Phase models and clustering in networks of oscillators with delayed coupling

We consider a general model for a network of oscillators with time delayed, circulant coupling. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to study the existence and stability of cluster solutions. Cluster solutions are phase locked solutions where the oscillators separate into groups. Oscillators within a group are synchronized while those in different groups are phase-locked. We give model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies

A Moving Bump in a Continuous Manifold: A Comprehensive Study of the Tracking Dynamics of Continuous Attractor Neural Networks

Understanding how the dynamics of a neural network is shaped by the network structure, and consequently how the network structure facilitates the functions implemented by the neural system, is at the core of using mathematical models to elucidate brain functions. This study investigates the tracking dynamics of continuous attractor neural networks (CANNs). Due to the translational invariance of neuronal recurrent interactions, CANNs can hold a continuous family of stationary states. They form a continuous manifold in which the neural system is neutrally stable. We systematically explore how this property facilitates the tracking performance of a CANN, which is believed to have clear correspondence with brain functions. By using the wave functions of the quantum harmonic oscillator as the basis, we demonstrate how the dynamics of a CANN is decomposed into different motion modes, corresponding to distortions in the amplitude, position, width or skewness of the network state. We then develop a perturbative approach that utilizes the dominating movement of the network's stationary states in the state space. This method allows us to approximate the network dynamics up to an arbitrary accuracy depending on the order of perturbation used. We quantify the distortions of a Gaussian bump during tracking, and study their effects on the tracking performance. Results are obtained on the maximum speed for a moving stimulus to be trackable and the reaction time for the network to catch up with an abrupt change in the stimulus.Comment: 43 pages, 10 figure

Evolutionary robotics and neuroscience

No description supplie

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback

A Neural Model of Biased Oscillations in Aplysia Head-Waving Behavior

A long-term bias in the exploratory head-waving behavior of Aplysia can be induced using bright lights as an aversive stimulus: coupling onset of the lights with head movements to one side results in a bias away from that side (Cook & Carew, 1986). This bias has been interpreted as a form of operant conditioning, and has previously been simulated with a neural network model based on associative synaptic facilitation (Raymond, Baxter, Buonomano, & Byrne, 1992). In this article we simulate the head-waving behavior using a recurrent gated dipole, a nonlinear dynamical neural model that has previously been used to explain various data including oscillatory behavior in biological pacemakers. Within the recurrent gated dipole, two channels operate antagonistically to generate oscillations, which drive the side-to-side head waving. The frequency of oscillations depends on transmitter mobilization dynamics, which exhibit both short- and long-term adaptation. We assume that light onset results in a nonspecific increase in arousal to both channels of the dipole. Repeated pairing of arousal increments with activation of one channel (the "reinforced" channel) of the dipole leads to a bias in transmitter dynamics, which causes the oscillation to last a shorter time on the reinforced channel than on the non-reinforced channel. Our model provides a parsimonious explanation of the observed behavior, and it avoids some of the unexpected results obtained with the Raymond et al. model. In addition, our model makes predictions concerning the rate of onset and extinction of the biases, and it suggests new lines of experimentation to test the nature of the head-waving behavior.Office of Naval Research (N00014-92-J-4015, N00014-91-J-4100, N0014-92-J-1309); Air Force Office of Scientific Research (F49620-92-J-0499); A.P. Sloan Foundation (BR-3122

Statistical Mechanics of Recurrent Neural Networks I. Statics

A lecture notes style review of the equilibrium statistical mechanics of recurrent neural networks with discrete and continuous neurons (e.g. Ising, coupled-oscillators). To be published in the Handbook of Biological Physics (North-Holland). Accompanied by a similar review (part II) dealing with the dynamics.Comment: 49 pages, LaTe