Symmetry, bifurcations, and chaos in a distributed respiratory control system

A multivariate model is outlined for a distributed respiratory central pattern generator (RCPG) and its afferent control. Oscillatory behavior of the system depends on structure and symmetry of a matrix of phase-switching functions (F omega, phi) that control distribution of central excitation (CE) and inhibition (CI) within the circuit. The matrix diagonal (F omega) controls activation of CI variables as excitatory inputs are altered (e.g., central and afferent contributions to inspiratory off switch); off-diagonal terms (F phi) distribute excitations within the CI system and produce complex eigenvalues at the switching points between inspiration and expiration. For null F phi, phase switchings of saddle equilibria located at end expiration and end inspiration are overdamped all-or-nothing events; graded control of CI is seen for phi > 0. When coupling is significant (phi >> 0), CI dynamics become underdamped, admitting a domain of inputs where chaotic behavior is predictably observed. For the homogeneous RCPG (symmetric F omega, phi), CE oscillations are one-dimensional limit cycles (D = 1) or weakly chaotic (D approximately equal to 1). When perturbations from symmetry are significant, the distributed RCPG becomes partitioned where strongly chaotic oscillations (D > or = 2) and central apnea (D = 0) are seen more frequently. The equations provide means for mapping Silnikov bifurcations that alter the geometry and dimension of the breathing pattern and formalisms for discussing RCPG processing of afferent information.