Fig 1.
Biological processes and schematic representation of DA-D2 self-regulation.
A: Striatal DA activates D2 autoreceptors, which triggers a reaction cascade through the G protein Gα/β that subsequently increases DA reuptake and decreases DA release. VGCC represents the voltage-gated calcium channels and kV1, kV2 represent the voltage-gated potassium channels. This picture is adapted from [30] B: Schematic diagram of the Dopamine Ultradian Synaptic Regulation (DUSR) model. Variables are defined in Table 1.
Table 1.
Variables of the DUSR model.
Table 2.
The DUSR model parameters.
Fig 2.
Simulated output of the core DUSR model.
An ultradian rhythm (period = 4.0 h) for key output variables: extracellular dopamine (DAex), activated D2-autoreceptors (D2AR), dopamine transporter activity (TDA), and DAergic neuron firing rate (F). F is plotted as a ratio over the maximum firing rate Fmax. Vertical lines indicate the peak phases of DA; red circles on the plotted lines mark the peak phase of the respective variable. F of the release loop phase-leads D2AR and TDA of the removal loop.
Fig 3.
Local sensitivity of the simulated ultradian period.
Local period sensitivity analysis performed at the nominal period of 4.0 h using Eq (7); Nominal parameter values listed in Table 2. The period is in general most sensitive to parameters unique to the release loop, moderately sensitive to parameters shared by both feedback loops, less sensitive to parameters unique to the removal loop, and not sensitive to parameters not directly involved in DA feedback (β).
Fig 4.
Bifurcation analysis of the core DUSR model.
A: Bifurcation diagrams for nine parameters with top local period sensitivity. Black curves indicate the stable (solid) and unstable (dotted) steady states for DAex. Coloured curves identify the limit-cycle oscillatory behaviours (max, min of the oscillation; period in colours). Stable DA equilibrium is replaced with self-sustained oscillations marked at the supercritical Hopf bifurcation point (red circle) and re-emerges from a saddle-node bifurcation (yellow star). An infinite period solution appears at the moment of the saddle-node bifurcation, forming a saddle-node on invariant cycle (SNIC) bifurcation and is characterized by heightened sensitivity of both the presence and the period of the DA oscillation to parameter values. B: The corresponding ultradian period at parameter values within the stable limit-cycle range. The majority of the period flexibility occurs within ±2 h of the nominal period (4.0 h) when shifting individual parameters within their oscillatory range. DAex timecourses for a long period (12 h) and a short period (3 h) are shown above and below the parameter legend.
Fig 5.
Circadian regulation on the ultradian rhythm.
The effect of an inhibitory circadian-signal input (yellow line) Rcirc to V0 on extracellular dopamine concentration DAex (black line) over time. A: Circadian inputs of different waveforms consolidate ultradian DA episodes within a circadian active phase. B: Increasing the circadian signal’s strength substitutes its consolidating effect on the ultradian rhythm with a complete masking effect. C: Circadian-signal modification has a bidirectional effect on the ultradian rhythm. Depending on the DUSR system’s intrinsic oscillatory state, removal of circadian regulation results in either a more pronounced ultradian rhythm (upper panel) or the loss of quasi-ultradian episodes (lower panel).
Fig 6.
Behavioural activity output of the DUSR under circadian regulation.
Double-plotted activity defined as when DAex values exceed the previous day’s average, Eq (11). A: Behaviour activity for circadian signals of different waveforms and same amplitude. Red lines mark the peak circadian input and blue lines mark the trough. B: Behaviour activity output for increasing amplitudes of a symmetric sinusoidal circadian signal is increasingly refined within the subjective night.
Fig 7.
Phase-dependent effect of a transient excitatory pulse on the ultradian oscillation.
A: DAex behaviour in response to a single pulse input (Isens = 75, duration = 15 min) given at various phases. The black solid line represents the DAex trajectory without additional input. The dotted and solid coloured lines represent DAex trajectories during and after the pulse, with colours denoting the ultradian phase when the pulse is given. B: Phase response curve and amplitude response curve of DAex to a pulse input of various magnitude/duration combinations. The evoked phase shift and the percentage change of maximum-minimum DAex values within the immediate cycle are plotted against the timing of a centred pulse on the ultradian cycle.
Fig 8.
Circadian-ultradian interaction enhances responsiveness to transient stimulus.
Simulated DAex trajectories (black line) of the DUSR model with an inhibitory circadian signal (yellow curve) and a transient excitatory pulse (vertical red line). A: The effect of a transient excitatory pulse on the DUSR model is amplified by the circadian inhibitory signal. Plotted from top to bottom are the DAex response trajectories to (I) an excitatory pulse of the DUSR model, (II) DUSR with a constant inhibitory signal, (III) with a circadian inhibitory signal, (IV) with a masking circadian inhibitory signal, (V) with a circadian inhibitory signal and the ultradian DA feedback loop fixed so that D2AR and TDA are held constant and DAex decreases in scale. B: Effect of an identical transient excitatory pulse given at the same phase of on DAex under increasing circadian regulation. C: Effect of an identical transient excitatory pulse given at different circadian phases on DAex.