Grazing bifurcation and mode-locking in reconstructing chaotic dynamics with a leaky integrate-and-fire model

We examined the firing patterns of a chaotically forced leaky integrate-and-fire (LIF) model, and the validity of reconstructing input chaotic dynamics from an observed spike sequence. We generated inputs to the model from the Rössler system at various values of the bifurcation parameter, and carried out numerical simulations of the LIF model forced by each input. For both chaotic and periodic inputs, therotation numbers and the Lyapunov exponents were calculated to investigate the mode-locked behavior of the system. Similar behaviors as in the periodically forced LIF model were also observed in the chaotically forced LIF model. We observed (i)grazing bifurcation with the emergence of qualitatively distinct behaviors separated by a certain border in the parameter space, and (ii) modelocked regions where the output spike sequences are modelocked to the chaotic inputs. We found that thegrazing bifurcation is related to the reconstruction of chaotic dynamics with the LIF. Our results can explain why the shape of the partially reconstructed ISI attractor, which was observed in previous studies.

[1]  P. Bressloff,et al.  Mode locking and Arnold tongues in integrate-and-fire neural oscillators. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  J. Rinzel,et al.  INTEGRATE-AND-FIRE MODELS OF NERVE MEMBRANE RESPONSE TO OSCILLATORY INPUT. , 1981 .

[3]  P. C. Bressloff,et al.  Erratum: Mode locking and Arnold tongues in integrate-and-fire neural oscillators [Phys. Rev. E 60, 2086 (1999)] , 2001 .

[4]  S. Coombes,et al.  Mode locking in a periodically forced integrate-and-fire-or-burst neuron model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Timothy D. Sauer,et al.  Reconstruction of Integrate-and-Fire Dynamics , 1996 .

[6]  K. Pakdaman.  Periodically forced leaky integrate-and-fire model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[9]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[10]  F. Rhodes,et al.  Topologies and Rotation Numbers for Families of Monotone Functions on the Circle , 1991 .

[11]  Peter Grigg,et al.  ENCODING CHAOS IN NEURAL SPIKE TRAINS , 1998 .

[12]  A. Hodgkin.  The local electric changes associated with repetitive action in a non‐medullated axon , 1948, The Journal of physiology.

[13]  C. Morris,et al.  Voltage oscillations in the barnacle giant muscle fiber. , 1981, Biophysical journal.

[14]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[15]  C. Budd.  Non-Smooth Dynamical Systems and the Grazing Bifurcation , 1996 .

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

[17]  F. Rhodes,et al.  Rotation Numbers for Monotone Functions on the Circle , 1986 .

[18]  Hideyuki Suzuki,et al.  Analysis of neural spike trains with interspike interval reconstruction , 2000, Biological Cybernetics.

[19]  G. Ermentrout,et al.  Analysis of neural excitability and oscillations , 1989 .