Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

One-Dimensional Population Density Approaches to Recurrently Coupled Networks of Neurons with Noise

Mean-field systems have been previously derived for networks of coupled, two-dimensional, integrate-and-fire neurons such as the Izhikevich, adapting exponential (AdEx) and quartic integrate and fire (QIF), among others. Unfortunately, the mean-field systems have a degree of frequency error and the networks analyzed often do not include noise when there is adaptation. Here, we derive a one-dimensional partial differential equation (PDE) approximation for the marginal voltage density under a first order moment closure for coupled networks of integrate-and-fire neurons with white noise inputs. The PDE has substantially less frequency error than the mean-field system, and provides a great deal more information, at the cost of analytical tractability. The convergence properties of the mean-field system in the low noise limit are elucidated. A novel method for the analysis of the stability of the asynchronous tonic firing solution is also presented and implemented. Unlike previous attempts at stability analysis with these network types, information about the marginal densities of the adaptation variables is used. This method can in principle be applied to other systems with nonlinear partial differential equations.Comment: 26 Pages, 6 Figure

Coarse-Grained Analysis of Microscopic Neuronal Simulators on Networks: Bifurcation and Rare-events computations

We show how the Equation-Free approach for mutliscale computations can be exploited to extract, in a computational strict and systematic way the emergent dynamical attributes, from detailed large-scale microscopic stochastic models, of neurons that interact on complex networks. In particular we show how the Equation-Free approach can be exploited to perform system-level tasks such as bifurcation, stability analysis and estimation of mean appearance times of rare events, bypassing the need for obtaining analytical approximations, providing an "on-demand" model reduction. Using the detailed simulator as a black-box timestepper, we compute the coarse-grained equilibrium bifurcation diagrams, examine the stability of the solution branches and perform a rare-events analysis with respect to certain characteristics of the underlying network topology such as the connectivity degre

Limits and dynamics of stochastic neuronal networks with random heterogeneous delays

Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.Comment: Corrected misprint (useless stopping time) in proof of Lemma 1 and clarified a regularity hypothesis (remark 1

A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory

Noise-induced behaviors in neural mean field dynamics

The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. Asymptotic equations as the number of neurons tends to infinity (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions which generally makes their direct analysis difficult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numerically study the influence of noise on the collective behaviors, and compare these asymptotic regimes to simulations of the network. We observe that the mean field equations provide an accurate description of the solutions of the network equations for network sizes as small as a few hundreds of neurons. In particular, we observe that the level of noise in the system qualitatively modifies its collective behavior, producing for instance synchronized oscillations of the whole network, desynchronization of oscillating regimes, and stabilization or destabilization of stationary solutions. These results shed a new light on the role of noise in shaping collective dynamics of neurons, and gives us clues for understanding similar phenomena observed in biological networks