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September  2011, 16(2): 529-545. doi: 10.3934/dcdsb.2011.16.529

Firing control of ink gland motor cells in Aplysia californica

Xiangying Meng 1, , Quanbao Ji 2, and  John Rinzel 3,

1. 

Department of Dynamics and Control, Beihang University, Beijing 100191, China
2. 

Mathematical and Computational Department, Anhui Huainan Normal University, Anhui, 232007, China
3. 

Center for Neural Science and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States

Received  April 2010 Revised  November 2010 Published  June 2011

The release of ink in Aplysia californica occurs selectively to long-lasting stimuli. There is a good correspondence between features of the behavior and the firing pattern of the ink gland motor neurons. Indeed, the neurons do not fire for brief inputs and there is a delayed firing for long duration inputs. The biophysical mechanisms for the long delay before firing is due to a transient potassium current which activates rapidly but inactivates more slowly. Based on voltage-clamp experiments, a nine-variable Hodgkin-Huxley-like model for the ink gland motor neurons was developed by Byrne. Here, fast-slow analysis and two-parameter dynamical analysis are used to investigate the contribution of different currents and to predict various firing patterns, including the long latency before firing.
Keywords: transient potassium current, Bifurcation, neuron, oscillation, bistability..
Mathematics Subject Classification: Primary: 34C05, 34C23, 34G20; Secondary: 92C2.
Citation: Xiangying Meng, Quanbao Ji, John Rinzel. Firing control of ink gland motor cells in Aplysia californica. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 529-545. doi: 10.3934/dcdsb.2011.16.529
References:
[1]

P. Dayan and L. F. Abbott, "Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems," MIT Press, 2001.

[2]

Q. S. Lu, H. G. Gu, Z. Q Yang, X. Shi, L. X. Duan and Y. H. Zheng, Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis, Acta Mech. Sin., 24 (2008), 593-628.

[3]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction in nerve, J. Physiol. Lond., 117 (1952), 500-544.

[4]

J. Rinzel, Bursting oscillation in an excitable membrane model, In "Ordinary and Partial Differential Equations" (B. D. sleeman, R. J. Jarvis eds.), Springer, Berlin (1987), 267-281.

[5]

J. H. Byrne, Analysis of ionic conductance mechanisms in motor cells mediating inking behavior in aplysia californica, J. Neurophysiol., 43 (1980), 630-650.

[6]

J. H. Byrne, Quantitative aspects of ionic conductance mechanisms contributing to firing pattern of motor cells mediating inking behavior in aplysia californica, J. Neurophysiol., 43 (1980), 651-668.

[7]

J. H. Byrne, Simulation of the neural activity underlying a short-term modification of inking behavior in aplysis, Brain Res., 204 (1981), 200-203.

[8]

D. Golomb, K. Donner, L. Shacham, D. Shiosberg, Y. Amital, Y. and D. Hansel, Mechanisms of firing patterns in fast-spiking cortical interneurons, PLos Comp. Biol., 3 (2007), 1498-1512.

[9]

J. A. Connor, D. Walter and R. McKown, Neural repetitive firing, Modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons, Biophys. J., 18 (1977), 81-102.

[10]

M. E. Rush and J. Rinzel, The potassium A-current, low firing rates and rebound excitation in hodgin-huxley models, Bulletin of Mathematical Biology, 57 (1995), 899-929.

[11]

J. Rothman and P. B. Manis, The roles potassium currents play in regulating the electrical activity of ventral cochlear nucleus neurons, J. Neurophysiol., 89 (2003), 3097-3113.

[12]

P. O. Kanold, and P. B. Manis, A physiologically based model of discharge pattern regulation by transient K currents in cochlear nucleus pyramidal cells, J. Neurophysiol., 85 (2001), 523-538.

[13]

X. Y. Meng, Q. S. Lu and J. Rinzel, Control of firing patterns by two transient potassium currents: leading spike, latency, bistability,, J. Computl. Neurosci., (). 

[14]

J. F. Storm, Temporal integration by a slowly inactivating $K$+ current in hippocampal neurons, Nature, 336 (1988), 379-381.

[15]

P. B. Manis, Membrane properties and discharge characteristics of guinea pig dorsal cochlear nucleus neurons studied in vitro, J. Neurosci., 10 (1990), 2338-2351.

[16]

W. S. Rhode, P. H. Smith and D. Oertel, Physiological response properties of cells labeled intracellularly with horseradish peroxidase in cat dorsal cochlear nucleus, J. Comparative Neurol., 213 (1983), 426-447.

[17]

T. J. Carew and E. R. Kandel, Inking in Aplysia californica. I. Neural circuit of an all-or-none behavioral response, J. Neurophysiol., 40 (1977), 692-707.

[18]

E. Shapiro, J. Koester and J. H. Byrne, Aplysia ink release: locus of selective sensitivity to long-duration stimuli, J. Neurophysiol., 42 (1979), 1223-1232.

show all references

References:
[1]

P. Dayan and L. F. Abbott, "Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems," MIT Press, 2001.

[2]

Q. S. Lu, H. G. Gu, Z. Q Yang, X. Shi, L. X. Duan and Y. H. Zheng, Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis, Acta Mech. Sin., 24 (2008), 593-628.

[3]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction in nerve, J. Physiol. Lond., 117 (1952), 500-544.

[4]

J. Rinzel, Bursting oscillation in an excitable membrane model, In "Ordinary and Partial Differential Equations" (B. D. sleeman, R. J. Jarvis eds.), Springer, Berlin (1987), 267-281.

[5]

J. H. Byrne, Analysis of ionic conductance mechanisms in motor cells mediating inking behavior in aplysia californica, J. Neurophysiol., 43 (1980), 630-650.

[6]

J. H. Byrne, Quantitative aspects of ionic conductance mechanisms contributing to firing pattern of motor cells mediating inking behavior in aplysia californica, J. Neurophysiol., 43 (1980), 651-668.

[7]

J. H. Byrne, Simulation of the neural activity underlying a short-term modification of inking behavior in aplysis, Brain Res., 204 (1981), 200-203.

[8]

D. Golomb, K. Donner, L. Shacham, D. Shiosberg, Y. Amital, Y. and D. Hansel, Mechanisms of firing patterns in fast-spiking cortical interneurons, PLos Comp. Biol., 3 (2007), 1498-1512.

[9]

J. A. Connor, D. Walter and R. McKown, Neural repetitive firing, Modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons, Biophys. J., 18 (1977), 81-102.

[10]

M. E. Rush and J. Rinzel, The potassium A-current, low firing rates and rebound excitation in hodgin-huxley models, Bulletin of Mathematical Biology, 57 (1995), 899-929.

[11]

J. Rothman and P. B. Manis, The roles potassium currents play in regulating the electrical activity of ventral cochlear nucleus neurons, J. Neurophysiol., 89 (2003), 3097-3113.

[12]

P. O. Kanold, and P. B. Manis, A physiologically based model of discharge pattern regulation by transient K currents in cochlear nucleus pyramidal cells, J. Neurophysiol., 85 (2001), 523-538.

[13]

X. Y. Meng, Q. S. Lu and J. Rinzel, Control of firing patterns by two transient potassium currents: leading spike, latency, bistability,, J. Computl. Neurosci., (). 

[14]

J. F. Storm, Temporal integration by a slowly inactivating $K$+ current in hippocampal neurons, Nature, 336 (1988), 379-381.

[15]

P. B. Manis, Membrane properties and discharge characteristics of guinea pig dorsal cochlear nucleus neurons studied in vitro, J. Neurosci., 10 (1990), 2338-2351.

[16]

W. S. Rhode, P. H. Smith and D. Oertel, Physiological response properties of cells labeled intracellularly with horseradish peroxidase in cat dorsal cochlear nucleus, J. Comparative Neurol., 213 (1983), 426-447.

[17]

T. J. Carew and E. R. Kandel, Inking in Aplysia californica. I. Neural circuit of an all-or-none behavioral response, J. Neurophysiol., 40 (1977), 692-707.

[18]

E. Shapiro, J. Koester and J. H. Byrne, Aplysia ink release: locus of selective sensitivity to long-duration stimuli, J. Neurophysiol., 42 (1979), 1223-1232.

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