Loss of synchronization in partially coupled Hodgkin–Huxley equations

Golubitsky

2006

J. Math. Biol.

We use symmetry to study two central pattern generator (CPG) models for biped locomotion. The first one is a coupled four-cell network, proposed by Golubitsky, Stewart, Buono, and Collins, that models rhythms associated to legs. A classification based on symmetry shows that this network can produce periodic solutions with rhythms corresponding to the standard bipedal gaits of run, walk, hop, gallop, and skip, among others. Moreover, the four-cell model can produce two types of hop, two types of gallop, and three additional symmetry types of periodic solutions that have yet to be identified with the rhythms of known bipedal gaits. The second locomotor CPG network models interlimb coordination in bipeds (arms+legs). It is obtained by breaking the symmetry between fore and hind legs in an eight-cell CPG network for quadruped gaits, also proposed by Golubitsky et al. We match the rhythms of perturbed periodic solutions found in this eight-cell network with legs rhythms produced by the four-cell CPG model. We also compare patterns of oscillation of gaits of the eight-cell model with results on bipedal interlimb coordination in the literature, showing that the eight-cell model is a plausible network for modeling human interlimb coordination.We show numerical simulations of periodic solutions corresponding to the bipedal gaits in the two CPG models. These simulations use clamped Hodgkin-Huxley equations to model cell internal dynamics and partial linear coupling (where only the electrical potentials of different cells are coupled). We use synaptic coupling in the four-cell model and diffusive coupling in the eight-cell model.

“…The eigenvalues of H − (k i + k j + k l ), i = j = l have been computed in [22] for different values of I. This computation was used to obtain bifurcation to the four primary gaits.…”

Section: Example 22 We Consider Hodgkin-huxley Equationsmentioning

confidence: 99%

Central pattern generators for bipedal locomotion

Golubitsky

2006

J. Math. Biol.

“…The stability of the numerical solutions in section 2.3 are computed using the eigenvalues of matrices L 4 leg , L 65 , L 6 , L 5 . For more information see and Labouriau and Alves-Pinto (2004). Let H be the matrix of the linearization of the uncoupled Hodgkin-Huxley equations and let J 3 1 j il 3 i2l be the matrix with only one nonzero entry j 11 3 1.…”

Section: Bifurcation and Stability Of Gaitsmentioning

confidence: 99%

“…Eigenvalues of H 91k i 7k j 7k l 3, i 3 j 3 l, are computed in Labouriau and Alves-Pinto (2004). This may be used to compute bifurcation into the four primary gaits.…”

Section: Numerical Simulations In Two Cpg Models 1493mentioning

confidence: 99%

Numerical Simulations in Two CPG Models for Bipedal Locomotion

Journal of Vibration and Control

2007

Pinto and Golubitsky proposed a new model for a central pattern generator (CPG) for arm/leg coordination in bipeds, the network arm. arm is derived from a CPG quadruped model (quad) by breaking the symmetry between fore- and hind limbs. This paper explores numerically some of the solution properties of two general CPG models: arm and a CPG model for leg rhythms in bipeds (which is called leg). It is assumed that the internal dynamics of each cell is modeled by a space-clamped Hodgkin—Huxley equation and the coupling is only through the voltage term. Numerical simulations are done using XPPAUT. Secondary gaits in leg are obtained from bifurcation of the Poincaré map of a primary gait. Examples of secondary gaits are gallop, asymmetric hop, one-legged hop, and skip. In particular, the gallop is obtained by bifurcation from the primary gait two-legged jump. Numerical evidence of this is shown. Numerical simulations are given of periodic solutions in arm identified with arm/leg coordination patterns in bipeds.

“…Networks of coupled cells may arise as models of animal and robot locomotion, speciation, visual perception, electric power grids, internet communication, secure communication [7][8][9][10][11]20,21,24,26,28,30,31], and many others.…”

Section: Introductionmentioning

confidence: 99%

Exciting dynamical behavior in a network of two coupled rings of Chen oscillators

2014

Nonlinear Dyn

We study exotic patterns appearing in a network of coupled Chen oscillators. Namely, we consider a network of two rings coupled through a "buffer" cell, with Z 3 × Z 5 symmetry group. Numerical simulations of the network reveal steady states, rotating waves in one ring and quasiperiodic behavior in the other, and chaotic states in the two rings, to name a few. The different patterns seem to arise through a sequence of Hopf bifurcations, period-doubling, and halving-period bifurcations. The network architecture seems to explain certain observed features, such as equilibria and the rotating waves, whereas the properties of the chaotic oscillator may explain others, such as the quasiperiodic and chaotic states. We use XPPAUT and MATLAB to compute numerically the relevant states.