Dynamics of Deterministic and Stochastic Paired Excitatory-Inhibitory Delayed Feedback

Neural Computation

Volumn 15, Issue 12, 2003, Pages 2779-2822

  Laing, Carlo R ^a   Longtin, André ^a  

^a UNIVERSITY OF OTTAWA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ACTION POTENTIAL; ALGORITHM; ANIMAL; ARTICLE; ARTIFACT; BIOLOGICAL MODEL; CENTRAL NERVOUS SYSTEM; COMPUTER SIMULATION; EXCITATORY POSTSYNAPTIC POTENTIAL; FEEDBACK SYSTEM; HUMAN; NERVE CELL; NERVE CELL INHIBITION; NERVE TRACT; PHYSIOLOGY; STATISTICS; SYNAPTIC TRANSMISSION;

ACTION POTENTIALS; ALGORITHMS; ANIMALS; ARTIFACTS; CENTRAL NERVOUS SYSTEM; COMPUTER SIMULATION; EXCITATORY POSTSYNAPTIC POTENTIALS; FEEDBACK; HUMANS; MODELS, NEUROLOGICAL; NEURAL INHIBITION; NEURAL PATHWAYS; NEURONS; STOCHASTIC PROCESSES; SYNAPTIC TRANSMISSION;

EID: 0242635711     PISSN: 08997667     EISSN: None     Source Type: Journal    
DOI: 10.1162/089976603322518740     Document Type: Article

References (65)

1. an der Heiden, U. (1980). Analysis of neural networks. Berlin: Springer-Verlag.
2. Bastian, J. (1990). Descending control of electroreception. I. Properties of nucleus praeeminentialis neurons projecting indirectly to the electrosensory lateral line lobe. J. Neurosci., 10, 1226-1240.
3. Belair, J., Campbell, S. A., & van den Driessche, P. (1996). Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math., 56, 245-255.
4. Berman, N. J., & Maler, L. (1999). Neural architecture of the electrosensory lateral line lobe: Adaptations for coincidence detection, a sensory searchlight and frequency-dependent adaptive filtering. J. Exp. Biol., 202, 1243-1253.
5. Bressloff, P. C., & Coombes, S. (1999). Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays. Physica D, 130, 232-254.
6. Bressloff, P. C., & Coombes, S. (2000a). Dynamics of strongly coupled spiking neurons. Neural Comput., 12, 91-129.
7. Bressloff, P. C., & Coombes, S. (2000b). A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators. SIAM J. Appl. Math., 60, 820-841.
8. Brunel, N. (1996). Hebbian learning of context in recurrent neural networks. Neural Comput., 8, 1677-1710.
9. Cajal, R. S. (1909). Histologie du système nerveux de l'homme et des vertèbres. Paris: Maloine.
10. Campbell, S. A. (1999). Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops. Dynam. Contin. Discrete Impuls. Systems, 5, 225-235.
11. Contreras, D., Destexhe, A., Sejnowski, T. J., & Steriade, M. (1996). Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback. Science, 274, 771-774.
12. Crick, F., & Koch, C. (1998). Constraints on cortical and thalamic projections: The no-strong-loops hypothesis. Nature, 391, 245-250.
13. Dayan, P., & Abbott, L.F. (2001). Theoretical neuroscience. Cambridge, MA: MIT Press.
14. Destexhe, A. (1994). Oscillations, complex spatiotemporal behavior, and information transport in networks of excitatory and inhibitory neurons. Phys. Rev. E, 50, 1594-1606.
15. Diez-Martinez, O., & Segundo, J. P. (1983). Behavior of a single neuron in a recurrent inhibitory loop. Biol. Cybern., 47, 33-41.
16. Doiron, B., Chacron, M. J., Maler, L., Longtin, A., & Bastian, J. (2003). Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature, 421, 539-543.
17. Douglas, R. J., Koch, C., Mahowald, M., Martin, K. A. C., & Suarez, H. H. (1995). Recurrent excitation in neocortical circuits. Science, 269, 981-985.
18. Engelborghs, K., Luzyanina, T., & Samaey, G. (2001). DDE-BIFTOOL v. 2.00 user manual: A Matlab package for bifurcation analysis of delay differential equations (Tech. Rep. TW-330). Department of Computer Science, K. U. Leuven, Belgium.
19. Ermentrout, G. B. (1994). Reduction of conductance-based models with slow synapses to neural nets. Neural Comput., 6, 679-695.
20. Ermentrout, B. (1998). Linearization by noise. Neural Comput., 10, 1721-1729.
21. Ermentrout, G. B., & Kopell, N. (1998). Fine structure of neural spiking and synchronization in the presence of conduction delays. Proc. Natl. Acad. Sci. USA, 95, 1259-1264.
22. Ermentrout, G.B., Pascal, M., & Gutkin, B. (2001). The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators. Neural Comput., 13, 1285-1310.
23. Ernst, U., Pawelzik, K., & Geisel, T. (1995). Synchronization induced by temporal delays in pulse-coupled oscillators. Phys. Rev. Lett., 74, 1570-1573.
24. Eurich, C. W., Mackey, M. C., & Schwegler, H. (2002). Recurrent inhibitory dynamics: The role of state-dependent distributions of conduction delay times. J. Theor. Biol., 216, 31-50.
25. Fohlmeister, J. (1979). A theoretical study of neural adaptation and transient responses due to inhibitory feedback. Bull. Math. Biol., 41, 257-282.
26. Foss, J., Longtin, A., Mensour, B., & Milton, J. G. (1996). Phys. Rev. Lett., 76, 708-711.
27. Foss, J., & Milton, J. G. (2000). Multistability in recurrent neural loops arising from delay. J. Neurophysiol., 84, 975-985.
28. Giannakopoulos, F., & Zapp, A. (2001). Bifurcations in a planar system of differential delay equations modeling neural activity. Physica D, 159, 215-232.
29. Glass, L., & Mackey, M. C. (1988). From docks to chaos: The rhythms of life. Princeton, NJ: Princeton University Press.
30. Glass, L., & Malta, C. P. (1990). Chaos in multilooped negative feedback systems. J. Theor. Biol., 145, 217-223.
31. Golomb, D., & Ermentrout, G. B. (2002). Slow excitation supports propagation of slow pulses in networks of excitatory and inhibitory populations. Phys. Rev. E, 65, 061911.
32. Guglielmi, N., & Hairer, E. (1999) Order stars and stability for delay differential equations. Numer. Math., 83, 371-383.
33. Guillouzic, S., L'Heureux, I., & Longtin, A. (2000). Rate processes in a delayed, stochastically driven, overdamped system. Phys. Rev. E, 61, 4906-4914.
34. Hahnloser, R. H. R., Sarpeshkar, R., Mahowald, M. A., Douglas, R. J., & Seung, H. S. (2000). Digital selection and analogue amplification in a cortex-inspired silicon circuit. Nature, 405, 947-951.
35. Hansel, D., & Mato, G. (2001). Existence and stability of persistent states in large neuronal networks. Phys. Rev. Lett., 86, 4175-4178.
36. Hansel, D., & Sompolinsky, H. (1992). Synchronization and computation in a chaotic neural network. Phys. Rev. Lett., 68, 718-721.
37. Hayes, N. D. (1950). Roots of the transcendental equation associated with a certain differential-difference equation. J. London Math. Soc., 25, 226-232.
38. Herz, A.V.M., Li, Z., & van Hemmen, J. L. (1991). Statistical mechanics of temporal association in neural networks with transmission delays. Phys. Rev. Lett., 66, 1370-1373.
39. Hohn, N., & Burkitt, A. N. (2001). Shot noise in the leaky integrate-and-fire neuron. Phys. Rev. E, 63, 031902.
40. Kistler, W.M. and van Hemmen, J.L. (1999). Delayed reverberation through time windows as a key to cerebellar function. Biol. Cybern., 81, 373-380.
41. Knight, B. W. (2000). Dynamics of encoding in neuron populations: Some general mathematical features. Neural Comput., 12, 473-518.
42. Koch, C. (1999). Biophysics of computation. New York: Oxford University Press.
43. Kunysz, A. M., Shrier, A., & Glass, L. (1997). Bursting behavior during fixed-delay stimulation of spontaneously beating chick heart cell aggregates. Am. J. Physiol., 273, C331-C346.
44. Lansky, P., & Sacerdote, L. (2001). The Ornstein-Uhlenbeck neuronal model with signal-dependent noise. Phys. Lett. A, 285, 132-140.
45. Latham, P. E., Richmond, B. J., Nelson, P. G., & Nirenberg, S. (2000). Intrinsic dynamics in neuronal networks. I. Theory. J. Neurophysiol., 83, 808-827.
46. Longtin, A., & Milton, J. G. (1988). Complex oscillations in the human pupil light reflex with mixed delayed feedback. Math. Biosc., 90, 184-199.
47. Longtin, A., & Milton, J. G. (1989). Modelling autonomous oscillations in the human light reflex using non-linear delay-differential equations. Bull. Math. Biol., 51, 605-624.
48. Mackey, M. C., & Glass, L. (1977). Oscillations and chaos in physiological control systems. Science, 197, 287-289.
49. Mackey, M. C., & an der Heiden, U. (1982). The dynamics of recurrent inhibition. J. Math. Biol., 19, 211-225.
50. Markus, C., & Westerwelt, R. (1989). Stability of analog neural networks with delay. Phys. Rev. A, 39, 347-359.
51. Mensour, B., & Longtin, A. (1998). Power spectra and dynamical invariants for delay-differential and difference equations. Physica, 113, 1-25.
52. Murphy, P. C., Duckett, S. G., & Sillito, A. M. (1999). Feedback connections to the lateral geniculate nucleus and cortical response properties. Science, 286, 1552-1554.
53. Nelson, M. E. (1994). A mechanism for neuronal gain control by descending pathways. Neural Comput., 6, 242-254.
54. Pakdaman, K., Vibert, J. F., Boussard, E., & Azmy, N. (1996). Single neuron with recurrent excitation: Effect of the transmission delay. Neural Networks, 9, 797-818.
55. Plant, R. E. (1981). A FitzHugh differential-difference equation modeling recurrent neural feedback. SIAM J. Appl. Math., 40, 150.
56. Ricciardi, L. M. (1977). Diffusion processes and related topics in biology. Berlin: Springer-Verlag.
57. Seung, H. S., Lee, D. D., Reis, B. Y., & Tank, D. W. (2000). The autapse: A simple illustration of short-term analog memory storage by tuned synaptic feedback. J. Comput. Neurosci., 9, 171-185.
58. Shampine, L. F., & Thompson, S. (2000). Solving DDEs with MATLAB. Available on-line: http://www.radford.edu/̃thompson/webddes/.
59. Shayer, L. P., & Campbell, S. A. (2000). Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl. Math., 61, 673-700.
60. Sompolinsky, H., & Kanter, I. (1986). Temporal association in asymmetric neural networks. Phys. Rev. Lett., 57, 2861-2864.
61. Stépán, G. (1989). Retarded dynamical systems: Stability and characteristic functions. Longman Scientific and Technical.
62. Stark, L. (1968). Neurological control systems. New York: Plenum Press.
63. Traub, R. D., & Miles, R. (1991). Neural networks of the hippocampus. Cambridge: Cambridge University Press.
64. Wang, X. J. (1999). Synaptic basis of cortical persistent activity: The importance of NMDA receptors to working memory J. Neuroscience, 19(21), 9587.
65. Wilson, H. R., & Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J., 12, 1-24.