Dissection and reduction of a modeled bursting neuron

Journal of Computational Neuroscience

Volumn 3, Issue 3, 1996, Pages 199-223

  Butera, R J ^a   Clark, J W ^a   Byrne, J H ^b  

^a Rice University   (United States)

^b UNIVERSITY OF TEXAS MEDICAL SCHOOL   (United States)

Author keywords

Bifurcation; Bursting; Model reduction; Nonlinear dynamics

Indexed keywords

BIFURCATION (MATHEMATICS); ELECTROPHYSIOLOGY;

% REDUCTIONS; ACTION-POTENTIALS; BIFURCATION; BURSTING; BURSTING NEURONS; LOW-ORDER MODELING; LOWER ORDER MODELS; MODEL REDUCTION; SLOW WAVE; VARIABLE MODEL;

TRANSIENT ANALYSIS;

ACTION POTENTIAL; ARTICLE; COMPUTER SIMULATION; NERVE CELL; NERVE CELL MEMBRANE POTENTIAL; NERVE POTENTIAL; NERVE STIMULATION; NONLINEAR SYSTEM; PRIORITY JOURNAL;

EID: 0029768812     PISSN: 09295313     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00161132     Document Type: Article

References (39)

1. Alevizos A, Skelton M, Weiss KR, Koester J (1991) A comparison of bursting neurons in Aplysia. Biological Bulletin 180:269-275.
2. Baer SM, Rinzel J, Carrillo H (1995) Analysis of an autonomous phase model for neuronal parabolic bursting. Journal of Mathematical Biology 33:309-333.
3. Bertram R (1993) A computational study of the effects of serotonin on a molluscan burster neuron. Biological Cybernetics 69:257-267.
4. Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bulletin of Mathematical Biology 57:413-439.
5. Butera RJ, Clark JW, Canavier CC, Baxter DA, Byrne JH (1995) Analysis of the effects of modulatory agents on a modeled bursting neuron: Dynamic interactions between voltage and calcium dependent systems. Journal of Computational Neuroscience 2:19-44.
6. Canavier CC, Baxter DA, Clark JW, Byrne JH (1993) Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of post-synaptic activity. Journal of Neurophysiology 69:2252-2257.
7. Canavier CC, Clark JW, Byrne JH (1991) Simulation of the bursting activity of neuron R15 in Aplysia: Role of ionic currents, calcium balance, and modulatory transmitters. Journal of Neurophysiology 66:2107-2124.
8. 2+ current. Journal of Theoretical Biology 142:305-315.
9. Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic β-cell. Biophysical Journal 42:181-190.
10. Doedel EJ (1981) AUTO: A program for the automatic bifurcation and analysis of autonomous systems. Congressus Numerantium 30:265-284.
11. Dvořák I, Šiška J (1989) Analysis of metabolic systems with complex slow and fast dynamics. Bulletin of Mathematical Biology 51:255-274.
12. Ermentrout GB (1994) Reduction of conductance-based models with slow synapses to neural nets. Neural Computation 6:679-695.
13. Hairer E, Wanner G (1990) Solving Ordinary Differential Equations II. Stiff and Differential-algebraic Problems. Springer Series in Computational Mathematics. Springer-Verlag, New York.
14. Hale J, Koçak H (1991) Dynamics and Bifurcations. Vol. 3 of Texts in Applied Mathematics. Springer-Verlag, New York.
15. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction excitation in nerve. Journal of Physiology (London) 117:500-544.
16. Junge D, Stephens CL (1973) Cyclic variation of potassium conductance in a burst generating neurone in Aplysia. Journal of Physiology (London) 235:155-181.
17. Keizer J, Mangus G (1989) ATP-sensitive potassium channel and bursting in the pancreatic beta-cell: A theoretical study. Biophysical Journal 56:229-242.
18. Kepler TB, Abbott LF, Marder E (1992) Reduction of conductancebased neuron models. Biological Cybernetics 66:381-387.
19. Kononenko NI (1994) Dissection of a model for membrane potential oscillations in bursting neuron of snail, Helix pomata. Comparative Biochemistry and Physiology, Part A 107A:323-332.
20. LoFaro T, Kopell N, Marder E, Hooper SL (1994) Subharmonic coordination in networks of neurons with slow conductances. Neural Computation 6:69-84.
21. Mathieu PA, Roberge FA (1971) Characteristics of pacemaker oscillations in Aplysia neurons. Canadian Journal of Pharmacology and Physiology 49:787-795.
22. Morris C, Lecar H (1981) Voltage oscillations in the barnacle muscle fiber. Biophysical Journal 35:193-213.
23. Pavlidis T (1973) Biological Oscillators: Their Mathematical Analysis. Academic Press, New York.
24. Pinsker HM (1977) Aplysia bursting neurons as endogenous oscillators. I. Phase-response curves for pulsed inhibitory synaptic input. Journal of Neurophysiology 40:527-543.
25. Plant RE, Kim M (1975) On the mechanism underlying bursting in the Aplysia abdominal ganglion R15 cell. Mathematical Biosciences 26:357-375.
26. Plant RE, Kim M (1976) Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophysical Journal 16:227-244.
27. Rheinboldt W, Burkardt J (1983a) Algorithm 596: A program for a locally parameterized continuation process. ACM Transactions on Mathematical Software 9:236-241.
28. Rheinboldt W, Burkardt J (1983b) A locally parameterized continuation process. ACM Transactions on Mathematical Software 9:215-235.
29. Rinzel J (1985) Bursting oscillations in an excitable membrane model. In: BD Sleeman, D Jones, eds. Ordinary and Partial Differential Equations, Vol. 1151 of Lecture Notes in Mathematics, Springer-Verlag, Berlin.
30. Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. In: E Teramoto, M Yamaguti, eds. Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, Vol. 71 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin.
31. Rinzel J, Ermentrout GB (1989) Analysis of neural excitability and oscillations. In: C Koch, I Segev, eds. Methods in Neuronal Modeling, MIT Press, Cambridge MA.
32. Rinzel J, Lee YK (1987) Dissection of a model for neuronal parabolic bursting. Journal of Mathematical Biology 25:653-675.
33. Rowat PF, Selverston AI (1993) Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network. Journal of Neurophysiology 70:1030-1053.
34. Rush ME, Rinzel J (1994) Analysis of bursting in a thalamic neuron model. Biological Cybernetics 71:281-291.
35. Sherman A, Rinzel J, Keizer J (1988) Emergence of organized bursting in clusters of pancreatic β-cells by channel sharing. Biophysical Journal 54:411-425.
36. Skinner FK, Kopell N, Marder E (1994) Mechanisms of oscillation and frequency control in reciprocally inhibitory model neural networks. Journal of Computational Neuroscience 1:69-87.
37. Smolen P, Terman D, Rinzel J (1993) Properties of a bursting model with two slow inhibitory variables. SIAM Journal of Applied Mathematics 53:861-892.
38. Strumwasser F (1971) The cellular basis of behavior in Aplysia. Journal of Psychiatric Research 8:237-257.
39. Wilson WA (1982) Patterned bursting discharge of invertebrate neurons. In: DO Carpenter, ed. Cellular Pacemakers. Vol. 1, J. Wiley, New York.