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From lamprey to salamander: an exploratory modeling study on the architecture of the spinal locomotor networks in the salamander

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Abstract

The evolutionary transition from water to land required new locomotor modes and corresponding adjustments of the spinal “central pattern generators” for locomotion. Salamanders resemble the first terrestrial tetrapods and represent a key animal for the study of these changes. Based on recent physiological data from salamanders, and previous work on the swimming, limbless lamprey, we present a model of the basic oscillatory network in the salamander spinal cord, the spinal segment. Model neurons are of the Hodgkin–Huxley type. Spinal hemisegments contain sparsely connected excitatory and inhibitory neuron populations, and are coupled to a contralateral hemisegment. The model yields a large range of experimental findings, especially the NMDA-induced oscillations observed in isolated axial hemisegments and segments of the salamander Pleurodeles waltlii. The model reproduces most of the effects of the blockade of AMPA synapses, glycinergic synapses, calcium-activated potassium current, persistent sodium current, and \(h\)-current. Driving segments with a population of brainstem neurons yields fast oscillations in the in vivo swimming frequency range. A minimal modification to the conductances involved in burst-termination yields the slower stepping frequency range. Slow oscillators can impose their frequency on fast oscillators, as is likely the case during gait transitions from swimming to stepping. Our study shows that a lamprey-like network can potentially serve as a building block of axial and limb oscillators for swimming and stepping in salamanders.

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Notes

  1. Note that this limb oscillator is a caricature. Here, we are interested only in the frequency range of oscillations. In the salamander, as in tetrapods in general, the limb CPG architecture might actually comprise many individual oscillators, e.g., for flexor and extensor muscle pairs. The architecture of the limb CPG is a separate and complex topic.

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Acknowledgments

A. B. receives financial supported from the Swiss initiative in systems biology: SystemsX.ch. J.-M.C. receives grants from the European Community (LAMPETRA Grant: FP7-ICT-2007-1-216100) and the Fondation pour la Recherche Médicale (DBC 20101021008). D. R. receives salary support from the Groupe de Recherche sur le Système Nerveux Central (GRSNC) and the Fonds de la Recherche en Santé du Québec (FRSQ). A. J. I. acknowledges support from the European Community (LAMPETRA Grant: FP7-ICT-2007-1-216100). A. B. acknowledges Jeremie Knüsel for critical comments on the manuscript and Daniele Colangelo for IT support; further more Richard Naud, Skander Mensi, Christian Pozzorini and Andrea Prunotto for fruitful discussions.

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Authors and Affiliations

  1. Biorobotics Laboratory, School of Engineering, École Polytechnique Fédérale de Lausanne, Station 14, 1015 , Lausanne, VD, Switzerland

    Andrej Bicanski & Auke Jan Ijspeert

  2. Groupe de Recherche sur le Système Nerveux Central, Département de Physiologie, Université de Montréal, Montréal, QC, H3C 3J7, Canada

    Dimitri Ryczko

  3. INSERM U862—Neurocentre Magendie, Motor System Diseases Team, Université de Bordeaux, 146 rue Leo Saignat, 33077 , Bordeaux Cedex, France

    Jean-Marie Cabelguen

Authors
  1. Andrej Bicanski
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  2. Dimitri Ryczko
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  3. Jean-Marie Cabelguen
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  4. Auke Jan Ijspeert
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Correspondence to Andrej Bicanski.

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This article forms part of a special issue of Biological Cybernetics entitled “Lamprey, Salamander Robots and Central Nervous System”.

Appendix: Equations and parameters

Appendix: Equations and parameters

Neurons are modeled as standard Hodgkin–Huxley neurons (Hodgkin and Huxley 1952) with three electrically coupled compartments, one for the soma, one for the dendrites and an initial compartment reminiscent of the axon hillock in order to transform spike-like membrane oscillations into full amplitude action potentials. The following equation governs the membrane potential U of individual compartments

$$\begin{aligned} C\frac{{\text{ d}}U}{{\text{ d}}t}=\sum \nolimits _i {(U_i -U)} g_\mathrm{core} +\sum \nolimits _j {I_j } +I_\mathrm{leak}, \end{aligned}$$
(1)

where the first sum is taken over the adjacent compartments. The sum over \(j\) indicates summation over compartmental currents. \(I_\mathrm{leak}\) is the leak current. \(C\) represents the capacitance and \(g_\mathrm{core}\) the electrical coupling to the adjacent compartments. The currents are modeled with the standard Hodgkin–Huxley equations of the type

$$\begin{aligned} I_j =g_j p^aq^b(U_i -E_\mathrm{rev}) \end{aligned}$$
(2)

Here \(g_{j}\) is the conductance of the ionic channel, \(p^{a}\) and \(q^{b}\) its activation and inactivation variables with their respective exponents and \(E_\mathrm{rev}\) the reversal potential of the charge carriers in question. The activation and inactivation variables are described either in terms of opening and closing rates \(\alpha \) and \(\beta \). E.g.,

$$\begin{aligned} \frac{{\text{ d}}p}{{\text{ d}}t}=\alpha _\mathrm{p} (U)(1-p)-\beta _\mathrm{p} (U)p, \end{aligned}$$
(3)

or in terms time constants \(\tau _\mathrm{p}\) and asymptotic values \(p_\mathrm{inf}\)

$$\begin{aligned} \frac{{\text{ d}}p}{{\text{ d}}t}=\frac{p_{\inf } -p}{\tau _\mathrm{p}} \end{aligned}$$
(4)

The equations for calculating \(\alpha \) and \(\beta \) are given below. Channels parameters are summarized in Table 2. For calcium concentration-dependent currents the activation is modeled with a concentration-dependent variable \(z\) for each calcium pool (Ca\(_\mathrm{N}\), Ca\(_\mathrm{L}\), Ca\(_\mathrm{NMDA}\) synapse)

$$\begin{aligned} z=\frac{[{\text{ Ca}}]}{B_z }, \end{aligned}$$
(5)

where \(B_{z}\) is a pool-specific parameter (modified from Huss et al. 2007). The concentrations are modeled with an equation of the type

$$\begin{aligned} \frac{{\text{ d}}[Ca]}{{\text{ d}}t}={\text{ AI}}-{\text{ B[Ca]}}, \end{aligned}$$
(6)

where the parameters \(A\) and \(B\) determine the calcium inflow and decay, respectively. When the calcium inflow scaled with strength of the descending drive the following heuristic scaling factor was used

$$\begin{aligned} A(f_\mathrm{RS} )=\frac{f_\mathrm{RS}^{2.4} }{f_\mathrm{RS{-}base}^{2.4} } \end{aligned}$$
(7)

where \(f_\mathrm{RS}\) is the average spiking frequency of the RS population and \(f_\mathrm{RS{-}base}\) the reference value.

Synaptic communication is implemented as follows. Upon detection of a spike the synaptic conductance is raised instantaneously to the value given in Table 1 and then decays exponentially (cf. Table 1). The electrotonic properties of the dendrites attenuate an AMPA EPSP elicited in the dendrite by roughly 40 % in the soma. (EPSP amplitude 0.74 mV in dendrite vs. 0.45 mV in soma).

Channel kinetics were computed with the following equations. Channel parameters (cf. Table 2) are linked to their respective equations by the numbers in their subscripts, which indicate the equation number. Here, \(a(U)\) is a placeholder for the calculated quantity.

$$\begin{aligned} a(U)=\frac{A(U-B)}{1-\exp ((B-U)/C)} \end{aligned}$$
(8)
$$\begin{aligned} a(U)=\frac{A(B-U)}{1-\exp ((U-B)/C)} \end{aligned}$$
(9)
$$\begin{aligned} a(U)=\frac{A}{1+\exp ((B-U)/C)} \end{aligned}$$
(10)
$$\begin{aligned} a(U)=A\exp ((U-B)/C) \end{aligned}$$
(11)
$$\begin{aligned} a(U)=A\exp (-(U-B)/C) \end{aligned}$$
(12)
$$\begin{aligned} a(U)=\frac{1}{1+\exp ((U-B)/C)} \end{aligned}$$
(13)
$$\begin{aligned} a(U)=\frac{1}{1+\exp ((B-U)/C)} \end{aligned}$$
(14)
$$\begin{aligned} a(U)=A+B*\exp (-(C-U)^2/D^2) \end{aligned}$$
(15)

The coupled membrane potential equation was solved using the reverse Euler method [Dayan and Abbott (2001) and references therein] while the simpler equations are solved with a Runge–Kutta 4 implementation.

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Bicanski, A., Ryczko, D., Cabelguen, JM. et al. From lamprey to salamander: an exploratory modeling study on the architecture of the spinal locomotor networks in the salamander. Biol Cybern 107, 565–587 (2013). https://doi.org/10.1007/s00422-012-0538-y

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