Mode locking and Arnold tongues in integrate-and-fire neural oscillators.

An analysis of mode-locked solutions that may arise in periodically forced integrate-and-fire (IF) neural oscillators is introduced based upon a firing map formulation of the dynamics. A q:p mode-locked solution is identified with a spike train in which p firing events occur in a period qDelta, where Delta is the forcing period. A linear stability analysis of the map of firing times around such solutions allows the determination of the Arnold tongue structure for regions in parameter space where stable solutions exist. The analysis is verified against direct numerical simulations for both a sinusoidally forced IF system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current. This approach is extended to the case of two synaptically coupled IF oscillators, showing that mode-locked states can exist for some self-consistently determined common period of repetitive firing. Numerical simulations show that such solutions have a bursting structure where regions of spiking activity are interspersed with quiescent periods before repeating. The influence of the synaptic current upon the Arnold tongue structure is explored in the regime of weak coupling.

[1]  J. Rogers Chaos , 1876, Molecular Vibrations.

[2]  Rene,et al.  THE AMERICAN JOURNAL OF PHYSIOLOGY. , 1897, Science.

[3]  H E M Journal of Neurophysiology , 1938, Nature.

[4]  H. Kalmus Biological Cybernetics , 1972, Nature.

[5]  G. Ermentrout n:m Phase-locking of weakly coupled oscillators , 1981 .

[6]  Haim Sompolinsky,et al.  STATISTICAL MECHANICS OF NEURAL NETWORKS , 1988 .

[7]  M. V. Rossum,et al.  In Neural Computation , 2022 .

[8]  Physical Review Letters 63 , 1989 .

[9]  Albrecht Rau,et al.  Statistical mechanics of neural networks , 1992 .

[10]  R. Calabrese,et al.  Motor-pattern-generating networks in invertebrates: modeling our way toward understanding , 1992, Trends in Neurosciences.

[11]  J. Deuchars,et al.  Temporal and spatial properties of local circuits in neocortex , 1994, Trends in Neurosciences.

[12]  Sauer,et al.  Reconstruction of dynamical systems from interspike intervals. , 1994, Physical review letters.

[13]  T. Sejnowski,et al.  Reliability of spike timing in neocortical neurons. , 1995, Science.

[14]  David K. Campbell,et al.  Piecewise linear models for the quasiperiodic transition to chaos. , 1995, Chaos.

[15]  Kaplan,et al.  Subthreshold dynamics in periodically stimulated squid giant axons. , 1996, Physical review letters.

[16]  R. MacKay,et al.  Transition to topological chaos for circle maps , 1996 .

[17]  L. Abbott,et al.  Synaptic Depression and Cortical Gain Control , 1997, Science.

[18]  H. Markram,et al.  The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[19]  G. Lord,et al.  Intrinsic modulation of pulse-coupled integrate-and-fire neurons , 1997 .

[20]  P. Bressloff,et al.  PHYSICS OF THE EXTENDED NEURON , 1997 .

[21]  G D Lewen,et al.  Reproducibility and Variability in Neural Spike Trains , 1997, Science.

[22]  André Longtin,et al.  Interspike interval attractors from chaotically driven neuron models , 1997 .

[23]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[24]  André Longtin,et al.  Stochastic and Deterministic Resonances for Excitable Systems , 1998 .

[25]  James P. Keener,et al.  Mathematical physiology , 1998 .

[26]  P. Bressloff,et al.  Desynchronization, Mode Locking, and Bursting in Strongly Coupled Integrate-and-Fire Oscillators , 1998 .

[27]  P. Bressloff,et al.  Spike train dynamics underlying pattern formation in integrate-and-fire oscillator networks , 1998 .

[28]  Carson C. Chow Phase-locking in weakly heterogeneous neuronal networks , 1997, cond-mat/9709220.

[29]  J. D. Hunter,et al.  Resonance effect for neural spike time reliability. , 1998, Journal of neurophysiology.

[30]  S. Coombes Liapunov exponents and mode-locked solutions for integrate-and-fire dynamical systems , 1999 .

[31]  Taishin Nomura,et al.  SYNTHETIC ANALYSIS OF PERIODICALLY STIMULATED EXCITABLE AND OSCILLATORY MEMBRANE MODELS , 1999 .

[32]  P. Bressloff,et al.  Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays , 1999 .

[33]  T. Deguchi,et al.  International Journal of Modern Physics B, ❢c World Scientific Publishing Company , 2001 .

[34]  October I Physical Review Letters , 2022 .