A mathematical modeling study of inter-segmental coordination during stick insect walking

Abstract

The biomechanical conditions for walking in the stick insect require a modeling approach that is based on the control of pairs of antagonistic motoneuron (MN) pools for each leg joint by independent central pattern generators (CPGs). Each CPG controls a pair of antagonistic MN pools. Furthermore, specific sensory feedback signals play an important role in the control of single leg movement and in the generation of inter-leg coordination or the interplay between both tasks. Currently, however, no mathematical model exists that provides a theoretical approach to understanding the generation of coordinated locomotion in such a multi-legged locomotor system. In the present study, I created such a theoretical model for the stick insect walking system, which describes the MN activity of a single forward stepping middle leg and helps to explain the neuronal mechanisms underlying coordinating information transfer between ipsilateral legs. In this model, CPGs that belong to the same leg, as well as those belonging to different legs, are connected by specific sensory feedback pathways that convey information about movements and forces generated during locomotion. The model emphasizes the importance of sensory feedback, which is used by the central nervous system to enhance weak excitatory and inhibitory synaptic connections from front to rear between the three thorax-coxa-joint CPGs. Thereby the sensory feedback activates caudal pattern generation networks and helps to coordinate leg movements by generating in-phase and out-of-phase thoracic MN activity.

Appendices

Appendix

I summarized some central notations used in this paper in the following table:

Table 1 Some notations used in the paper

Interneuron model using the persistent sodium current

The ordinary differential equations for this model are

$$ \begin{array}{*{20}{c}} {{C_m}v\prime = - {I_{NaP}} - {I_L} - {I_{syn}} - {I_{app}},} \hfill \\{h\prime = ({h_\infty }(v) - h)/{\tau_h}(v),} \hfill \\{s\prime = \alpha (1 - s){s_\infty }(v) - \beta s,} \hfill \\\end{array} $$

(A1)

with associated functions

$$ \begin{array}{*{20}{c}} {{I_{syn}} = {g_{syn}}s(v - {e_{vyn}}),} \hfill \\{{I_{NaP}} = {g_{nap}}{m_\infty }(v)h(v - {e_{nap}}),} \hfill \\{{I_L} = {g_l}(v - {e_l}),} \hfill \\{{I_{app}} = {g_{app}}v,} \hfill \\{{h_\infty }(v) = 1/(1 + \exp ((v - {\theta_h})/{\sigma_h})),} \hfill \\{{s_\infty }(v) = 1/(1 + \exp ((v - {\theta_{syn}})/{\sigma_{syn}})),} \hfill \\{{\tau_h}(v) = 1/(\varepsilon \cosh ((v - {\theta_h})/{\sigma_h}/2)),} \hfill \\{{m_\infty }(v) = 1/(1 + \exp ((v - {\theta_m})/{\sigma_m})),} \hfill \\\end{array} $$

where

$$ \begin{array}{*{20}{c}} {{{\hbox{C}}_{\rm{m}}} = 0.{21,}\,{{\hbox{g}}_{\rm{nap}}} = {10,}\,{{\hbox{g}}_{\rm{l}}} = {2}{.8,}\,{{\hbox{e}}_{\rm{nap}}} = {50,}\,{{\hbox{e}}_{\rm{l}}} = - {65,}\,{{\hbox{v}}_{\rm{syn}}} = - {80,}\,{\theta_{\rm{m}}} = - {37,}\,{\sigma_{\rm{m}}} = - {6,}\,{\theta_{\rm{h}}} = - {30,}\,{\sigma_{\rm{h}}} = {6,}} \hfill \\{\varepsilon = {0}{.01,}\,{\theta_{\rm{syn}}} = - {43,}\,{\sigma_{\rm{syn}}} = - {0}{.1,}\,{{\hbox{g}}_{\rm{syn}}} = {1,}\,{{\hbox{g}}_{\rm{app}}} = {0}{.19,}\,\alpha = {1,}\,\beta = {1}{.}} \hfill \\\end{array} $$

Since I assumed the synaptic coupling to be fast, I set $s = {s_\infty }$ in the simulations.

The following assumptions are made on system (A1):

(H1) For i ∈ {1,2} and fixed s ∈ I _s , the v-nullcline, $\{ ({v_i},{h_i}):{F_i}({v_i},{h_i},{s_j}) = 0\},$ defines a cubic shaped curve, composed of left, middle, and right branches, in the (v _i ,h _i ) phase plane.

Dropping the subscript j from s _j , denote the branches of F _i = 0 by $v = {v^i}_L(h,s)$,$v = {v^i}_M(h,s)$, $v = {v^i}_R(h,s)$, with ${v^i}_L < {v^i}_M < {v^i}_R$ for each (h,s) on which all three functions are defined. The variable s corresponds to the synaptic input received by the cell, driven by the voltage of the other cell. The drive current to each cell, ${g_{app,i}}({v_i} - {v_{app}})$, is also treated as synaptic but is independent of the other cell in the network.

Specifically, under (H1), for fixed s ∈ I _s , the left branch $({v_L}(h,s),h)$ meets the middle branch $({v_M}(h,s),h)$ in the left knee of the v-nullcline, while the middle branch meets the right branch $({v_R}(h,s),h)$ in the right knee. For each s ∈ I _s , let ${p_{LK}}(s) = ({v_{LK}}(s),{h_{LK}}(s))$ denote the left knee of the v-nullcline and, similarly, let ${p_{RK}}(s) = ({v_{RK}}(s),{h_{RK}}(s))$ denote the right knee of the v-nullcline.

The dotted and dashed cubic shaped curves in Fig. 3(a) represent the v-nullclines for s=s_max and s=0, respectively.

(H2) For i ∈ {1,2}, the h-nullcline, $\{ ({v_i},{h_i}):{g_i}({v_i},{h_i}) = 0\}$, is a monotonic curve in the (v _i ,h _i ) plane. For fixed $s \in [0,{s_{\max }}[$, the h-nullcline intersects F _i = 0 at a unique point ${p_{FP}}(s) = ({v_{FP}}(s),{h_{FP}}(s))$ on the right branch of the corresponding v-nullcline.

A cell is defined as excitable if the intersection point of the h- and v-nullcline p _FP (0) lies on the left branch of the v-nullcline, $\{ (v,h):v = {v_L}(h,0)\}$, as oscillatory if p _FP (0) lies on the middle branch of the v-nullcline, $\{ (v,h):v = {v_M}(h,0)\}$; in this case, the cell will intrinsically oscillate, yielding a reduced representation of bursting activity and finally, a cell is tonic if p _FP (0) lies on the right, most depolarized branch of the v-nullcline, $\{ (v,h):v = {v_R}(h,0)\}$, yielding a reduced representation of tonic spiking, which will be the case in this modeling study. The latter case is illustrated in Fig. 3(a), in which the h-nullcline (solid monotonic curve) intersects each v-nullcline in a unique point on the right branch for s<s_max.

One further restriction has to be made on the system (A1):

(H3) For i ∈ {1,2} and s = s _max, the h-nullcline intersects F _i = 0 either at a unique point p _FP,R (s _max) on the right branch of the corresponding v-nullcline or at three different points one on the left, one on the middle and one on the right branch of the corresponding v-nullcline, p _FP,L (s _max), p _FP,M (s _max), p _FP,R (s _max), depending on the extra drive g _app,i

These two cases are illustrated in Figs. 2(a) and 3(a), respectively.

In a neuron, a bursting solution alternates repeatedly between silent phases of relatively constant, low voltage and active phases featuring voltage spikes, which are rapid voltage oscillations of significant amplitude. A model of the form (A1) can be obtained from a model bursting neuron by omitting spike-generating currents but maintaining a current that allows for transitions to an elevated voltage state. In this model, a bursting solution consists of an oscillation composed of silent phases, with ${v_i} \approx {v^i}_L(h,s)$, alternating with active phases, with ${v_i} \approx {v^i}_R(h,s)$.

This situation bears similarities to the activity of non-spiking interneurons in the stick insect central neural networks during rhythmic activity, which are known to participate in rhythm generation in a walking stick insect (Büschges et al. 1994).

2.1 Slow and fast subsystems of the interneuron model

In our previous work (Daun et al. 2009) we established conditions under which periodically oscillating solutions can be constructed for system (A1) under the structural hypotheses (H1), (H2) and (H3) and the parameter choices given above, in the singular limit of ε ↓ 0. Results on geometric singular perturbation theory suggest that this construction will yield the existence of nearby oscillating solutions for ε > 0 sufficiently small (Mischenko et al. 1994).

For system (A1), there are associated fast and slow subsystems. The fast subsystem is obtained by setting ε = 0 directly and thus it takes the form

$$ \begin{array}{*{20}{c}} {{C_m}{v_i}\prime = {F_i}({v_i},{h_i},{s_j}),j \ne i,} \hfill \\{{h_i}\prime = 0,} \hfill \\{{s_i}\prime = \alpha {s_\infty }({v_i})(1 - {s_i}) - \beta {s_i}.} \hfill \\\end{array} $$

(A2)

Recall that i ∈ {1,2}, so Eq. (A2) is a system of six equations.

To define various slow subsystems, set τ = εt and let “dot” denote differentiation with respect to τ. Under this rescaling of time, system (A1) becomes, with i ∈ {1,2}

$$ \begin{array}{*{20}{c}} {\varepsilon {C_m}{{\dot{v}}_i} = {F_i}({v_i},{h_i},{s_j}),j \ne i,} \hfill \\{{{\dot{h}}_i} = {g_i}({v_i},{h_i}),} \hfill \\{\varepsilon {{\dot{s}}_i} = \alpha {s_\infty }({v_i})(1 - {s_i}) - \beta {s_i}.} \hfill \\\end{array} $$

(A3)

The slow subsystems are obtained from system (A3) by setting ε = 0, solving the algebraic equations, and inserting the results into the h-equation. This process yields, for each i ∈ {1,2},

$$ {\dot{h}_i} = {g_i}({v_X}({h_i},{s_j}),{h_i}),j \ne i, $$

(A4)

for X ∈ {L,M,R}. In Eq. (A4), s _j depends on v _j and hence is a function of h _j .

Consider the limit of σ _syn ↓ 0 in the equation ${s_\infty }(v) = 1/(1 + \exp ((v - {\theta_{syn}})/{\sigma_{syn}}))$. Since the branch v _M (h,s) is unstable with respect to the fast subsystem, there are four distinct slow subsystems (A4) that could theoretically be relevant. Two of these are obtained when cell i is silent and cell j active for i = 1 or i = 2, and each of these takes the form

$$ {\dot{h}_i} = {G_L}({h_i}): = g({v_L}({h_i},{s_{\max }}),{h_i}), $$

(A5)

$$ {\dot{h}_j} = {G_R}({h_j}): = g({v_R}({h_j},0),{h_j}). $$

(A6)

The other two subsystems involve the cases that both cells are silent or active. A key point is that the singular solution consists of a concatenation of solutions of systems (A2) and (A5)–(A6) for i = 1,2. Projected to each (v,h)-plane, the solutions to system (A2) consist of jumps between branches of v-nullclines for different values of s, while the solutions to the slow subsystems take the form of pieces of these nullclines. Although the slow subsystems above correspond to σ _syn ↓ 0, solutions obtained in this limit persist for small $|{\sigma_{syn}}| > 0$ for the persistent sodium model.

Simplified motoneuron model

The ordinary differential equation of this 1-dimensional model is given by

$$ {C_m}v\prime = - {I_{NaP}} - {I_L} - {I_{syn}} - {I_{app}}, $$

with associated functions and parameters as above except

$$ {I_{NaP}} = {g_{nap}}{m_\infty }(v){h_\infty }(v)(v - {e_{nap}}). $$

The coupling between the interneurons forming the half-center CPG and the motoneurons as well as the intra- and inter-leg couplings are as described in the main text.