PROPAGATING COALITIONS IN NETWORKS OF NONLINEAR OSCILLATORS (2006)

A brain can perform pattern recognition tasks that are not yet possible for an electronic computer. We continue here our investigation of electronic circuits that are inspired by knowledge of structures in a brain. These circuits are oscillators that are motivated by principles of neuroscience, but yet are constructible as micro circuits, and possibly as nano-circuits. Populations of such oscillators can exhibit patterns in their output frequencies. This may be in response to an external signal being applied across the population or in response to internal waves that propagate through the population. We have investigated the former phenomenon where coalitions of oscillators, classified by their output frequency, form in response to a common driving signal. The resulting patterns have been used to characterize the input signal and they serve as a basis for comparison of signals and other pattern recognition tasks. In this paper, we investigate pattern formation that results from the propagation of synchronized activity waves within the population of oscillators. The novelty here is in the model: We derive a nonlinear wave equation to describe networks of oscillators, and simulate solutions to demonstrate some basic results. We use knowledge of how a brain works to design networks of electronic circuits that can perform pattern recognition tasks. The circuits are inspired by electronic circuits that have been derived from Hodgkin's and Huxley's seminal work (Hodgkin, 1948, 1952; Huxley, 1952). Their work shows that (1) there are mechanisms to maintain a membrane potential in the absence of stimulation, (2) there is negative differential resistance partly due to the opening and closing of ionic channels, (3) there is a homeostatic mechanism that returns the membrane potential to rest, and (4) charge accumulates on both sides of the membrane. A useful model of a neuron focuses on an Action Potential Generator (APG) region that can be described in terms of the membrane potential in that region (Hoppensteadt, 2004). There is substantial experimental evidence that the electrical dynamics of an APG is similar to that of a mechanical pendulum (Hodgkin, 1948; Hoppensteadt, 1997; Izhikevich, 2004). For reference here, an oscillator is a device, possibly at rest, that can sustain regular oscillations when stimulated with constant input. For example, we refer to a pendulum, even at rest, as being an oscillator since it can perform regular oscillations when a constant torque is applied to the support point. Its frequencies might not be apparent in the absence of external forcing. The state of a pendulum is naturally described in terms of an angle variable. The novelty of this work is in the derivation from neuroscience principles of a nonlinear wave equation (4) that supports isolated waves of synchronization through a population of oscillators, and we provide an illustrative simulation of its solution structure in a two dimensional array. Pendulum equations appear in many investigations ranging from neuroscience to quan- tum mechanics, and the angle variable that they govern are related to integral of a voltage. For example, in quantum mechanics the dynamics of a Josephson junction (Feynman, et