Abstract
The goal of the study was to investigate the influence of asymmetric coupling, between the soma and dendrites, on the nonlinear dynamic behaviour of a two-compartment model. We used a recently published method for generating reduced two-compartment models that retain the asymmetric coupling of anatomically reconstructed motor neurons. The passive input-output relationship of the asymmetrically coupled model was analytically compared to the symmetrically coupled case. Predictions based on the analytic comparison were tested using numerical simulations. The simulations evaluated the nonlinear dynamics of the models as a function of coupling parameters. Analytical results showed that the input resistance at the dendrite of the asymmetric model was directly related to the degree of coupling asymmetry. In contrast, a comparable symmetric model had identical input resistances at both the soma and dendrite regardless of coupling strength. These findings lead to predictions that variations in dendritic excitability, subsequent to changes in input resistance, might change the current threshold and onset timing of the plateau potential generated in the dendrite. Since the plateau potential underlies bistable firing, these results further predicted that asymmetric coupling might alter nonlinear (i.e. bistable) firing patterns. The numerical simulations supported analytical predictions, showing that the fully bistable firing pattern of the asymmetric model depended on the degree of coupling asymmetry and its correlated dendritic excitability. The physiological property of asymmetric coupling plays an important role in generating and stabilizing the bistability of motor neurons by interacting with the excitability of dendritic branches.
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References
Ballou, E. W., Smith, W. B., et al. (2006). Measuring dendritic distribution of membrane proteins. Journal of Neuroscience Methods, 156(1–2), 257–266.
Bennett, D. J., Hultborn, H., et al. (1998). Synaptic activation of plateaus in hindlimb motoneurons of decerebrate cats. Journal of Neurophysiology, 80(4), 2023–2037.
Bennett, D. J., Li, Y., et al. (2001). Plateau potentials in sacrocaudal motoneurons of chronic spinal rats, recorded in vitro. Journal of Neurophysiology, 86(4), 1955–1971.
Booth, V., & Rinzel, J. (1995). A minimal, compartmental model for a dendritic origin of bistability of motoneuron firing patterns. Journal of Computational Neuroscience, 2(4), 299–312.
Booth, V., Rinzel, J., et al. (1997). Compartmental model of vertebrate motoneurons for Ca2+-dependent spiking and plateau potentials under pharmacological treatment. Journal of Neurophysiology, 78(6), 3371–3385.
Brunel, N. (2003). Dynamics and plasticity of stimulus-selective persistent activity in cortical network models. Cerebral Cortex, 13(11), 1151–1161.
Bui, T. V., Ter-Mikaelian, M., et al. (2006). Computational estimation of the distribution of L-type Ca(2+) channels in motoneurons based on variable threshold of activation of persistent inward currents. Journal of Neurophysiology, 95(1), 225–241.
Carlin, K. P., Jiang, Z., et al. (2000). Characterization of calcium currents in functionally mature mouse spinal motoneurons. The European Journal of Neuroscience, 12(5), 1624–1634.
Carlin, K. P., Jones, K. E., et al. (2000). Dendritic L-type calcium currents in mouse spinal motoneurons: implications for bistability. The European Journal of Neuroscience, 12(5), 1635–1646.
Carlin, K. P., Bui, T. V., et al. (2009). Staircase currents in motoneurons: insight into the spatial arrangement of calcium channels in the dendritic tree. The Journal of Neuroscience, 29(16), 5343–5353.
Carnevale, N. T., & Johnston, D. (1982). Electrophysiological characterization of remote chemical synapses. Journal of Neurophysiology, 47(4), 606–621.
Cotel, F., Antri, M., et al. (2009). Identified ankle extensor and flexor motoneurons display different firing profiles in the neonatal rat. The Journal of Neuroscience, 29(9), 2748–2753.
Cullheim, S., Fleshman, J. W., et al. (1987a). Membrane area and dendritic structure in type-identified triceps surae alpha motoneurons. The Journal of Comparative Neurology, 255(1), 68–81.
Cullheim, S., Fleshman, J. W., et al. (1987b). Three-dimensional architecture of dendritic trees in type-identified alpha-motoneurons. The Journal of Comparative Neurology, 255(1), 82–96.
Dechter, R. (2003). Constraint processing (p. 481). San Francisco: Morgan Kaufmann Publishers.
Doiron, B., Laing, C., et al. (2002). Ghostbursting: a novel neuronal burst mechanism. Journal of Computational Neuroscience, 12(1), 5–25.
Donohue, D. E., & Ascoli, G. A. (2008). A comparative computer simulation of dendritic morphology. PLoS Computational Biology, 4(5), e1000089.
Egorov, A. V., Hamam, B. N., et al. (2002). Graded persistent activity in entorhinal cortex neurons. Nature, 420(6912), 173–178.
Eken, T., & Kiehn, O. (1989). Bistable firing properties of soleus motor units in unrestrained rats. Acta Physiologica Scandinavica, 136(3), 383–394.
Elbasiouny, S. M., Bennett, D. J., et al. (2005). Simulation of dendritic CaV1.3 channels in cat lumbar motoneurons: spatial distribution. Journal of Neurophysiology, 94(6), 3961–3974.
Grande, G., Bui, T. V., et al. (2007). Estimates of the location of L-type Ca2+ channels in motoneurons of different size: a computational study. Journal of Neurophysiology, 97, 4023–4035.
Gutman, A. (1991). Bistability of dendrites. International Journal of Neural Systems, 1, 291–304.
Hausser, M., Spruston, N., et al. (2000). Diversity and dynamics of dendritic signaling. Science, 290(5492), 739–744.
Heckman, C. J., Lee, R. H., et al. (2003). Hyperexcitable dendrites in motoneurons and their neuromodulatory control during motor behavior. Trends in Neurosciences, 26(12), 688–695.
Holmes, W. R., & Rall, W. (1992). Estimating the electrotonic structure of neurons with compartmental models. Journal of Neurophysiology, 68(4), 1438–1452.
Hounsgaard, J., & Mintz, I. (1988). Calcium conductance and firing properties of spinal motoneurones in the turtle. The Journal of Physiology, 398, 591–603.
Hounsgaard, J., & Kiehn, O. (1989). Serotonin-induced bistability of turtle motoneurones caused by a nifedipine-sensitive calcium plateau potential. The Journal of Physiology, 414, 265–282.
Hounsgaard, J., & Kiehn, O. (1993). Calcium spikes and calcium plateaux evoked by differential polarization in dendrites of turtle motoneurones in vitro. The Journal of Physiology, 468, 245–259.
Hounsgaard, J., Hultborn, H., et al. (1988). Bistability of alpha-motoneurones in the decerebrate cat and in the acute spinal cat after intravenous 5-hydroxytryptophan. The Journal of Physiology, 405, 345–367.
Jaffe, D. B., & Carnevale, N. T. (1999). Passive normalization of synaptic integration influenced by dendritic architecture. Journal of Neurophysiology, 82(6), 3268–3285.
Jones, K. E., Carlin, K. P., Rempel, J., et al. (2000). Simulation techniques for localising and identifying the kinetics of calcium channels in dendritic neurons. Neurocomputing, 32, 173–180.
Katz, B., & Miledi, R. (1963). A Study of Spontaneous Miniature Potentials in Spinal Motoneurones. Journal de Physiologie, 168, 389–422.
Kiehn, O. (1991). Plateau potentials and active integration in the ‘final common pathway’ for motor behaviour. Trends in Neurosciences, 14(2), 68–73.
Kiehn, O., & Eken, T. (1997). Prolonged firing in motor units: evidence of plateau potentials in human motoneurons? Journal of Neurophysiology, 78(6), 3061–3068.
Kim, H., Major, L. A., et al. (2009). Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites. Journal of Computational Neuroscience, 27(3), 321–336.
Larkum, M. E., Rioult, M. G., et al. (1996). Propagation of action potentials in the dendrites of neurons from rat spinal cord slice cultures. Journal of Neurophysiology, 75(1), 154–170.
Larkum, M. E., Zhu, J. J., et al. (1999). A new cellular mechanism for coupling inputs arriving at different cortical layers. Nature, 398(6725), 338–341.
Lee, R. H., & Heckman, C. J. (1996). Influence of voltage-sensitive dendritic conductances on bistable firing and effective synaptic current in cat spinal motoneurons in vivo. Journal of Neurophysiology, 76(3), 2107–2110.
Lee, R. H., & Heckman, C. J. (1998a). Bistability in spinal motoneurons in vivo: systematic variations in rhythmic firing patterns. Journal of Neurophysiology, 80(2), 572–582.
Lee, R. H., & Heckman, C. J. (1998b). Bistability in spinal motoneurons in vivo: systematic variations in persistent inward currents. Journal of Neurophysiology, 80(2), 583–593.
Lee, R. H., & Heckman, C. J. (1999). Paradoxical effect of QX-314 on persistent inward currents and bistable behavior in spinal motoneurons in vivo. Journal of Neurophysiology, 82(5), 2518–2527.
Lee, R. H., & Heckman, C. J. (2000). Adjustable amplification of synaptic input in the dendrites of spinal motoneurons in vivo. The Journal of Neuroscience, 20(17), 6734–6740.
Li, Y., & Bennett, D. J. (2003). Persistent sodium and calcium currents cause plateau potentials in motoneurons of chronic spinal rats. Journal of Neurophysiology, 90(2), 857–869.
Li, W. C., Soffe, S. R., et al. (2006). Persistent responses to brief stimuli: feedback excitation among brainstem neurons. The Journal of Neuroscience, 26(15), 4026–4035.
Llinas, R., & Sugimori, M. (1980). Electrophysiological properties of in vivo Purkinje cell somata in mammalian cerebellar slices. Journal de Physiologie, 305, 171–195.
Mainen, Z. F., & Sejnowski, T. J. (1996). Influence of dendritic structure on firing pattern in model neocortical neurons. Nature, 382(6589), 363–366.
Major, G., Evans, J. D., et al. (1993). Solutions for transients in arbitrarily branching cables: I. Voltage recording with a somatic shunt. Biophysical Journal, 65(1), 423–449.
Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35(1), 193–213.
Pinsky, P. F., & Rinzel, J. (1994). Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons. Journal of Computational Neuroscience, 1(1–2), 39–60.
Rall, W., & Rinzel, J. (1973). Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model. Biophysical Journal, 13(7), 648–687.
Rinzel, J., & Ermentrout, B. (1998). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling: From ions to networks (pp. 251–291). Cambridge, MA: MIT Press.
Rinzel, J., & Rall, W. (1974). Transient response in a dendritic neuron model for current injected at one branch. Biophysical Journal, 14(10), 759–790.
Schwindt, P. C., & Crill, W. E. (1980). Properties of a persistent inward current in normal and TEA-injected motoneurons. Journal of Neurophysiology, 43(6), 1700–1724.
Simon, M., Perrier, J. F., et al. (2003). Subcellular distribution of L-type Ca2+ channels responsible for plateau potentials in motoneurons from the lumbar spinal cord of the turtle. The European Journal of Neuroscience, 18(2), 258–266.
Steriade, M. (1999). Coherent oscillations and short-term plasticity in corticothalamic networks. Trends in Neurosciences, 22(8), 337–345.
Stuart, G., Spruston, N., et al. (1997). Action potential initiation and backpropagation in neurons of the mammalian CNS. Trends in Neurosciences, 20(3), 125–131.
Thurbon, D., Luscher, H. R., et al. (1998). Passive electrical properties of ventral horn neurons in rat spinal cord slices. Journal of Neurophysiology, 80(1), 2485–2502.
Tsai, K. Y., Carnevale, N. T., et al. (1994). Efficient mapping from neuroanatomical to electrotonic space. Network: Computation in Neural Systems, 5(1), 21–46.
Zengel, J. E., Reid, S. A., et al. (1985). Membrane electrical properties and prediction of motor-unit type of medial gastrocnemius motoneurons in the cat. Journal of Neurophysiology, 53(5), 1323–1344.
Acknowledgements
The study was supported by the Natural Sciences and Engineering Research Council of Canada [NSERC] with salary support for KEJ from the Alberta Heritage Foundation for Medical Research [AHFMR].
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Appendix
Appendix
The system equations of the asymmetric two-compartment model in Fig. 1 were derived based on the previous dimensionless reduced model (Booth and Rinzel 1995), in which Morris-Lecar membrane excitability (Morris and Lecar 1981) was employed to produce bistable firing patterns.
The membrane potential at the somatic compartment, V S (t):
where \( {n_{S\infty }}\left( {{V_S}} \right) = 0.5\left[ {1 + \tanh \left\{ {\left( {{V_S} - {v_{3S}}} \right)/{v_{4S}}} \right\}} \right] \) and \( {\tau_S}\left( {{V_S}} \right) = \left[ {\cosh \left\{ {\left( {{V_S} - {v_{3S}}} \right)/\left( {2{v_{4S}}} \right)} \right\}} \right]{^{ - 1}} \)
The membrane potential at the dendritic compartment, V D (t):
where \( {n_{D\infty }}\left( {{V_D}} \right) = 0.5\left[ {1 + \tanh \left\{ {\left( {{V_D} - {v_{3D}}} \right)/{v_{4D}}} \right\}} \right] \) and \( {\tau_D}\left( {{V_D}} \right) = \left[ {\cosh \left\{ {\left( {{V_D} - {v_{3D}}} \right)/\left( {2{v_{4D}}} \right)} \right\}} \right]{^{ - 1}} \)
Regular firing was mediated by lumped inward (G Na ∙m S∞ ) and outward (G K,S ∙n S ) conductances at the somatic compartment. Similarly the activation of plateau potential was regulated by lumped inward (G Ca ∙m D∞ ) and outward (G K,D ∙n D ) conductances at the dendritic compartment. Definitions and standard values of membrane parameters in the system equations were provided in Table 1.
Glossary
- DDVA
-
Direction Dependant Voltage Attenuation
- ASD
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voltage Attenuation factor from Soma to Dendrites
- ADS
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voltage Attenuation factor from Dendrites to Soma
- PIC
-
Persistent Inward Current
- CI
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Characteristic Index
- TTP
-
Time To onset of Plateau potential
- TES
-
Time to End of Somatic spiking
- DSF
-
Difference in Spiking Frequency
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Kim, H., Jones, K.E. Asymmetric electrotonic coupling between the soma and dendrites alters the bistable firing behaviour of reduced models. J Comput Neurosci 30, 659–674 (2011). https://doi.org/10.1007/s10827-010-0284-x
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DOI: https://doi.org/10.1007/s10827-010-0284-x