Abstract
Weakly coupled phase oscillators and strongly coupled relaxation oscillators have different mechanisms for creating stable phase lags. Many oscillations in central pattern generators combine features of each type of coupling: local networks composed of strongly coupled relaxation oscillators are weakly coupled to similar local networks. This paper analyzes the phase lags produced by this combination of mechanisms and shows how the parameters of a local network, such as the decay time of inhibition, can affect the phase lags between the local networks. The analysis is motivated by the crayfish central pattern generator used for swimming, and uses techniques from geometrical singular perturbation theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bizzi, E., Giszter, S.F., Loeb, E., Mussa-Ivaldi, F.A., Saltiel, P.: Modular organization of motor behavior in the frog's spinal cord. Trends in Neuroscience 18, 442–446 (1995)
Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana University Mathematics Journal 21(3), 193–225 (1971)
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eqns. 31, 53–98 (1979)
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, New York, 1977
Izhikevich, E.M.: Phase equations for relaxation oscillations. Siam J. Appl. Math 60, 1789–1805 (2000)
Jones, S.R.: Rhythms in the neocortex and in CPG neurons: A dynamical systems analysis. Ph.D., Boston University, 2001
Jones, S.R., Mulloney, B., Kaper, T.J., Kopell, N.: Coordination of cellular pattern-generating circuits that control movements: the sources of stable differences in intersegmental phases. J. Neurosci. 15 (2), 283–298 (2003)
Kopell, N.: Chains of coupled oscillators. Brain Theory and Neural Networks M. A. Arbib (ed.) MIT Press, Cambridge, 1995
Kopell, N., Ermentrout, G.B.: Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. In: B. Elsevier, (ed.) Fiedler Handbook on Dynamical Systems, Vol 2. Toward applications 2002, pp 3–54
Levinson, N.: A second order differential equation with singular solutions. Ann. Math. 50, 127–153 (1949)
Levinson, N.: Perturbations of discontinuous solutions of nonlinear systems of differential equations. Acta Mathematica 82, 71–106 (1950)
Mishchenko, E.F., Kolesov, Yu.S., Kolesov, A.Yu., Rozov, N.Kh.: Asymptotic methods in singularly perturbed systems. Consultants bureau, New York and London, 1994
Mishchenko, E.F., Rozov, N.Kh.: Differential equations with small parameters and relaxation oscillations. Plenum Press, New York, 1980
Murchinson, D., Chrachri, A., Mulloney B.: A separate local pattern-generating circuit controls the movement of each swimmeret in crayfish. J. Neurophysiol. 70, 2620–2631 (1993)
Nipp, K.: Invariant manifolds of singularly perturbed ordinary differential equations. Z.A.M.P., 36, 309–320 (1985)
Rand, R.H., Cohen, A.H., Holmes, P.J.: Systems of coupled oscillators as models of centeral pattern generators. In: A.H., Cohen, S., Grillner, S., Rossignol, (eds.) Neural Control of Rhythmic Movements in Vertebrates Wiley, New York, 1987, pp. 369–413
Rowat, P.F., Selverston, A.I.: Oscillatory mechanisms in pairs of neurons connected with fast inhibitory synapses. J. Comput. Neurosci. 4 (2), 103–127 (1997)
Rubin, J.E., Terman, D.: Geometric analysis of population rhythms in synaptically coupled neuronal networks. Neural Comp. 12(3), 597–648 (2000a)
Rubin, J.E., Terman, D.: Analysis of clustered firing patterns in synaptically coupled networks of oscillators. J. Math. Biol. 41, 513–545 (2000b)
Skinner, F.K., Kopell, N., Marder, E.: Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J. Comp. Neuro. 1, 69–87 (1994)
Skinner, F.K., Mulloney, B.: Intersegmental coordination of limb movements during locomotion: Mathematical models predict circuits that drive swimmeret beating. J. Neuro. Sci. 18 (10), 3831–3842 (1998)
Somers, D., Kopell, N.: Rapid Synchronization through fast threshold modulation. Biol. Cybern 68 (5), 393–407 (1993)
Somers, D., Kopell, N.: Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J. Math. Biol. 33 (3), 261–280 (1995)
Taylor, A.L., Cottrell, G.W., Kristan, W.B. Jr.: Analysis of oscillations in a reciprocally inhibitory network with synaptic depression. Neural. Comp. 14 (3), 561–581 (2002)
Tikhonov, A.N.: On the dependence of the solutions of differential equations on small parameter. Mat. Sb. 31, 575–586 (1948)
Wang, X.-J., Rinzel, J.: Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural. Comp. 4, 84–97 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jones, S., Kopell, N. Local network parameters can affect inter-network phase lags in central pattern generators. J. Math. Biol. 52, 115–140 (2006). https://doi.org/10.1007/s00285-005-0348-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-005-0348-0