View source for LEGION: locally excitatory globally inhibitory oscillator networks

==Basic Oscillator Model==
[[Image:LEGION_Fig1.png|thumb|300px|right|f2|Enabled state of a [[relaxation oscillator]]. The arrows indicate the direction of motion, and double arrows indicate jumping.]]
[[Image:LEGION_Fig2.png|thumb|300px|right|f3|Excitable state of a [[relaxation oscillator]].]]
In the standard formulation, Terman and Wang (1995) define the basic oscillator of a LEGION network as a reciprocally connected pair of excitatory variable <math>x</math> and inhibitory variable <math>y\ :</math>
:<math>
\begin{matrix}\dot{x}&=&3x-x^3+2-y+I\\\dot{y}&=&\varepsilon(\alpha(1+\tanh(x/\beta))-y)
\end{matrix}
</math>
Here, <math>I</math> denotes the external stimulation to the oscillator. The parameter <math>\varepsilon</math> is positive and it is chosen so that <math>0<\varepsilon<<1\ .</math> In this case, the above equations define a typical [[relaxation oscillator]] with two time scales induced by <math>\varepsilon\ .</math> The <math>x</math>-nullcline (i.e., the set defined by the condition <math>\dot{x}=0</math>) is a cubic function. The <math>y</math>-nullcline is a sigmoidal function with parameters <math>\alpha\,</math> and <math>\beta\,\ .</math> 

If <math>I > 0\ ,</math> the two nullclines intersect only at a point along the middle branch of the cubic, and in this case the oscillator produces a stable [[limit cycle]] and is called ''enabled''.  The limit cycle alternates between an ''active phase'' of relatively high <math>x</math> values and a ''silent phase'' of relatively low <math>x</math> values. This is illustrated in <figref>LEGION_Fig1.png</figref>. Within each of the two phases the oscillator exhibits near steady-state behavior, and in the silent phase it tracks the left branch (LB) of the cubic and in the active phase the right branch (RB). In contrast to the behavior within each phase, the transition between the two phases takes place rapidly, and it is referred to as ''jumping''. Such alternations between fast change and slow change are characteristic of relaxation oscillations (van der Pol 1926). The parameter <math>\alpha\,</math> determines the relative durations of the two phases - a larger <math>\alpha\,</math> produces a relatively shorter active phase.

If <math>I<0\ ,</math> the two nullclines of the above equations intersect at a stable [[equilibrium]] point on the LB of the cubic (see <figref>LEGION_Fig2.png</figref>). In this case no oscillation occurs, and the oscillator is called ''[[Excitability|excitable]]''. Whether the oscillator is enabled or excitable depends on external stimulation, and hence oscillations defined by the equations are stimulus-dependent.

The relaxation oscillator model defined above may be interpreted either as a model of [[action potential]] generation, where <math>x</math> represents the [[membrane potential]] of a neuron and <math>y</math> represents a recovery variable (e.g., the level of activation of [[ion channels]]), or an [[Bursting|oscillating burst]] of neuronal spikes where <math>x</math> represents the envelope of the burst. <figref>LEGION_Fig3.png</figref> shows a typical trace of <math>x</math> activity, akin to a spike train. In fact, the equations are dynamically very similar to standard neuronal models, including the [[FitzHugh-Nagumo model]] and the [[Morris-Lecar model]]. All of these models can be viewed as simplifications of the classic [[Hodgkin-Huxley model]].
[[Image:LEGION_Fig3.png|thumb|300px|right|f4|Activity of a relaxation oscillator with respect to time.]]