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

Values of the kinetic parameters used to simulate some of the dynamics of the adenylate energy system.

<p>Values of the kinetic parameters used to simulate some of the dynamics of the adenylate energy system.</p

Emergence of oscillations in the AEC (Scenario II).

<p>Different oscillatory behavior appears when varying r₂, controlling the ADP time delays. (a) For r₂ = 37 s the AEC periodically oscillates with a very low relative amplitude of 0.045. (b–c) Existence of complex AEC oscillatory patterns for: (b) r₂ = 72 s and (c) r2 = 94 s. (d–e) AEC transitions between different oscillatory behavior and steady state patterns for several r₂ values. (d) 50 s, 27 s, 30 s, 32 s, 33 s, 72 s, 52 s. (e) 50 s, 27 s, 30 s, 32 s, 34 s, 36 s, 33 s, 36 s, 38 s, 40 s.</p

AEC dynamics under low production of ATP.

<p>AEC values as a function of time. At very small values (), which represents a strong reduction of the ATP synthesis due to low substrate intake, the dynamic of the adenylate energy system shows a steady state behavior that slowly starts to descend, in a monotone way, up to reach the lowest energy values (AEC ∼0.59) at which the steady state loses stability and oscillatory patterns emerge with a decreasing trend. Finally, when the maximum of the energy charge oscillations reaches a very small value (AEC ∼0.28) the adenylate system suddenly collapses after 12,000 seconds of temporal evolution.</p

Dynamical solutions of Scenario I.

<p>For  = 1.02 (normal activity for the ATP synthesis), periodic oscillations emerge. (a) ATP concentrations. (b) ADP concentrations. (c) AMP concentrations. (d) The Gibbs free energy change for ATP hydrolysis to ADP. (e) The total adenine nucleotide (TAN) pool. It can be observed that ATP and ADP oscillate in anti-phase (the ATP maximum concentration corresponds to the ADP minimum concentration). Likewise, it is noted that the total adenine nucleotide pool shows very small amplitude of only 0.27 and a period around 65 s. (f) ATP transitions between different periodic oscillations and a steady state pattern for several values of (0.97, 1.08, 1.02, 0.97). Maxima and minima values per oscillation are shown in y-axis.</p

Numerical analysis for the model of the adenylate energy system.

<p>a–c: (cf. Scenario I in text) In y-axis we are plotting the max and the min of the different variables α, β and γ. For situations with no oscillations (stable fixed point colored in solid black lines) the max and the min are coincident. For situations with oscillations, the max and the min of the oscillations are plotted separately; in blue we are coloring the max of the oscillation, in red, its minimum value. is the control parameter. The numerical integration shows simple solutions. For small values () the adenine nucleotide concentrations present different stable steady states which lose stability at a Hopf bifurcation at ∼1. For , the attractor is a stable limit cycle. d–f: (Scenario II) The delay r₂ is the control parameter. The numerical bifurcation analysis reveals that the temporal structure is complex, emerging 5 Hopf bifurcations as well as a secondary bifurcation of Neimark-Sacker type. Two pairs of Hopf bifurcations are connected in the parameter space. A third supercritical Hopf bifurcation occurs at r₂∼71.94, rapidly followed by another Hopf bifurcation, subcritical, at r₂∼72.83. This marks the beginning of the region where the system is multi-stable. The last Hopf bifurcation, born at r₂∼72.83, which is subcritical exhibiting the presence of several Torus bifurcations, occurs on a branch of limit cycles when a pair of complex-conjugated Floquet multipliers, leave the unit circle. Branches of stable (resp. unstable) steady states are represented by solid (resp. dashed) black lines; branches of stable (resp. unstable) limit cycles are represented by the max of the oscillation in blue and the minimum in red and by solid (resp. dashed). Hopf bifurcation points are black dots labeled H; Torus bifurcation points are blue dots labeled TR. The bifurcation parameters (Scenario I) and r₂ (Scenario II) are represented on the horizontal axis. The max and min values of each variable are represented on the vertical axis.</p

Elemental biochemical processes involved in the energy status of cells.

<p>The synthesis sources of ATP are coupled to energy-consumption processes through a network of enzymatic reactions which, interconverting ATP, ADP and AMP, shapes a permanent cycle of synthesis-degradation for the adenine nucleotides. This dynamic functional structure defines the elemental processes of the adenylate energy network, a thermodynamically open system able to accept, store, and supply energy to cells.</p