Contributors:Antonio Politi, Lawrence D. Gaspers, Andrew P. Thomas, Thomas Höfer
IP3 turnover time controls feedbacks. (A) Positive-feedback model. Dynamics of [Ca2+]c, [IP3], and the fraction of open IP3Rs (solid, dashed, and dot-dashed lines, respectively) during an oscillation period; the fraction of open IP3Rs is given by [rcp/(c+Ka)/(p+Kp)]3 (see Eq. 6). Fast IP3 turnover yields a pronounced spike (left panel, τp=0.1 s), whereas slow IP3 turnover supports only a small-amplitude response (right panel, τp=15 s). (B) The negative-feedback model shows the opposite behavior, with a small-amplitude response for fast IP3 turnover (left panel, τp=0.1s) and a sharp spike for slow IP3 turnover (left panel, τp=15s). (C) Positive-feedback model. Bifurcation diagram showing the maxima and minima of the [Ca2+]c oscillations as a function of the stimulus for different values of the IP3 turnover. The bifurcation diagrams for different values of τp are compared by plotting them against the product V¯PLC=VPLCτp;; in this way, the steady-state concentrations of Ca2+ and IP3 are identical for a given V¯PLC (solid and dashed lines indicate stable and unstable states, respectively; the stability of the steady state is shown for τp=15s). Both amplitude and range of stimuli leading to oscillations increase with faster IP3 turnover. (D) The corresponding bifurcation diagrams for the negative-feedback model show the opposite behavior. The amplitude and range of stimuli leading to oscillations increase with slower IP3 turnover. (E, F) Dependence of frequency encoding on IP3 turnover in the positive and negative feedback models, respectively. Shown are the differences ΔT between the largest (for low stimulation) and smallest (for high stimulation) oscillation period.
... Agonist-induced IP3 and Ca2+ oscillations in the positive and negative feedback models. (A) Positive feedback model with Ca2+ activation of PLC. Changes in [Ca2+]c, [IP3], and in the fraction of active receptors r (top, middle, and bottom panels) after stepwise increases in the agonist concentration (arrowheads), modeled by an increase in the maximal rate of PLC (VPLC=0.3μM/s for toscillations (thick lines) and the [Ca2+]c steady states (thin lines) as a function of the stimulus (VPLC). Solid and dashed lines indicate stable and unstable states, respectively. HB, Hopf bifurcation; HC, homoclinic bifurcation; SN, saddle-node bifurcation; FB, saddle node of periodics. (D) Bifurcation diagram for negative feedback model. PD, period doubling; TR, torus bifurcation. Between PD and HB1 and TR and FB there exist complex oscillations (omitted for clarity). The parameter values used are listed in Table 1.
... Complex responses to an IP3 buffer in the positive-feedback model. (A) Bifurcation diagram showing the region of oscillations as function of stimulus (VPLC) and IP3 buffer concentration (gray-shaded area; the solid lines indicate the locus where the steady state becomes unstable via a Hopf bifurcation). In region I, regular oscillations have a decreased rate of [Ca2+]c rise with increased [IP3 buffer] as shown in Fig. 6 A. In region II, the IP3 buffer abolishes the Ca2+ oscillations completely, as shown in Fig. 6 C. In region III, bursting [Ca2+]c oscillations are observed (the lower boundary of this region is determined by a period doubling bifurcation, dotted line). We have indicated an additional region IV, which is characterized by oscillations persisting even at high [IP3 buffer]. The parameters are as in Fig. 6, with ε=2 When the strength of the Ca2+ plasma-mem... Control coefficients for the oscillation period. (A, B) Positive and negative feedback models, respectively, in the absence of Ca2+ fluxes across the plasma membrane (ε=0); control coefficients of Ca2+ exchange across the ER membrane (Cer, solid line), IP3 metabolism (Cp, dashed line), and IP3R dynamics (Cr, dotted line) as function of the period of the oscillations. A positive period control coefficient signifies that a slowing of the corresponding process increases the oscillation period. (C, D) Period control coefficients in the positive and negative feedback models, respectively, in the presence of plasma-membrane fluxes of Ca2+(ε=1). The dash-dotted line indicates the control exerted by Ca2+ exchange across the plasma membrane (Cpm).
... Frequency encoding of agonist stimulus. (A) Positive feedback: oscillation periods observed at different stimulation strengths (varying VPLC). Increasing the half-saturation constant of PLC for Ca2+, KPLC, from 0 (no functional positive feedback) to 0.2μM (functional feedback) greatly enhances frequency encoding. (B) Negative feedback. Increasing the amount of IP3K relative to IP3P (η) enhances frequency encoding. (C, D) The feedback effects shown in panels A and B are preserved when plasma-membrane fluxes of Ca2+ are included in the models (ε=1). (E) Range of oscillation periods, ΔT=Tmax−Tmin, in the presence (+) and absence (w/o) of positive feedback for two different strengths of the plasma-membrane Ca2+ fluxes (ε=1,4). (F) Range of oscillation periods in the presence (−) and absence (w/o) of negative feedback for two different strengths of the plasma-membrane Ca2+ fluxes (ε=1,4). We have found that the IP3K has an impact on the oscillation period only when the Ca2+ fluxes between ER and cytoplasm are comparatively slow and the IP3R is less sensitive to Ca2+ activation. To expose the period effect of the negative feedback, we have chosen different parameter values than in the positive feedback model (see Table 1).
Contributors:Ramaswamy, Rajesh, Sbalzarini, Ivo F.
Mesoscopic oscillatory reaction systems, for example in cell biology, can exhibit stochastic oscillations in the form of cyclic random walks even if the corresponding macroscopic system does not oscillate. We study how the intrinsic noise from molecular discreteness influences the frequency spectrum of mesoscopic oscillators using as a model system a cascade of coupled Brusselators away from the Hopf bifurcation. The results show that the spectrum of an oscillator depends on the level of noise. In particular, the peak frequency of the oscillator is reduced by increasing noise, and the bandwidth increased. Along a cascade of coupled oscillators, the peak frequency is further reduced with every stage and also the bandwidth is reduced. These effects can help understand the role of noise in chemical oscillators and provide fingerprints for more reliable parameter identification and volume measurement from experimental spectra.,Scientific Reports, 1,ISSN:2045-2322,
Contributors:Alexandre Hyafil, Anne-Lise Giraud, Lorenzo Fontolan, Boris Gutkin
Phase–Amplitude Coupling (PAC) and Amplitude–Amplitude Coupling (AAC). (A) Dense-spiking continuous fast oscillation (FO). Plot shows FO amplitude as a function of slow oscillation (SO) phase for distinct FO frequencies (from 10Hz to 100Hz) for the dense-spiking continuous FO model of Figure 1E. Red, high FO power; blue, low FO power. Peak FO frequency varies throughout SO cycle (PFC) but the overall FO power remains constant. (B) Sparse-spiking continuous FO. PAC is present in a model identical to that of panel A with simply an increased level of noise, producing sparse-spiking FO oscillations. (Left) Raster plot showing that the number of FO spikes at each cycle varies as a function of SO phase. (Right) FO amplitude as a function of SO phase, showing strong PAC with maximal FO amplitude close to SO peak, on top of PFC. (C) Sparse-spiking continuous FO with adaptation. The model in (B) was modified by introducing an adaptation current, slowing down the dynamics of the FO oscillation in response to SO modulation. PAC plot shows that maximal FO power occurs at a later SO cycle compared with the no-adaptation model of (B). (D) Dual ING/PING alternation. (Leftmost and middle-left) The FO network alternates within a SO cycle between a period of pyramidal-interneuron-generated oscillations (PING mechanism) and another period of faster interneuron-generated oscillations (ING) (this is obtained by a SO modulation applied to the inhibitory instead of the excitatory population). (Middle-right) FO amplitude as a function of SO phase. PAC occurs for two distinct FO frequencies, with maximal FO power for these two frequencies in antiphase. Such a pattern with distinct frequencies (pointing to distinct oscillation-generation mechanisms) should not be confounded with the case of a frequency modulation within a single FO frequency band, as shown in (A). (Rightmost) Data from rat hippocampus recordings showing a similar alternation between slow gamma and fast gamma within a theta cycle (adapted from ). (D) Positive AAC for positive asymmetric SO. (Left panel) The cross-frequency coupling (CFC) network with sparse-spiking FO oscillations was simulated with a SO sinusoid which amplitude was modulated at a 1Hz rate, whereby an increase in SO amplitude is accompanied by a general increase in SO level (green curve, left panel). This leads to general increase of FO amplitude on top of an increase of PAC strength. (Middle panel) FO amplitude as a function of SO phase, separately for high SO amplitude cycles (red curve) and low SO amplitude cycles (black curve). PAC was present as for the non-modulated SO sinusoid (Figure 3B), but here PAC modulation increased with higher SO amplitude. (Right panel) FO amplitude as a function of SO amplitude, showing strong positive AAC. Figure S1 in the supplemental information online shows the absence of AAC for a symmetric SO. (F) Negative AAC for negative asymmetric SO. Same model as in (E) but here the increase in SO amplitude is accompanied by a general decrease in SO level. This in turn leads to general decrease of FO amplitude on top of an increase of PAC strength, hence a negative correlation between SO and FO amplitudes.... cross-frequency coupling... neural oscillations... Phase–Frequency Coupling (PFC) and Phase–Phase Coupling (PPC). (A) Continuous fast oscillation (FO). (Top) PPC in the continuous FO model of Figure 1E. The top panel shows the histogram of FO phases as a function of slow oscillation (SO) phase for a simulation of 30 s. Darker areas show higher phase concentration. The overall largely homogeneous pattern indicates that very limited PPC emerges, which is confirmed by computing the FO phase-locking factor (PLF) as a function of SO phase : low PLF indicates that SO phase had virtually no influence on FO phase. The green line shows the outline of the SO modulating signal. (Bottom) PFC in the same model. The black curve shows the frequency of FO bursts as a function of SO phase. (B) Intermittent oscillatory/quiescent FO. (Top) PPC in the intermittent oscillatory/quiescent FO model of Figure 1F. The top panel shows very strong FO phase concentration throughout the SO cycle, with nearly non-existent phase change during the quiescent period (i.e., frequency vanishes) and rapid phase dynamics in the oscillatory period. FO-PLF remains high during the whole SO cycle, with highest values during the quiescent period. (Bottom) PFC in the same model. FO frequency is modulated during the oscillatory period and vanishes during the quiescent period. (C) Intermittent oscillatory/asynchronous FO displaying PFC without PPC. (Top) PPC in the intermittent oscillatory/asynchronous FO model of Figure 1G. PPC hardly emerges in this network because no phase-resetting occurs during the asynchronous phase. (Bottom) PFC in the same model. FO frequency is modulated during the oscillatory period and vanishes during the asynchronous period. (D) Type II FO displaying PPC without PFC. (Top left) Raster plot of a network of a single class of interconnected excitatory neurons (adaptive quadratic integrate-and-fire) receiving rhythmic SO impulse. (Bottom left) Nearly symmetrical phase response curve (PRC) of FO neurons showing phase lead for later phases and phase lag for earlier phase, corresponding to a type II oscillator (Box 1). (Top right) Strong PPC in the network because type II oscillators phase-lock easily to external excitatory input. (Lower right) PFC is absent in the same network. Because of the symmetrical PRC, on average there is no acceleration or slowing down of the FO because phase leads and delays cancel out, and FO remains at a constant frequency throughout the SO cycle.
... (A,B) Spiking frequency of type I and type II neurons as a function of driving-current magnitude. (A) Cells with type I membrane excitability can spike at any arbitrary frequency once passed beyond the bifurcation point. (B) When a type II neuron goes through the bifurcation point it starts firing from a finite frequency. (C) Phase response curve of a type I neuronal oscillator: any small excitatory input induces a phase advance within the spiking cycle of the oscillator. (D) Phase response curve of a type II neuron: an excitatory pulse either advances or delays the next spike depending on the particular phase at which the input arrives within the spiking cycle. (E) When solicited with an external sinusoidal input (SO), individual neurons from Aplysia display Arnold tongue maps (i.e., different forms of m:n coupling depending on the input frequency and amplitude). Experimental results are in fair agreement with model simulations (adapted from ). (F,G) Phase–amplitude coupling modeled through a set of Wilson–Cowan equations to represent the activity of a specific population of interneurons in rat hippocampus (adapted from ). Experimenters recorded the response of excitatory and inhibitory neurons to sinusoidal inputs induced using optogenetics tools. (F) Schematic illustration of the excitatory–inhibitory hippocampal network and spontaneous FO visible in E and I population activity in absence of external noise. (G) Phase portrait of the FO model showing the stable limit cycle of the system (blue), the nullclines (i.e., the points where either the excitatory or the inhibitory activity would be constant; black: dashed, excitatory; solid, inhibitory), and the isochrones (the set of initial points from which the dynamics would evolve towards oscillations having the same phase; red). Phase and amplitude modulations of FO can be inferred from the phase portrait, given a specific set of initial conditions.
... Functions of Cross-Frequency Coupling (CFC). (A) Multi-item/sequence representation. (a) Multi-item representations through CFC is envisaged for working memory (Top, in hippocampus and prefrontal cortex) [54,116], spatial memory for previously and subsequently visited items (in hippocampus) [60,61], and visual attention to a series of items in visual field (Bottom, in visual cortex) . (b) Putative CFC architecture for multi-item representation. Intertwined CFC network with a common population of pyramidal neurons connecting to two populations of distinct types of inhibitory neurons, each responsible for the generation of one of the two oscillations (slow, SO; and fast, FO). Some models also include direct connections between the two inhibitory populations [6,20]. (c) Multi-item representation relies on intermittent/quiescent CFC regime: the nested FO oscillation only occurs during a defined phase range, with one item being represented at each FO burst. AAC (through asymmetrical SO) allows modulating the phase range of FO and thus the number of represented items. (B) Long-distance communication. (a) Putative ... Architectures for Cross-Frequency Neural Coupling. (A–D) Cross-frequency coupling (CFC) architectures. (A) Intertwined oscillators. Neural oscillations in distinct frequency bands are generated by partially overlapping neural populations. In the depicted example, excitatory neurons (represented by the triangle) participates in the generation of both a slow oscillation (SO) and a fast oscillation (FO), whereas separate inhibitory populations (represented by green and orange circles) are involved in the generation of respectively SO and FO. CFC arises though the dynamics of the neural population common to both oscillations. Intermingled CFC has been proposed to explain the emergence theta/gamma coupled oscillations in hippocampus [6,30,100], where a common excitatory population is coupled to fast-spiking (FS) cells (generating gamma oscillations) and oriens-lacunosum moleculare (O-LM) cells (generating theta oscillations). (B) Bidirectional coupling. Segregated populations are implicated in the generation of the SO and FO (in the depicted example, a population of excitatory and inhibitory neurons for both oscillations), and coupling is mediated by reciprocal coupling between SO neurons and FO neurons. This architecture was used in the first computational model of coupled theta–gamma oscillations in hippocampus, which featured two coupled inhibitory subpopulations with distinct GABA decay time . Moreover, precise spiking and local field potential (LFP) dynamics of in vitro cortical slices was explained by a sophisticated model whereby a beta1 rhythm concatenates two bidirectionally coupled oscillations, a gamma rhythm generated in superficial layers and a beta2 rhythm generated in deep layers [102,103]. (C) Unidirectional coupling. Distinct populations are implicated in the generation of SO and FO, and coupling arises though one population projecting onto the other population (here SO to FO). In a recent model of speech perception in auditory cortex, coupled theta–gamma oscillations were modeled by two separate excitatory–inhibitory subpopulations responsible for the generation of both rhythms , with the theta module projecting onto the gamma module. (D) Sensory entrainment. Sensory entrainment of neural oscillators is a special case of unidirectional coupling where a neural oscillator is modulated by slow modulations in sensory stimulus (e.g., visual rhythmic movements or amplitude-modulated sounds). This model has been tested in the context of visual processing: a PING module generating broad gamma rhythm responds to visual activity experimentally recorded in monkey thalamus that carries strong slow modulations . (E–G) Temporal dynamics of FO. (E) Continuous FO. (Left) Schematic phase diagram of FO dynamical state: continuous FO occurs when SO modulation shapes a trajectory within the region of existence of FO. (Right) The continuous model of CFC from Fontolan et al.  (see their Figure 5) was transformed into a mathematically equivalent model of quadratic-and-fire neurons and simulated (with membrane white noise of variance 0.01). SO is modeled as a simple modulatory signal while FO is composed of a population of pyramidal and inhibitory cells generating gamma oscillations through the PING mechanism. (Top right) SO modulating signal. (Bottom right) raster plot of FO spikes: pyramidal neurons (n=50, light orange); inhibitory neurons (n=50, dark orange). FO spiking occurs throughout the SO cycle. (F) Oscillation/quiescent-state intermittent FO. (Left) This type of intermittent FO occurs when SO modulation shapes a trajectory between the oscillatory and quiescent regions of FO. (Right) The intermittent model of CFC from  (see their Figure 7) was transformed into network of quadratic-and-fire neurons and simulated. FO alternates between a period of gamma oscillations when excitation from SO is large enough, and a quiescent state when SO signal is lower. (G) Oscillation/asynchronous-state intermittent FO. (Left) This type of intermittent FO occurs when SO modulation shapes a trajectory between the oscillatory and asynchronous regimes of FO. (Right) We modified the parameters of the gamma network of  (constant current, synaptic conductance, and membrane noise) such that the network alternates within SO cycle between a period of strong oscillation and a period of asynchronous spiking activity.
Contributors:Claudia Scheffzük, Valeriy I. Kukushka, Alexei L. Vyssotski, Andreas Draguhn, Adriano B.L. Tort, Jurij Brankačk
Power spectral content (A) and theta phase coupling of fast oscillations (B) vary strongly across the sleep-wake cycle. Five different states are shown: active waking (aWk, black solid lines and black filled bars), quiet waking (qWk, black dotted lines, black open bars), Non REM sleep (NREM, blue solid lines, blue filled bars), tonic REM sleep (toREM, red solid lines, red filled bars) and phasic REM sleep (phREM, red dotted line, red open bars). (A) The upper row shows mean power spectra for theta, gamma and fast gamma frequency ranges (N = 10). Notice prominent theta, gamma and partly fast gamma power peaks present only in the three “theta states” (aWk, toREM and phREM) but not in the two “non-theta states” (qWk, NREM). The middle row depicts peak frequencies (means ± S.E.M.). The lower row depicts band power values. The “theta states” differ significantly both in frequency and power of theta, gamma and fast gamma from the “non-theta states” (*: significances in reference to aWk; #: in reference to toREM; +: in reference to phREM; §: in reference to qWk). (B) Upper left: Theta phase coupling strength for frequencies between 20 and 200 Hz. Upper middle and right panels: Theta-gamma coupling strength is significantly larger during “theta states” compared to “non-theta states”. Theta-fast gamma coupling is significantly larger during REM sleep states compared to aWk, qWk and NREM. Lower panels: peak frequency values for theta phase (left), gamma (middle) and fast gamma amplitudes (means ± S.E.M.; symbols of significances as in A). Baseline data during aWk, toREM and phREM have been previously published in Brankačk et al. (2012).
... Diazepam (DZ) decreases the frequency of fast oscillations amplitude-modulated by theta. (A) Gamma amplitude frequency modulated by theta significantly decreased in active waking (aWk, p frequency decreased in aWk (p frequencies associated with maximal theta-phase coupling are shown. Significances: *p < 0.05, **p < 0.005, ***p < 0.0005 compared to vehicle.
... Cross-frequency coupling... Diazepam (DZ) differentially alters power peak frequencies (left panels) and band power (right panels) across behavioral states. DZ decreased peak frequencies of theta, gamma and fast gamma oscillations (A, C, E) in all vigilance states, while its effect on band power (B, D, F) changed differentially, depending on frequency range and vigilance state. Means (±S.E.M.) of ten animals are shown. (B) Theta band power decreased only in aWk (p = 0.006). (D) Gamma band power increased in aWk (p = 0.0003), was largely unchanged in toREM and decreased in phREM (p frequency was estimated by the method illustrated in Supplementary Fig. 1 and described in Methods. Significances: *p < 0.05, **p < 0.005, ***p < 0.0005 compared to vehicle.
... Neuronal oscillations... Diazepam (DZ) causes a global slowing of neocortical EEG frequencies, for both slow (0–20 Hz) and high (20–160 Hz) frequency ranges in all three vigilance states with prominent theta oscillations: active waking (aWk), tonic REM (toREM) and phasic REM (phREM). Significances: **: p < 0.005, ***: p < 0.0005 compared to vehicle. See Methods for details.
... Diazepam (DZ) decreases the frequencies of maximal cross-frequency coupling but leaves coupling strength largely unaffected. (A) Average heat maps of comodulation strength calculated from 30 s episodes during active waking (aWk), tonic REM (toREM) and phasic REM (phREM) of ten mice, 30 min after treatment with DZ or vehicle. Warm colors represent high coupling strength, cold colors low coupling between theta phase frequency (abscissa) and the amplitude of faster oscillations (ordinate). (B) Theta coupling strength versus amplitude frequency calculated at phase frequencies of maximal coupling are shown for three theta states and different treatments, as labeled. Only theta-gamma coupling during aWk decreased significantly after DZ.