Waves and bumps in neuronal networks with axo-dendritic synaptic interactions

Physica D: Nonlinear Phenomena

Volumn 178, Issue 3-4, 2003, Pages 219-241

  Coombes, S ^a   Lord, G J ^b   Owen, M R ^a  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

^b HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Axo dendritic synapses; Hamiltonian fourth order systems; Integral equations; Neuronal networks; Patterns; Space dependent delays; Traveling waves

Indexed keywords

CONTINUUM MECHANICS; HAMILTONIANS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS;

AXO-DENDRITIC SYNAPTIC INTERACTIONS;

NEURAL NETWORKS;

EID: 0037446036     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(03)00002-2     Document Type: Article

References (50)

1. Golomb D., Amitai Y., Propagating neuronal discharges in neocortical slices: computational and experimental study. J. Neurophysiol. 78:1997;1199-1211.
2. Miles R., Traub R.D., Wong R.K.S., Spread of synchronous firing in longitudinal slices from the CA3 region of hippocampus. J. Neurophysiol. 60:1995;1481-1496.
3. Kim U., Bal T., McCormick D.A., Spindle waves are propagating synchronized oscillations in the ferret LGNd in vitro. J. Neurophysiol. 74:1995;1301-1323.
4. Roberts A., How does a nervous system produce behaviour? A case study in neurobiology. Sci. Prog. Oxford. 74:1990;31-51.
5. B.W. Connors, Y. Amitai, Generation of epileptiform discharges by local circuits in neocortex, in: P.A. Schwartzkroin (Ed.), Epilepsy: Models, Mechanisms and Concepts, Cambridge University Press, Cambridge, 1993, pp. 388-424.
6. Gray C.M., Synchronous oscillations in neuronal systems: mechanisms and functions. J. Comput. Neurosci. 1:1992;11-38.
7. Colby C.L., Duhamel J.R., Goldberg M.E., Occulocentric spatial representation in parietal cortex. Cereb. Cortex. 5:1995;470-481.
8. Golomb D., Wang X.J., Rinzel J., Propagation of spindle waves in a thalamic slice model. J. Neurophysiol. 75:1996;750-769.
9. Rinzel J., Terman D., Wang X.J., Ermentrout B., Propagating activity patterns in large-scale inhibitory neuronal networks. Science. 279:1998;1351-1355.
10. Pinto D.J., Ermentrout G.B., Spatially structured activity in synaptically coupled neuronal networks. I. Travelling fronts and pulses. SIAM J. Appl. Math. 62:2001;206-225.
11. C.R. Laing, W.C. Troy, B. Gutkin, G.B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J. Appl. Math. 63 (2002) 62-97.
12. Pinto D.J., Ermentrout G.B., Spatially structured activity in synaptically coupled neuronal networks. II. Lateral inhibition and standing pulses. SIAM J. Appl. Math. 62:2001;226-243.
13. J. Huguenard, D.A. McCormick, Electrophysiology of the neuron, Oxford University Press, Oxford, 1996.
14. Wilson H.R., Cowan J.D., A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik. 13:1973;55-80.
15. Nunez P.L., The brain wave equation: a model for the EEG. Math. Biosci. 21:1974;279.
16. Amari S., Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern. 27:1977;77-87.
17. Jirsa V.K., Haken H., Field theory of electromagnetic brain activity. Phys. Rev. Lett. 77:1996;960-963.
18. Bressloff P.C., Coombes S., Dynamics of strongly-coupled spiking neurons. Neural Comput. 12:2000;91-129.
19. Ermentrout G.B., Neural nets as spatio-temporal pattern forming systems. Rep. Prog. Phys. 61:1998;353-430.
20. J. Rinzel, G.B. Ermentrout, Analysis of neural excitability and oscillations, in: C. Koch, I. Segev (Eds.), Methods in Neuronal Modeling, MIT Press, Cambridge, MA, 1989, pp. 135-170.
21. E.L. White, Cortical Circuits: Synaptic Organization of the Cerebral Cortex. Structure, Function, and Theory, Birkhauser, Basel, 1989.
22. Ermentrout G.B., McLeod J.B., Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edinburgh A. 123:1993;461-478.
23. K. Schumacher, Travelling-front solutions for integrodifferential equations II, in: W. Jäger, H. Rost, P. Tautu (Eds.), Lecture Notes in Biomathematics: Biological Growth and Spread, vol. 38, Springer, Berlin, 1980.
24. Kelso J.A.S., Bressler S.L., Buchanan S., Deguzman G.C., Ding M., Fuchs A., Holroyd T., A phase-transition in human brain and behaviour. Phys. Lett. A. 169:1992;134-144.
25. Wu J., Zou X., Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Diff. Eqs. 135:1997;315-357.
26. Elmer C.E., Van Vleck E.S., Analysis and computation of travelling wave solutions of bistable differential-difference equations. Nonlinearity. 12:1999;771-798.
27. E.J. Doedel, A.R. Champneys, T.R. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X.J. Wang, AUTO97 Continuation and bifurcation software for ordinary differential equations, 1997. http://indy.cs.concordia.ca/auto/main.html.
28. Beyn W.J., The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 9:1990;379-405.
29. Doedel E., Friedman M.J., Kunin B.I., Successive continuation for locating connecting orbits. Numer. Algorithms. 14:1997;103-124.
30. Sandstede B., Convergence estimates for the numerical approximation of homoclinic solutions. IMA J. Numer. Anal. 17:1997;437-462.
31. Liu Y.H., Wang X.J., Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron. J. Comput. Neurosci. 10:2001;25-45.
32. Bressloff P.C., New mechanism for neural pattern formation. Phys. Rev. Lett. 76:1996;4644-4647.
33. J.D. Murray, Mathematical Biology, Springer, Berlin, 1989.
34. P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer, Berlin, 1979.
35. Bressloff P.C., Coombes S., Physics of the extended neuron. Int. J. Mod. Phys. B. 11:1997;2343-2392.
36. Cross M.C., Hohenberg P.C., Pattern formation outside of equilibrium. Rev. Mod. Phys. 65:1993;851-1112.
37. Champneys A.R., Homoclinic orbits in reversible systems and their applications in mechanics. Physica D. 112:1998;158-186.
38. Woods P.D., Champneys A.R., Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation. Physica D. 129:1999;147-170.
39. Hunt G.W., Lord G.J., Champneys A.R., Homoclinic and heteroclinic orbits underlying the post-buckling of axially compressed cylindrical shells. Comput. Meth. Appl. Mech. Eng., Spec. Issue Comput. Meth. Bifurcat. Theor. 170:1999;239-251.
40. Hunt G.W., Peletier M.A., Champneys A.R., Woods P.D., Ahmer Wadee M., Budd C.J., Lord G.J., Cellular buckling in long structures. Theme Issue Localization. 21:2000;3-29.
41. Swindale N.V., The development of topography in the visual cortex: a review of models. Network. 7:1996;161-274.
42. Bressloff P.C., Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physica D. 155:2001;83-100.
43. D. Terman, L. Zhang, Asymptotic stability of traveling wave solutions of integral-differential equations arising from neuronal networks, preprint, 2002. http://www.math.ohio-state.edu/̃terman/.
44. Taylor J.G., Neural bubble dynamics in two dimensions: foundations. Biol. Cybernet. 80:1999;393-409.
45. Werner H., Richter T., Circular stationary solutions in two-dimensional neural fields. Biol. Cybernet. 85:2001;211-217.
46. Li C., Jasper H.H., Microelectrode studies of the electrical activity of the cerebral cortex in the cast. J. Physiol. 121:1953;117-140.
47. Koch C., Cable theory in neurons with active, linearized membranes. Biol. Cybernet. 50:1984;15-33.
48. Segev I., Rall W., Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations. Trends Neurosci. 21(11):1998;453-460.
49. Bressloff P.C., Coombes S., Spike train dynamics underlying pattern formation in integrate-and-fire oscillator networks. Phys. Rev. Lett. 81:1998;2384-2387.
50. Laing C., Chow C.C., Stationary bumps in networks of spiking neurons. Neural Comput. 13:2001;1473-1494.