Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops

We consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop, when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate-and-fire neuron model. We show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations. We observe that the number of fixed points of the so-called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns. We demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns.

[1]  Foss,et al.  Multistability and delayed recurrent loops. , 1996, Physical review letters.

[2]  Masahiko Morita,et al.  Associative memory with nonmonotone dynamics , 1993, Neural Networks.

[3]  D. Hansel,et al.  Existence and stability of persistent states in large neuronal networks. , 2001, Physical review letters.

[4]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[5]  K. Ikeda,et al.  High-dimensional chaotic behavior in systems with time-delayed feedback , 1987 .

[6]  Michael C. Mackey,et al.  Solution multistability in first-order nonlinear differential delay equations. , 1993, Chaos.

[7]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[8]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

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

[10]  B. Richmond,et al.  Intrinsic dynamics in neuronal networks. I. Theory. , 2000, Journal of neurophysiology.

[11]  Wulfram Gerstner,et al.  SPIKING NEURON MODELS Single Neurons , Populations , Plasticity , 2002 .

[12]  J. Milton,et al.  Epilepsy: multistability in a dynamic disease , 2000 .

[13]  Jianhong Wu,et al.  Existence and attraction of a phase-locked oscillation in a delayed network of two neurons , 2001, Differential and Integral Equations.

[14]  Jianhong Wu,et al.  Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network , 1999 .

[15]  Jianhong Wu,et al.  Slowly Oscillating Periodic Solutions for a Delayed Frustrated Network of Two Neurons , 2001 .

[16]  R. Traub,et al.  Neuronal Networks of the Hippocampus , 1991 .

[17]  Jianhong Wu,et al.  Connecting Orbits from Synchronous Periodic Solutions to Phase-Locked Periodic Solutions in a Delay Differential System , 2000 .

[18]  Shigeru Shinomoto,et al.  Reverberating activity in a neural network with distributed signal transmission delays. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  J. Milton,et al.  Multistability in recurrent neural loops arising from delay. , 2000, Journal of neurophysiology.

[20]  John W. Clark,et al.  Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation , 1997, Biological Cybernetics.

[21]  Jianhong Wu,et al.  The Asymptotic Shapes of Periodic Solutions of a Singular Delay Differential System , 2001 .

[22]  D. A. Baxter,et al.  Multiple modes of activity in a model neuron suggest a novel mechanism for the effects of neuromodulators. , 1994, Journal of neurophysiology.

[23]  Robert M. May,et al.  Dynamical diseases , 1978, Nature.

[24]  U. an der Heiden,et al.  The dynamics of recurrent inhibition , 1984, Journal of mathematical biology.

[25]  Jianfeng Feng,et al.  Is the integrate-and-fire model good enough?--a review , 2001, Neural Networks.

[26]  J. Milton,et al.  NOISE, MULTISTABILITY, AND DELAYED RECURRENT LOOPS , 1996 .

[27]  R. Stoop,et al.  Chaotic Spike Patterns Evoked by Periodic Inhibition of Rat Cortical Neurons , 1997 .

[28]  Michael C. Mackey,et al.  The dynamics of production and destruction: Analytic insight into complex behavior , 1982 .

[29]  Wulfram Gerstner,et al.  Spiking Neuron Models: An Introduction , 2002 .