Abstract
In this article, physiological and pathological forms of excitability are studied in a two-dimensional electrical model of excitable cell endowed with a generic inward persistent conductance. Bifurcation analysis of the model is performed as a function of the maximal inward persistent conductance, the input current, or the voltage dependency of the activation function. Several discharge modes are exhibited, including: (1) a basic mode that corresponds to a resting potential and production of action potential; (2) bistability between resting potential and self-sustained spiking; (3) a pacemaker mode of discharge; and (4) bistability between resting potential and plateau potential. These behaviours can be compared to experimentally described physiological and pathological forms of excitability that depend upon inward persistent conductances. In the results obtained, attractors allow for a qualitative description of physiological and pathological states. However, it is not possible to obtain an unambiguous identification of particular 'physiological attractors' or 'pathological attractors'. In the perspective of the theory of dynamical systems, we suggest that pathological states can be modelled in two different ways, i.e. by bifurcation (as in the present model) or by perturbation. We also highlight some other theoretical concepts that may be relevant to a theoretical description of pathology.
Similar content being viewed by others
REFERENCES
Arvanitaki, A. (1939). Les variations graduées de la polarisation des systèmes excitables, Paris: Hermann.
Av-Ron E., H. Parnas and L.A. Segel (1991). A minimal biophysical model for an excitable and oscillatory neuron. Biological Cybernetics 65: 487-500.
Bargas, J. and E. Galarraga (1995). Ion channels: keys to neuronal specialization. In Arbib A. M. The handbook of brain theory and neural networks. The MIT Press, Cambridge, Mass.
Bennet, P.B., K. Yazawa, N. Makita, A.L. George Jr (1995). Molecular mechanism for an inherited cardiac arrhythmia. Nature 376: 683-685.
Camperi, M. and X.J. Wang (1998). A model of visuospatial working memory in prefrontal cortex: recurrent network and cellular bistability. The Journal of Computational Neurosciences, in press.
Cannon, S.C. (1996). Sodium channel defects in myotonia and periodic paralysis. Annual Review of Neuroscience 19: 141-164.
Cannon, S.C. and D.P. Corey (1993). Loss of sodium channel inactivation by Anemone toxin (ATX II) mimics the myotonic state in hyperkalemic periodic paralysis. The Journal of Physiology 466: 501-520.
Cannon, S.C., R.H. Brown and D.P. Corey (1993). Theoretical reconstruction of myotonia and paralysis caused by incomplete inactivation of sodium channels. Biophysical Journal 65: 270-288.
Connors, B.W., M.J. Gutnick and D.A. Prince (1982). Electrophysiological properties of neocortical neurones in vitro. The Journal of Neurophysiology 48(6): 1302-1320.
Conway, B.A., H. Hultborn, O. Kiehn and I. Mintz (1988). Plateau potentials in-motoneurones induced by intravenous injection of L-DOPA and clonidine in the spinal cat. The Journal of Physiology 405: 369-384.
Cowan, R.L. and C.J. Wilson (1994). Spontaneous firing patterns and axonal projections of single corticostriatal neurones in the rat medial agranular cortex. The Journal of Neurophysiology 71(1): 17-32.
Delord, B., A. Klaassen, Y. Burnod, R. Costalat and E. Guigon (1997). Bistable behaviour in a neocortical neuron model. Neuroreport 8: 1019-1023.
Dickson, C.T., A.R. Mena and A. Alonsa (1997). Electroresponsiveness of medial enthorinal cortex layer III neurones in vitro. Neuroscience 81(4): 937-950.
FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. The Biophysical Journal 1: 455-466.
French, C.R., P. Sah, K.J. Buckett and P.W. Gage (1990). A voltage-dependent persistent sodium current in mammalian hippocampal neurones. The Journal of General Physiology 95: 1139-1157.
Goldman-Rakic, P.S. and L.D. Selemon (1997). Functional and anatomical aspects of prefrontal cortex pathology in schizophrenia. Schizophrenia Bulletin 23(3): 437-458.
Guigon, E., B. Dorizzi, Y. Burnod and W. Schultz. (1995). Neural correlates of learning in the prefrontal cortex of the monkey: a predictive model. Cerebra Cortex 5: 135-147.
Hodgkin, A.L. (1948). The local electric changes associated with repetitive action in a non-medulated axon. The Journal of Physiology 107: 165-181.
Hounsgaard, J., H. Hultborn, B. Jespersen and O. Kiehn (1984). Intrinsic membrane properties causing a bistable behaviour of a-motoneurons. Experimental Brain Research 55: 391-394.
Joos, G. and D.D. Joseph (1990). Elementary Stability and Bifurcation Theory. 2nd edition. Springer-Verlag, New-York, Berlin, Heidelberg.
Klink, R.M. and A. Alonso (1993). Ionic mechanisms for the subthreshold oscillations and differential electroresponsiveness of medial entorhinal cortex layer II neurones. The Journal of Neurophysiology 70: 144-157.
Kuznetsov, Y.A. (1995). Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, 112. Springer-Verlag. New York, Berlin, Heidelberg.
Lampl, I., P. Schwindt and W. Crill (1998). Reduction of cortical pyramidal neuron excitability by the action of phsnytoin on persistent Na+ current. The Journal of Pharmacology and Experimental Therapeutics 284(1): 228-237
Lavin, A. and A.A. Grace (1998). Dopamine modulates the hyperpolarized state of prefrontal cortical neurones. Society of Neurosciences Abstracts 139.12: 351.
Lipowsky, R., T. Gillessen and C. Alzheimer (1996). Dendritic Na+ channels amplify EPSPs in hippocampal CA1 pyramidal cells. The Journal of Neurophysiology 76(4): 2181-2191
Llinàs, R.R. (1988). The intrinsic electrophysiological properties of mammalian neurones: insights into central nervous function. Science 242: 1654-1664.
Nagumo, J.S., S. Arimoto and S. Yoshizawa (1962). An active pulse transmission line simulating nerve axon. Proceedings of the Institute of Radio Engineers 50: 2061-2070.
Nielsen, J. and H. Hultborn (1993). Regulated properties of motoneurons and primary afferents: new possible spinal mechanisms underlying spasticity. In: Spasticity: mechanisms and management (Eds eiThilmann A.F., Burke D.J., Rymer W.Z.) Springer-Verlag.
Okubo, Y., T. Suhara, K. Suzuki, K. Kobayashi, O. Inoue, O. Terasaki, Y. Someya, T. Sassa, Y. Sudo, E. Matsushima, M. Iyo, Y. Tateno and M. Toru (1997). Decreased prefrontal dopamine D1 receptors in schizophrenia revealed by PET. Nature 385: 634-636.
Rinzel, J. (1985). Excitation dynamics: insights from simplified membrane models. Federation proceedings 44(15): 2944-2946.
Rinzel, J. (1990). Electrical excitability of cells, theory and experiment: review of the Hodgkin-Huxley foundation and an update. The Bulletin of Mathematical Biology 52(1): 5-23.
Rinzel, J. and G.B. Ermentrout (1989). Analysis of excitability and oscillations. In Koch C., Segev I. (eds). Methods in neuronal modeling, The MIT Press, Cambridge, Mass.
Tunnicliff, G. (1996). Basis of antiseizure action of phenytoin. General Pharmacology 27(7): 1091-1097.
Turrigiano, G., L.F. Abbot and E. Marder (1994). Activity-dependent changes in the intrinsic properties of cultured neurones. Science 264: 974-977.
Wang, X.J. and J. Rinzel (1995). Oscillatory and bursting properties of neurones. In Arbib A. M. The handbook of brain theory and neural networks. The MIT Press, Cambridge, Mass.
Wang, L. and I. Ross (1990). Synchronous neural networks of nonlinear threshold elements with hysteresis. Proceedings of the National Academy of Sciences 87: 988-992.
Yang, C.R. and K.J. Seamans (1996). Dopamine D1 receptors actions in layers V-VI rat prefrontal cortex neurones in vitro: modulation of dendritic-somatic signal integration. The Journal of Neuroscience 16(5): 1922-1935.
Yang, C.R., J.K. Seamans and N. Gorelova (1996). Electrophysiological and morphological properties of layers V-VI principal pyramidal cells in rat prefrontal cortex in vitro. The Journal of Neuroscience 16(5): 1904-1921.
Yuen, G.L., P.E. Hockherger and J.C. Houk (1995). Bistability in cerebellar purkinje cell dendrites modelled with high-threhold calcium and delayed-rectifier potassium channels. Biological Cybernetics 73(4): 375-388.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Delord, B. Attractors and Pathological Aspects in Excitable Cells. Acta Biotheor 47, 239–252 (1999). https://doi.org/10.1023/A:1002694822715
Issue Date:
DOI: https://doi.org/10.1023/A:1002694822715