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System State
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v w \(a\): \(v\): \(w\): Parameters \(a_0\): \(p\): \(T_s\): |
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Initializing... |
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The dynamical system are described by the equations \begin{equation} \frac{d v}{dt} = v - \frac{v^3}{3} - w + I_\mathrm{ext}\left(t\right), \end{equation} \begin{equation} \frac{d w}{dt} = \frac{1}{\tau}\left(v + a\right), \end{equation} and \begin{equation} \frac{d a}{dt} = \frac{1}{\tau_a}\left(a_c - p w - a\right) \end{equation} where \(a_c = \left(1-p\right)a_0 + p a_0^3/3\). The control parameters are \(a_0\) and \(p\). It is an adaptive oscillator that rest at a fixed point in the absence of stimulus. The periodic input stimulus current \(I_\mathrm{ext}\) has an amplitude 1.
The attractor manifold as shown in translucent cyan is defined at the zero adaptation limit (adaptation time \(\tau_a = \infty\)). The cylinder part is spanned by the limit cycle in \(v\)-\(w\) plane, at fixed \(a<1\) while the curvy line is extended by the fixed point at \(a≥1\). The state of the system in the \(a\)–\(v\)–\(w\) space is presented by the tiny red pearl. The box frame shown in the 3D space represents the unit volume in the first octant.
The trajectory of the system is shown in blue in the absense of external stimulus. In the presence of periodic stimulus, the trajectory is shown in red when \(I_\mathrm{ext}=1\) for the first half of a stimulus period and in olive when \(I_\mathrm{ext}=0\) in the second half of a stimulus period.
Blue trace: Reduced membrane potential \(v\). Green trace: Voltager-gated restoring ion channel current \(w\). Pink trace: Stimulus current \(I_\mathrm{ext}\).
The visualized simulation is written in JavaScript using WebGL. The third party JS libraries used include gl-matrix.js for matrix operations and hand.js for pointer event normalization.