An inter-segmental network model and its use in elucidating gait-switches in the stick insect

Abstract

Animal locomotion requires highly coordinated working of the segmental neuronal networks that control the limb movements. Experiments have shown that sensory signals originating from the extremities play a pivotal role in controlling locomotion patterns by acting on central networks. Based on the results from stick insect locomotion, we constructed an inter-segmental model comprising local networks for all three legs, i.e. for the pro-, meso- and meta-thorax, their inter-connections and the main sensory inputs modifying their activities. In the model, the local networks are uniform, and each of them consists of a central pattern generator (CPG) providing the rhythmic oscillation for the protractor-retractor motor systems, the corresponding motoneurons (MNs), and local inhibitory interneurons (IINs) between the CPGs and the MNs. Between segments, the CPGs are connected cyclically by both excitatory and inhibitory pathways that are modulated by the aforementioned sensory inputs. Simulations done with our network model showed that it was capable of reproducing basic patterns of locomotion such as those occurring during tri- and tetrapod gaits. The model further revealed a number of elementary neuronal processes (e.g. synaptic inhibition, or changing the synaptic drive at specific neurons) that in the simulations were necessary, and in their entirety sufficient, to bring about a transition from one type of gait to another. The main result of this simulation study is that exactly the same mechanism underlies the transition between the two types of gait irrespective of the direction of the change. Moreover, the model suggests that the majority of these processes can be attributed to direct sensory influences, and changes are required only in centrally controlled synaptic drives to the CPGs.

Appendix

1.1 Parameter values used in the CPG interneuron model and inhibitory interneuron model

1.1.1 CPG interneurons

${\bf I}_{\textbf{NaP}}$: g_NaP = 10 nS, E _Na  = 50 mV, V _hm  = − 37 mV, γ _m  = − 1/6 mV^− 1, V _hh  = − 30 mV, γ _h  = 1/6 mV^− 1, V _τh  = − 30 mV, γ _τ  = 1/12 mV^− 1, ε = 0.0023; I _L : g _L  = 2.8 nS, E _L  = − 65 mV; C _m  = 0.9154 pF; ${\bf I}_{\textbf{syn}}$: V _hs  = − 43 mV, γ _s  = − 10 mV^− 1

1.1.2 Inhibitory interneurons

ϵ = 0.01 and C _m  = 0.21 pF, all other parameter values are the same as above.

1.1.3 Synapses and central inputs

As far as the reversal potentials of I _syn and I _app are concerned, their values are set in accordance with the properties of the individual synapses and central inputs. Thus E _syn = 0 mV for excitatory, and E _syn = − 80 mV for inhibitory synapses. The same applies to E _app.

The actual values of the maximal conductances of the mutually inhibitory synapses between CPG neurons (e.g. C1–C2) are the same for all CPG neurons: g _synCPG  = 1.0 nS.

For the excitatory modulatory synapses, we have uniformly g _syn = 0.2 nS, for the inhibitory ones: g _syn = 0.1 nS. The conductance values of the central inputs I _app to the CPGs are given in the text.

For all excitatory synapses on IINs from CPG neurons: g _syn = 0.5 nS. All central inhibitory inputs to IINs have g _d1 ≡ g _d2 ≡ g _app = 1.6 nS.

1.2 Details and parameter values of the motoneuron model

1.2.1 Mathematical description of the constituent currents

The general equation for the voltage-gated currents that enter the current balance equation

$$ C_{m} dV/dt = -(I_{Na}+I_{K}+I_{q}+I_{L}+I_{\rm syn}+I_{\rm app}) \label{eq:cb2} $$

(11)

reads as

$$ I_{x}=g_{x}^{}m_{x}^{p}h_{x}^{}(V-E_{x}). \label{eq:Ix} $$

(12)

where x = Na,K,q,L (I _syn and I _app are of the same form as in the CPG neuron model: Eqs. (4) and (5)). The parameters g _x and E _x have the same meaning as before (cf. Section CPG neuron model). The (in)activation variables h _x and m _x , respectively, obey the differential equation of the form:

$$ dy/dt=\alpha_{y}(V)(1-y)-\beta_{y}(V)y \label{eq:y} $$

(13)

where y stands for h _x or m _x . The coefficients of Eq. (13) are nonlinear functions of the membrane potential V. They can have different forms depending on the type of the membrane current (cf. Traub et al. 1991). The parameter p is an integer.

1.2.2 Synapses and central inputs

For all inhibitory synapses on MNs from IINs: g _syn = 0.2 mS/cm².

All central excitatory inputs to MNs have g _MN  ≡ g _app = 0.2 mS/cm².