Firing patterns in a conductance-based neuron model: Bifurcation, phase diagram, and chaos

Biological Cybernetics

Volumn 107, Issue 1, 2013, Pages 15-24

  Qi, Y ^a   Watts, A L ^a   Kim, J W ^a,b   Robinson, P A ^a,b,c  

^a UNIVERSITY OF SYDNEY   (Australia)

^b WOOLCOCK INSTITUTE OF MEDICAL RESEARCH   (Australia)

^c UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Action potential; Conductance based neuron model; Linear stability analysis; Period doubling adding route to chaos

Indexed keywords

ACTION POTENTIALS; FIRING PATTERNS; HODGKIN-HUXLEY MODELS; INSTABILITY ANALYSIS; NEOCORTICAL NEURONS; NEURAL FIELDS; NEURAL FIRING; NEURON MODEL; NUMERICAL COMPUTATIONS; SUBTHRESHOLD; UNDERLYING MECHANISM;

ELECTROPHYSIOLOGY; LINEAR STABILITY ANALYSIS; PHASE DIAGRAMS;

NEURONS;

ACTION POTENTIAL; ARTICLE; NERVE CELL; NONLINEAR SYSTEM; PHYSIOLOGY; THEORETICAL MODEL;

ACTION POTENTIALS; MODELS, THEORETICAL; NEURONS; NONLINEAR DYNAMICS;

MLCS; MLOWN;

EID: 84872914340     PISSN: 03401200     EISSN: 14320770     Source Type: Journal    
DOI: 10.1007/s00422-012-0520-8     Document Type: Article

References (44)

1. Alligood KT, Sauer TD, Yorke JA (1997) Chaos, an introduction to dynamical systems. Springer, New York
2. 2+ channel plays a crucial role in stabilizing sleep. Proc Natl Acad Sci USA 102: 1743-1748
3. Bezanilla F (2008) Ion channels: from conductance to structure. Neuron 60: 456-468
4. Chay TR (1985) Chaos in a three-variable model of an excitable cell. Physica D 15: 233-242
5. Chay TR, Rinzel J (1985) Bursting, beating, and chaos in an excitable membrane model. Biophys J 47: 357-366
6. Connor JA, Walter D, McKown R (1977) Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from Crustacean axons. Biophys J 18: 81-102
7. Deco G, Jirsa VK, Robinson PA, Breakspear M, Friston K (2008) The dynamic brain: from spiking neurons to neural masses and cortical fields. Public Libr Sci Comput Biol 4:e1000092
8. Doi S, Kumagai S (2005) Generation of very slow neural rhythms and chaos near the Hopf bifurcation in single neuron models. J Comput Neurosci 19: 325-356
9. Ermentrout GB (1994) Reduction of conductance-based models with slow synapses to neural nets. Neural Comput 6: 679-695
10. Ermentrout GB (1998a) Linearization of f-I curves by adaptation. Neural Comput 10: 1721-1729
11. Ermentrout GB (1998b) Neural networks as a spatio-temporal pattern-forming systems. Rep Prog Phys 61: 353-430
12. Ermentrout 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
13. Gerstner W, Kistler W (2002) Spiking neuron models: single neurons, populations, plasticity. Cambridge University Press, Cambridge
14. Guckenheimer J, Oliva R (2002) Chaos in the Hodgkin-Huxley model. SIAM J Appl Dyn Syst 1: 105-114
15. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117: 500-544
16. Holden AV, Fan YS (1992) From simple to simple bursting oscillatory behaviour via chaos in the Rose-Hindemarsh model for neuronal activity. Chaos Soliton Fractals 2: 221-236
17. Izhikevich EM (2005) Which model to use for cortical spiking neurons. IEEE Trans Neural Netw 15: 1063-1070
18. Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT, Gambridge
19. Izhikevich EM, Desai NS, Walcott EC, Hoppensteadt FC (2003) Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 26: 161-167
20. Lample I, Yarom Y (1997) Subthreshold oscillations and resonant behavior: two manifestations of the same mechanism. Neuroscience 78: 325-341
21. Markram H, Toledo-Rodrigurez M, Wang Y, Gupta A, Silberberg G, Wu C (2004) Interneurons of the neocortical inhibitory system. Nat Rev Neurosci 5: 793-807
22. McCormick DA, Bal T (1997) Sleep and arousal: thalamocortical mechanisms. Annu Rev Neurosci 20: 185-215
23. Naud R, Marcille N, Clopath C, Gerstner W (2008) Firing patterns in the adaptive exponential integrate-and-fire model. Biol Cybern 99: 335-347
24. Pedroarena C, Llinas R (1997) Dendritic calcium conductances generate high-frequency oscillation in thalamocortical neurons. Proc Natl Acad Sci USA 94: 724-728
25. Rabinovich MI, Abarbanel HDI (1998) The role of chaos in neural systems. Neuroscience 87: 5-14
26. Rabinovich MI, Varona P, Selverston AI, Abarbanel HDI (2006) Dynamical principles in neuroscience. Rev Mod Phys 78: 1213-1265
27. Rinzel JM (1985) Excitation dynamics: insights from simplified membrane models. Fed Proc 44: 2944-2946
28. Rinzel JM, Ermentrout GB (1989) Analysis of neuronal excitability. In: Koch C, Segev I (eds) Methods of Neuronal Modeling, MIT Press, Cambridge, MA
29. Robinson PA, Kim JW (2012) Spike, rate, field, and hybrid methods for treating neuronal dynamics and interactions. J Neurosci Methods 205: 283-294
30. Robinson PA, Wu H, Kim JW (2008) Neural rate equations for bursting dynamics derived from conductance-based equations. J Theor Biol 250: 663-672
31. Rose RM, Hindmarsh JL (1985) The assembly of ionic currents in a thalamic neuron I: the three-dimensional model. Proc R Soc Lond B 237: 267-288
32. Rubin J, Wechselberger M (2007) Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model. Biol Cybern 97: 5-32
33. Rubin J, Wechselberger M (2008) The selection of mixed-mode oscillations in a Hodgkin-Huxely model with multiple timescales. Chaos 18: 015105-015112
34. Shriki O, Hansel D, Sompolinsky H (2003) Rate models for conductance-based cortical neuronal networks. Neural Comput 15: 1809-1841
35. Staba RJ, Wilson CL, Fried I, Engel J (2002) Single neuron burst firing in the human hippocampus during sleep. Hippocampus 12: 724-734
36. Steriade M, Timofeev I, Grenier F (2001) Natural waking and sleep states: a view from inside neocortical neurons. J Neurophysiol 85: 1965-1985
37. Terman D (1991) Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J Appl Math 51: 1418-1450
38. Touboul JD, Brette R (2008) Dynamics and birfurcations of the adaptive exponential integrate-and-fire model. Biol Cybern 99: 319-334
39. Touboul JD, Ermentrout GB (2011) Finite-size and correlation-induced effects in mean-field dynamics. J Comput Neurosci 31: 453-484
40. Wang XJ (1993) Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. Physica D 62: 263-274
41. Wilson HR (1999a) Simplified dynamics of human and mammalian neocortical neurons. J Theor Biol 200: 375-388
42. Wilson HR (1999b) Spikes, decisions, and actions. Oxford University Press, Oxford
43. Wu H, Robinson PA, Kim JW (2011) Firing responses of bursting neurons with delayed feedback. J Comput Neurosci 31: 61-71
44. Yang Z, Lu Q, Li L (2006) The genesis of period-adding bursting without bursting-chaos in the Chay model. Chaos Soliton Fractals 27: 689-697