Local/Global Analysis of the Stationary Solutions of Some Neural Field Equations

Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighboring brain areas. Understanding the properties of these solutions is essential to advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular, infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviors of the solutions. We also provide a generic finite dimensional appr...

[1]  M. Goldberg,et al.  Oculocentric spatial representation in parietal cortex. , 1995, Cerebral cortex.

[2]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[3]  H. Sompolinsky,et al.  13 Modeling Feature Selectivity in Local Cortical Circuits , 2022 .

[4]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[5]  Oren Shriki,et al.  Rate Models for Conductance-Based Cortical Neuronal Networks , 2003, Neural Computation.

[6]  Olivier D. Faugeras,et al.  Absolute Stability and Complete Synchronization in a Class of Neural Fields Models , 2008, SIAM J. Appl. Math..

[7]  Tian Ma,et al.  Bifurcation Theory and Applications , 2005 .

[8]  P. Bressloff,et al.  Breathers in two-dimensional neural media. , 2005, Physical review letters.

[9]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[10]  M. Golubitsky,et al.  Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. , 2001, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[11]  G. Ermentrout,et al.  Existence and uniqueness of travelling waves for a neural network , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Gaute T. Einevoll,et al.  Localized activity patterns in two-population neuronal networks , 2005 .

[13]  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.

[14]  V. Hutson.  Integral Equations , 1967, Nature.

[15]  S. Amari.  Dynamics of pattern formation in lateral-inhibition type neural fields , 1977, Biological Cybernetics.

[16]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[17]  P. Goldman-Rakic,et al.  Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex. , 1989, Journal of neurophysiology.

[18]  S. Amari,et al.  Existence and stability of local excitations in homogeneous neural fields , 1979, Journal of mathematical biology.

[19]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

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

[21]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[22]  Paul C. Bressloff,et al.  Spontaneous symmetry breaking in self–organizing neural fields , 2005, Biological Cybernetics.

[23]  P C Bressloff,et al.  Oscillatory waves in inhomogeneous neural media. , 2003, Physical review letters.

[24]  Carlo R. Laing,et al.  PDE Methods for Nonlocal Models , 2003, SIAM J. Appl. Dyn. Syst..

[25]  C. Bowden.  Waves , 2011 .

[26]  Paul C. Bressloff,et al.  Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn , 2000, Neural Computation.

[27]  J A Kelso,et al.  Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  Masayasu Mimura,et al.  Layer oscillations in reaction-diffusion systems , 1989 .

[29]  Paul C. Bressloff,et al.  Traveling Pulses and Wave Propagation Failure in Inhomogeneous Neural Media , 2008, SIAM J. Appl. Dyn. Syst..

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

[31]  G. Choquet,et al.  Cours d'analyse , 2022 .

[32]  Olivier D. Faugeras,et al.  A Constructive Mean-Field Analysis of Multi-Population Neural Networks with Random Synaptic Weights and Stochastic Inputs , 2008, Front. Comput. Neurosci..

[33]  P. Bressloff.  Traveling fronts and wave propagation failure in an inhomogeneous neural network , 2001 .

[34]  Xiaofeng Xu,et al.  For Trilinos tutorial. , 2010 .

[35]  J. Cowan,et al.  A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue , 1973, Kybernetik.

[36]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[37]  Paul C. Bressloff.  Weakly Interacting Pulses in Synaptically Coupled Neural Media , 2005, SIAM J. Appl. Math..

[38]  Olivier D. Faugeras,et al.  Persistent Neural States: Stationary Localized Activity Patterns in Nonlinear Continuous n-Population, q-Dimensional Neural Networks , 2009, Neural Computation.

[39]  Boris S. Gutkin,et al.  Multiple Bumps in a Neuronal Model of Working Memory , 2002, SIAM J. Appl. Math..

[40]  Hansjörg Kielhöfer,et al.  Bifurcation theory : an introduction with applications to PDEs , 2004 .

[41]  J. Cowan,et al.  Large Scale Spatially Organized Activity in Neural Nets , 1980 .

[42]  Peter Dayan,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems , 2001 .

[43]  Paul C. Bressloff,et al.  An Amplitude Equation Approach to Contextual Effects in Visual Cortex , 2002, Neural Computation.

[44]  R. Desimone,et al.  Neural Mechanisms of Visual Working Memory in Prefrontal Cortex of the Macaque , 1996, The Journal of Neuroscience.

[45]  P. Matthews,et al.  Dynamic instabilities in scalar neural field equations with space-dependent delays , 2007 .

[46]  P. Bressloff.  Spatially periodic modulation of cortical patterns by long-range horizontal connections , 2003 .

[47]  Axel Hutt,et al.  Stability and Bifurcations in Neural Fields with Finite Propagation Speed and General Connectivity , 2004, SIAM J. Appl. Math..

[48]  Paul H. Rabinowitz,et al.  Some global results for nonlinear eigenvalue problems , 1971 .

[49]  Viktor K Jirsa,et al.  Neural field dynamics with heterogeneous connection topology. , 2007, Physical review letters.

[50]  Jonathan E. Rubin,et al.  Sustained Spatial Patterns of Activity in Neuronal Populations without Recurrent Excitation , 2004, SIAM J. Appl. Math..