Evaluation of Kinetic Properties of Dendritic Potassium Current in Ghostbursting Model of Electrosensory Neurons

Takaaki Shirahata

Kagawa School of Pharmaceutical Sciences, Tokushima Bunri University, Sanuki, Japan.

DOI: 10.4236/am.2015.61013                

Abstract

A ghostbursting model is a mathematical model (a system of coupled nonlinear ordinary differential equations) that is based on the Hodgkin-Huxley formalism. The ghostbursting model describes bursting similar to the in vitro bursting of electrosensory neurons of weakly electric fish. Doiron and coworkers have focused on two system parameters of the model: maximal conductance of the dendritic potassium current and the current injected into the somatic compartment . They performed bifurcation analysis and revealed that the -parameter space was divided into three dynamical states: quiescence, periodic tonic spiking, and bursting. The present study focused on a third system parameter: the time constant of dendritic potassium current inactivation . A computer simulation of the model revealed how the dynamical states of the -parameter space changed in response to variations of .

Keywords

Mathematical Model, Bifurcation, Ghostbursting, Time Constant

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Hodgkin, A.L. and Huxley, A.F. (1952) A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. Journal of Physiology, 117, 500-544. http://dx.doi.org/10.1113/jphysiol.1952.sp004764
[2]	Bungay, S.D. and Campbell, S.A. (2009) Modelling a Respiratory Central Pattern Generator Neuron in Lymnaea stagnalis. Canadian Applied Mathematics Quarterly, 17, 283-291.
[3]	Ermentrout, G.B. and Terman, D. (2010) Mathematical Foundations of Neuroscience. Springer, New York. http://dx.doi.org/10.1007/978-0-387-87708-2
[4]	Teka, W., Tabak, J., Vo, T., Wechselberger, M. and Bertram, R. (2011) The Dynamics Underlying Pseudo-Plateau Bursting in a Pituitary Cell Model. Journal of Mathematical Neuroscience, 1, 12. http://dx.doi.org/10.1186/2190-8567-1-12
[5]	Blagoev, K.B., Shukla, K. and Levine, H. (2013) We Need Theoretical Physics Approaches to Study Living Systems. Physical Biology, 10, Article ID: 040201. http://dx.doi.org/10.1088/1478-3975/10/4/040201
[6]	Holmes, P., Eckhoff, P., Wong-Lin, K.F., Bogacz, R., Zacksenhouse, M. and Cohen, J.D. (2010) The Physics of Decision Making: Stochastic Differential Equations as Models for Neural Dynamics and Evidence Accumulation in Cortical Circuits. Exner, P., Ed., XVIth International Congress on Mathematical Physics: Prague, 3-8 August 2009, 123-142. http://dx.doi.org/10.1142/9789814304634_0006
[7]	Gieschke, R. and Serafin, D. (2014) Development of Innovative Drugs via Modeling with MATLAB: A Practical Guide. Springer, Heidelberg. http://dx.doi.org/10.1007/978-3-642-39765-3
[8]	Doiron, B., Laing, C., Longtin, A. and Maler, L. (2002) Ghostbursting: A Novel Neuronal Burst Mechanism. Journal of Computational Neuroscience, 12, 5-25. http://dx.doi.org/10.1023/A:1014921628797
[9]	Shirahata, T. (2012) Analysis of the Electrosensory Pyramidal Cell Bursting Model for Weakly Electric Fish: Model Prediction under Low Levels of Dendritic Potassium Conductance. Acta Biologica Hungarica, 63, 313-320. http://dx.doi.org/10.1556/ABiol.63.2012.3.1
[10]	Vo, T., Tabak, J., Bertram, R. and Wechselberger, M. (2014) A Geometric Understanding of How Fast Activating Potassium Channels Promote Bursting in Pituitary Cells. Journal of Computational Neuroscience, 36, 259-278. http://dx.doi.org/10.1007/s10827-013-0470-8
[11]	Bianchi, D., Marasco, A., Limongiello, A., Marchetti, C., Marie, H., Tirozzi, B. and Migliore, M. (2012) On the Mechanisms Underlying the Depolarization Block in the Spiking Dynamics of CA1 Pyramidal Neurons. Journal of Computational Neuroscience, 33, 20-225. http://dx.doi.org/10.1007/s10827-012-0383-y
[12]	Atamanyk, J. and Langford, W.F. (2003) A Compartmental Model of Cheyne-Stokes Respiration. In: Ruan, S., Wolkowicz, G.S.K. and Wu, J., Eds., Dynamical Systems and Their Applications in Biology (Fields Institute Communications Volume 36), American Mathematical Society, Providence, 1-16.
[13]	Stiefs, D., Gross, T., Steuer, R. and Feudel, U. (2008) Computation and Visualization of Bifurcation Surfaces. International Journal of Bifurcation and Chaos, 18, 2191-2206. http://dx.doi.org/10.1142/S0218127408021658