Stability and bifurcations in neural fields with axonal delay and general connectivity

A stability analysis is presented for neural field equations in the presence of axonal delays and for a general class of connectivity kernels and synap- tic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the delays play a crucial role in non-stationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the kernel. Bounds are determined for the frequencies of bifurcating periodic solutions. A perturbative scheme is used to calculate the types of bifurca- tions leading to spatial patterns, oscillatory solutions, and traveling waves. For high transmission speeds a simple method is derived that allows the de- termination of the bifurcation type by visual inspection of the Fourier trans- forms of the connectivity kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible second order systems with combinations of Gaussian excitatory and inhibitory connections.

[1]  P. Robinson,et al.  Modal analysis of corticothalamic dynamics, electroencephalographic spectra, and evoked potentials. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Peter A. Robinson,et al.  Unified neurophysical model of EEG spectra and evoked potentials , 2002, Biological Cybernetics.

[3]  P. Goldman-Rakic.  Cellular basis of working memory , 1995, Neuron.

[4]  H. Haken,et al.  A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics , 1997 .

[5]  P. Bressloff.  Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons , 2000, Journal of mathematical biology.

[6]  Thomas Wennekers,et al.  Dynamic Approximation of Spatiotemporal Receptive Fields in Nonlinear Neural Field Models , 2002, Neural Computation.

[7]  H. Schuster,et al.  A model for neuronal oscillations in the visual cortex , 1990, Biological Cybernetics.

[8]  P. Nunez,et al.  Neocortical Dynamics and Human EEG Rhythms , 1995 .

[9]  J. Lance,et al.  Current concepts of migraine pathogenesis , 1993, Neurology.

[10]  C. Uhl.  Analysis of neurophysiological brain functioning , 1999 .

[11]  P. Nunez.  Toward a quantitative description of large-scale neocortical dynamic function and EEG , 2000, Behavioral and Brain Sciences.

[12]  E. G. Jones,et al.  Thalamic oscillations and signaling , 1990 .

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

[14]  A. Grinvald,et al.  Spatio-Temporal Dynamics of Odor Representations in the Mammalian Olfactory Bulb , 2002, Neuron.

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

[16]  P. Bressloff,et al.  PHYSICS OF THE EXTENDED NEURON , 1997 .

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

[18]  H. Schuster,et al.  A model for neuronal oscillations in the visual cortex , 1990, Biological Cybernetics.

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

[20]  G. Lord,et al.  Waves and bumps in neuronal networks with axo-dendritic synaptic interactions , 2003 .

[21]  Michael Bestehorn,et al.  Activity dynamics in nonlocal interacting neural fields. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[23]  Jian-Young Wu,et al.  Propagating Activation during Oscillations and Evoked Responses in Neocortical Slices , 1999, The Journal of Neuroscience.

[24]  G. Blasdel,et al.  Voltage-sensitive dyes reveal a modular organization in monkey striate cortex , 1986, Nature.

[25]  W Singer,et al.  Visual feature integration and the temporal correlation hypothesis. , 1995, Annual review of neuroscience.

[26]  G. Ermentrout,et al.  The evolution of synaptically generated waves in one- and two-dimensional domains , 2002 .

[27]  P. Tass,et al.  Sscillatory Cortical Activity during Visual Hallucinations , 1997, Journal of biological physics.

[28]  Thomas Wennekers,et al.  Pattern formation in intracortical neuronal fields , 2003, Network.

[29]  Viktor K. Jirsa,et al.  Spatiotemporal forward solution of the EEG and MEG using network modeling , 2002, IEEE Transactions on Medical Imaging.

[30]  G. Ermentrout,et al.  Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity , 2000, Network.

[31]  Lance Jw,et al.  Current concepts of migraine pathogenesis. , 1993 .

[32]  T Hervé,et al.  Diffusion and innovation rates for multidimensional neuronal data with large spatial covariances , 2000, Network.

[33]  A. Grinvald,et al.  Real-time optical imaging of naturally evoked electrical activity in intact frog brain , 1984, Nature.

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

[35]  James J. Wright,et al.  The electroencephalogram and cortical neural networks , 1992 .

[36]  Peter König,et al.  Mechanisms to synchronize neuronal activity , 2001, Biological Cybernetics.

[37]  James J. Wright,et al.  Propagation and stability of waves of electrical activity in the cerebral cortex , 1997 .

[38]  Walter J. Freeman,et al.  Characteristics of the Synchronization of Brain Activity imposed by Finite conduction Velocities of axons , 2000, Int. J. Bifurc. Chaos.

[39]  P. Goldman-Rakic,et al.  Multiple visual areas in the posterior parietal cortex of primates. , 1993, Progress in brain research.

[40]  B. Salzberg,et al.  Optical Recording of Impulses in Individual Neurones of an Invertebrate Central Nervous System , 1973, Nature.