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Synaptic patterning of left-right alternation in a computational model of the rodent hindlimb central pattern generator

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Abstract

Establishing, maintaining, and modifying the phase relationships between extensor and flexor muscle groups is essential for central pattern generators in the spinal cord to coordinate the hindlimbs well enough to produce the basic walking rhythm. This paper investigates a simplified computational model for the spinal hindlimb central pattern generator (CPG) that is abstracted from experimental data from the rodent spinal cord. This model produces locomotor-like activity with appropriate phase relationships in which right and left muscle groups alternate while extensor and flexor muscle groups alternate. Convergence to this locomotor pattern is slow, however, and the range of parameter values for which the model produces appropriate output is relatively narrow. We examine these aspects of the model’s coordination of left-right activity through investigation of successively more complicated subnetworks, focusing on the role of the synaptic architecture in shaping motoneuron phasing. We find unexpected sensitivity in the phase response properties of individual neurons in response to stimulation and a need for high levels of both inhibition and excitation to achieve the walking rhythm. In the absence of cross-cord excitation, equal levels of ipsilateral and contralateral inhibition result in a strong preference for hopping over walking. Inhibition alone can produce the walking rhythm, but contralateral inhibition must be much stronger than ipsilateral inhibition. Cross-cord excitatory connections significantly enhance convergence to the walking rhythm, which is achieved most rapidly with strong crossed excitation and greater contralateral than ipsilateral inhibition. We discuss the implications of these results for CPG architectures based on unit burst generators.

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Notes

  1. Equivalently described as simultaneous ipsilateral flexor-extensor antisynchrony and contralateral flexor-extensor synchrony.

  2. For the networks examined in this paper, the period of the burst cycle of each coupled RGN stabilized within a few cycles (typically fewer than four) to a value close (within 0.1–5%) to the period of the uncoupled RGN burst cycle, regardless of whether the phase differences between component neurons converged. The period T here can be taken as the period of the individual (uncoupled) reference burst or as the (stable) period of the coupled burst without any qualitative difference in the results, and only a very slight quantitative difference in the measured phase offsets. The results in the next section use the coupled burst period.

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Acknowledgements

We thank Ole Kiehn for valuable discussions during the course of the research and useful comments on earlier versions of this paper. This work was supported by NIH CRCNS grant 1R01NS050943, DOE grant DE-FG02-93ER25164, and NSF FIBR grant 0425878 subcontract SA4554-10295PG.

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Correspondence to William Erik Sherwood.

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Appendix: CPG model equations

Appendix: CPG model equations

Membrane voltage for model RGN cells is determined by spiking Na +  and K +  currents, a persistent sodium current, and a leak current:

$$ \dot{V} = -(I_{\mathrm{Na}} + I_{\mathrm{K}} + I_{\mathrm{NaP}} + I_{\mathrm{L}} - I_\mathrm{app})/C $$
(4)

while the equation governing MN and CIN cell membrane voltage omits the persistent sodium current:

$$ \dot{V} = -(I_{\mathrm{Na}} + I_{\mathrm{K}} + I_{\mathrm{L}} - I_\mathrm{app})/C $$
(5)

Membrane currents have the forms

$$ I_{\mathrm{Na}} = g_{\mathrm{Na}} m^3_{\infty}(V)(1-n)(V-V_{\mathrm{Na}}) $$
(6)
$$ I_{\mathrm{K}} = g_{\mathrm{K}} n^4 (V-V_{\mathrm{K}}) $$
(7)
$$ I_{\mathrm{NaP}} = g_{\mathrm{NaP}} m_{{\mathrm{NaP}_\infty}}(V)h(V-V_{\mathrm{Na}}) $$
(8)
$$ I_{\mathrm{L}} = g_{\mathrm{L}} (V-V_{\mathrm{L}}) $$
(9)

and dynamic activation variables are governed by equations of the form

$$ \dot{y} = (y_{\infty}(V) -y)/\tau_{y}(V) $$
(10)

for y ∈ {h, n}.

Membrane voltage-dependent steady state equations for the (in)activation of the various currents have the form

$$ x_{\infty}(V) = (1+\exp( (V-\theta_{x}) /k_{x}))^{-1} $$
(11)

and the equations for voltage-dependent time constants have the form

$$ \tau_{y}(V) = \overline{\tau}_{y}/\cosh [(V - \theta_{y})/(2 k_{y})] $$
(12)

where x ∈ {m, m NaP, h, n} and y ∈ {h, n}.

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Sherwood, W.E., Harris-Warrick, R. & Guckenheimer, J. Synaptic patterning of left-right alternation in a computational model of the rodent hindlimb central pattern generator. J Comput Neurosci 30, 323–360 (2011). https://doi.org/10.1007/s10827-010-0259-y

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