Graphical model showing how a tipping point for cortical spreading depression can arise.

<p>a) Three equilibriums may occur at intersection points where the rate of generation of new pulses (sigmoidal curve) equals the rate of decay (dashed line) of neural pulses. Activity increases when the generation of new pulses exceeds the decay of pulses (sections I and III) and decreases in the other sections (sections II and IV). It can be seen from the arrows representing this direction of change that the middle intersection point is a repellor that marks the border between the basins of attraction of the two alternative stable states. b) Increasing base-line excitability promotes the generation of new pulses causing the unstable equilibrium (open dot) and the stable normal state (left hand solid dot) to move closer together. This reduces resilience of the normal state in the sense that a smaller perturbation is needed to invoke a shift to the Aura state (horizontal dashed arrows in panel). c) Plotting how the intersection points representing equilibriums move as a function of base-line excitability, a catastrophe fold arises. The fold bifurcation point (F) marks the loss of stability of the normal state.</p

Stability landscape interpretation of how resilience of the normal mode of brain activity can be lost at high levels of base-line excitability as determined by genetically coded or other physiological conditions.

<p>The catastrophe fold at the base plane corresponds to the one depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0072514#pone-0072514-g002" target="_blank">figure 2c</a>.</p

Causal structure that may lead to a tipping point for autonomous firing, as illustrated by the minimal model.

<p>Causal structure that may lead to a tipping point for autonomous firing, as illustrated by the minimal model.</p

Temperature dependence of denitrification rates at constant low dissolved oxygen levels.

<p>Denitrification rates measured in vegetated freshwater microcosms, as compared to the fitted Arrhenius equation (dotted black line; equation 1), the modelled denitrification rates (black line; equations 2–4 with parameters from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018508#pone-0018508-t002" target="_blank">Table 2</a>), and modelled denitrification with O₂ fixed at 1 mg l⁻¹(dashed blue line; equation 2).</p

Model parameter values.

<p>Model parameter values.</p

Body size data of Australian diving beetles

Excel file containing 1) body sizes of individuals of diving beetle species collected in the field, 2) body size ranges for additional beetle species, taken from Watts and Humphreys 200

Schematic overview of major direct and indirect effects of temperature on denitrification.

<p>Schematic overview of major direct and indirect effects of temperature on denitrification.</p

Example systems with alternative stable states in space.

<p>(<i>a</i>) Shallow lake: clear water with Chara vegetation vs. turbid water (photo by Ruurd Noordhuis). (<i>b</i>) Salt marsh: vegetation vs. bare marshland (photo by Johan van de Koppel). (<i>c</i>) Musselbed: mussels vs. bare soil (photo by Andre Meijboom).</p

Critical size of a local disturbance and the speed of a travelling wave as a function of the maximal mortality rate <i>c</i>.

<p><i>(a)</i> On an infinitely sized landscape, disturbances smaller than the critical size <i>Δx</i> (in m) are repaired, while larger disturbances will initiate a propagating wave that travels through the landscape with <i>(b)</i> a constant wave speed (in m d⁻¹). The thick dashed line represents the Maxwell point. The thin dashed lines represent the two fold bifurcations in a non-spatial system. Left of the Maxwell point the entire landscape was initially set to the low biomass state, and the disturbance was set to the high biomass state. Right of the Maxwell point the landscape was initially set to the high biomass state, and the disturbance was set to the low biomass state (indicated by the small upper panels). In this model, an <i>n</i>-fold increase in diffusion rate leads to a </p><p></p><p></p><p></p><p><mi>n</mi></p><p></p><p></p><p></p>-fold increase in both critical disturbance size and wave speed.<p></p

Appendix C. Reliability in identification of increasing trends in variance and autocorrelation.

Reliability in identification of increasing trends in variance and autocorrelation