This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with ℤ_{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.

Animals use many different patterns of locomotion, known as

It is generally believed, with much circumstantial evidence and a growing amount of more specific evidence, that gait patterns in mammals, reptiles, and indeed many other animals are determined by a relatively simple network of neurons in the spinal cord, known as a Central Pattern Generator (CPG). In a series of papers, Kopell and Ermentrout [

Collins and Stewart [

Golubitsky _{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.

These results are model-independent, in the sense that no specific equations are employed to model the CPG dynamics. The analysis applies to all such equations, provided a Hopf bifurcation of the appropriate kind occurs and appropriate nondegeneracy conditions are satisfied, as they are generically. Buono [

Golubitsky et al. [_{2} × ℤ_{2} symmetry. See

Here we revisit this four-node CPG, interpreting it as a coupled cell network in the sense of [

We consider local bifurcations from a fully synchronous equilibrium _{u}_{u}

We focus on one approach to these bifurcations which seems a reasonable way to model gaits, and reveals an elegant underlying structure having a richer group of symmetries than the rate equations themselves. Specifically, we consider the

In this context, we derive explicit conditions on connection strengths for the first local bifurcation to be one of four symmetry types of steady-state bifurcation, or one of the four symmetry types of Hopf bifurcation identified by Pinto and Golubitsky [

The specific form of rate equations, in which the argument of the gain function is a linear combination of state variables, implies that the four regions corresponding to primary gaits are connected polyhedra. They all have the same shape and size: each is a truncated pyramid with an equilateral triangle as base. They fit together to form a hollow tetrahedron centred at the origin, as in

The four primary gait regions are relatively large when

Despite the nonlinear form of the equations, it turns out to be possible to derive much of the bifurcation behaviour analytically. The possibility of doing this stems from a pleasant property of a gain function commonly used in rate equations, combined with linearity of the argument of the gain function in the equations. The rate equations themselves have a symmetry group 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.

Rate models in a network with the same symmetries are analysed in Diekman

We do not discuss the stability of the bifurcating states here, to avoid complicating an already lengthy paper. However, simulations suggest that all of the symmetry types of states that we describe can exist stably for some ranges of parameters.

Other types of dynamics can occur in this model, including apparent chaos, but the parameter regions concerned seem to be small. We do not discuss more exotic possibilities here. Curtu [_{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

This is the first in a projected series of papers. Others will discuss stability issues and investigate quadruped and hexapod gaits from a similar point of view. New features enter into the analysis, in part because the eigenvalues of the connection matrix can now be complex.

In Section 2 we introduce gain functions, the connection matrix, and rate models. We establish some key properties of the family of gain functions used here. Section 3 describes the CPG network used to model biped gaits, sets up the corresponding rate equations, and discusses parameter symmetries used later to simplify the calculations. We also describe the four primary gaits hop, jump, run, and walk in the context of symmetric Hopf bifurcation, following Pinto and Golubitsky [

Section 4 analyses the fully synchronous steady states

Section 5 reviews basic existence results for local bifurcation (steady-state and Hopf) in symmetric ODEs. Section 6 summarises basic data for primary gaits: eigenvalues and eigenvectors of the connection matrix, symmetries of bifurcating branches for steady-state and Hopf bifurcation. We state the main theorem in Section 7.

Section 8 examines how the eigenvalues of the Jacobian evaluated at a fully synchronous equilibrium

Section 10 analyses which eigenvalue of the connection matrix is largest, as the connection strengths vary. The largest eigenvalue is closely associated with the first bifurcation to a state with the corresponding symmetry type. This principle is exploited in the proof of the main theorem, but first we discuss a symmetry group

We briefly discuss degeneracies in the connection strengths in Section 13. Most of these correspond to transitions between different first bifurcations, that is, to mode interactions. We observe that in some cases disjoint sets of nodes decouple, a feature of rate models that is non-generic in general symmetric dynamical systems. Finally, Section 14 shows simulations of the rate equations giving the four primary gaits.

In a rate model, the state of node

The original choice of

The function
^{−1} given by

Let

We also note that each member of this family of gain functions is symmetric about its inflection point. Specifically,

Indeed,

We specify the rate model in terms of the _{ij}_{i}

The model

Following Pinto and Golubitsky [

This network has symmetry group _{2} 〈_{2} 〈

There are also parameter symmetries, of two kinds. First: any permutation of the four nodes produces the same network topology, but the connection strengths are permuted. For example

We employ the abbreviated notation of ^{T}. Let ^{E}^{E}^{H}^{H}

These are the same equations as

Here the only parameter that changes is ^{E}_{1}

For the network of

This network is bidirectional (identical arrows point both ways) so

To preserve symmetry, we take all inputs _{j}_{j}

Here the parameters are:

The equations inherit the symmetry of the network, so they are equivariant under the action of _{2} × ℤ_{2} defined by
^{−1} to define a left action, but in this case every element of

_{2} × ℤ_{2}-symmetric system at Hopf bifurcation, as classified in Pinto and Golubitsky [_{2} × ℤ_{2} on the critical (purely imaginary) eigenspace [

This system can also undergo steady-state bifurcation, either symmetry-preserving or symmetry-breaking. This possibility is examined in Section 8.

As a preliminary step, we analyse fully synchronous steady states in the rate

The main result generalises to any homogeneous network. When the network is homogeneous, the diagonal

Let the gain function be as in

The function

We prove:

^{L}, I^{U} depending on^{L}^{u}, and two solutions (one of multiplicity 2) when I^{L}, I^{U}

Consider a fully synchronous equilibrium at which all nodes have

Apply

The geometry of
^{L}, ^{U}^{L}^{U} ,^{L}^{U}

The derivative of

The tangencies with lines of slope

Solve

From ^{±}

Solutions are real when

Now substitute in

In ^{L}^{U}

If we fix a value of ^{U}^{−}. As we show later, there is always a Hopf bifurcation before that point. However, that does not rule out a

For each (

Geometrically, the image of ^{σ}

This maps
^{σ}^{σ}

^{σ}

^{σ}

^{σ}^{−}) → (−∞,I^{U}) ^{−}

The proof is a restatement of the analysis in Theorem 1 and

We can therefore use ^{U}

By Lemma 1, if
^{U} ,^{U}

This proposition is decisive because the geometry of the bifurcation curves is much simpler in terms of

Local bifurcations that are not the first can be converted from ^{σ}

Our main goal is to analyse the time-periodic gait patterns generated by the four-cell CPG of

However, steady-state bifurcations also occur in the rate equations. These correspond to various “stand” gaits, where the animal is motionless, which may not be of great interest. (However, there is a difference between passive standing and “getting ready” to initiate a periodic gait, which is one way to interpret these symmetry-breaking equilibria.) At any rate, the analysis has to take them into account. We discussed the bifurcation of synchronous (that is, symmetry-preserving) steady states in Section 5. Here we review general theorems about symmetry-breaking local bifurcations. We state the Equivariant Branching Lemma for the existence of steady-state bifurcations with specified symmetry types. We also summarise the Equivariant Hopf Theorem and specify our notational conventions. This helps to avoid potential confusion in the later analysis, because several different conventions exist in the literature. We follow [

Let G be a group (technically, a compact Lie group, which includes all finite groups) acting linearly on ℝ^{n}^{n}^{n} ×^{n}

It follows that if

Assume that as _{0} there is a path of equilibria {(_{0}_{0}) for which the Jacobian

Steady-state bifurcation in the presence of symmetry is governed by the Equivariant Branching Lemma of Cicogna [

The ^{n}

Conversely, if Σ is a subgroup of

This space is invariant under the normaliser _{g}_{g}

If ^{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

Proposition 3.2 of Chapter XIII of [

_{(x0,λ0)} _{0} = _{0}). _{0} _{0}

A proof of the Equivariant Branching Lemma of Cicogna [

_{(x0,x0)} _{0} _{0}

_{0}, λ_{0}),

Note that it is the action of Σ on _{0} that is required to have a 1-dimensional fixed-point space, not the action on ℝ^{n}.

The Equivariant Hopf Theorem provides information about the spatio-temporal symmetries of bifurcating branches of periodic states. It also applies in circumstances where the classical Hopf bifurcation theorem does not, namely multiple eigenvalues caused by symmetry.

Let
_{2}_{π}^{n}

Then ^{1} acts on
_{2}_{π}

A map _{2}_{π}

So, transformation of ^{1} for which

W ≅ V ⊕

Proposition 1.4 of [

_{(x0,λ0)} _{0} = _{0}) _{iω} _{iω}

It is clear that _{i}_{ω}^{1}-action. Define the

We can define an action of
^{1} on _{i}_{ω}

By equivariance, ^{1} acts on _{i}_{ω}

We say that the eigenvalues ±i_{(}_{x}_{,λ(}_{x}_{))} for _{0}, such that

The Equivariant Hopf Theorem of [

_{(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

Then there exists a unique branch of small-amplitude time-periodic solutions to _{0}, λ_{0}), with period near

Notice that we use the linearised action of _{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.

In the generic case where the critical eigenspace is

The group

When _{2} × ℤ_{2}) a

Bifurcation depends on the eigenvalues of the Jacobian, and we will see that these are closely related to the eigenvalues of the connection matrix _{2} × ℤ_{2} on ℝ^{4} is:

By equivariance or direct verification, these are the eigenvectors of

The next proposition is trivial but important. Equalities between eigenvalues of

As well as

The first of these conditions is equivalent to

Pinto and Golubitsky [_{2} × ℤ_{2}-symmetric network. We reproduce that table in the notation of this paper as _{2} × ℤ_{2}. Since the group is abelian, this is also the normaliser of Fix(Σ). Recall that

The same four isotropy groups determine the possible actions on a critical eigenspace _{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

We can now state the main theorem of this paper. It provides a complete characterisation of the regions of parameter space for which the first bifurcation is of a given type, as illustrated in

Theorem 3

Assume that k < K, that is, ε < abg/4. Suppose that μ_{P} ≠ k, K and none of the conditions

We develop the proof of this theorem in stages in the following sections.

We now derive conditions for local bifurcations of given symmetry type (that is, spatial isotropy subgroup) H, J, R, or W. Subject to technical genericity conditions, a local bifurcation occurs when the Jacobian _{u}

It turns out that for any rate model with _{u}_{u}

Write the variables in the order
^{T} is an eigenvector of J_{u} with eigenvalue λ if and only if

Moreover, every eigenvector is of this form.

Let ^{n}^{T} is an eigenvector of _{u}

Then

This suggests taking

Therefore, if _{u}

Finally, we must show that every eigenvector is of this form. If an arbitrary [^{T} is an eigenvector of _{u}_{u}

We discuss this condition in Theorem 10, and prove that at any local bifurcation it holds only when _{P}. Generically in the connection strengths it does not occur.

Let

In fact, the subsequent analysis of local bifurcations shows that

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

Generically in connection-strengths, the eigenvalues _{P} are distinct as P ranges over the four patterns.

Generically in ^{T} of the eigenvectors implies that distinct patterns P, Q lead to distinct eigenvectors _{p},_{q}, even if the associated eigenvalues are equal. This is the case even when there are degeneracies in the connection strengths (

Corollary 2 shows that in the local bifurcation analysis for nondegenerate connection strengths, Theorem 4 determines all eigenvalues and eigenvectors of _{u}

We seek conditions for the corresponding

Set

This leads to

Then use

We seek conditions for the corresponding eigenvalues

By

Substitute in

In the simulations of Section 14,

We can now derive a necessary condition for Hopf bifurcation from a synchronous equilibrium, for the primary gaits.

_{P} be the eigenvalue for the primary gait P, where P = H, J, R, or W. Then the necessary condition

Let

Also

Since _{2} × ℤ_{2}) we obtain 8 eigenvectors by this method, except when there is a solution of multiplicity 2 to

We now use

These plots illustrate the main possibilities and motivate the subsequent analysis.

Note that

These plots share some general features. The curves for jump and walk extend towards _{P} and in ^{σ} in ^{U}^{−} and ^{L}^{+} in

In some of these plots, there is no Hopf bifurcation for some parameter values. For example, this is the case for

Local bifurcation from a fully synchronous state _{u}_{P} of the connection matrix

This follows directly from _{H} _{J} if and only if

The conditions ^{3} = {(^{3} and the complement of their union comprises values of (

It is convenient to draw these regions by considering constant-

In three dimensions the structure of this partition is tetrahedral, and we can describe it in terms of the tetrahedron

For any constant _{P} = _{P} of

_{H} is the face with vertices (−1, 1, 1), (1, −1, 1), (1, 1, −1)

_{J} is the face with vertices (−1, 1, 1), (1, −1, 1), (−1, −1, −1)

_{R} is the face with vertices (−1, 1, 1), (1, 1, −1), (−1, −1, −1)

_{W} is the face with vertices (1, −1, 1), (1, 1, −1), (−1, −1, −1)

Routine calculations, or an examination of

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

In more detail, _{P} is the interior of an infinite triangular pyramid obtained by extending to infinity the pyramid with base _{P} and vertex the origin. These regions are related by rigid motions, corresponding to symmetries of

A direct derivation in three dimensions is sketched in Remark 7 below.

As _{2} × ℤ_{2} of the CPG network, and also richer than the parameter symmetries
^{3} acting on

The CPG

The symmetry group
^{4} of the tetrahedron acts on the four vertices by permuting them. We will see that this action of the tetrahedral group does

The action of
^{4} on nodes induces a permutation of the three pairs of opposite edges, that is, of the symbols (_{2} × ℤ_{2}. See for example Rotman [_{2} × ℤ_{2} of

There is an induced action of
_{4}, also with kernel
_{4} acts as rigid motions of

Since
_{4} acts by parameter symmetries, the partition of ℝ^{3} into regions in

These parameter symmetries explain the
_{3} symmetry of _{2} × ℤ_{2} × ℤ_{2} changes the signs of _{3} permutes them. Geometrically,

Four of these forms are the eigenvalues of _{1} = _{2} = −_{3} = −_{4} =

By definition,
_{P} : P = H, J, R, W}. It is easy to prove that
^{4}, and it permutes these eigenvalues faithfully.

We now describe a general representation-theoretic context that explains this structure, and can be used for other networks with an abelian symmetry group.

Suppose that _{j}_{j}_{2} × ℤ_{2}.) Let

When _{j}_{j}

If

For the four-cell network, _{2} × ℤ_{2} and

We take

The extension of 〈−id〉 splits, with complement the subgroup of signed permutations having an even number of minus signs. This is the group isomorphic to
^{4} that acts on the eigenvalues and creates the tetrahedral symmetry in the space of connection strengths

_{G}_{2}. Thus

We now complete the proof of Theorem 3.

By Proposition 2 we can derive the geometry of bifurcation varieties for first bifurcation by employing

First, we set up some notation for the various bifurcation varieties and curves that arise. Let P be a gait pattern (P = H, J, R, W for hop, jump, run, walk respectively). We assume that

The remaining parameters define the parameter space ℝ^{3} = {(^{4} = {(

By

By

By Theorem 6 Hopf bifurcation occurs if and only if

For fixed (

Which of these bifurcations occurs first (or none) is determined by the relative positions of the eight curves. Since they have explicit, simple equations, it is straightforward to derive necessary and sufficient conditions. The tetrahedral symmetry

For each (

Then

By

For given (

Necessary conditions affecting this ordering are:

This follows immediately from

We divide ℝ^{3} into four infinite pyramidal regions _{P} as in

These regions are determined by the inequalities on ^{3} and the boundary of their union is given by the degeneracy conditions

To find the conditions for a given type of first bifurcation, we consider type H and then appeal to the tetrahedral symmetry group

Define three positive constants, of which

_{P} ≥

_{P} ≥

_{P} ≤

_{P} < K.

_{P} > K.

The function

_{P} ≥

_{P} ≥

_{P} ≤ _{P}.

Let

_{P}<

_{P}>

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

For all four primary gaits P, the sets
_{P}, and do so in exactly the same manner for each gait type.

_{P} be as in _{P} the first local bifurcation is:

_{P} <

_{P} <

_{P} >

Let P = H and specialise to the region _{H}.

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

If _{H} > _{P} <

Finally, the first bifurcation is steady-state of type H when _{H} >

This proves the lemma for region _{H}. Apply the group
_{P} when P = J, R, W.

Theorem 3 is an explicit statement of this lemma in terms of the connection strengths, so the proof is complete.

We now complete the geometry of

For fixed

The “no bifurcation” region is

_{P} defines a family of planes _{P} =

These vectors correspond to the vertices of
_{P} = _{P} < _{P} = _{q}

The four vectors ^{4}, and are analogous to trilinear coordinates {(_{1}, _{2}, _{3}) : _{1} + _{2}_{3} = 0} in ℝ^{3}, see Loney [_{4}. The eigenvectors for the patterns H, J, R, W are mutually orthogonal (and become orthonormal if they are multiplied by

We now discuss degenerate bifurcations, which occur when one or more of the degeneracy conditions _{u}

From the geometry of

Three main types of mode interaction occur along a generic path in (

Transition from one primary Hopf mode to a different primary Hopf mode: change of gait.

Transition from one primary steady-state mode to a different primary steady-state mode: change of equilibrium type.

When _{P} = _{P} =

We discuss transitions (1, 2) first. Rate equations have a special form, even for more general gain functions
_{ij}

If

If

The decoupling implies that the dynamics depends on the initial conditions. In particular, phase shifts between the corresponding pairs of nodes can be arbitrary. This behaviour would be non-generic in a general dynamical system, even one with ℤ_{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.

Cancellations involving more nodes can also occur, for example when

This type of decoupling is a general feature of all rate models.

There can be a similar “approximate decoupling” near a point of Hopf bifurcation when nodes are in antisynchrony. Here the linearised eigenfunctions are very close to

A transition of type (3) is associated with the other potential source of degeneracy in the associated eigenvalues _{u}

Suppose the bifurcation is steady-state. By

Substitute in

On the other hand, if the bifurcation is Hopf then

Substitute in

By Lemma 8, Γ =

At a Hopf bifurcation

In other words, the degeneracy in _{P} =

Theorem 4 characterises _{u}_{u}_{2} symmetry. So we conjecture that the above transition may be associated with such a bifurcation.

From

I am grateful to Marty Golubitsky for introducing me to rate equations and for many helpful conversations.

The author declares no conflict of interest.

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

Tetrahedral partition of connection-space. (

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

Graph of gain function in the typical case

Solutions of

(^{U} , I^{L}

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Hopf curves. Parameter values are:

Regions of (

Typical set of bifurcation curves for fixed (

Regions of (

Regions of (

Hop.

Run.

Jump.

Walk.

Spatio-temporal isotropy subgroups of the primary gaits.

Σ = ker |
Fix(Σ) | |||
---|---|---|---|---|

hop | ℤ_{2}(_{2}〈 |
{( |
H | |

jump | ℤ_{2}〈 |
{( |
J | |

run | ℤ_{2}( |
{( |
R | |

walk | ℤ_{2}〈 |
{( |
W |

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

| |||||
---|---|---|---|---|---|

_{P} | |||||

| |||||

Hop | |||||

Jump | |||||

Run | |||||

Walk | |||||

| |||||

Type H steady | |||||

Type J steady | − | ||||

Type R steady | − | ||||

Type W steady |