| Title | KERNELS OF AN INTEGRO-DIFFERENTIAL EQUATION FROM AN INITIAL PULSE |
| Publication Type | Journal Article |
| Year of Publication | 2007 |
| Authors | Botelho, F, Jamison, JE, Murdock, JA |
| Secondary Title | Communications in Applied Analysis |
| Volume | 11 |
| Issue | 2 |
| Start Page | 327 |
| Pagination | 337 |
| Date Published | 04/2005 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 45J05 |
| Abstract | We consider an integro-differential equation, proposed in the literature as a model of neuronal activity. We establish conditions under which an initial activity function exhibiting localized pattern formation completely characterizes the system. We also investigate how such an initial activity determines more complicated pattern formations.
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| URL | http://www.acadsol.eu/en/articles/11/2/12.pdf |
| Short Title | Kernels of an Integro-Differential Equation |
| Refereed Designation | Refereed |
| Full Text | References[1] S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biological Cybernetics, 27 (1977), 77-87.
[2] S. Amari, Mathematical Theory of Neural Networks, Sangyo-Tosho, Tokyo, 1978. [3] P. Blomquist, J. Wyller, and G. Einevoll, Localized activity patterns in two-population neuronal networks, Physica D, 206 (2005), 180-212. [4] F. Botelho, J. Jamison, and A. Murdock, Single pulse solutions for oscillatory coupling function in neural networks, Preprint (2006). [5] S. Coombes, G. Lord, and M. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178, 219-241. [6] Y. Guo and C. Chow, Existence and stability of standing pulses in neural networks: I Existence, SIAM Journal on Applied Dynamical Systems, 4 (2005), 217 -248. [7] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Inc., 1985. [8] E. Krisner, The link between integral equations and higher order ODEs, Journal of Mathematical Analysis and Applications, 291 (2004), 165-179. [9] J.A.H. Murdock, F. Botelho, and J. Jamison, Persistence of spatial patterns produced by neural field equations, Physica D, 215 (2006), 106-116. [10] C. Laing and W. Troy, PDE Methods for Nonlocal Models, 2004. [11] C. Laing, W. Troy, B. Gutkin, and B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J. Applied Math., 63 (2002), no. 1, 62-97. [12] C. Elphick, E. Meron, and E. Spiegel, Patterns of propagating pulses, SIAM J. Applied Math., 50 (1990), no. 2, 490-503. [13] S. Haykin, Neural Networks: A Comprehensive Foundation, Macmillan College Publ. Co. (1994). [14] H.R. Wilson and J.D. Cowan, A mathematical theory of the function dynamics of cortical and thalamic nervous tissue, Kybernetic, 13 (1973), 55-80. [15] K. Zhang, Representation of spatial orientation by the intrinsic dynamics of head-direction cell ensemble: A theory, J. Neurosci., 16 (1977), 2112-2126. [16] L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Diff. and Integral Equations, 16 (2003), no. 5, 513-536. [17] N. Pavel, Differential equations, flow invariance and applications, Research Notes in Mathematics, 113 (1994).
[18] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Springer Verlag, 7, 2000. [19] S.M. Stringer, T.P. Trappenberg, E.T. Rolls, and I.E.T. de Araujo, Self-organizing continuous attractor networks and path integration: One-dimensional models of head direction cells, Network: Computation in Neural Systems, 13 (2002), 217-242. [20] S. Funahashi, C.J. Bruce, and P.S. Goldman-Rakic, Mnemonic coding of visual space in the monkey’s dorsolateral prefrontal cortex, Journal of Neurophysiology, 61 (1989), 331-349. [21] H. Wilson and J. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), 55-80. |