15 research outputs found
Finite-size and correlation-induced effects in Mean-field Dynamics
The brain's activity is characterized by the interaction of a very large
number of neurons that are strongly affected by noise. However, signals often
arise at macroscopic scales integrating the effect of many neurons into a
reliable pattern of activity. In order to study such large neuronal assemblies,
one is often led to derive mean-field limits summarizing the effect of the
interaction of a large number of neurons into an effective signal. Classical
mean-field approaches consider the evolution of a deterministic variable, the
mean activity, thus neglecting the stochastic nature of neural behavior. In
this article, we build upon two recent approaches that include correlations and
higher order moments in mean-field equations, and study how these stochastic
effects influence the solutions of the mean-field equations, both in the limit
of an infinite number of neurons and for large yet finite networks. We
introduce a new model, the infinite model, which arises from both equations by
a rescaling of the variables and, which is invertible for finite-size networks,
and hence, provides equivalent equations to those previously derived models.
The study of this model allows us to understand qualitative behavior of such
large-scale networks. We show that, though the solutions of the deterministic
mean-field equation constitute uncorrelated solutions of the new mean-field
equations, the stability properties of limit cycles are modified by the
presence of correlations, and additional non-trivial behaviors including
periodic orbits appear when there were none in the mean field. The origin of
all these behaviors is then explored in finite-size networks where interesting
mesoscopic scale effects appear. This study leads us to show that the
infinite-size system appears as a singular limit of the network equations, and
for any finite network, the system will differ from the infinite system
Stochastic firing rate models
We review a recent approach to the mean-field limits in neural networks that
takes into account the stochastic nature of input current and the uncertainty
in synaptic coupling. This approach was proved to be a rigorous limit of the
network equations in a general setting, and we express here the results in a
more customary and simpler framework. We propose a heuristic argument to derive
these equations providing a more intuitive understanding of their origin. These
equations are characterized by a strong coupling between the different moments
of the solutions. We analyse the equations, present an algorithm to simulate
the solutions of these mean-field equations, and investigate numerically the
equations. In particular, we build a bridge between these equations and
Sompolinsky and collaborators approach (1988, 1990), and show how the coupling
between the mean and the covariance function deviates from customary
approaches
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
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On conductance-based neural field models
This technical note introduces a conductance-based neural field model that combines biologically realistic synaptic dynamics—based on transmembrane currents—with neural field equations, describing the propagation of spikes over the cortical surface. This model allows for fairly realistic inter-and intra-laminar intrinsic connections that underlie spatiotemporal neuronal dynamics. We focus on the response functions of expected neuronal states (such as depolarization) that generate observed electrophysiological signals (like LFP recordings and EEG). These response functions characterize the model's transfer functions and implicit spectral responses to (uncorrelated) input. Our main finding is that both the evoked responses (impulse response functions) and induced responses (transfer functions) show qualitative differences depending upon whether one uses a neural mass or field model. Furthermore, there are differences between the equivalent convolution and conductance models. Overall, all models reproduce a characteristic increase in frequency, when inhibition was increased by increasing the rate constants of inhibitory populations. However, convolution and conductance-based models showed qualitatively different changes in power, with convolution models showing decreases with increasing inhibition, while conductance models show the opposite effect. These differences suggest that conductance based field models may be important in empirical studies of cortical gain control or pharmacological manipulations
Large Deviations of a Spatially-Stationary Network of Interacting Neurons
In this work we determine a process-level Large Deviation Principle (LDP) for
a model of interacting neurons indexed by a lattice . The neurons
are subject to noise, which is modelled as a correlated martingale. The
probability law governing the noise is strictly stationary, and we are
therefore able to find a LDP for the probability laws governing the
stationary empirical measure generated by the neurons in a cube
of length . We use this LDP to determine an LDP for the neural network
model. The connection weights between the neurons evolve according to a
learning rule / neuronal plasticity, and these results are adaptable to a large
variety of neural network models. This LDP is of great use in the mathematical
modelling of neural networks, because it allows a quantification of the
likelihood of the system deviating from its limit, and also a determination of
which direction the system is likely to deviate. The work is also of interest
because there are nontrivial correlations between the neurons even in the
asymptotic limit, thereby presenting itself as a generalisation of traditional
mean-field models
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
Laws of large numbers and Langevin approximations for stochastic neural field equations
In this study we consider limit theorems for microscopic stochastic models of
neural fields. We show that the Wilson-Cowan equation can be obtained as the
limit in probability on compacts for a sequence of microscopic models when the
number of neuron populations distributed in space and the number of neurons per
population tend to infinity. Though the latter divergence is not necessary.
This result also allows to obtain limits for qualitatively different stochastic
convergence concepts, e.g., convergence in the mean. Further, we present a
central limit theorem for the martingale part of the microscopic models which,
suitably rescaled, converges to a centered Gaussian process with independent
increments. These two results provide the basis for presenting the neural field
Langevin equation, a stochastic differential equation taking values in a
Hilbert space, which is the infinite-dimensional analogue of the Chemical
Langevin Equation in the present setting. On a technical level we apply
recently developed law of large numbers and central limit theorems for
piecewise deterministic processes taking values in Hilbert spaces to a master
equation formulation of stochastic neuronal network models. These theorems are
valid for processes taking values in Hilbert spaces and by this are able to
incorporate spatial structures of the underlying model.Comment: 38 page