# Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator

## Abstract

**:**

_{2}× ℤ

_{2}symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.

## 1. Introduction

_{4}× ℤ

_{2}symmetry, so that two nodes are associated with each leg. A physiological interpretation was suggested: the two nodes associated with a given leg control the timing of flexor and extensor muscle groups.

_{2}× ℤ

_{2}symmetry. See Figure 1. They used Hopf bifurcation and the H/K Theorem of Buono and Golubitsky [16] to classify the spatio-temporal symmetries of periodic states for this symmetry class, and obtained a list of four primary states arising from Hopf bifurcation, plus seven secondary states predicted by the H/K Theorem and expected to arise through mode interactions between primary states.

_{u}= Df|

_{u}has either a zero eigenvalue (steady-state bifurcation) or a complex conjugate pair of purely imaginary eigenvalues (Hopf bifurcation). The symmetries of the network lead to four distinct symmetry classes of bifurcating branches in each case. The Hopf branches are the primary gaits.

**× ℤ**

_{2}_{2}of order 4, but the bifurcation behaviour is controlled by a larger group, the symmetries of a tetrahedron, of order 24. We describe how this group arises and why it determines many key aspects of the bifurcation behaviour. Ultimately, it is related to the eigenvalues of the connection matrix, which are closely related to the eigenvalues of the Jacobian for the rate equations, evaluated at a synchronous equilibrium.

_{2}symmetry) in considerable detail. Even in this case there is a richer range of dynamic behaviour than equilibria and periodic states arising from local bifurcation. Note, however, that they assume that certain connections are inhibitory, so the corresponding connection strength is negative. This restriction implies in particular that there is a unique fully synchronous steady state for all parameter values under consideration. This is not the case when the signs are not restricted, as we show in Section 4.

#### 1.1. Structure of the Paper

## 2. Gain Functions and Rate Models

^{−1}given by

_{ij}∈ ℝ is the strength of the connection from node j to node i (positive for excitation, negative for inhibition). Let g > 0 be a parameter that determines the strength of reduction of the activity variable by the fatigue variable. Assume that nodes have some kind of input, with the input to node i being I

_{i}. The rate equations are:

## 3. The CPG Network

_{2}〈ρ〉 × ℤ

_{2}〈τ〉 generated by

**Proposition 1**Suppose that a network is homogeneous, with connection matrix A defining a rate model, so every row of A has the same sum r. Then the rate Equation (11) are invariant under the parameter symmetry η for which

^{T}. Let y = a − x, so that y

^{E}= a − x

^{E}and y

^{H}= a − x

^{H}. Then

^{E}(there denoted u

_{1}) depends on I, is symmetric under rotation through 180°.

_{j}to be equal, so that I

_{j}= I for 1 ≤ j ≤ 4. We use I as bifurcation parameter and consider the connection strengths as subsidiary parameters; these are arbitrary, but held fixed as I varies. Explicit rate equations (written in a convenient order for later calculations, in which the activity variables and fatigue variables are collected together) are:

α = | strength of diagonal connection |

γ = | strength of medial connection |

β = | strength of lateral connection |

ε = | fast/slow dynamic timescale |

g = | strength of reduction of activity variable by fatigue variable |

I = | input |

_{2}× ℤ

_{2}defined by

^{−1}to define a left action, but in this case every element of G is equal to its inverse.) The subspaces ${\mathbb{R}}_{E}^{4}$ , ${\mathbb{R}}_{H}^{4}$ are G-invariant subspaces with isomorphic G-actions.

_{2}× ℤ

_{2}-symmetric system at Hopf bifurcation, as classified in Pinto and Golubitsky [21]. We have abbreviated their “two-legged hop” and “two-legged jump” to “hop” and “jump”. Which pattern occurs is determined by the representation of ℤ

_{2}× ℤ

_{2}on the critical (purely imaginary) eigenspace [17,18,38]. See Section 5.

## 4. Synchronous Steady States

**Theorem 1**Let be a homogeneous network whose connection matrix A has all row-sums equal to r. Let

^{L}, I

^{U}depending on σ for which there are three solutions when I

^{L}< I < I

^{u}, and two solutions (one of multiplicity 2) when I = I

^{L}, I

^{U}.

^{L}, I

^{U}depending on σ for which there are three solutions when I

^{L}< I < I

^{U}, and two solutions (one of multiplicity 2) when I = I

^{L}, I

^{U}. See Figure 5 when a = 1, b = 8, c = 1.

^{L}, I

^{U}against σ. Not surprisingly, we recognise the bifurcation curve of a cusp catastrophe surface, Zeeman [30], Poston and Stewart [29]. Figure 6b shows this surface. There are three solutions to Equation (12), that is, three fully symmetric equilibria, when (σ, I) lies inside the cusp; a unique solution when (σ, I) lies outside the cusp; two solutions (one with multiplicity 2) when (σ, I) lies on the curves except at the cusp point; and one solution (with multiplicity 3) when (σ, I) lies at the cusp point.

^{U}of the cusp curve, which corresponds to u

^{−}. As we show later, there is always a Hopf bifurcation before that point. However, that does not rule out a synchrony-breaking steady-state bifurcation occurring before the Hopf bifurcation. We consider this possibility in Section 12 and show that it does not occur.

**Remark 1**Curtu [26,27] study a rate model on a network with two identical nodes, and Theorem 1 applies to this network. However, in those papers there is a unique fully synchronous equilibrium for all I. The reason for this apparent discrepancy is that those papers assume an inhibitory connection, so σ < 0. Therefore, σ < c/2 since c > 0.

**The Cusp Catastrophe**. The geometry of the cusp catastrophe surface makes it relatively straightforward to understand how the first bifurcation from a synchronous equilibrium (as I varies quasistatically) depends on the connection strengths. See Zeeman [30] and Poston and Stewart [29] for further discussion in the context of the canonical cusp catastrophe. More precisely, the cusp geometry motivates the following procedure.

^{σ}is the constant-σ section of the cusp surface . Define a projection

^{σ}of the corresponding varieties in (σ, u)-space. Therefore, saddle-node (fold) bifurcations in the fully synchronous state u, which correspond to tangencies in Figure 5, are given by Equation (15), so that

^{σ}on suitable domains to rewrite this as

**Lemma 1**For given connection strengths (α, β,γ), let σ = α + β + γ − g. Then

- If$\sigma <\frac{4}{ab}$ then ξ
^{σ}: ℝ → (0, a; is a monotonic strictly increasing diffeomorphism. - If$\sigma =\frac{4}{ab}$ then ξ
^{σ}: ℝ → (0, a; is a monotonic strictly increasing homeomorphism. - If$\sigma >\frac{4}{ab}$ and ξ
^{σ}: (0,u^{−}) → (−∞,I^{U}) is a monotonic strictly increasing homeomorphism. It is a diffeomorphism when u < u^{−}.

^{U}, the saddle-node bifurcation on the lower layer of the cusp surface . We therefore have:

**Proposition 2**For any given α, β, γ the first local bifurcation as I increases quasistatically corresponds to the first local bifurcation as u increases.

^{U}, provided we consider quasistatic variation. But I

^{U}is itself a bifurcation point—here a saddle-node.

## 5. Review of Symmetric Steady-State and Hopf Bifurcation

^{n}. Denote the action of g ∈ G on x ∈ ℝ

^{n}by gx. Consider a G-equivariant ODE

^{n}× ℝ → ℝ

^{n}is smooth. Here x represents the state of the system and λ is a bifurcation parameter. The equivariance condition is

_{0}there is a path of equilibria {(x(λ), λ) : λ ∈ ℝ}. Local bifurcations (either steady-state or Hopf; occur at points (x

_{0}, λ

_{0}) for which the Jacobian

#### 5.1. Symmetric Steady-State Bifurcation

^{n}is

_{g}(Σ). Since Σ acts trivially, the effective action is that of N

_{g}(Σ)/Σ.

^{1}is a compact Lie group. Finite-dimensional real linear representations of compact Lie groups occur in three classes, distinguished by the space of commuting linear maps. By Schur's Lemma this is a division algebra over ℝ, hence is isomorphic to one of ℝ, ℂ, and the quaternions (x0210D;. If this algebra is (x0211D; we call the representation absolutely irreducible. See Adams [42] 3.22.

**Proposition 3**Let f be a G-equivariant bifurcation problem (19), and suppose that along a branch of steady states (x(λ),λ) the Jacobian Df|

_{(x0,λ0)}has a zero eigenvalue, where x

_{0}= x(λ

_{0}). Let E

_{0}be the corresponding real generalised eigenspace. Then generically E

_{0}is absolutely irreducible under the action of G.

**Lemma 2**Consider a G-equivariant bifurcation problem (19) such that Df|

_{(x0,x0)}has a zero eigenvalue for which E

_{0}is absolutely irreducible and the eigenvalue passes through zero with nonzero speed. Let Σ be an isotropy subgroup of G acting on E

_{0}for which

_{0}, λ

_{0}), with isotropy group Σ.

_{0}that is required to have a 1-dimensional fixed-point space, not the action on ℝ

^{n}.

#### 5.2. Symmetric Hopf Bifurcation

_{2}

_{π}, ${\mathcal{S}}_{2\pi}^{1}$ be the loop spaces of continuous and once-differentiable 2π-periodic maps ℝ → ℝ

^{n}. These are Banach spaces. Define the circle group

^{1}acts on

_{2}

_{π}and ${\mathcal{S}}_{2\pi}^{1}$ by

_{2}

_{π}or ${\mathcal{S}}_{2\pi}^{1}$ is fixed by (g, θ) if and only if

^{1}for which Equation (20) is valid. A representation W of G is said to be G-simple if either

- (1)
- W ≅ V ⊕ V where V is absolutely irreducible.
- (2)
- W is irreducible of type ℂ or ℍ.

**Proposition 4**Let f be a G-equivariant bifurcation problem (19), and suppose that along a branch of steady states (x(λ), λ) the Jacobian Df|

_{(x0,λ0)}has purely imaginary eigenvalues ±iω, where x

_{0}= x(λ

_{0}) and ω ≠ 0. Let E

_{iω}be the corresponding real generalised eigenspace. Then generically E

_{iω}is G-simple.

_{i}

_{ω}is always G-invariant. In the generic case it also supports a natural

^{1}-action. Define the restricted Jacobian

^{1}on E

_{i}

_{ω}by

^{1}acts on E

_{i}

_{ω}by

_{(}

_{x}

_{,λ(}

_{x}

_{))}for λ near λ

_{0}, such that

**Theorem 2**Consider a G-equivariant bifurcation problem (19) such that Df|

_{(x0,λ0)}has apair of purely imaginary eigenvalues ±iω for which E

_{iω}is G-simple and the eigenvalues cross the imaginary axis with nonzero speed. Let Σ be an isotropy subgroup of G×

^{1}acting on E

_{iω}for which

_{0}, λ

_{0}), with period near $\frac{2\pi}{\left|\omega \right|}$ , having spatio-temporal symmetry group Σ in loop space.

_{ℝ}Fix(Σ) is computed, and infer the existence of a periodic solution of the full nonlinear equation with the same group Σ as its spatio-temporal symmetry group in loop space. So the symmetries are exact, not linearised approximations.

_{2}× ℤ

_{2}) a G-simple critical eigenspace generically corresponds to a simple eigenvalue, so the classical Hopf bifurcation theorem applies. In particular, the bifurcating branch is locally unique. However, the Equivariant Hopf Theorem is a stronger result because the spatio-temporal isotropy subgroup provides additional information about the symmetries along the bifurcating branch. Similar remarks apply to steady-state bifurcation.

## 6. Data for Primary Gaits

_{2}× ℤ

_{2}on ℝ

^{4}is:

**Proposition 5**The eigenvalues of A are distinct unless

**Definition 1**A triple of connection strengths (α, β, γ) is degenerate if any of the following conditions is satisfied for some patterns P, Q:

_{2}× ℤ

_{2}-symmetric network. We reproduce that table in the notation of this paper as Table 1, using Σ in place of their K to denote the spatial (that is, pointwise) isotropy subgroup. For all primary gaits, the setwise isotropy subgroup H is ℤ

_{2}× ℤ

_{2}. Since the group is abelian, this is also the normaliser of Fix(Σ). Recall that = ℝ/2πℤ so π is half the period.

_{0}for steady-state bifurcation, in the generic case when this space has dimension 1. We say that such a steady-state bifurcation is of type P if the isotropy subgroup is that of the primary gait P (= H, J, R, or W). Steady-state bifurcation of type H is symmetry-preserving; the other three are symmetry-breaking.

## 7. Main Theorem

_{P}≠ k, K and none of the conditions (22) holds. Then the first local bifurcation (or its absence) is given by Table 2.

**Remark 2**The assumption that ε < abg/4 is natural because it is usual to assume that ε ≪ 1, leading to fast/slow dynamics. If ε > abg/4, no parameter values lead to Hopf bifurcation.

## 8. Local Bifurcation Analysis

_{u}at a fully synchronous equilibrium u has an eigenvalue on the imaginary axis. If it is zero, we get a steady-state bifurcation and the bifurcating branch consists of equilibria; if it is purely imaginary, we get a Hopf bifurcation and the branch consists of periodic states.

#### 8.1. Eigenvalues of the Jacobian

_{u}at a fully synchronous equilibrium u, and those of the connection matrix A. We therefore begin with a general theorem characterising the relation between eigenvalues and eigenvectors of J

_{u}and those of A.

**Theorem 4**Let υ be an eigenvector of A with eigenvalue μ. Then [(λ + 1)υ, υ]

^{T}is an eigenvector of J

_{u}with eigenvalue λ if and only if

^{n}. Then [w, υ]

^{T}is an eigenvector of J

_{u}with eigenvalue λ if and only if

_{u}. Solving Equation (26) yields:

^{T}is an eigenvector of J

_{u}with eigenvalue λ, conditions Equations (27) and (28) are valid. By Equation (28) w = (λ + 1)υ. Substituting for w in Equation (27) now proves that υ is an eigenvector of A, unless λ + 1 = 0; that is, λ = −1. We claim that −1 is not an eigenvalue of J

_{u}. If it is, Equation (28) implies that w = 0, and Equation (27) reduces to gΓυ = 0, implying that gΓ = 0. But the parameter g in Equation (5) is greater than 0, and Equation (25) implies that Γ > 0, a contradiction.

**Remark 3**Generically, the quadratic Equation (26) has two distinct solutions. The solutions coincide if

_{P}. Generically in the connection strengths it does not occur.

**Corollary 1**For any α,β,γ, and sufficiently small I, the unique synchronous steady state u is linearly stable.

**Remark 4**The Hopf bifurcation curves in Figures 7, 8, 9, 10, 11, 12, 13 and 14 below, corresponding to hop and run, have segments with negative I. This does not contradict stability of the steady state for sufficiently small I because these segments are not relevant when I varies quasistatically.

**Corollary 2**If (α, β, γ) is not degenerate in the sense of Definition 1, the Jacobian J

_{u}is semisimple (diagonalisable) over ℂ.

_{P}are distinct as P ranges over the four patterns.

^{T}of the eigenvectors implies that distinct patterns P, Q lead to distinct eigenvectors μ

_{p},μ

_{q}, even if the associated eigenvalues are equal. This is the case even when there are degeneracies in the connection strengths (α,β,γ).

_{u}.

#### 8.2. Conditions for Steady-State Bifurcation

**Theorem 5**Suppose that μ is real. Then the eigenvalue λ (evaluated at the synchronous equilibrium u) is zero if and only if the following equivalent conditions hold:

#### 8.3. Conditions for Hopf Bifurcation

**Theorem 6**Suppose that μ is real. Then the eigenvalue λ (evaluated at the synchronous equilibrium u) is purely imaginary if and only if

**Theorem 7**Let μ

_{P}be the eigenvalue for the primary gait P, where P = H, J, R, or W. Then the necessary condition (38) for Hopf bifurcation to gait P is valid if and only if

_{2}× ℤ

_{2}) we obtain 8 eigenvectors by this method, except when there is a solution of multiplicity 2 to Equation (26), which is non-generic as discussed in Theorem 10 below.

## 9. Plots of Hopf Bifurcation Curves

_{P}and in σ. The hop and run curves fold over with a self-intersection. This is a consequence of the map ξ

^{σ}in Equation (18). The upper branch of the curve corresponds to the lower sheet of the cusp surface, and the lower branch of the curve corresponds to the upper sheet of the cusp surface: this follows since I

^{U}is defined by u

^{−}and I

^{L}is defined by u

^{+}in Equation (13).

## 10. Dominant Eigenvalues

_{u}. However, Equation (26) implies that the overall “skeleton” for the partition of (α, β, γ)-space, according to the type of the first bifurcation (relative to I, or equivalently to u), is determined by which of the four eigenvalues μ

_{P}of the connection matrix A is largest. We discuss this issue first.

**Theorem 8**Assume that the degeneracy conditions (22) do not hold. Then the largest eigenvalue of A is:

_{H}> μ

_{J}if and only if α+β+γ > −α+β−γ, that is, α + γ > 0. Five similar comparisons establish the stated conditions.

**Remark 5**If any of the degeneracy conditions (22) is valid, two or more eigenvalues become equal.

^{3}= {(α,β,γ)} into four open regions. Their closure is ℝ

^{3}and the complement of their union comprises values of (α,β,γ) satisfying at least one of the degeneracy conditions (22).

_{P}= c, for P = H, J, R, W, are parallel to the four faces of . In fact, since the vertices of c are (±c, ±c, ±c) with 1 or 3 minus signs, they contain the faces of c . Define the face F

_{P}of corresponding to gait P as follows:

- (1)
- F
_{H}is the face with vertices (−1, 1, 1), (1, −1, 1), (1, 1, −1) - (2)
- F
_{J}is the face with vertices (−1, 1, 1), (1, −1, 1), (−1, −1, −1) - (3)
- F
_{R}is the face with vertices (−1, 1, 1), (1, 1, −1), (−1, −1, −1) - (4)
- F
_{W}is the face with vertices (1, −1, 1), (1, 1, −1), (−1, −1, −1)

**Theorem 9**For each gait pattern P, the region R

_{P}of (α, β, γ) -space in which μ

_{P}is larger than the other three eigenvalues of A is the positive cone

_{P}is the interior of an infinite triangular pyramid obtained by extending to infinity the pyramid with base F

_{P}and vertex the origin. These regions are related by rigid motions, corresponding to symmetries of . Figure 2b illustrates the three-dimensional geometry involved.

## 11. Tetrahedral Structure

_{2}× ℤ

_{2}of the CPG network, and also richer than the parameter symmetries

^{3}acting on α, β, γ. It simplifies the proof of Theorem 3.

^{4}of the tetrahedron acts on the four vertices by permuting them. We will see that this action of the tetrahedral group does not fully explain the tetrahedral structure of the first bifurcations, although it goes part way. Instead, we require a subtler (though related) action. We first discuss the above action.

^{4}on nodes induces a permutation of the three pairs of opposite edges, that is, of the symbols (α, β, γ). This arises via the standard homomorphism

_{2}× ℤ

_{2}. See for example Rotman [43] page 42. Here comprises the parameter symmetries that fix the parameters, that is, the symmetry group ℤ

_{2}× ℤ

_{2}of .

_{4}, also with kernel , and

_{4}acts as rigid motions of

_{4}acts by parameter symmetries, the partition of ℝ

^{3}into regions in Figures 15, 16 and 17 are preserved. This action preserves the face of the large tetrahedron that forms the base of the “hop” region, and permutes the other three regions.

_{3}symmetry of Figure 15, but not the tetrahedral symmetry. To see how this arises, we consider a different group acting on the set of linear forms ±α ± β ± γ that include the four eigenvalues of A. This is the wreath product of order 48, Hall [44]. Here the base group ℤ

_{2}× ℤ

_{2}× ℤ

_{2}changes the signs of α, β, γ, and

_{3}permutes them. Geometrically, is the symmetry group of rigid motions of the cube with vertices (±1, ±1, ±1).

_{1}= α + β + γ,μ

_{2}= −α + β − γ, μ

_{3}= −α − β + γ, μ

_{4}= α − β − γ. These are the forms for which the number of minus signs is even. The group has a subgroup of order 24 that changes signs in pairs. Geometrically, is the symmetry group of rigid motions of the tetrahedron with vertices (±1, ±1, ±1) in which the number of minus signs is 0 or 2.

_{P}: P = H, J, R, W}. It is easy to prove that ≅

^{4}, and it permutes these eigenvalues faithfully.

#### 11.1. Representation-Theoretic Generalities

_{j}are irreducible representations. We further assume that the representations on each summand are non-isomorphic, so that these irreducibles are also the isotypic components of X, that is, the sums of all isomorphic irreducible subspaces. Theorem 3.5 of Chapter XII of [18] proves that the isotypic components are invariant under any matrix A that commutes with G. So A leaves each irreducible component X

_{j}invariant. (In particular, the connection matrix A has this property when G = ℤ

_{2}× ℤ

_{2}.) Let A denote the algebra of all matrices that commute with the G-action.

_{j}∈ X

_{j}is a common eigenvector for the actions of all g ∈ G, and also for A. That is,

**Proposition 6**Suppose that B intertwines so there exists Ã satisfying Equation (46). Let υ be any eigenvector of Ã with eigenvalue μ̃. Then Bυ is an eigenvector of A with eigenvalue μ̃.

_{2}× ℤ

_{2}and

^{4}that acts on the eigenvalues and creates the tetrahedral symmetry in the space of connection strengths α, β, γ.

**Remark 6**The tetrahedral group is not a symmetry group of the rate equations in the sense of equivariance. More generally, it is not induced by a parameter symmetry. This follows since the first bifurcation to a hop gait is a saddle-node, whereas the first bifurcation to a jump, run, or walk gait can be shown to be a pitchfork, as suggested by the normalizer symmetry N

_{G}(Σ)/Σ ≅ ℤ

_{2}. Thus consists of symmetries of the bifurcation varieties, but not of the dynamics (even allowing changes of parameters).

## 12. Proof of the Main Theorem

^{3}= {(α, β, γ)}. We also consider ℝ

^{4}= {(α, β, γ, u)}.

**Definition 2**For each gait pattern P define

**Lemma 3**For given (α, β, γ) and distinct patterns P, Q:

^{3}into four infinite pyramidal regions R

_{P}as in Equation (45), observing that

^{3}and the boundary of their union is given by the degeneracy conditions (22).

**Lemma 4**If ε < abg/4 then

**Lemma 5**Suppose that ε < abg/4. Let P be a gait pattern.

- (1)
- ≅ ∅ ⇔ μ
_{P}≥ k - (2)
- ≅ ∅ ⇔ μ
_{P}≥ m - (3)
- ≅ ∅ ⇔ k < μ
_{P}≤ K - (4)
- ${h}_{\text{\Rho}}^{\alpha \beta \gamma}<{s}_{\text{\Rho}}^{\alpha \beta \gamma}$ if and only if μ
_{P}< K.${h}_{\text{\Rho}}^{\alpha \beta \gamma}>{s}_{\text{\Rho}}^{\alpha \beta \gamma}$ if and only if μ_{P}> K.

- (1)
- ${\mathscr{H}}_{\text{\Rho}}\ne \phantom{0}/\iff \frac{{\mu}_{\text{\Rho}}}{1+\mathit{\varepsilon}}\ge \frac{4}{ab}$, which is equivalent to μ
_{P}≥ k. - (2)
- ${S}_{\text{\Rho}}\ne \phantom{0}/\iff {\mu}_{\text{\Rho}}\ge g+\frac{4}{ab}$ , which is equivalent to μ
_{P}≥ m. - (3)
- ${\overline{\mathscr{H}}}_{\text{\Rho}}\ne \phantom{0}/$ requires μ
_{P}≤ K by Equation (50), and also ${\mathscr{H}}_{\text{\Rho}}\ne \phantom{0}/$ , so k < μ_{P}. - (4)
- Let $s={s}_{\text{\Rho}}^{\alpha \beta \gamma}$ and $h={h}_{\text{\Rho}}^{\alpha \beta \gamma}$. From Equations (48) and (49)$$\begin{array}{cc}\hfill {\mu}_{\text{\Rho}}-g-\frac{{\mu}_{\text{\Rho}}}{1+\mathit{\varepsilon}}& =\frac{ba}{a}\left(\frac{1}{s(a-s)}-\frac{1}{h(a-h)}\right)\hfill \\ \hfill \frac{1}{1+\mathit{\varepsilon}}\left({\mu}_{\text{\Rho}}+\mathit{\varepsilon}{\mu}_{\text{\Rho}}-{\mu}_{\text{\Rho}}-g\right)& =\frac{ba}{a}\left(\frac{h(a-h)-s(a-s)}{s(a-s)h(a-h)}\right)\hfill \\ \hfill \frac{1}{1+\mathit{\varepsilon}}(\mathit{\varepsilon}{\mu}_{\text{\Rho}}-g)& =\frac{ba}{a}\left(\frac{{s}^{2}-{h}^{2}+a(h-s)}{s(a-s)h(a-h)}\right)\hfill \\ \hfill \frac{\mathit{\varepsilon}}{1+\mathit{\varepsilon}}\left({\mu}_{\text{\Rho}}-\frac{1+\mathit{\varepsilon}}{\mathit{\varepsilon}}g\right)& =\frac{ba}{a}\left(\frac{(s-h)(s+h-a)}{s(a-s)h(a-h)}\right)\hfill \\ \hfill sgn({\mu}_{\text{\Rho}}-|K)& =sgn(h-s)\hfill \end{array}$$

_{P}< K, and h > s when μ

_{P}> K, as claimed.

**Lemma 6**The group permutes the sets , , and . The permutations act on the patterns {H, J, R, P} according to the

_{4}-action on the corresponding eigenvalues of A.

_{P}, and do so in exactly the same manner for each gait type.

**Corollary 3**The partition of (α, β, γ) -space into regions that determine the first local bifurcation as I increases from 0 is preserved by the action of . It therefore has tetrahedral symmetry.

**Lemma 7**Let P be a gait pattern. Let R

_{P}be as in Equation (51). Then for (α, β, γ) ∈ R

_{P}the first local bifurcation is:

- (1)
- None if μ
_{P}< k. - (2)
- Hopf of type P if k < μ
_{P}< K. - (3)
- Steady-state of type P if μ
_{P}> K.

_{H}.

_{q}< μ

_{H}when (α, β, γ) ∈ R

_{H}. Therefore, μ

_{q}< k for all four patterns Q when (α, β, γ) ∈ R

_{H}. By Lemma 5 (2), bearing Lemma 4 in mind, there are no local bifurcations in R

_{H}when μ

_{H}< k.

_{H}> k there exists a Hopf bifurcation of type H. By Lemma 5 (1,3,4) this is the first bifurcation provided that k < μ

_{P}< K.

_{H}> K by Lemma 5 (2,4).

_{H}. Apply the group to deduce the other cases R

_{P}when P = J, R, W.

#### 12.1. Plot of Gait Regions

**Remark 7**An alternative approach is possible. We have derived the geometry of first bifurcations using two-dimensional sections in order to make it easier to find parameter values with given behaviour. There is, however, a more direct approach in three dimensions. For each pattern P, the linear form μ

_{P}defines a family of planes μ

_{P}= q for constant q. These planes are the orthogonal complements of four vectors:

_{P}= q are parallel to the opposite faces. Regions in which μ

_{P}< q are open half-spaces bounded by such planes. It is then easy to check that when q = K the planes are the faces of , extended to infinity, and when q = k they are the faces of , extended to infinity For distinct patterns P and Q the planes μ

_{P}= μ

_{q}determine the transitions between first bifurcation to P and first bifurcation to Q, as in Section 13 below. The rest of the geometry then follows.

^{4}, and are analogous to trilinear coordinates {(x

_{1}, x

_{2}, x

_{3}) : x

_{1}+ x

_{2}+ x

_{3}= 0} in ℝ

^{3}, see Loney [45] chapter III. The analogy can be developed further by introducing connections of strength δ from each node to itself in Figure 1. Now the connection matrix A becomes A + δI

_{4}. The eigenvectors for the patterns H, J, R, W are mutually orthogonal (and become orthonormal if they are multiplied by $\frac{1}{2}$). Now , are three-dimensional sections δ = 0 of the corresponding four-dimensional decomposition.

## 13. Degeneracies

_{u}even if they occur. However, the condition (53) below might signal such a Jordan block.

- (1)
- Transition from one primary Hopf mode to a different primary Hopf mode: change of gait.
- (2)
- Transition from one primary steady-state mode to a different primary steady-state mode: change of equilibrium type.
- (3)
- When μ
_{P}= K there is a transition between a primary Hopf mode of symmetry-type P and a steady-state mode of the same symmetry-type. Also, when μ_{P}= k there is a transition between a primary Hopf mode of symmetry-type P and a fully synchronous steady-state mode. These transitions correspond to the onset or cessation of gait P.

_{ij}is zero, and if moreover the corresponding cells j are in synchrony, then their combined input into cell i is zero. That is, cell i is decoupled from the cells j. We therefore have:

- (1)
- If α = − β and the gait is of types H or P, nodes 1, 2 decouple from 3, 4.
- (2)
- If α = −γ and the gait is of types H or J, nodes 1, 3 decouple from 2, 4.
- (3)
- If β = −γ and the gait is of types H or W, nodes 1, 4 decouple from 2, 3.

_{2}× ℤ

_{2}symmetry. It follows that the behaviour of mixed modes (secondary bifurcations) in rate models might not resemble that of generic mixed modes in dynamical systems with given symmetry.

_{u}, namely Equation (31). For convenience we repeat the equation here: the two solutions for λ coincide for given μ when

**Lemma 8**If Equation (53) holds at a point of local bifurcation then

**Theorem 10**If Equation (53) holds at a point of local bifurcation then μ = K.

_{P}= K. By Lemma 5(3) this occurs when the frequency ω of a Hopf bifurcation tends to zero. That is, we find an infinite-period or “blue sky” bifurcation. Typically, this is associated with a homoclinic orbit and arises via a Takens-Bogdanov bifurcation [46]. The restriction of the Jacobian to the critical eigenspace at such a point is not semisimple but nilpotent, which is consistent with the linear degeneracy implied by Equation (53).

_{u}, and the trace of J

_{u}varies continuously with parameters, so coincidence of the values of λ implies that there exists a nontrivial 2 × 2 Jordan block for that value of λ. Diekman et al. [36] find a Takens-Bogdanov bifurcation in the “reduced network” for a rate model with two nodes and ℤ

_{2}symmetry. So we conjecture that the above transition may be associated with such a bifurcation.

## 14. Simulations

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Central Pattern Generator (CPG) network for bipedal gaits, with explicit connections corresponding to a rate model.

**Figure 2.**Tetrahedral partition of connection-space. (

**a**) The two tetrahedra

**(img), (img)**; (

**b**) one of the four primary Hopf regions, in this case the hop gait. Base triangle left unshaded to show interior; (

**c**) Extending the pyramid to define the corresponding steady-state regions, which extend to infinity in the direction indicated.

**Figure 3.**Phase shift patterns for the four primary bipedal gaits. The numbers 0 and 1/2 indicate relative phases of the nodes.

**Figure 4.**Graph of gain function in the typical case a = 1, b = 8, c = 1. Here x runs horizontally and the vertical axis represents (x).

**Figure 5.**Solutions of Equation (12) when a=1, b = 8,c = 1. Here, u runs horizontally and the vertical axis represents the values of (u)(curve) and I + σu (straight lines).

**Figure 6.**(

**a**) The curves I

^{U}, I

^{L}plotted against σ. Parameter values in the gain function are a = 1, b = 8, c = 1; (

**b**) the cusp catastrophe surface in (σ, I, u)-space.

**Figure 7.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = 0.4, β = 0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 8.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = 0.4, β = −0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 9.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = −0.4, β = 0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 10.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = −0.4, β = −0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 11.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = 1, β = 0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 12.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = 1,β = −0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 13.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α = −1, β = 0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 14.**Hopf curves. Parameter values are: a = 1, b = 8, c = 1; g = 1.8, ε = 0.67, α =− 1, β = −0.7. Hop = red, jump = green, run = magenta, walk = blue, cusp = black.

**Figure 16.**Typical set of bifurcation curves for fixed (α, β), showing u against γ. Hop = red, jump = green, run = magenta, walk = blue. Solid lines = Hopf, dashed lines = steady. Parameter values: a = 1, b = 8, c= 1; g = 1.8, ε = 0.67, α = 1, β = 0.7.

**Figure 17.**Regions of (α, β) values corresponding to the four gaits when the “gap” region is excluded. (

**a**) γ > 0; (

**b**) γ < 0. Inner rectangle is empty when γ > k; outer rectangle is empty when γ > K.

**Figure 18.**Regions of (α, β) values corresponding to first bifurcation to the four types of Hopf and four types of steady-state bifurcation, or to no bifurcation. (

**a**) γ > 0; (

**b**) γ < 0. Inner rectangle is empty when γ > k; outer rectangle is empty when γ > K.

**Figure 19.**Hop. Top: nodes 1–4; Bottom: all nodes. Parameter values: a = 1, b = 8, c = 1;ε = 0.67, g = 1.8; α = 0.5, γ = 0.8, β = 0.6; I = 0.8.

**Figure 20.**Run. Top: nodes 1–4. Bottom: all nodes. Parameter values: a = 1, b = 8, c = 1; ε = 0.67, g = 1.8; α = −0.5,γ = 0.8, β = −0.6; I = 1.1.

**Figure 21.**Jump. Top: nodes 1–4. Bottom: all nodes. Parameter values: a = 1, b = 8, c = 1;ε = 0.67, g = 1.8; α = −0.5, γ= −0.8, β = 0.6; I = 1.1.

**Figure 22.**Walk. Top: nodes 1–4. Bottom: all nodes. Parameter values: a = 1, b = 8, c = 1; ε = 0.67, g = 1.8; α = 0.5, γ = −0.8, β = −0.6; I = 1.1.

Gait | Twist ϕ | Σ = ker ϕ | Fix(Σ) | Type |
---|---|---|---|---|

hop | ρ ↦ 0, τ ↦ 0 | ℤ_{2}(ρ) x ℤ_{2}〈τ〉 | {(x, x, x, x )} | H |

jump | ρ ↦π, τ ↦ 0 | ℤ_{2}〈ρ〉 | {(x, y, x, y)} | J |

run | ρ ↦ 0, τ ↦ π | ℤ_{2}(τ) | {(x, x, y, y)} | R |

walk | ρ ↦ π, τ ↦ π | ℤ_{2}〈ρτ〉 | {(x, y, y, x)} | W |

**Table 2.**Necessary and sufficient conditions for each type of first bifurcation when no degeneracy condition holds.

Name | Parameters in A | Conditions involving k, K | |||
---|---|---|---|---|---|

None | μ < _{P}k for all patterns P | ||||

all other types: | |||||

Hop | α + γ>0 | α + β > 0 | γ + β > 0 | k<α+β +γ<K | |

Jump | α + γ<0 | β>γ | β > α | k < −α + β − γ < K | |

Run | α + β < 0 | γ > β | γ > α | k < −α − β + γ < K | |

Walk | γ + β<0 | α > γ | α > β | k < α − β − γ < K | |

Type H steady | α + γ>0 | α + β > 0 | γ + β > 0 | α +β +γ > K | |

Type J steady | α + γ < 0 | β > γ | β > α | −α + β − γ > K | |

Type R steady | α + β < 0 | γ > β | γ > α | −α − β + γ > K | |

Type W steady | γ + β < 0 | α > γ | α > β | α − β − γ > K |

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**MDPI and ACS Style**

Stewart, I. Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator. *Symmetry* **2014**, *6*, 23-66.
https://doi.org/10.3390/sym6010023

**AMA Style**

Stewart I. Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator. *Symmetry*. 2014; 6(1):23-66.
https://doi.org/10.3390/sym6010023

**Chicago/Turabian Style**

Stewart, Ian. 2014. "Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator" *Symmetry* 6, no. 1: 23-66.
https://doi.org/10.3390/sym6010023