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

We study a stochastic nonlocal partial differential equation, 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 study 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, i.e. reducing the dimensionality of the system. 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, and the effects of changing parameters on this potential can easily be studied. We demonstrate this approach in terms of (a) an experience-based 'intelligent' choice of the coarse variable and (b) a variable obtained through data-mining direct simulation results, using a diffusion map approach.

[1]  Boris S. Gutkin,et al.  The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators , 2001, Neural Computation.

[2]  Boris S. Gutkin,et al.  Noise delays onset of sustained firing in a minimal model of persistent activity , 2004, Neurocomputing.

[3]  R. Taylor,et al.  The Numerical Treatment of Integral Equations , 1978 .

[4]  B. Ermentrout.  Neural networks as spatio-temporal pattern-forming systems , 1998 .

[5]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[6]  Lawrence Sirovich,et al.  On the Simulation of Large Populations of Neurons , 2004, Journal of Computational Neuroscience.

[7]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Bard Ermentrout,et al.  Spatially Structured Activity in Synaptically Coupled Neuronal Networks: II. Lateral Inhibition and Standing Pulses , 2001, SIAM J. Appl. Math..

[9]  Bruce W. Knight,et al.  Dynamics of Encoding in Neuron Populations: Some General Mathematical Features , 2000, Neural Computation.

[10]  Nicolas Le Roux,et al.  Learning Eigenfunctions Links Spectral Embedding and Kernel PCA , 2004, Neural Computation.

[11]  Bard Ermentrout,et al.  Pattern Formation in a Network of Excitatory and Inhibitory Cells with Adaptation , 2004, SIAM J. Appl. Dyn. Syst..

[12]  S. Coombes,et al.  Bumps, breathers, and waves in a neural network with spike frequency adaptation. , 2005, Physical review letters.

[13]  William H. Press,et al.  Numerical recipes , 1990 .

[14]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[15]  M. Shelley,et al.  An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Carlo R. Laing,et al.  On the application of “equation-free modelling” to neural systems , 2006, Journal of Computational Neuroscience.

[17]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: multiscale methods. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Moss,et al.  Postponement of Hopf bifurcations by multiplicative colored noise. , 1987, Physical review. A, General physics.

[19]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[20]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .

[21]  H. Sompolinsky,et al.  Theory of orientation tuning in visual cortex. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Ronald R. Coifman,et al.  Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck Operators , 2005, NIPS.

[23]  Ioannis G Kevrekidis,et al.  Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation. , 2005, The Journal of chemical physics.

[24]  Bard Ermentrout,et al.  Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses , 2001, SIAM J. Appl. Math..

[25]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[26]  Yacine Aït-Sahalia.  Transition Densities for Interest Rate and Other Nonlinear Diffusions , 1999 .

[27]  Eugene M. Izhikevich,et al.  Phase Equations for Relaxation Oscillators , 2000, SIAM J. Appl. Math..

[28]  André Longtin,et al.  Noise-induced stabilization of bumps in systems with long-range spatial coupling , 2001 .

[29]  Ioannis G Kevrekidis,et al.  Variable-free exploration of stochastic models: a gene regulatory network example. , 2006, The Journal of chemical physics.

[30]  Ioannis G Kevrekidis,et al.  Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations. , 2006, The Journal of chemical physics.

[31]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[32]  C. W. Gear,et al.  'Coarse' integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods , 2002 .

[33]  David Saunders,et al.  Phase resetting and coupling of noisy neural oscillators , 2006, Journal of Computational Neuroscience.

[34]  Stephen Coombes,et al.  Evans Functions for Integral Neural Field Equations with Heaviside Firing Rate Function , 2004, SIAM J. Appl. Dyn. Syst..

[35]  B. Nadler,et al.  Diffusion maps, spectral clustering and reaction coordinates of dynamical systems , 2005, math/0503445.

[36]  Ioannis G Kevrekidis,et al.  Coarse nonlinear dynamics and metastability of filling-emptying transitions: water in carbon nanotubes. , 2005, Physical review letters.

[37]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[38]  Gradisek,et al.  Analysis of time series from stochastic processes , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.