Estimating statistics of neuronal dynamics via Markov chains

Biological Cybernetics

Volumn 84, Issue 1, 2001, Pages 31-40

  Froyland, Gary ^a   Aihara, Kazuyuki ^b,c  

^a UNIVERSITY OF PADERBORN   (Germany)

^b UNIVERSITY OF TOKYO   (Japan)

^c JAPAN SCIENCE AND TECHNOLOGY AGENCY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

ACTION POTENTIAL; ALGORITHM; ARTICLE; BIOLOGICAL MODEL; NERVE CELL; PHYSIOLOGY; PROBABILITY;

ACTION POTENTIALS; ALGORITHMS; MARKOV CHAINS; MODELS, NEUROLOGICAL; NEURONS;

EID: 0035219080     PISSN: 03401200     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00007979     Document Type: Article

References (35)

1. Abeles M (1982) Role of the cortical neuron: integrator or coincidence detector? Isr J Med Sci 18: 83-92
2. Aertsen AMHJ, Gerstein GL, Habib MK, Palm G (1989) Dynamics of neuronal firing correlation: modulation of "effective connectivity". J Neurophysiol 61(5): 900-917
3. Aihara K (1995) Chaos in axons. In: Arbib MA (ed) The handbook of brain theory and neural networks. MIT Press, Cambridge, Mass., pp 183-185
4. Aihara K, Numajiri T, Matsumoto G, Kotani M (1986) Structure of attractors in periodically forced neural oscillators. Phys Lett A 116(7): 313-317
5. Aihara K, Takabe T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 144(6/7): 333-340
6. Blum JR, Rosenblatt JL (1967) On the moments of recurrence time. J Math Sci 2: 1-6
7. Dellnitz M, Junge O (1998) An adaptive subdivision technique for the approximation of attractors and invariant measures. Comput Visual Sci 1: 63-68
8. Francke H, Plachky D, Thomsen W (1985) A finitely additive version of Poincaré's recurrence theorem. In: Christopeit N, Helmes K, Kohlmann M (eds) Stochastic differential systems -Proceedings of the 3rd Bad Honnef Conference. (Lecture Notes in Control and Information Science) Springer, Berlin Heidelberg New York
9. Freeman WJ (2000) Neurodynamics: an exploration in mesoscopic brain dynamics. Springer, London Berlin Heidelberg
10. Froyland G (2000) Extracting dynamical behaviour via Markov models. In: Mees A (ed) Nonlinear dynamics and statistics: Proceedings, Newton Institute. Birkhäuser, Basel, pp 283-324
11. Fujii H, Itoh H, Aihara K, Ichinose N, Tsukada M (1997) Dynamical cell assembly hypothesis - theoretical possibility of spatio-temporal coding in the cortex. Neural Netw 9: 1303-1350
12. Glass L, Mackey MC (1988) From clocks to chaos. Princeton University Press, Princeton, N.J.
13. Hayashi H, Ishizuka S (1992) Chaotic nature of bursting discharges in the Onchidium pacemaker neuron. J Theor Biol 156: 269-291
14. Holden AV (1976) Models of the stochastic activity of neurons. Springer, Berlin Heidelberg New York
15. Ichinose N, Aihara K, Judd K (1998) Extending the concept of isochrons from oscillatory to excitable systems for modeling an excitable neuron. Int J Bifurcation Chaos 8: 2375-2385
16. Judd K, Aihara K (1993) Pulse propagation networks: a neural network model that uses temporal coding by action potentials. Neural Netw 6(2): 203-215 Errata. Neural Netw 7: 1491, 1994
17. Judd K, Aihara K (2000) Generation, recognition and learning of recurrent signals by pulse propagation networks. Int J Bifurcation Chaos, in press
18. Kac M (1947) On the notion of recurrence in discrete stochastic processes. Bull Am Math Soc 53: 1002-1010
19. Kemeny JG, Snell JL (1976) Finite Markov chains. Springer, Berlin Heidelberg New York
20. Kitazawa S, Kimura T, Yin P-B (1998) Cerebellar complex spikes encode both destinations and errors in arm movement. Nature 392: 494-497
21. Kobayashi Y, Kawano K, Takemura A, Inoue Y, Kitama T, Gomi H, Kawato M (1998) Temporal firing patterns of purkinje cells in the cerebellar ventral paraflocculus during ocular following responses in monkeys II. Complex spikes. J Neurophysiol 80: 832-848
22. König P, Engel AK, Singer W (1996) Integrator or coincidence detector? The role of the cortical neuron revisited. Trends Neurosci 19: 130-137
23. Lasota A, Mackey MC (1994) Chaos, fractals, and noise. Stochastic aspects of dynamics, 2nd edn. (Applied Mathematical Sciences vol 97) Springer, Berlin Heidelberg New York
24. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261: 459-467
25. Mees A, Aihara K, Adachi M, Judd K, Ikeguchi T, Matsumoto G (1992) Deterministic prediction and chaos in squid axon response. Phys Lett A 169(1/2): 41-45
26. Petersen K (1983) Ergodic theory. (Cambridge Studies in Advanced Mathematics vol 2) Cambridge University Press, Cambridge
27. Ricciardi LM (1995) Diffusion models of neuron activity. In: Arbib MA (ed) The handbook of brain theory and neural networks. MIT Press, Cambridge, Mass., pp 299-304
28. Segundo JP, Altshuler E, Stiber M, Garfinkel A (1991) Periodic inhibition of living pacemaker neurons (I) locked, intermittent, messy, and hopping behaviours. Int J Bifurcation Chaos 1: 549-581
29. Shadlen MN, Newsome WT (1998) The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. J Neurosci 18: 3870-3896
30. Shinomoto S, Sakai Y, Funahashi S (1999) The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Comput 11: 935-951
31. Softky WR, Koch C (1993) The highly irregular firing of cortical cells is inconsistent with temporal integration of random ESPSs. J Neurosci 13: 334-350
32. Tuckwell HC (1989) Stochastic processes in the neurosciences. SIAM, Philadelphia, Pa.
33. Winfree A (1974) Patterns of phase compromise in biological cycles. J Math Biol 1: 73-95
34. Winfree A (1980) The geometry of biological time. Springer, Berlin Heidelberg New York
35. Wolfowitz J (1967) The moments of recurrence time. Proc Am Math Soc 18: 613-614