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The role of phase shifts of sensory inputs in walking revealed by means of phase reduction

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Detailed neural network models of animal locomotion are important means to understand the underlying mechanisms that control the coordinated movement of individual limbs. Daun-Gruhn and Tóth, Journal of Computational Neuroscience 31(2), 43–60 (2011) constructed an inter-segmental network model of stick insect locomotion consisting of three interconnected central pattern generators (CPGs) that are associated with the protraction-retraction movements of the front, middle and hind leg. This model could reproduce the basic locomotion coordination patterns, such as tri- and tetrapod, and the transitions between them. However, the analysis of such a system is a formidable task because of its large number of variables and parameters. In this study, we employed phase reduction and averaging theory to this large network model in order to reduce it to a system of coupled phase oscillators. This enabled us to analyze the complex behavior of the system in a reduced parameter space. In this paper, we show that the reduced model reproduces the results of the original model. By analyzing the interaction of just two coupled phase oscillators, we found that the neighboring CPGs could operate within distinct regimes, depending on the phase shift between the sensory inputs from the extremities and the phases of the individual CPGs. We demonstrate that this dependence is essential to produce different coordination patterns and the transition between them. Additionally, applying averaging theory to the system of all three phase oscillators, we calculate the stable fixed points - they correspond to stable tripod or tetrapod coordination patterns and identify two ways of transition between them.

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Acknowledgements

We would like to thank Philip Holmes for useful discussions in the course of the work. This research was supported by Deutsche Forschungsgemeinschaft (DFG) grants GR3690/2-1, GR3690/4-1, DA1953/5-2, and by BMBF-NSF grant 01GQ1412.

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Correspondence to Azamat Yeldesbay.

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Action Editor: Frances K. Skinner

Appendices

Appendix A: Parameters of CPG neurons

All 6 CPG neurons have the same parameters, and their values are the same, except for those of the g a p p s. The CPG neurons are labeled by the numbers 1 to 6 as shown in Fig. 1. Parameters of the system Eq. (1):

Parameters of I N a P : g N a = 10.0 nS, E N a = 50.0 mV, V h m = − 37.0 mV, γ m = − 0.1667 mV, V h h = − 30.0 mV, γ h = 0.1667mV− 1, V h τ = − 30.0 mV, γ τ = 0.0833mV− 1, ε = 0.0023 ms.

Parameters of I L : g L = 2.8 nS, E L = − 65.0 mV; and C m = 0.9154 pF.

Parameters of I a p p : E a p p = 0.0 mV. The values of the parameters gapp1 and gapp2 are given in Table 4.

Table 4 Parameters of the driving current I a p p

Appendix B: Synaptic connections

Here, in Tables 5 and 6 the synaptic connections in the inter-segmental network model are exhibited.

Table 5 Table of synaptic connections from CPG neuron C i in column i to CPG neuron C j in row j
Table 6 Synaptic connection types

Appendix C: Phase response curve of a CPG

The phase response curve (PRC) of a single CPG can be calculated by perturbing the system (Eq. (1) without external input) at different phases of the oscillatory period and finding the resulting phase shift elicited by the perturbation. In Fig. 17, an example of a PRC of a single CPG is illustrated.

Fig. 17
figure 17

Calculation of PRC for a single CPG is done using the perturbation method. The top panel shows the membrane potential (V1) of a CPG neuron (Eq. (1)). The bottom panel displays the periodic unperturbed orbit and its perturbation in the Vh plane. Red line: unperturbed (free running) system, blue line: perturbed system, empty square: before application of the perturbation, filled square: after application of the perturbation. The points with zero phase are labeled with ∗ in the unperturbed case and with a circle in the perturbed system. To get a clear effect, the strength of perturbation was chosen to be high (10.0mV)

If the perturbations to an oscillatory system become infinitesimally small at every phase of the oscillatory period, the resulting curve of responses to these perturbations is then called the infinitesimal phase response curve (iPRC) of the system. Its value, at every phase of the oscillatory period, can be calculated directly by solving the adjoint problem derived from the system of equations that describe the oscillator (Ermentrout 1996; Hoppensteadt and Izhikevich 1997; Izhikevich 2007). This approach goes back to Malkin (1949, 1959). He considered a periodic oscillator \(\dot {X}=f(X)\) forced by a time-dependent input p(t), that obeys the following equation.

$$ \dot{X} = f(X) + \epsilon p(t), $$
(22)

where 𝜖 ≪ 1. He formulated the following theorem:

Theorem 1

Malkin’s theorem. If the unperturbed system Eq. (22)(𝜖 = 0) has a limit cycle with period T, then the phase of the system (22)is described by the equation

$$ \dot{\varphi} = 1 + \epsilon Z(\varphi) \cdot p(t), $$
(23)

where a T-periodic function Z is the solution to the linear adjoint equation

$$ Z^{\prime} = - \left( Df\right)^{\top} \cdot Z, $$
(24)

subject to the normalization condition

$$ \frac{1}{T}{{\int}_{0}^{T}} Z(t)\cdot f(x(t)) \mathrm{d}t = 1. $$
(25)

Here, Z is the iPRC and x(t)is the solutionto the unperturbed system Eq. (22)(𝜖 = 0)on the limit cycle.

If the system \(\dot {X}=f(X)\) has a periodic limit cycle with period T, then by rescaling the time, we can transform this equation system to a boundary value problem (BVP) of the following form

$$ \left\{ \begin{array}{l} \dot{x}- T f(x) = 0, \\ x(0) - x(1) = 0, \end{array}\right. $$
(26)

where the solution x(t) is a periodic function of period 1.

To find the iPRC of a single CPG, the system (Eq. (1) without external input) and the corresponding adjoint problem (Eq. (24)) were transformed to a BVP of the form Eq. (26) and solved together with the normalization conditions (Eq. (25)) using AUTO (Doedel et al. 2007). The iPRC calculated thus is a vector of the same size as the system Eq. (1). The iPRC is almost zero in the active state of the retractor neuron of the CPG, and non-zero during its quiescent phase during which the leg is in the swing phase. This phase is thus prone to perturbations by an external input, especially just before its end.

Appendix D: Regions of existence of coordination patterns

We displayed several coordination patterns in Fig. 18. Depending on the sequence of stepping legs (front-middle-hind or hind-middle-front), there can be two different types of the tetrapod coordination pattern. They are denoted in Fig. 18 as tetrapod 1 and tetrapod 2. In our model, we consider ipsilateral legs (e.g. only R1, R2, and R3), only. In this case, only a possible overlap of the swing phases of the front and hind leg will distinguish the tripod coordination pattern from the tetrapod one. Thus, by determining the phase differences between the CPGs at which the swing phases of the legs overlap, we can determine the type of the coordination pattern.

Fig. 18
figure 18

Tripod and tetrapod coordination patterns. R1, R2, R3: right front, middle, and hind leg; L1, L2, L3: left front, middle, and hind leg. In the panels on the left hand side, the black bars denote swing phases of the legs. The panels on the right hand side show the phase relations between the legs: lines of the same color connect the legs that have simultaneous swing phases. Tetrapod 1 and tetrapod 2 differ in the direction of swing phase sequence, as indicated by arrows

We consider first the phase relations between two CPGs (Fig. 19). As described in Sections 3.4 and 4.1, the CPG switches from protraction to retraction at phase φ = 0, and from retraction to protraction at phase φ = r0. Here, r0 is the duty factor of an isolated CPG. As pointed out earlier, the kinematics of the legs are determined by the activity of the CPG neurons. Hence, the stance and swing phase of the legs became the intervals 0 ≤ φ < r0 and r0φ < 1, respectively (see Fig. 8). The relative position of the swing phases of two CPGs with phases φ j and φ k with and without overlap are illustrated in Fig. 19a and b. The phase relation of two CPGs in the (φ k ,φ j ) plane are displayed in Fig. 19c. Note that the variables φ j and φ k in this figure are periodic with a normalized period 1. From Fig. 19, we conclude that the swing phases of the CPGs won’t overlap, if 1 − r0𝜃r0 (the blue areas), where 𝜃 = φ k φ j . By contrast, if 𝜃 < 1 − r0 or 𝜃 > r0, which is the same as |𝜃| < 1 − r0, then the swing phases of two oscillators will overlap. In Fig. 19, deeper shades of red mean larger overlap. Perfect overlap occurs along the diagonal (𝜃 = 0).

The condition for the existence of the tetrapod coordination pattern is that the swing phases of any two legs should not overlap. Taking into account the definition of 𝜃1 = φ1φ2, and 𝜃2 = φ3φ2, we can write

$$\begin{array}{@{}rcl@{}} 1-r_{0} &\leq& \theta_{1} \leq r_{0},\\ 1-r_{0} &\leq& \theta_{2} \leq r_{0},\\ 1-r_{0} &\leq& \theta_{2}-\theta_{1} \leq r_{0}. \end{array} $$
(27)

For the tripod coordination pattern, as mentioned before, the swing phases of the hind leg and the front leg overlap. Thus the last condition in Eq. (27) will be substituted by the condition for overlap:

$$ |\theta_{2}-\theta_{1}|<1-r_{0}. $$
(28)

Putting the conditions Eq. (27) and Eq. (28) together, we obtain the regions of existence of the two different coordination patterns, as illustrated in Fig. 20. The solutions to Eq. (17) in the lower blue region correspond to the coordination pattern tetrapod 1, whereas those in the upper blue triangle to the coordination pattern tetrapod 2 (see Fig. 18). The solutions in the shaded red region correspond to the tripod coordination pattern. The swing phases increasingly overlap toward the center (indicated by deeper red color in Fig. 19). Perfect overlap is achieved on the diagonal red dashed line. The white area surrounding the regions of existence of the coordination patterns represents solutions in which the swing phases of neighboring legs overlap. If the stance phase of the CPG oscillator shortens, that is the duty factor r0 decreases, the regions of existence of tetrapod shrink. When the condition 1 − r0 = 2r0 − 1 (cf. the definition of the region borders in Fig. 20) is fulfilled, i.e. r0 = 2/3, the tetrapod solution becomes extinct.

Fig. 19
figure 19

Phase relations between the activities of two CPGs with and without overlapping swing phases. a and b Illustration of the relative position of the swing phases (black bars). c The phase differences between the activities of the two CPG oscillators with phases φ k and φ j in the (φ k ,φ j ) plane. The swing phases constitute the transparent gray rectangle (r0,1] × (r0,1] of the (φ k ,φ j ) plane. The dashed lines parallel to the diagonal are the lines of constant phase difference: 𝜃 = φ k φ j = const. In the blue regions, the swing phases of the two oscillatory activities with a constant phase difference do not overlap. The regions of overlap are drawn in varying shades of red. The deeper the shade of the red color, the greater is the overlap. Note that the region boundaries reflect the periodicity of phase variables φ k and φ j

Fig. 20
figure 20

Illustration of the regions of existence of the coordination patterns in the (𝜃1,𝜃2) plane (torus). The blue triangles are the regions of existence of tetrapod coordination patterns of type 1 and type 2 (see Fig. 18). The shaded red area is the region of existence of the tripod coordination pattern, where the swing phases of the front and hind leg overlap. The increasingly deep red color indicates increasing overlap. On the diagonal red dashed line, the overlap is perfect. The corresponding relations between 𝜃1 and 𝜃2 on the region boundaries (gray dotted lines) are given by Eq. (27)

To demonstrate the meaning of the regions of coordination patterns we projected the trajectories of the stable tetrapod and tripod coordination patterns obtained by simulation of the non-averaged phase oscillators system Eq. (15) in Section 4.3 onto the (𝜃1,𝜃2) plane as shown in Fig. 21. One can see that the trajectories for a given coordination pattern do not leave the corresponding region in the plane. Note that the trajectories in the (𝜃1,𝜃2) coordinate system can cross and overlap since they are projections of the three dimensional system Eq. (15) onto a two dimensional surface.

Fig. 21
figure 21

Projections of the trajectories of the stable tetrapod (left panels) and tripod (right panels) coordination patterns shown in Fig. 13 onto the (𝜃1,𝜃2) plane. a and b Time courses of φ1 (red), φ2 (green), and φ3 (blue) of the non-averaged system Eq. (15). c and d Time courses of the phase differences 𝜃1 = φ1φ2 (red) and 𝜃2 = φ3φ2 (blue) calculated by subtraction of the appropriate phase angles from the upper panels. e and f Projection of the trajectories (black) onto the (𝜃1,𝜃2) plane enlarged around the corresponding region. The blue and red areas are the regions of the tetrapod and tripod coordination patterns, respectively (see Fig. 20)

Appendix E: The structure of the bifurcation diagram

The averaged coupling functions (Eqs. (14) and (15)) can be approximated as sums

$$\begin{array}{@{}rcl@{}} H^{i}(\theta,{\Delta}^{i}) &=& g^{i} \sum\limits_{j = 1}^{N_{j}} s(\theta+\tau_{j}) \mathrm{y}({\Delta}^{i}+\tau_{j}) F^{i}(\tau_{j}) {\Delta}\tau, \end{array} $$
(29)
$$\begin{array}{@{}rcl@{}} H^{e}(\theta,{\Delta}^{e}) \!&=&\!g^{e} \sum\limits_{j = 1}^{N_{j}} s(\theta+\!\tau_{j}) \mathrm{y}(\theta\,+\,{\Delta}^{e}\,+\,\tau_{j}) F^{e}(\tau_{j}) {\Delta}\tau, \end{array} $$
(30)

where τ j ∈ [0, 1], Δτ is a small increment, and the functions Fi(τ) = −Z(τ) [V (τ) − Ei] and Fe(τ) = −Z(τ) [V (τ) − Ee] are depicted in Fig. 22a.

We introduce a new variable α = 1 − τ (or τ = −α due to periodicity). The functions Fe,i(−α) have a large amplitude for 0 < α < α, where α≪ 1 and Fe,i(−α) = 0. The functions Fe,i change the sign for α < α < 1 − r0 (see Fig. 22a inset panel). For simplicity we assume that Fi,e(−α) = 0 for r0 < α ≤ 1, s(φ) = 1 for 0 < φr0, and y(φ) = 1 for 0 < φr y .

For a given value of α the term in the sum in Eq. (30) is not zero if both s(𝜃α)≠ 0 and y(𝜃 + Δeα)≠ 0. In the (Δe,𝜃) plane (torus) these conditions are fulfilled inside the regions depicted in Fig. 22b with dashed red and solid blue borders, respectively. The regions are filled with light gray color and the intersections of these regions are drawn with dark gray color. Summing up all intersection regions over the values of α within the range [0, 1 − r0], where Fe(−α)≠ 0, gives the regions whose boundaries are shown in Fig. 22b by dash-dotted black lines. Thus, inside this region He(𝜃e)≠ 0.

The sum Eq. (30), which is the convolution of three functions s, y and Fe, is positive for 0 < α < α due to the large positive amplitude of Fe. By contrast, within the range where the larger amplitude is not included (α < α < 1 − r0), the sum is negative. As α→ 0 the regions in the (Δe,𝜃) plane where He is positive or negative approach the ones depicted in Fig. 22c with blue and red colors, respectively. The white region is where He(𝜃e) ≈ 0.

Similarly, the function Hi(𝜃i) is a convolution of the functions s(𝜃α), y(Δiα), and Fi(−α) (see Eq. (29)). Since Δi = 1/8, the condition y(Δiα)≠ 0 reduces the range of α to [0,Δi]. Thus, from the condition s(𝜃α)≠ 0 we find the range 0 ≤ 𝜃r0 + Δi where Hi(𝜃)≠ 0. Again the large negative amplitude of the function Fi(−α) dominates for 0 < α < α. Therefor, for the limit as α = 0, the function Hi is negative for 0 < 𝜃r0 and positive for r0 < 𝜃r0 + Δi (see the right side of Fig. 22c).

Fig. 22
figure 22

Illustration of the explanation of the structure of the bifurcation diagram Fig. 11. a Example of functions Fe(τ) (red line) and Fi(τ) (blue line). Inset panel shows the enlargement around [r0,1]. The vertical dotted lines denote r0 and α (Fe,i(−α) = 0). b The regions where s(𝜃α)≠ 0 (light gray with dashed red boundaries) and y(𝜃 + Δeα)≠ 0 (light gray with solid blue boundaries) in the (𝜃e) plane for a given value of α. The intersection of these regions are depicted with dark gray color. The regions bounded by black dash-dotted lines are the regions where He≠ 0, obtained by changing the value of α within [0,1 − r0]. c The regions in the (𝜃e) plane where the function He > 0 (blue color with solid boundaries), He < 0 (red color with dashed boundaries), and He ≈ 0 (white region) in the limiting case as α = 0. The corresponding expressions for the boundaries of the regions (black dotted lines) are given. The intervals of 𝜃 when Hi > 0, Hi < 0, and Hi ≈ 0 are marked on the right side

Taking into account that |He| > |Hi| and that the functions He and Hi in Eq. (19) have negative sign, we can find the approximate expressions for the stable and unstable branches in the bifurcation diagram Fig. 11a for the averaged phase difference equation of the uni-directionally coupled oscillators model (Eq. (19)):

  • The central stable branch (the solution types 2 and 3) and the oblique unstable solution branch are the boundaries of the condition y(𝜃 + Δe)≠ 0: 𝜃 + Δe = 1 and 𝜃 + Δe = r y . Stability of the former branch is provided by the negative Hi (cf. Fig. 6).

  • The horizontal stable (the solution type 1) and unstable branch are the boundaries of the condition s(𝜃)≠ 0: 𝜃 = 0 and 𝜃 = r0.

  • In the upper left corner of the bifurcation diagram Fig. 11a (𝜃 ∈ [r0, 1] and Δe ∈ [0, 1 − r0]) the function Hi dominates. Thus, the branch of the solution type 4 is the boundary of the condition s(𝜃)≠ 0. The unstable branch above is bounded by the condition s(𝜃 − (r0 + Δi))≠ 0: 𝜃 = r0 + Δi.

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Yeldesbay, A., Tóth, T. & Daun, S. The role of phase shifts of sensory inputs in walking revealed by means of phase reduction. J Comput Neurosci 44, 313–339 (2018). https://doi.org/10.1007/s10827-018-0681-0

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