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Publisher: Society for Industrial and Applied Mathematics
Languages: English
Types: Article
Subjects:
Identifiers:doi:10.1137/040605953
In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.
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    • [1] J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability analysis of travelling waves, Journal uf¨r die reine und angewandte Mathematik, 410 (1990), pp. 167-212.
    • [2] S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields , Biological Cybernetics, 27 (1977), pp. 77-87.
    • [3] N. D. Aparicio, S. J. A. Malham, and M. Oliver, Numerical evaluation of the Evans function by Magnus integration, BIT, submitted (2004).
    • [4] N. J. Balmforth, R. V. Craster, and S. J. A. Malham, Unsteady fronts in an autocatalytic system, Proceedings of the Royal Society of London A, 455 (1999), pp. 1401-1433.
    • [5] P. C. Bressloff, Traveling fronts and wave propagation failure in an inhomogeneous neural network, Physica D, 155 (2001).
    • [6] P. C. Bressloff and S. E. Folias, Front-bifurcations in an excitatory neural network, SIAM Journal on Applied Mathematics, submitted (2004).
    • [7] P. C. Bressloff, S. E. Folias, A. Prat, and Y. X. Li, Oscillatory waves in inhomogeneous neural media, Physical Review Letters, 91 (2003), p. 178101.
    • [8] T. J. Bridges, G. Derks, and G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework, Physica D, 172 (2002), pp. 190-216.
    • [9] F. Chen, Travelling waves for a neural network, Electronic Journal of Differential Equations, 2003 (2003), pp. 1-4.
    • [10] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), pp. 125-160.
    • [11] Z. Chen, G. B. Ermentrout, and J. B. McLeod, Traveling fronts for a class of nonlocal convolution differential equation, Applicable Analysis, 64 (1997), pp. 235-253.
    • [12] R. D. Chervin, P. A. Pierce, and B. W. Connors, Propagation of excitation in neural network models, Journal of Neurophysiology, 60 (1988), pp. 1695-1713.
    • [13] S. Coombes, G. J. Lord, and M. R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178 (2003), pp. 219-241.
    • [14] D. Cremers and A. V. M. Herz, Traveling waves of excitation in neural field models: equivalence of rate descriptions and integrate-and-fire dynamics , Neural Computation, 14 (2002), pp. 1651-1667.
    • [15] G. B. Ermentrout, Neural nets as spatio-temporal pattern forming systems, Reports on Progress in Physics, 61 (1998), pp. 353-430.
    • [16] G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proceedings of the Royal Society of Edinburgh, 123A (1993), pp. 461-478.
    • [17] J. Evans, Nerve axon equations: IV The stable and unstable impulse, Indiana University Mathematics Journal, 24 (1975), pp. 1169-1190.
    • [18] S. E. Folias and P. C. Bressloff, Breathing pulses in an excitatory neural network, SIAM Journal on Applied Dynamical Systems, submitted (2004).
    • [19] D. Golomb and Y. Amitai, Propagating neuronal discharges in neocortical slices: Computational and experimental study, Journal of Neurophysiology, 78 (1997), pp. 1199-1211.
    • [20] A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations, Nonlinearity, 7 (1994), pp. 805-835.
    • [21] P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves, Discrete and Continuous Dynamical Systems - Series A, 4 (2004), pp. 837-855.
    • [22] A. Hutt, M. Bestehorn, and T. Wennekers, Pattern formation in intracortical neuronal fields , Network, 14 (2003), pp. 351-368.
    • [23] M. A. P. Idiart and L. F. Abbott, Propagation of excitation in neural network models, Network, 4 (1993), pp. 285- 294.
    • [24] C. K. R. T. Jones, Stability of the traveling wave solutions of the FitzHugh-Nagumo system, Transactions of the American Mathematical Society, 286 (1984), pp. 431-469.
    • [25] T. Kapitula, N. Kutz, and B. Sandstede, The Evans function for nonlocal equations, Indiana University Mathematics Journal, to appear (2004).
    • [26] T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques, SIAM Journal on Mathematical Analysis, 33 (2002), pp. 1117-1143.
    • [27] , Eigenvalues and resonances using the Evans function, Discrete and Continuous Dynamical Systems, 10 (2004), pp. 857-869.
    • [28] U. Kim, T. Bal, and D. A. McCormick, Spindle waves are propagating synchronized oscillations in the ferret LGNd in vitro, Journal of Neurophysiology, 74 (1995), pp. 1301-1323.
    • [29] C. R. Laing and W. C. Troy, PDE methods for nonlocal models, SIAM Journal on Applied Dynamical Systems, 2 (2003), pp. 487-516.
    • [30] Y. Li and K. Promislow, The mechanism of the polarization mode instability in birefringent fiber optics , SIAM Journal on Mathematical Analysis, 31 (2000), pp. 1351-1373.
    • [31] R. Miles, R. D. Traub, and R. K. S. Wong, Spread of synchronous firing in longitudinal slices from the CA3 region of Hippocampus, Journal of Neurophysiology, 60 (1995), pp. 1481-1496.
    • [32] R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philosophical Transactions of the Royal Society London A, 340 (1992), pp. 47-94.
    • [33] , Asymptotic stability of solitary waves, Communications in Mathematical Physics, 164 (1994), pp. 305-349.
    • [34] D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. Travelling fronts and pulses, SIAM Journal on Applied Mathematics, 62 (2001), pp. 206-225.
    • [35] , Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses, SIAM Journal on Applied Mathematics, 62 (2001), pp. 226-243.
    • [36] B. Sandstede, Handbook of Dynamical Systems II, Elsevier, 2002, ch. Stability of travelling waves, pp. 983-1055.
    • [37] J. G. Taylor, Neural 'bubble' dynamics in two dimensions: foundations, Biological Cybernetics, 80 (1999), pp. 393- 409.
    • [38] H. Werner and T. Richter, Circular stationary solutions in two-dimensional neural fields , Biological Cybernetics, 85 (2001), pp. 211-217.
    • [39] H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), pp. 55-80.
    • [40] J.-Y. Wu, L. Guan, and Y. Tsau, Propagating activation during oscillations and evoked responses in neocortical slices, Journal of Neuroscience, 19 (1999), pp. 5005-5015.
    • [41] L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16 (2003), pp. 513-536.
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