Terman-Wang OscillatorTerman-Wang Oscillator
Neurophysiological experiments have demonstrated that neural oscillations occur in the visual cortex, olfactory cortex, auditory cortex and other brain areas. Such oscillations may underlie the mechanism by which we are able to bind together information in disparate regions of the brain to form coherent mental objects. More specifically, we may solve this "binding problem" by an oscillatory correlation mechanism. In such a scheme, neuronal oscillators representing features of the same object are synchronized, and are desynchronised from oscillators representing features of different objects.
The Terman-Wang oscillator belongs to the same family of "relaxation" oscillators as the van der Pol oscillator. It is defined as a feedback loop between an excitatory variable x and an inhibitory variable y:
dx/dt = 3x-x^3+2-y+I
dy/dt = eta(gamma(1+tanh(x/beta)-y))
Here, eta, gamma and beta are constant parameters, and I is the external input to the oscillator.
When analysing a dynamical system, it is informative to consider the nullclines, which can be obtained by setting dx/dt and dy/dt to zero. For the Terman-Wang oscillator,
3x-x^3+2-y+I = 0 (x-nullcline)
eta(gamma(1+tanh(x/beta)-y)) = 0 (x-nullcline)
If I>0, these nullclines intersect only at a point along the middle of the x-nullcline (a cubic) when beta is small. In this case, the oscillator produces a stable periodic oscillation (so long as eta is sufficiently small) - a neural oscillation. The periodic solution alternates between near-steady-states of relatively high x values (the "active phase") and relatively low x values (the "silent phase"). Compared to motion in each phase, the transition between phases takes place rapidly (a process called "jumping").
If I<0, the nullclines intersect at a stable fixed point. In this case, no oscillation occurs.
Sliders are provided for the external input to the oscillator, and for each of the parameters eta, gamma and beta (1). Press the Start simulation button (2) to start the simulation for the parameter values shown. The activity of the oscillator is shown in two ways - a plot of the phase plane (x plotted against y) (3) and a plot in which the excitatory variable x is plotted against time (4). In the phase plane, the point corresponding to the current value of x and y is shown as a red dot (5). The rapid jumping between the active phase and silent phase should be clearly apparent in the phase plane.
Play with the parameters and get a feel for the behaviour that this interesting dynamical system can exhibit. The Preset menu will set the parameters to illustrate three kinds of behaviour - a fixed point (no activity), a slow oscillation and a fast oscillation.
You may also choose whether to display the nullclines in the phase-plane plot (6).
1. What happens to the x-nullcline when you change the input? What happens to the y-nullcline? Is this what you would expect from the equations above?
D. Terman and D. L. Wang (1995) Global competition and local cooperation in a network of neural oscillators. Physica D, 81, pp. 148-176.
See also the demonstrations for the Wang neural oscillator network (wangNetwork) and vowel segregation using neural oscillations (vowelSeg).
Produced by: Guy J. Brown
Release date: June 22 1998
Permissions: This demonstration may be used and modified freely by anyone. It may be distributed in unmodified form.