Coarse-grained dynamics of an activity bump in a neural field model

Abstract

Abstract.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.

Reduced models for binocular rivalry

“…We do this by processing data like that shown in Fig. 2 to extract estimates of the terms involved in stochastic DEs (SDEs) for χ and Φ using the techniques in Gradišek et al (2000) (see also Friedrich et al (2000), Laing et al (2007), van Mourik et al (2006)). These SDEs will form our macroscopic model, and are assumed to linearly combine purely deterministic and purely stochastic components, i.e.…”

“…It is similar in spirit to centre manifold calculations in the neighbourhood of a bifurcation, or to the use of approximate inertial manifolds (Jolly et al 1990;Rega and Troger 2005). We will perform this data-mining (or "manifold learning") using the recently-developed diffusion map approach Erban et al 2007;Laing et al 2007;Nadler et al 2006).…”

“…The procedure can be thought of as a nonlinear generalization of principal component analysis (Jolliffe 2002) and further details can be found in Erban et al (2007) and Laing et al (2007).…”

Diffusion maps, clustering and fuzzy Markov modeling in peptide folding transitions

“…8 By contrast, in a good representation the projection provides us with suitable "reaction coordinates" 9 along which the motion has minimal memory effects; 10 even driven motion in such coordinates is expected to stay close to equilibrium, thus minimizing dissipation effects because of coupling to unrelaxed motions transverse to the chosen reaction coordinate. Examples of work motivated by these ideas and making use of biased sampling in reduced coordinates include accelerating stochastic simulations 11,12 and systematically identifying low dimensional parametrizations of a biomolecule's free energy surface. 13 An alternative approach to obtaining reduced sets of good global reduction coordinates is to group states locally (as cells in conformation space) and to summarize the dynamics in terms of transitions between such groups.…”

Nonlinear machine learning and design of reconfigurable digital colloids

“…Using the L-method 53 , we identified a gap in the eigenvalue spectrum beyond λ 3 , prompting the construction of a two-dimensional landscape in the top two non-trivial diffusion map eigenvectors (Ψ 2 , Ψ 3 ). The remaining (1.4 × 10 7 − 941) configurations were projected into this two-dimensional intrinsic manifold using the Nyström extension [62][63][64] . By constructing a diffusion map over microstates collected from simulations spanning all values of Λ, we can project each simulation trajectory into a shared intrinsic manifold and quantify the impact of Λ upon the morphology, stability, and kinetics of the digital colloid within a unified low-dimensional basis 18 .…”