Dynamics of a single QIF neuron (<i>N</i> = 1 in Eq (1)) in the bistable regime (a)–(d), and in the tonic regime (e)–(h).

(a): Sketch (not to scale) of the bifurcation diagram of steady states (curve) and periodic solutions (cylinder) of the single QIF neuron subject to a constant input K1 = η1 (ε = 0). A stable quiescent state (down state) coexists with a stable tonic firing solution (up state), separated by an unstable equilibrium (dashed curve). A homoclinic bifurcation is present when K1 = h. In this intrinsically bistable regime (K1 = η1 ∈ (h, 0)), the cell selects the up or down state depending on initial conditions. (b): When 0 ε ≪ 1, K1(t) = η1 + A sin(εt) becomes a slowly varying quantity, oscillating around the value of η1 (ellipses on the K1 axes) with amplitude A, and transitions between the up and the down phases become possible. The onset between phases is determined by a family of canard solutions 1–2 (see text); in the bistable regime they appear in the down-down (green), and down-up (purple) transitions. (c): Time profiles of two solutions for the system with slow input K1(t), displaying a down-down and down-up transition, containing a canard segment (1–2). (d) The solutions in (c) are plotted in the variables (V1, K1), and superimposed on the curve of equilibria of the ε = 0 system (grey parabola), providing evidence of canard behaviour (1–2), and part of the orbits greyed out to enhance visibility. Parameters: ε = 0.01, J = 6, τs = 0.3, η1 = −0.2; A values are reported in the panels. (e): Sketch of the bifurcation diagram of steady states and periodic solutions with constant input (ε = 0) in the tonic regime k1 = η1 > 0). In this regime the cell displays solely the firing solution (up state). (f): When 0 ε ≪ 1 transitions between the up and the down phases become possible, mediated by canard solutions which are possible as up-up and up-down transitions (3–4), but not vice-versa. (g): Time profiles of two solutions in the tonic regime, with slow input K1(t), displaying an up-up and up-down transition, containing a canard segment (3–4). (h): The solutions in (f) are plotted in the variables (V1, K1), and superimposed on the curve of equilibria of the ε = 0 system (grey parabola), providing evidence of canard behaviour (3–4), and part of the orbits greyed out to enhance visibility. Parameters: ε = 0.01 J = 6, τs = 0.3, η1 = 0.5; A values are reported in the panels.</p

Bifurcation diagrams for the deterministic reaction rate equations.

<p>The diagrams are constructed using XPPAUT for equations (1)–(13) and the parameter values given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002396#s2" target="_blank">Results</a>. Numbers of reporter protein molecules produced are plotted against the natural logarithm of the external signal , in the a) autoregulated and b) constitutive cases, showing a bistable and graded response respectively. Bold lines denote stable solutions and dashed lines denote unstable solutions.</p

Simplified diagram of the KZW stochastic TCS model.

<p>(After <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002396#pcbi.1002396-Kierzek1" target="_blank">[15]</a>.)</p

Chemical reactions in the PhoBR <i>E. coli</i> stochastic kinetic model [15].

<p>Reactions included in the KZW stochastic model <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002396#pcbi.1002396-Kierzek1" target="_blank">[15]</a>, together with the associated propensity functions and values of the stochastic rate constants. Values given are for the autoregulated, slow transcription case. The value of varies with the external signal in our simulations. d-mRNA-RR, d-RR, d-mRNA-HK, d-HK, d-mRNA-Rep and d-Rep are degradation products of mRNA-RR, RR, mRNA-HK, HK, mRNA-Rep and Rep respectively. prom-HK is the promoter region of the HK gene.</p

Results of Kierzek, Zhou and Wanner [15].

<p>Histograms of 10,000 realisations at time , as fractions of largest value: for a) autoregulated RR gene & fast transcription of HK; b) autoregulated RR gene & slow transcription of HK; c) constitutively expressed RR gene & fast transcription of HK; d) constitutively expressed RR gene & slow transcription of HK. (Adapted from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002396#pcbi.1002396-Kierzek1" target="_blank">[15]</a>. Reproduced by permission of The Royal Society of Chemistry <a href="http://pubs.rsc.org/en/content/articlelanding/2010/mb/b906951h" target="_blank">http://pubs.rsc.org/en/content/articlelanding/2010/mb/b906951h</a> DOI: 10.1039/B906951H.</p

Correspondence between chemical species and model variables.

<p>Correspondence between chemical species and model variables.</p

Timescales of persistence for approximate basal solution.

<p>Steady states are shown for the mean number of phosphorylated RR dimer molecules, , in equation (70), found using the equation-free numerical approach and poor man's continuation, starting from the lowest value of the external signal and an approximate basal solution or a zero solution and continuing towards higher signal values. The mean number of molecules produced is plotted against the natural logarithm of the external signal in the a) autoregulated fast transcription ( and ), b) autoregulated slow transcription ( and ), c) constitutively expressed fast transcription ( and ) and d) constitutively expressed slow transcription ( and ) cases. Where the two curves coincide, the points are plotted and the points are underlying.</p

Graded response in the constitutively expressed fast transcription case.

<p>The equation-free tracking of the induced state in the constitutively expressed fast transcription case shows a graded response for in one run. This behaviour is lost upon increasing the time horizon to (see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002396#pcbi-1002396-g006" target="_blank">Fig. 6c</a>).</p

Humber region bio-based economy FCM from first workshop.

<p>Thickness of the links denotes the strength of the influence.</p

Modified FCM from the Humber Environmental Managers’ Meeting showing the addition of a link from international instability to fossil fuel price.

<p>Thickness of the links denotes the strength of the influence.</p