidal and prototypical neurons for phase transitions in the external pallidum

The external pallidum (GPe) plays a central role for basal ganglia functions and dynamics and, consequently, has been included in most computational studies of the basal ganglia. These studies considered the GPe as a homogeneous neural population. However, experimental studies have shown that the GPe contains at least two distinct cell types (prototypical and arkypallidal cells). In this work, we provide in silico insight into how pallidal heterogeneity modulates dynamic regimes inside the GPe and how they affect the GPe response to oscillatory input. We derive a mean-field model of the GPe system from a microscopic spiking neural network of recurrently coupled prototypical and arkypallidal neurons. Using bifurcation analysis, we examine the influence of the intra-pallidal connectivity on the GPe dynamics. We find that under healthy conditions, the inhibitory coupling determines whether the GPe is close to either a bi-stable or an oscillatory regime. Furthermore, we show that oscillatory input to the GPe, arriving from subthalamic nucleus or striatum, leads to characteristic patterns of cross-frequency coupling observed at the GPe. Based on these findings, we propose two different hypotheses of how dopamine depletion at the GPe may lead to phase-amplitude coupling between the parkinsonian beta rhythm and a GPe-intrinsic gamma rhythm. Finally, we show that these findings generalize to realistic spiking neural networks of sparsely coupled type-I excitable GPe neurons

Investigating the role of fast-spiking interneurons in neocortical dynamics

PhD ThesisFast-spiking interneurons are the largest interneuronal population in neocortex. It is well documented that this population is crucial in many functions of the neocortex by subserving all aspects of neural computation, like gain control, and by enabling dynamic phenomena, like the generation of high frequency oscillations. Fast-spiking interneurons, which represent mainly the parvalbumin-expressing, soma-targeting basket cells, are also implicated in pathological dynamics, like the propagation of seizures or the impaired coordination of activity in schizophrenia. In the present thesis, I investigate the role of fast-spiking interneurons in such dynamic phenomena by using computational and experimental techniques. First, I introduce a neural mass model of the neocortical microcircuit featuring divisive inhibition, a gain control mechanism, which is thought to be delivered mainly by the soma-targeting interneurons. Its dynamics were analysed at the onset of chaos and during the phenomena of entrainment and long-range synchronization. It is demonstrated that the mechanism of divisive inhibition reduces the sensitivity of the network to parameter changes and enhances the stability and exibility of oscillations. Next, in vitro electrophysiology was used to investigate the propagation of activity in the network of electrically coupled fast-spiking interneurons. Experimental evidence suggests that these interneurons and their gap junctions are involved in the propagation of seizures. Using multi-electrode array recordings and optogenetics, I investigated the possibility of such propagating activity under the conditions of raised extracellular K+ concentration which applies during seizures. Propagated activity was recorded and the involvement of gap junctions was con rmed by pharmacological manipulations. Finally, the interaction between two oscillations was investigated. Two oscillations with di erent frequencies were induced in cortical slices by directly activating the pyramidal cells using optogenetics. Their interaction suggested the possibility of a coincidence detection mechanism at the circuit level. Pharmacological manipulations were used to explore the role of the inhibitory interneurons during this phenomenon. The results, however, showed that the observed phenomenon was not a result of synaptic activity. Nevertheless, the experiments provided some insights about the excitability of the tissue through scattered light while using optogenetics. This investigation provides new insights into the role of fast-spiking interneurons in the neocortex. In particular, it is suggested that the gain control mechanism is important for the physiological oscillatory dynamics of the network and that the gap junctions between these interneurons can potentially contribute to the inhibitory restraint during a seizure.Wellcome Trust