The dopamine bifurcation map.
The delay-difference model that started the HCS. Ramp the presynaptic-activity parameter q and
watch a single orbit settle, bifurcate, turn chaotic — and then leave through infinity and come back from
the other side.
The model
One equation, three slowly drifting parameters.
The map is a delay-difference equation originally written to model the dynamics of midbrain dopaminergic neurons — part of an effort to understand on–off phenomena in Parkinson's disease and catastrophe-theory accounts of schizophrenia. Each iterate is computed from the one before it:
q is a measure of presynaptic activity; s and δ are the model's other
two parameters.
Nothing is held fixed. Starting from q = 0, s = 0.35, δ = 7.6 and
x = 0, every step lowers q by Q / 50 000 while s grows by
0.000178 and δ by 0.000162. So the horizontal axis of the bifurcation map is time and parameter at
once: a single orbit, driven slowly through parameter space, draws its own bifurcation diagram. The slider sets
Q, how fast q is pushed down.
Clearing the fractions shows where the trouble comes from:
1 + q + s is positive the denominator can never vanish and the orbit stays bounded. Once
the ramp drives it negative, the denominator is zero at
x(n) = 3.8 · (1 ± √(−(1 + q + s) / 3.46 s)) — two poles. An iterate that lands near either one is
thrown to an enormous value, positive on one side of the pole and negative on the other.
The dashed line marked 1+q+s = 0 on the bifurcation map is the step where that happens. Hover
past it and the two poles appear as dashed lines on the phase portrait.
Reading the plots
The same orbit, two ways.
-
Bifurcation map (right). Every dot is one iterate,
x(n+1), plotted against its step numbern; the second row of tick labels givesqat that step. A single line is a fixed point, two interleaved lines a period-2 cycle, and so on; a smeared band is chaos. -
Lagged phase portrait (left). The same iterates plotted as
(x(n), x(n+1))— each point's position is the previous value against the next. This traces the map's own curve, and fixed points sit on the faint diagonalx(n+1) = x(n). - Colour is time. The prismatic spectrum used in the plotter: red early, through yellow, green and blue, to magenta late. The excursions through infinity happen in the yellow and green stretch.
- Linear axis: the interesting part falls off the chart. With the linear axis, any iterate beyond the range is pinned to the edge as a short tick, and the readout counts how many there are. The linear plot can only show that they left.
-
HCS axis: nothing is lost. Switch the axis to HCS and every value is mapped with
θ = 180 · (x/s) / (1 + |x/s|). The top and bottom edges are +∞ and −∞, which on the HCS circle are the same point. A branch that leaves through the top edge and returns from the bottom is a single continuous excursion through infinity. In the phase portrait both axes wrap this way, so its opposite edges join and the picture is really drawn on a torus.
source to Dopamine bifurcation, and lag-embed it into as many nested circles as
you like. The ramp there is this page's default, Q = 60.