Asymmetric electrotonic coupling between the soma and dendrites alters the bistable firing behaviour of reduced models

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.

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):

$$ {C_m}\mathop {{{V_S}}}\limits^\bullet = - {G_{Na}}{m_{S\infty }}\left( {{V_S} - {E_{Na}}} \right) - {G_{K,S}}{n_S}\left( {{V_S} - {E_K}} \right) - {G_{m,S}}\left( {{V_S} - {E_{Leak}}} \right) - \frac{{{G_C}}}{p}\left( {{V_S} - {V_D}} \right) + {I_S} $$

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$$ {m_{S\infty }}\left( {{V_S}} \right) = 0.5\left[ {1 + \tanh \left\{ {\left( {{V_S} - {v_{1S}}} \right)/{v_{2S}}} \right\}} \right] $$

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$$ \mathop {{{n_S}}}\limits^\bullet = {\phi_S}\left\{ {{n_{S\infty }}\left( {{V_S}} \right) - {n_S}} \right\}/{\tau_S}\left( {{V_S}} \right) $$

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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):

$$ {C_m}\mathop {{{V_D}}}\limits^\bullet = - {G_{Ca}}{m_{D\infty }}\left( {{V_D} - {E_{Ca}}} \right) - {G_{K,D}}{n_D}\left( {{V_D} - {E_K}} \right) - {G_{m,D}}\left( {{V_D} - {E_{Leak}}} \right) - \frac{{{G_C}}}{{1 - p}}\left( {{V_D} - {V_S}} \right) $$

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$$ {m_{D\infty }}\left( {{V_D}} \right) = 0.5\left[ {1 + \tanh \left\{ {\left( {{V_D} - {v_{1D}}} \right)/{v_{2D}}} \right\}} \right] $$

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$$ \mathop {{{n_D}}}\limits^\bullet = {\phi_D}\left\{ {{n_{D\infty }}\left( {{V_D}} \right) - {n_D}} \right\}/{\tau_D}\left( {{V_D}} \right), $$

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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

A_SD

voltage Attenuation factor from Soma to Dendrites

A_DS

voltage Attenuation factor from Dendrites to Soma

PIC

Persistent Inward Current

CI

Characteristic Index

TTP

Time To onset of Plateau potential

TES

Time to End of Somatic spiking

DSF

Difference in Spiking Frequency