Modeling the Organization of Spinal Cord Neural Circuits Controlling Two-Joint Muscles

Abstract

The activity of most motoneurons controlling one-joint muscles during locomotion are locked to either extensor or flexor phase of locomotion. In contrast, bifunctional motoneurons, controlling two-joint muscles such as posterior biceps femoris and semitendinosus (PBSt) or rectus femoris (RF), express a variety of activity patterns including firing bursts during both locomotor phases, which may depend on locomotor conditions. Although afferent feedback and supraspinal inputs significantly contribute to shaping the activity of PBSt and RF motoneurons during real locomotion, these motoneurons show complex firing patterns and variable behaviors under the conditions of fictive locomotion in the immobilized decerebrate cat, i.e., with a lack of patterned supraspinal and afferent inputs. This suggests that firing patterns of PBSt and RF motoneurons are defined by neural interactions inherent to the locomotor central pattern generator (CPG) within the spinal cord. In this study, we use computational modeling to suggest the architecture of spinal circuits representing the locomotor CPG and the connectivity pattern of spinal interneurons defining the behavior of bifunctional PBSt and RF motoneurons. The proposed model reproduces the complex firing patterns of these motoneurons during fictive locomotion under different conditions including spontaneous deletions of flexor and extensor activities and provides insights into the organization of spinal circuits controlling locomotion in mammals.

Appendix

All neurons were modelled in the Hodgkin-Huxley style. Motoneurons had two compartments: soma and dendrite and were described based on the previous models (Booth et al. 1997; Rybak et al. 2006a) . The membrane potentials of motoneuron soma (V _(S) ) and dendrite (V _(D) ) obey the following differential equations:

$$ \begin{aligned}& C\times \frac{d{{V}_{(S)}}}{dt}=-{{I}_{Na(S)}}-{{I}_{K(S)}}-{{I}_{A(S)}}-{{I}_{CaN(S)}}-{{I}_{K,Ca(S)}}-{{I}_{L(S)}}-{{I}_{C(S)}}\; \\& C\times \frac{d{{V}_{(D)}}}{dt}=-{{I}_{NaP(D)}}-{{I}_{CaN(D)}}-{{I}_{CaL(D)}}-{{I}_{K,Ca(D)}}-{{I}_{L(D)}}-{{I}_{C(D)}}-{{I}_{SynE}}-{{I}_{SynI}}\, \end{aligned} $$

(5.A1)

where C is the membrane capacitance and t is time (C = 1 μF/cm²), subscripts S and D indicate the soma or dendrite compartments, respectively.

The dendrite-soma coupling currents (with conductance g _C ) for soma I _C(S) and dendrite I _C(D) are described as follows:

$$ \begin{aligned}& {{I}_{C(S)}}=\frac{{{g}_{C}}}{p}\times ({{V}_{(S)}}-{{V}_{(D)}}); \\& {{I}_{C(D)}}=\frac{{{g}_{C}}}{1-p}\times ({{V}_{(D)}}-{{V}_{(S)}}),\end{aligned} $$

(5.A2)

where p is the parameter defining the ratio of somatic surface area to total surface area (p = 0.1); g _C  = 0.1 mS/cm².

The following currents (with the corresponding maximal channel conductances) are included into motoneuron soma compartments (Booth et al. 1997; Rybak et al. 2006a): fast sodium, I _Na (maximal conductance ${{\bar{g}}_{Na}}$ = 120 mS/cm²); persistent sodium, I _K ( = ${{\bar{g}}_{NaP}}$ 100 mS/cm²); calcium-N, I _CaN (${{\bar{g}}_{CaN}}$ = 14 mS/cm²); calcium-dependent potassium, I _{K, Ca} (${{\bar{g}}_{CaL}}$ = 2 mS/cm²), and leakage, I _L (g _L  = 0.51 mS/cm²) currents. In addition, based on evidence of the presence of the transient (rapidly inactivating) potassium current in the spinal cord interneurons and motoneurons (Safronov and Vogel 1995) this current has been also included in our motoneuron models (I _A with maximal conductance ${{\bar{g}}_{A}}$ = 200 ± 40 mS/cm²). The following currents (with the corresponding maximal channel conductances) are included into motoneuron dendritic compartment: persistent sodium, I _NaP (${{\bar{g}}_{NaP}}$ = 0.1 mS/cm²); calcium-N, I _CaN (${{\bar{g}}_{CaN}}$ = 0.3 mS/cm²); calcium-L (I _CaL , ${{\bar{g}}_{CaL}}$ = 0.33 mS/cm²), calcium-dependent potassium, I _{K, Ca} (${{\bar{g}}_{K,Ca}}$ = 0.8 mS/cm²), and leakage, I _L (g _L  = 0.51 mS/cm²) currents.

Interneurons are simulated as single-compartment models. The neurons within the RG-F, RG-E, PF-F, PF-E, In-eF, and In-eE populations contain fast sodium, I _Na ; persistent sodium, I _NaP ; delayed-rectifier potassium, I _K ; and leakage, I _L currents:

$$C \times \frac{{dV}}{{dt}} = - {I_{Na}} - {I_K} - {I_L} - {I_{SynE}} - {I_{SynI}}{\kern 1pt} $$

(5.A3)

The maximal channel conductances for neurons in theses populations are as follows: g _L  = 0.51 mS/cm²; ${{\bar{g}}_{Na}}$ = 150 mS/cm² in RG-F and RG-E neurons and 120 mS/cm² in the PF-F, PF-E, In-eF, and In-eE populations; ${{\bar{g}}_{NaP}}$ = 1.25 mS/cm² in RG-F, RG-E, In-eF, and In-eF neurons and 0.1 mS/cm² in the PF-F and PF-E populations; ${{\bar{g}}_{K}}$ = 5 mS/cm² in the RG-F and RG-E populations and 10 mS/cm² in the PF-F, PF-E, In-eE, and In-eF populations.

For simplicity, all other interneurons contain only minimal set of ionic currents:

$$ C\times \frac{dV}{dt}=-{{I}_{Na}}-{{I}_{K}}-{{I}_{L}}-{{I}_{SynE}}-{{I}_{SynI}}$$

(5.A4)

with the following maximal conductances: ${{\bar{g}}_{Na}}$ = 120 mS/cm²; ${{\bar{g}}_{K}}$ = 10 mS/cm²; g _L  = 0.51 mS/cm².

The ionic currents included into the modelled neurons are described as follows:

$$ \begin{aligned}& {{I}_{Na}}={{{\bar{g}}}_{Na}}\times m_{Na}^{3}\times {{h}_{Na}}\times ({{V}_{{}}}-{{E}_{Na}}); \\& {{I}_{NaP}}={{{\bar{g}}}_{NaP}}\times {{m}_{NaP}}\times {{h}_{NaP}}\times (V-{{E}_{Na}}); \\& {{I}_{K}}={{{\bar{g}}}_{K}}\times m_{K}^{4}\times ({{V}_{{}}}-{{E}_{K}}); \\& {{I}_{A}}={{{\bar{g}}}_{A}}\times (0.6\times m_{A1}^{4}\times {{h}_{A1}}+0.4\times m_{A2}^{4}\times {{h}_{A2}})\times (V-{{E}_{K}}); \\& {{I}_{CaN}}={{{\bar{g}}}_{CaN}}\times m_{CaN}^{2}\times {{h}_{CaN}}\times ({{V}_{{}}}-{{E}_{Ca}}); \\& {{I}_{CaL}}={{{\bar{g}}}_{CaL}}\times {{m}_{CaL}}\times (V-{{E}_{Ca}}); \\& {{I}_{K,Ca}}={{{\bar{g}}}_{K,Ca}}\times m_{K,Ca}^{{}}\times ({{V}_{{}}}-{{E}_{K}}); \\& {{I}_{L}}={{g}_{L}}\times ({{V}_{{}}}-{{E}_{L}}),\end{aligned} $$

(5.A5)

where V is the membrane potential of the corresponding neuron compartment (soma, V _(S) , or dendrite, V _(D) ) in two-compartment models, or the neuron membrane potential V in one-compartment models; E _Na , E _K , E _Ca , and E _L are the reversal potentials for sodium, potassium, calcium, and leakage current respectively; variables m and h with indexes indicating ionic currents represent, respectively, the activation and inactivation variables of the corresponding ionic channels.

The reversal potential values in the model are as follows: E _Na  = 55 mV; E _K= − 80 mV; E _Ca= 80 mV; E _L  = − 64 ± 0.64 mV in RG-F and RG-E neurons, E _L  = − 65 ± 0.325 mV in Inrg-F and Inrg-E interneurons and motoneurons, and E _L  = − 68 ± 0.34 mV in all other neurons.

Activation m and inactivation h of voltage-dependent ionic channels (e.g., Na, NaP, K, A, CaN, CaL) are described by the following differential equations:

$$\begin{aligned}& {{\tau }_{mi}}(V)\times \frac{d}{dt}{{m}_{i}}={{m}_{\infty i}}(V)-{{m}_{i}}; \\ & {{\tau }_{hi}}(V)\times \frac{d}{dt}{{h}_{i}}={{h}_{\infty i}}(V)-h, \\ \end{aligned}$$

(5.A6)

where i identifies the name of the channel, m _∞i (V) and h _∞i (V) represent the voltage-dependent steady-state activation and inactivation respectively, and τ _mi (V) and τ _hi (V) define the corresponding time constants (see their descriptions in Table 5.A1). Activation of the sodium channels is considered to be instantaneous (τ _mNa  = τ _mNaP  = 0, see (Booth et al. 1997; Butera et al. 1999)) .

Activation of the Ca²⁺ -dependent potassium channels is also considered instantaneous and described as follows (Booth et al. 1997) :

$$ {{m}_{K,Ca}}=\frac{Ca}{Ca+{{K}_{d}}}, $$

(5.A7)

where Ca is the Ca²⁺ concentration within the corresponding compartment of motoneuron, and K _d defines the half-saturation level of this conductance.

The kinetics of intracellular Ca²⁺ concentration (Ca, described separately for each compartment) is modelled according to the following equation (Booth et al. 1997) :

$$ \frac{d}{dt}C{{a}_{{}}}=f\times (-\alpha \times {{I}_{Ca}}-{{k}_{Ca}}\times Ca), $$

(5.A8)

where f defines the percent of free to total Ca²⁺; α converts the total Ca²⁺ current, I _Ca , to Ca²⁺ concentration; k _Ca represents the Ca²⁺ removal rate.

The synaptic excitatory (I _synE with conductance g _synE and reversal potential E _SynE= −10 mV) and inhibitory (I _synI with conductance g _synI and reversal potential E _SynI= −70 mV) currents are described as follows:

$$ \begin{aligned}& {{I}_{SynE}}={{g}_{SynE}}\times ({{V}_{{}}}-{{E}_{SynE}}); \\& {{I}_{SynI}}={{g}_{SynI}}\times ({{V}_{{}}}-{{E}_{SynI}}). \end{aligned} $$

(5.A9)

The excitatory (g _SynE ) and inhibitory synaptic (g _SynI ) conductances are equal to zero at rest and may be activated (opened) by the excitatory or inhibitory inputs to neuron i respectively:

$$\begin{array}{*{20}{l}}{{g_{SynEi}}(t) = {{\bar g}_E} \times \sum\limits_{j{\rm{\backslash kern}}1pt} {\sum\limits_{{t_{kj}} < t} {S\{ {w_{ji}}\} \times \exp ( - (t - {t_{kj}})/{\tau _{SynE}})} } + {{\bar g}_{Ed}} \times \sum\limits_m {S\{ {w_{dmi}}\} \times {d_{mi}}} ;}\\{{g_{SynIi}}(t) = {{\bar g}_I} \times \sum\limits_{j{\rm{\backslash kern}}1pt} {\sum\limits_{{t_{kj}} < t} {S\{ - {w_{ji}}\} \times \exp ( - (t - {t_{kj}})/{\tau _{SynI}})} } + {{\bar g}_{Id}} \times \sum\limits_m {S\{ - {w_{dmi}}\} \times {d_{mi}}} ,}\end{array}$$

(5.A10)

where the function S{x} = x, if x ≥ 0, and 0 if x < 0. According to equation (5.A10), the excitatory and inhibitory synaptic conductances have two terms: the first term describes the effects of inputs from other neurons in the network (excitatory and inhibitory respectively), and the second one describes effects of inputs from external drives d _mi (see also Rybak et al. 1997) . Each spike arriving to neuron i from neuron j at time t _kj increases the excitatory synaptic conductance by ${{\bar{g}}_{E}}\times {{w}_{ji}}$ if the synaptic weight w _ji  > 0, or increases the inhibitory synaptic conductance by -${{\bar{g}}_{I}}\times {{w}_{ji}}$ if the synaptic weight w _ji  < 0. ${{\bar{g}}_{E}}$ = 0.05 mS/cm² and ${{\bar{g}}_{I}}$ = 0.05 mS/cm² are the parameters defining an increase in the excitatory or inhibitory synaptic conductance, respectively, produced by one arriving spike at |w _ji | = 1. τ _SynE  = 5 ms and τ _SynE  = 15 ms are the decay time constants for the excitatory and inhibitory conductances respectively. In the second terms of equations (5.A10), ${{\bar{g}}_{Ed}}$ = ${{\bar{g}}_{Id}}$ = 1 mS/cm² is the parameter defining the increase in the excitatory synaptic conductance, produced by external input drive d _mi= 1 with a synaptic weight of |w _dmi | = 1. All synaptic weights used in the model can be found in Table 5.A2. The values of input drives used in particular simulation are shown in Table 5.A3.