1,493 research outputs found
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
Coarse-grained dynamics of an activity bump in a neural field model
We study a stochastic nonlocal PDE, arising in the context of modelling
spatially distributed neural activity, which is capable of sustaining
stationary and moving spatially-localized ``activity bumps''. This system is
known to undergo a pitchfork bifurcation in bump speed as a parameter (the
strength of adaptation) is changed; yet increasing the noise intensity
effectively slowed the motion of the bump. Here we revisit the system from the
point of view of describing the high-dimensional stochastic dynamics in terms
of the effective dynamics of a single scalar "coarse" variable. We show that
such a reduced description in the form of an effective Langevin equation
characterized by a double-well potential is quantitatively successful. The
effective potential can be extracted using short, appropriately-initialized
bursts of direct simulation. We demonstrate this approach in terms of (a) an
experience-based "intelligent" choice of the coarse observable and (b) an
observable obtained through data-mining direct simulation results, using a
diffusion map approach.Comment: Corrected aknowledgement
Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes
We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions
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