Introduction

Biological clocks generate endogenous physiological and behavioral oscillations on multiple timescales that persist in the absence of any external timing cues under constant conditions (Lamont and Amir, 2010). Such clocks are essential to generate internal rhythms and to permit their synchronization to external zeitgebers at timescales from milliseconds to years. These synchronized endogenous multiscale clocks thus provide both a continuous time frame for physiology and behavior - the organism’s “present time” - and enable organisms to predict and respond to both regular, periodic environmental changes (Stengl and Schneider, 2024). Even though endogenous clocks are widespread in prokaryotes and eukaryotes (Géron et al., 2024; Hall and Rosbash, 1993; Millar, 2016; Thaiss et al., 2014), current knowledge about how multiscale clocks are linked over widely diverging timescales remains sparse.

Insect circadian clock neurons, which are tightly connected to visual brain circuits, are amongst the best studied biological clocks (Hall and Rosbash, 1993; Nishiitsutsuji-Uwo and Pittendrigh, 1968; Reinhard et al., 2024). They regulate physiological and behavioral oscillations that underly sleep-wake rhythms tied to the 24 h light-dark cycles on Earth (Dubowy and Sehgal, 2017). Circadian clock neurons possess a molecular clockwork of transcriptional-translational feedback loops (TTFLs), which generate endogenous oscillations of clock gene mRNA and protein levels with a cycle period of ∼24 h (Hardin, 2011). It is assumed for both insects and mammals that this TTFL comprises the master clockwork that regulates all physiological and behavioral rhythms in the circadian time range through genetically controlled output pathways (Hastings et al., 2020).

Molecular TTFL clockworks were not only found in central brain clock neurons but also in sensory neurons, such as olfactory receptor neurons (ORNs) in insect antennae, where they play a role in odor-dependent circadian behaviors like feeding, pollination, reproduction, and navigation (Flecke et al., 2010; Flecke and Stengl, 2009; Krishnan et al., 1999; Merlin et al., 2009, 2006; Page and Koelling, 2003; Plautz et al., 1997; Sauman and Reppert, 1998; Schendzielorz et al., 2015; Schuckel et al., 2007; Tanoue et al., 2004; Zhou et al., 2005). Little is known about how the TTFL clockwork of insect ORNs regulates circadian rhythms in chemosensory sensitivity that underlie the circadian control of these different behaviors.

At night, male nocturnal Manduca sexta hawkmoths are very sensitive to the sex pheromone blend that their conspecific females release in an intermittent pattern (Itagaki and Conner, 1988). The intermittency is a prerequisite to maintain the male’s search flight, which is suspended at pulse frequencies >30 Hz (Baker et al., 1985; Baker and Vogt, 1988; Stengl, 2010). The pheromone pulses are resolved of up to 3 Hz by pheromone-specific ORNs in the antenna’s long trichoid sensilla (Lee and Strausfeld, 1990; Marion-Poll and Tobin, 1992). Pheromone-specificity is provided by odor receptor (OR) subunits in the ciliary membranes of ORNs that heteromerize with the conserved olfactory receptor co-receptor Orco (Nakagawa and Vosshall, 2009; Stengl and Funk, 2013; Wicher and Miazzi, 2021). While the various roles of Orco are still under debate, it is known that in heterologous expression systems Orco homomers form a slow, leaky, non-specific cation channel, which depolarizes the cells (Jones et al., 2011; Sargsyan et al., 2011; Sato et al., 2008; Wicher et al., 2008). In vivo tip recordings of long trichoid sensilla in M. sexta confirmed that Orco acts as spontaneously opening cation channel, acting as a pacemaker channel, because it promotes ultradian membrane potential oscillations that underly the spontaneous action potential activity (Nolte et al., 2013). These potential oscillations turn the ORNs into ultradian temporal filters that are sensitive to the frequency of the pheromone pulses they encounter. Pheromone pulses that arrive at a similar frequency as the endogenous sub-threshold membrane potential oscillations would synchronize these oscillations and would be more likely to elicit action potential trains. Therefore, this pacemaker channel-dependent mechanism would allow for active sensing if endogenous membrane potential oscillations are in the physiological range necessary for tracking intermittent pheromone pulses.

It is not known whether the Orco-dependent membrane potential oscillations also occur at circadian timescales, and whether or how they are linked to ultradian potential oscillations. One hypothesis is that the molecular TTFL clockwork of ORNs exclusively controls endogenous circadian rhythms of the membrane potential via circadian expression of the pacemaker channel Orco. Alternatively, the plasma membrane of ORNs could by itself constitute an autonomous clock that is entrained by but does not rely on the TTFL clock. Constant levels of ion channels that do not require daily degradation and daily transcription via the TTFL clock, but with modulation of open-time probability by posttranslational modifications, would constitute a more economical and faster adjustable mechanism to generate endogenous membrane potential oscillations (Stengl and Schneider, 2024). This would facilitate active sensing, the rapid tuning to detect behaviorally relevant zeitgeber signals in the insect’s environment.

Here, we performed a combination of experimental and computational modeling studies to investigate whether Orco plays a role in active sensing at the circadian and/or ultradian timescale via generation of endogenous membrane potential oscillations. We hypothesize that Orco is part of a post-transcriptional feedback loop (PTFL) clockwork in the plasma membrane, rather than being TTFL controlled. To capture naturally occurring, physiologically relevant processes in insect ORNs we performed in vivo long-term tip recordings of the spontaneous spiking activity of pheromone-sensitive trichoid sensilla of intact but restrained hawkmoth males.

We corroborated the experimental data with a Hodgkin-Huxley (HH) based computational ORN model. In general, the widely used deterministic HH models do not include circadian control of ion channels. Moreover, the highly dynamic, bursting firing patterns of ORNs cannot be described by the traditional deterministic modeling of neuronal activity. This constitutes a drawback because stochasticity plays a crucial role in shaping firing patterns which are influenced not only by external stimuli but also by the inherent noise involved in channel gating. Therefore, to address the above points realistically, we developed a Langevin formulation of the HH model, which explicitly accounts for the noise introduced by the random opening and closing of ion channels and circadian modulation of the Orco channel conductivity. By integrating channel noise, we could explore how circadian regulation of ionic currents contributes to the variability of firing patterns and how this noise interacts with external stimuli to influence olfactory signal processing.

Our model aims to highlight the advantages of using a stochastic framework for modeling the circadian activity of ORNs. The combined experimental and modeling efforts generate a more complete picture and predictions concerning these autonomous processes in sensory neurons, which are required for future exploration of physiologically relevant anticipatory behaviors that are driven by biological clocks.

Methods

Animals

Nocturnal M. sexta hawkmoths were bred and raised from eggs in the rearing facility at the University of Kassel. Males and females were separated during pupal stages. Males were housed isolated from the females to avoid exposure to female pheromones. All experiments were performed with adult males reared in a long-day photoperiod (L:D 17:7 h to prevent diapause) at 25 °C with relative humidity of about 55%. Larvae were fed with an artificial diet modified after Bell and Joachim (1976); adult moths could feed on sugar solution with added vitamins (Roth) ad libitum (Riffell et al., 2008).

Solutions

Ringer compositions were taken from (Pézier et al., 2007) and contained (in mM): 6.4 KCl, 12 MgCl₂, 1 CaCl₂, 12 NaCl, 10 HEPES, 340 glucose for hemolymph ringer (HLR) at 450 mosmol/kg, and 172 KCl, 3 MgCl₂, 1 CaCl₂, 25 NaCl, 10 HEPES, 22.5 glucose for sensillum lymph ringer (SLR) at 475 mosmol/kg, both at pH 6.5. The Orco antagonist OLC15 (N-(4-butylphenyl)-2-((4-ethyl-5-(2-pyridinyl)-4H-1,2,4-triazol-3-yl)thio)acetamide (Chen and Luetje, 2012); kindly provided by Dr. D. Wicher, Max Planck Institute for Chemical Ecology, Jena, Germany) was dissolved in DMSO to a concentration of 100 mM (stock solution stored at 4-8 °C) before being diluted in SLR to a final concentration of 50 µM immediately before the start of the recording. DMSO percentage in the final solution was 0.05%.

Electrophysiology

The spontaneous electrical activity of the unstimulated pheromone-sensitive ORNs in a single long trichoid sensillum was recorded extracellularly as in vivo tip recording (Kaissling et al., 1987). 1-2-d-old male M. sexta that were raised in isolation from females and pheromone exposure were fixed in a custom-made Teflon holder with adhesive tape. The right antenna used for recordings was oriented with its dorsal surface facing up and immobilized near the base with dental wax (Boxing wax strips, KerrHawe SA). Electrodes (Ag/AgCl wire in a glass pipette filled with SLR for the recording electrode and HLR for the reference electrode) were pulled (DMZ-Universal Puller, Zeitz Instruments) from borosilicate capillaries (OD:1.56mm, ID: 1.17mm, without filament; Science Products) to a tip diameter of about 2 μm. After cutting off 20 distal annuli of the antenna with micro scissors, the reference electrode was inserted into the antennal lumen up to the second annulus from the cut end. The cut was covered with electrode gel (GE Medical Systems Information Technology) to prevent desiccation. Electrodes were moved with manual micromanipulators (MMJ, Märzhäuser Wetzlar; and Leitz, Leica Microsystems) or with motorized micromanipulators (Microstar, Scientifica) with a joystick controller (Scientifica). Under microscopic control, some tips of the long trichoid sensilla of the distal second annulus were cut off with sharpened Dumont forceps, and the recording electrode slipped over a single sensillum of an upper sensillar row.

ORN activity was amplified 200-fold (custom built amplifier with 10¹² Ω input impedance, or ELC-01 MX in combination with DPA-2FX, npi, or EXT 10-2F, npi), low pass filtered at 1.3 kHz, digitized at 20 kHz with a Digidata (1550A or 1550B, Molecular Devices), and saved for offline analysis with Clampex 10 (pCLAMP suite, Molecular Devices).

All tip recording experiments were performed at room temperature (21-27 °C) and a relative humidity of 35-60%. We recorded ORN activity either in long-day conditions (17:7 LD) or constant darkness (DD) to examine the free-running peripheral ORN clock. For LD experiments, the light regime continued as in the rearing facility during the experiment. For DD and OLC15 experiments, the lights were turned off at the beginning of the recording, i.e. all animals were exposed to the same zeitgeber until the beginning of the experiment. To examine the effect of Orco on the spontaneous action potential (AP) activity pattern, we used the recording electrode to infuse OLC15 in DD conditions.

Data analysis

To characterize the spontaneous ORN spiking activity, we first detected spikes with a self-written program in Python. Using the scipy.signal package and find_peaks function, we applied a Bessel filter to the raw voltage trace to facilitate spike detection within the following conditions: spike full width (3-15 ms), threshold search for amplitude of spike (0.5-2.5 mV) and threshold prominence (0.1). Two or more consecutive spikes with intervals ≤ 50 ms were considered a burst (Dolzer et al., 2001). We then calculated the following attributes in bins of either 10 min or 1 h for further analysis: instantaneous spike frequency (1 / inter-spike interval (ISI)), inter-burst interval (IBI, time between two consecutive bursts with no single spikes in between), inter-event interval (IEI, time between two events, i.e. two consecutive single spikes, two consecutive bursts, a burst and a consecutive spike, and vice versa), burst duration (time from first to last spike in a burst), percentage of all spikes in the experiment that are members of burst, and mean number of spikes per burst (Figure 1). All statistics were done with those binned values.

Figure 1:

Hawkmoth behavioral and physiological rhythms desynchronize across the population in the absence of zeitgebers like pheromone stimulation and light-dark cycles. Therefore, it was necessary to phase align the recordings obtained in DD and OLC15. To achieve this, we used an optimization procedure to calculate the optimal temporal lags for each dataset, aiming to maximize the overall cross-correlation between the calculated binned spike frequency series. The optimization was performed using simulated annealing, which provided a set of time shifts (lags) for each series. These lags were applied to shift the frequency data in a manner that maximized the synchronization between the different subjects’ data, resulting in a “subjective ZT” with subjective ZT 0 at the first maximum of the binned ISI for each individual. The objective function used for optimization was the negative sum of pairwise correlations between all series after applying the shifts. The resulting alignment allowed us to compare the frequency patterns across the different conditions.

Once the data were aligned, we computed the mean frequency across all subjects for each bin and each condition. Then, the first high activity period was identified by finding the maximum in the mean frequency series. Subsequently, we identified the second high activity period as the time 24 hours after the first one and the first low activity period was found by locating the minimum in the mean frequency series between the two high activity periods.

Besides the mean frequency, we also searched for significant oscillations in IBI, IEI, burst duration, percentage of spikes that are part of burst, and mean number of spikes per burst. For this we found the maximum value of these statistics within a 3 h range centered in the first low activity period and compared it with the values at the second high activity period.

Continuous wavelet and Fourier analysis were performed using PyBoat (Mönke et al., 2020) to examine the time series of neuronal firing frequency. The continuous wavelet method allows us to detect and characterize rhythmic patterns within the data by identifying frequency components that change over time. Specifically, we used wavelet analysis to determine whether the neuron exhibits circadian and ultradian rhythms. Unlike traditional Fourier analysis, which provides an overall frequency composition but assumes stationarity, wavelet analysis is particularly useful for biological signals because it can reveal how these frequencies evolve dynamically. By applying this approach, we could assess not only the presence of ultradian rhythms but also whether their strength or occurrence fluctuated in a circadian manner, indicating modulation by the 24-hour cycle.

Real-time quantitative polymerase chain reaction (qPCR)

We followed the qPCR methodology detailed in (Schneider et al., 2025). Briefly, male moths were isolated and cultured until the second day after eclosion, and the antennae from four individuals were collected at each zeitgeber time (ZT) 1, ZT9, and ZT17 using liquid nitrogen, with three biological replicates per time point. Total RNA was extracted using TRIzol® Reagent (Invitrogen) and quantified for concentration and purity with the NanoDrop ND-1000 spectrophotometer (Thermo Fisher Scientific). First-strand complementary DNA (cDNA) was synthesized using the PrimeScript™ RT reagent Kit with gDNA Erase (Takara Bio Inc.). Primers (F: AAGCACGTGGTCAGATTGGT, R: GTAAGCCAGGAGCGTCAGAG) for Orco (Gene ID: LOC115452348) were designed using the Primer-BLAST online program of NCBI. The qPCR was conducted following the protocol of TB Green® Premix Ex Taq™ II (Tli RNase H Plus) (Takara Bio Inc.), with the program: 95 °C for 30 s, followed by 40 cycles of 95 °C for 5 s, and 60 °C for 30 s. Relative expression levels of Orco were calculated using the 2^-ΔΔCt method (Livak and Schmittgen, 2001).

Statistics

All statistics were done with self-written python codes, SigmaPlot 12 (Systat Software), or JASP (version 0.19.03, University of Amsterdam) with a significance level α = 0.05. Unless noted otherwise, data were checked for normal distribution with Anderson-Darling or Shapiro-Wilk tests and for equal variance with Levene’s test. If one of these assumptions failed, we used non-parametric tests, otherwise we used parametric tests.

To assess the statistical significance of observed differences in neuronal activity of the aligned data between ZTs, we applied the Wilcoxon signed-rank test to pairwise comparisons between the first low activity period and the subsequent high activity period and discarded the transient high activity at the beginning of most recordings.

For other data, we used an ANOVA on ranks with either Tukey’s or Dunn’s post-hoc test for multiple comparisons as indicated.

ORN Model

In this study, we introduce a single-compartment Hodgkin-Huxley (HH) type model of the spontaneous activity of M. sexta ORNs. The model contains 5 sets of ion channels that were previously identified in electrophysiological recordings (Dolzer et al., 2021): the fast sodium (Na⁺) and potassium (K⁺) channels responsible for the generation of spikes, low voltage-activated calcium channels (LVA) responsible for the initiation of bursts, calcium-gated potassium channels (BK) responsible for the termination of bursts, and the Orco channel, which is a voltage-independent non-specific, cAMP-gated channel. A schematic of the model is provided in Figure 2A. We modeled Orco as a ZT-dependent conductance to represent the effect of the oscillations in cAMP concentration that were identified in these cells (Schendzielorz et al., 2014) (Figure 2B, C). For simplicity we chose to use a sine function to directly represent the oscillations in Orco conductivity, but similar results can be achieved by modeling it as an output of a Goodwin model of the circadian clock via cAMP concentration.

Figure 2:

Each type of ion channel possesses one or more gates that can open and close individually with transition rates α and β for opening and closing, respectively, of activation gates (n, m) and inactivation gates (h). As a result, each possible combination of open and closed gates defines a possible state in which each ion channel can be found. A schematic of the Markov chain depicting all possible states and transitions for each ion channel is shown in Figure 2D. Each ion channel can conduct current only in the state where all its gates are open.

First, we constructed a deterministic 22-dimensional model to serve as the mean field of the Langevin model following the methodology proposed by Pu and Thomas (2020). Here, one dimension tracks the membrane potential, another the intracellular Ca²⁺ concentration, and the remaining 20 dimensions track the number of channels that are in a specific state of the Markov chains. Considering the conservation of the total number of ion channels, each type of ion channel required one less dimension than the total possible states.

The complete Langevin model adds to the mean field an independent noise source for each transition in the Markov chain. These noise sources are biophysically motivated since the transition between ion channel states is a stochastic process, with the probability of transition determined by the kinetics of each kind of ion channel. The variability in the spiking behavior of a neuron emerges from this non-deterministic gating of individual ion channels. The detailed mathematical description of the model can be found in the Supplementary Material.

Results

We tested the hypothesis that sensory neurons, such as insect ORNs, perform active, anticipatory sensing based upon an endogenous, plastic plasma membrane clock with the pacemaker channel Orco as the core element. We focused on pheromone-sensitive long trichoid sensilla of male M. sexta hawkmoth antennae which exhibit daily rhythms in pheromone sensitivity and temporal resolution that are controlled by a circadian clock. We performed minimally invasive in vivo tip-recordings over the course of several days to search for daily rhythms in spontaneous spiking activity as a measure of endogenous membrane potential oscillations.

Pheromone-sensitive hawkmoth ORNs exhibit Orco-dependent circadian rhythms in spontaneous spiking activity

The ORNs that innervate the pheromone-sensitive long trichoid sensilla on male M. sexta antennae display spontaneous spiking patterns of irregular bursts with interspersed single action potentials in short-term recordings (Dolzer et al., 2001). In the present study we documented the same pattern in long-term recordings over the course of several days. This enabled us to search for daily and circadian rhythmicity in spontaneous ORN spiking activity in the absence of pheromone stimulation.

The spontaneous spiking patterns revealed clear daily rhythms in 5 of 11 animals under long-day conditions (17:7 h LD) (Figure 3A). The maxima in spontaneous activity occurred mostly during the dark phase when nocturnal hawkmoths are active (Figure 3A (top), Bi light blue arrow), and minima occurred during the light phase when hawkmoths rest/sleep (Figure 3A (bottom), Bi dark blue arrow). High spiking activity in the dark phase had a mean maximum frequency of ∼3 Hz (theta range: 0.5 – 4 Hz) (Figure 3Bii). In contrast, the minimum spiking frequencies during the hawkmoth’s resting phase were below 0.5 Hz (Figure 3A (bottom)). Since ORN firing patterns had a clear daily rhythm in LD both in individual animals (Figure 3Bi) and averaged across animals (Figure 3Bii, Biii), we examined whether these rhythms depended on cycling light cues or whether they persisted in constant darkness (DD), as expected for an endogenous, circadian clock-driven rhythm.

Figure 3:

In DD, the spontaneous ORN firing pattern remained rhythmic (7 of 10 animals), albeit with less robustness: The single peak of high spiking activity that was observed at the same ZT across several days in LD divided into multiple bouts of high spiking activity in the activity phase in 7 of 10 animals (Figure 3Ci). Also, in DD conditions, the high spiking activity phase shifted into the subjective day due to the individual endogenous circadian periods of about 23.5 h +/-2.8h (21.27 h in Figure 3Ci). Thus, averaging the instantaneous spiking activity across all DD animals did not depict clear circadian rhythms (Figure 3Cii). Therefore, we phase-aligned the mean frequency of spontaneous activity (see Methods). After alignment, circadian rhythms were exposed across DD animals (Figure 3Ciii), comparable to the daily rhythms in LD (cf. Figure 3 Biii and Ciii).

Next, we examined whether Orco, a leaky, non-specific cation channel, is the dominant depolarizing pacemaker current of ORNs that drives circadian rhythms in spontaneous spiking activity. Infusion of the Orco antagonist OLC15 into the sensillum lymph obliterated circadian rhythms and attenuated the spontaneous activity in several, but not all experiments (N = 8 of 12) (Figure 3Di). This attenuation resulted in a linear decrease in spiking activity over several days (Figure 3Dii, Diii). Even after phase-aligning the individual experiments, no circadian rhythm was present in the average across all OLC15-treated animals (Figure 3Diii).

Because OLC15 is not membrane-permeable on its own it was infused with low concentrations (0.05%) of DMSO through the recording electrode into the sensillum lymph and, therefore, effectiveness increased over time. Hence, we fit the spiking activity of each animal and in each condition with linear regression lines over the whole recording time to compare the slopes of the fits (Figure 3E). In LD, the mean ± SD slope was 0.004 ± 0.006 Hz/h, indicating that the average spiking activity remained similar throughout the duration of the recording. In DD, the slope was −0.003 ± 0.018 Hz/h and was not different from LD. In contrast, the mean slope during OLC15 treatment was −0.007 ± 0.022 Hz/h, significantly smaller than in LD (ANOVA on ranks with Dunn’s post-hoc test; H(2) = 7.681, p = 0.021). The negative slope corroborated the finding that spiking activity decreased over time in the presence of the Orco blocker.

Activity rhythms between hawkmoths desynchronize and have only weak phase coupling in the absence of pheromone stimulation, even in LD cycles (Ghosh et al., 2024). Therefore, instead of comparing fixed ZT or circadian time (CT) intervals, we compared different attributes of spontaneous activity between periods of high and low activity, for DD conditions after phase alignment (see Methods; Figure 4). Despite the high variability in ORN spiking between individuals, the mean frequency of spontaneous ORN activity was significantly higher during the high activity phase than during low activity in both LD and DD recordings (Figure 4Ai, Aii). In LD but not in DD, the frequency of bursting increased significantly during the high activity phase (Figure 4Bi, Bii). The percentage of spikes that are part of bursts was less variable and lower during the high activity phase in LD but not significantly different in DD (Figure 4Ci, Cii). The mean burst duration was less variable during the high than the low activity phase in LD, but not significantly shorter in any condition (Figure 4Di, Dii). Furthermore, both in LD and DD, the mean inter-event frequencies were significantly higher during the high activity phase (Figure 4Ei,Eii), indicating a significantly higher overall spontaneous spiking activity during the hawkmoth’s activity phase. The mean number of spikes per burst, and intra-burst frequencies did not differ significantly between high or low activity periods in both LD and DD conditions (Figure 4Fi, Fii, Gi, Gii). The addition of Orco antagonist OLC15 deleted any significant differences of spontaneous activity found in LD and DD (Figure 4Aiii-Giii). In conclusion, the mean frequency of spontaneous spiking and the frequency of bursting expressed daily modulation, and are both most likely controlled via a circadian clock that targets the leak channel Orco.

Figure 4:

Orco imposes circadian modulation on the ultradian rhythms in spontaneous ORN activity

After establishing that Orco influences the circadian changes in ORN spiking activity between the animal’s resting and activity phase we examined if Orco also influences ORN spiking rhythms on the largely different ultradian timescales in the frequency range between 0.01-1000 Hz. In all conditions, two broad bands of ultradian instantaneous frequencies were evident: one upper band of >10 to ∼100 Hz and a lower band between 0.1 to ≤10 Hz (Figure 5). The upper band primarily represents the frequencies of spikes within a burst, while the lower band represents single spikes and frequencies between bursts.

Figure 5:

In LD, two types of daily/circadian modulation of the ultradian frequencies occurred (Figure 5A). The upper higher frequency band was mostly modulated in prevalence (its frequency of occurrence) but not in its frequency range. The lower band was modulated both in prevalence and frequency range, which resulted in a sinusoidal change in the mean of the frequency band over 24 h. The low and high frequency bands appeared to merge during the activity phase around ZT 0 in the animals that showed clear circadian rhythms (N = 5 of 11 in LD), when both bands had maximum prevalence. During the hawkmoth’s activity phase, higher frequencies of up to ∼10 Hz occurred, while frequencies dropped below 1 Hz during the resting phase.

The circadian modulation of both frequency bands was maintained in constant darkness (Figure 5B) but phase-shifted with respect to the projected ZT due to the endogenous circadian period of each individual. Again, this indicates that a clock is modulating the spontaneous ORN spiking activity over the course of a day.

The addition of the Orco antagonist OLC15 deleted any circadian modulation of either frequency band. Interestingly, the instantaneous frequencies in both bands still showed correlated changes in prevalence and merging, but now at ultradian periods only.

Additional Fourier analysis (see Methods) of spontaneous spiking activity over the course of several days revealed faster ultradian and slower infradian rhythms, ranging from 0.6 h to 38 h, in addition to the circadian (24 ± 4 h) rhythm (Figure 6Ai, Bi, Ci). Some rhythms (factorials of 24 h, like 12 h, 6 h, etc.) appeared to be harmonics of the circadian rhythm, which were not excluded in our analysis. Most animals expressed ultradian rhythms of 2-4 h in LD as well as DD conditions. With OLC15 the occurrence of circadian rhythmicity decreased from 70% to 33.3 %, while the prevalence of infradian periods increased from 0% to 33.3 % and ultradian periods remained the same. These results were corroborated by continuous wavelet analysis, which provided time-resolved spectral power across different rhythmic periods (Figure 6Aii, Bii, Cii).In control LD and DD conditions, wavelet power for ultradian periods increased during times of elevated firing rate and decreased during periods of low activity (Figure 6Aii and Bii, middle and bottom panels). This dynamic pattern indicates that ultradian rhythms were not constant over time but were instead modulated by the circadian rhythm, becoming more prominent during circadian peaks in neuronal activity. Such circadian gating of ultradian power was clearly visible in the wavelet spectrograms as alternating bands of higher and lower ultradian power aligned with daily activity cycles. Following Orco blockade with OLC15, this modulation was disrupted (Figure 6Cii middle panel): although ultradian components persisted (Figure 6Cii bottom panel), they no longer showed systematic variation with the circadian cycle, suggesting a decoupling of ultradian rhythms from circadian control.

Figure 6:

Orco expression is not under the control of the molecular TTFL clockwork

So far, our results indicated a prominent role of Orco in the circadian modulation of ORN spiking activity. Since it is the general opinion that the master circadian clock that dominates all physiological and behavioral circadian rhythms functions via transcriptional control we examined whether Orco transcription shows a circadian rhythm, as previously found for clock genes in hawkmoth antennae (Schneider et al., 2025; Schuckel et al., 2007). However, qPCR of whole male hawkmoth antennae revealed no significant circadian rhythm of Orco expression in DD (Figure 7; ANOVA on ranks, H(2) = 5.60, p = 0.061). Thus, it is possible that Orco is controlled via a posttranslational circadian clockwork mechanism that is associated directly with the plasma membrane (Stengl and Schneider, 2024; Stengl and Schröder, 2021).

Figure 7:

Circadian changes in Orco conductance are sufficient to modulate spiking activity in a computational model of a hawkmoth ORN

Although Orco is a leaky ion channel, its maximum conductivity is modulated by cAMP (Getahun et al., 2013; Martín et al., 2001; Sargsyan et al., 2011; Wicher et al., 2008). In hawkmoth antennae, cAMP levels oscillate throughout the day (Dolzer et al., 2021; Flecke et al., 2010; Flecke and Stengl, 2009; Schendzielorz et al., 2014, 2015), which makes cAMP modulation of Orco a putative mechanism for the Orco-dependent circadian modulation of ORN spiking activity, integrating Orco in a circadian PTFL based membrane clockwork (Stengl and Schneider, 2024).

To test the hypothesis that Orco is a cAMP-gated, leaky ion channel, we constructed an ORN model (Methods and Supplementary Material) and tuned the parameters to provide a quantitative match with the biological data. The developed model provides a unified description of the experimental observations. For instance, the ORN model yields irregular spiking activity with a bimodal distribution, consisting of both bursts and isolated spikes, successfully reproducing the experimental recordings (Figure 8A). This suggests a unified mechanism underlying the initiation of both phenomena: If the pacemaker-driven depolarization reaches the Na⁺ threshold a firing process begins, and at this point randomness determines if other slower ion channels responsible for bursting are sufficiently activated to sustain a burst of several spikes.

Figure 8:

Additionally, the ORN model replicates the distribution of the number of spikes per burst, which follows a decreasing exponential trend in both experimental and simulated results (Figure 8B).The model also successfully reproduces the circadian modulation of spiking activity: the maximum firing rate occurs when the Orco conductance is maximal, whereas the maximum percentage of spikes that are part of bursts correlates with the lowest Orco conductance, resulting in antiphase oscillations, as were observed in the experimental recordings (Figure 9).

Figure 9:

The frequency prevalences in the model over several days clustered in two distinct frequency bands: a top band that corresponds to the intra-burst frequencies and a bottom band, corresponding to inter-burst frequencies (Figure 10), in agreement with the biological experiments (Figure 5A, B). When these circadian oscillations in Orco conductance were included, inter-burst frequencies showed significant circadian variations, whereas intra-burst frequencies remained relatively constant, reminiscent of the biological data. Notably, the simulation captured the experimentally observed “merging” effect, where the low frequency band approaches the top band during high-activity periods (Figure 10).

Figure 10:

This merging of spiking and bursting frequencies suggests that the system approached a bifurcation, transitioning from a bursting regime to tonic spiking activity. Further evidence for the near bifurcation condition of the neurons is the distribution of number of spikes per burst that followed a decreasing exponential trend in contrast to a constant value, or a bounded range of spikes per burst, which would be expected from a standard bursting neuron model (Figure 8B). The observation that the circadian modulation of spiking activity requires only one parameter, namely changes in the Orco conductivity, to describe the circadian effects on the ORN spiking pattern indicates the crucial role of Orco channel in circadian control.

Discussion

We searched for a membrane-bound clockwork component in insect ORNs that allows for sensing of regular pheromone patterns to guide mating flights. For this, we recorded interlinked circadian and ultradian frequencies in the spontaneous spiking activity of the pheromone-sensitive long trichoid sensilla of male Manduca sexta hawkmoths with in vivo long-term tip recordings. We found that Orco, which serves several important functions in insect olfaction (Stengl and Funk, 2013; Wicher and Miazzi, 2021), acts as a pacemaker channel that controls the circadian modulation of the spiking activity without affecting the frequency ranges that are relevant for the primary fast, phasic pheromone response (Dolzer et al., 2003; Nolte et al., 2016; Schneider et al., 2025). While circadian rhythmicity disappeared in the presence of the Orco blocker OLC15, ultradian activity rhythms remained. We confirmed with computational modeling that the circadian modulation of open-time probability of Orco as a leaky pacemaker channel was sufficient to account for our experimental results. Since we observed no evidence for circadian transcriptional control of Orco, our findings are consistent with the hypothesis that the plasma membrane constitutes an endogenous post-translational feedback loop (PTFL) clockwork. In this clockwork, circadian changes in cyclic nucleotide levels (Schendzielorz et al., 2015) modulate Orco conductance rather than transcription and insertion of additional Orco channels in the membrane.

The expression of a pacemaker channel that does not require daily degradation and transcription would constitute an economic and rapidly adjustable mechanism to generate endogenous membrane potential oscillations that are tuned to detect behaviorally relevant zeitgeber signals in the insect’s environment (Stengl and Schneider, 2024; Stengl and Schröder, 2021). The present study provides the first evidence for a systems view of timing that involves a PTFL membrane clock in hawkmoth ORNs, instead of the prevailing hierarchical view that assumes all endogenous circadian oscillations are outputs of the genetic TTFL clock (Stengl and Schneider, 2024; Stengl and Schröder, 2021).

Orco acts as a pacemaker channel to drive the circadian rhythm in spontaneous ORN spiking activity

A common motif of spontaneously active neurons is the expression of pacemaker channels that open at hyperpolarized potentials and thus depolarize the neuron (Bose et al., 2014; Cochet-Bissuel et al., 2014; Das et al., 2016; Golowasch et al., 2017; Lüthi and McCormick, 1998; Ratliff et al., 2021; Robinson and Siegelbaum, 2003; Sharma et al., 2023). The resting membrane potential of most neurons lies close to the negative equilibrium potential for potassium ions due to potassium leak channels (Goldstein et al., 2001; Patel and Honoré, 2001; Talley et al., 2001). When the pacemaker-driven depolarization reaches spike thresholds, voltage-gated Na⁺, K⁺ and Ca²⁺ channels promote spiking. Voltage- and Ca²⁺-gated K⁺ channels then repolarize the neuron to negative membrane potentials, re-starting the cycle

The core element of the different frequencies of endogenous oscillations that needs to be identified is the respective pacemaker channel (Stengl and Schneider, 2024). The finding of distinct rhythms in spontaneous spiking activity under constant conditions are indicative of underlying endogenous membrane potential oscillations. Therefore, different sets of voltage- and Ca²⁺-dependent ion channels with antagonistic effects on the membrane potential appear to underlie different ultradian oscillation frequencies of spontaneous activity that still need to be further elucidated (Dolzer et al., 2021; Stengl, 1994, 1993; Zufall et al., 1991). While this study did not aim to unravel the overall generation of spontaneous activity in the ORNs, we could demonstrate that Orco acts as a circadian pacemaker channel that drives the circadian modulation of the spiking activity.

The constitutively expressed Orco is a core element of a circadian PTFL clockwork in hawkmoth ORNs

In hawkmoth ORNs, at least two types of pacemaker channels exist: the highly conserved, slow, leaky Orco, which we focus on in this study, and an HCN type channel (Butterwick et al., 2018; Dolzer et al., 2021; Nolte et al., 2016, 2013; Sato et al., 2008; Stengl and Funk, 2013; Stengl and Schneider, 2024; Stengl and Schröder, 2021; Wicher et al., 2008; Wicher and Miazzi, 2021).

When expressed in vitro in heterologous expression systems, Drosophila Orco homomers assemble into leaky, non-specific cation channels with a reversal potential around 0 mV, constitutively leaking Na⁺, K⁺, and Ca²⁺ (Sato et al., 2008; Wicher et al., 2008; Wicher and Miazzi, 2021). In Drosophila, Orco is regulated by Ca²⁺/calmodulin and phosphorylation/de-phosphorylation at multiple sites (Cao et al., 2016; Getahun et al., 2016; Guo and Smith, 2017; Mukunda et al., 2016, 2014; Sargsyan et al., 2011). In Drosophila, the Ca²⁺/calmodulin binding site controls the localization of the OR-Orco complex in the ciliary membrane of ORNs and depends on previous odor stimulation (Cao et al., 2016; Guo and Smith, 2022; Jain et al., 2021). After phosphorylation of Orco by protein kinase C (PKC), a cyclic nucleotide binding site becomes available, and subsequent binding of cAMP or cGMP increases the open-time probability (Getahun et al., 2013; Wicher et al., 2008). This indicates two complementary mechanisms by which Orco conductance can be regulated: by controlling the number of Orco channels in the membrane or by modulating the open-time probability of existing channels through posttranslational modifications.

Since blocking Orco in M. sexta decreased the spontaneous activity and deleted its circadian modulation, Orco is apparently the main target for modulation of circadian activity in hawkmoth ORNs. Since Orco transcription levels were constant, posttranslational modifications of Orco that modulate the open-time probability provide a mechanistic explanation. Increases in cAMP and cGMP levels sensitize or adapt the ORNs to pheromone, respectively (Dolzer et al., 2021, 2008; Flecke et al., 2010, 2006; Redkozubov, 2000; Schendzielorz et al., 2015, 2012; Ziegelberger et al., 1990). While cAMP levels are under circadian control and peak during the activity phase, cGMP levels oscillate in a daily rhythm and peak during the resting phase in insects (Schendzielorz et al., 2014, 2015). We propose that cAMP and cGMP binding modulate Orco with opposing effects, consistent with the physiological needs of the hawkmoth during its sleep-wake cycle: cAMP binding increases Orco conductance while cGMP binding decreases it.

Our hypothesis that a circadian PTFL membrane clock in hawkmoth ORNs controls circadian activity rhythms predicts that the pacemaker channel Orco is part of a membrane signalosome. The protein complexes of this signalosome would comprise the pacemaker channel that generates the specific circadian frequency of endogenous potential oscillations, linked to oscillations of second messenger cascades that determine physiological setpoints of the circadian sleep-wake cycles. Thus, the circadian cAMP oscillations could result from a PTFL membrane clock comprising of a signalosome that minimally consists of voltage-gated Ca²⁺-channels, Orco, and membrane-anchored adenylyl cyclases (ACs). Since the membrane potential directly influences intracellular Ca²⁺ levels, the AC activity could be periodically upregulated by the circadian oscillations of the endogenous membrane potential oscillations. Alternatively, AC expression could be under the control of the molecular TTFL clock, which would put Orco under indirect TTFL control. This possibility should be further explored in hawkmoth antennae. Instead of, or in addition to, the voltage-gated Ca²⁺ channels the signalosome could also work if it contained octopamine receptors that cause circadian rises in cAMP levels through activation of G_αs, as found in hawkmoths (Dacks et al., 2006; Flecke et al., 2010). Since octopamine levels in the hemolymph exhibit circadian oscillations and octopamine sets a physiological state of heightened alertness in insects, the circadian PTFL clock in ORNs would synchronize with endogenous physiological rhythms via octopamine receptor activation (Flecke and Stengl, 2009; Schendzielorz et al., 2015).

Circadian and ultradian rhythms in spontaneous ORN activity resemble the physiologically relevant timescales for active sensing

Moths are more sensitive to pheromone stimulation during their activity phase than at rest, apparently due to antennal circadian clocks (Flecke et al., 2010; Flecke and Stengl, 2009; Merlin et al., 2007, 2006; Schendzielorz et al., 2015; Schuckel et al., 2007; Silvegren et al., 2005). The greater the distance from the female, the lower the pheromone concentration. The increased pheromone sensitivity during the activity phase enhances the male’s chance to detect a calling female.

Pheromone concentration is encoded in the frequency of the first, phasic burst of spikes of the characteristic triphasic spiking sequence of ORNs in response to brief pheromone pulses, which is independent of Orco (Dolzer et al., 2003; Nolte et al., 2013; Schneider et al., 2025).

However, through the cAMP-dependent increase in Orco conductance during the activity phase, the ORN membrane potential is closer to spike threshold and fewer pheromone molecules are necessary to produce a receptor current that brings the membrane voltage over that threshold. Thus, larger Orco conductance increases the likelihood of an ORN response to pheromone stimulation without changing the typical triphasic spiking pattern of the response.

In the odor transduction of Drosophila, OR-Orco activation drives the initial receptor current response. In contrast, Orco is not part of an OR-Orco odor-gated receptor ion channel complex in hawkmoths but controls the spontaneous ORN activity and the late, long-lasting pheromone response (LLPR). The LLPR begins several hundred milliseconds after the initiation of the pheromone-dependent transduction current and correlates with the time course of pheromone-induced rises in cGMP levels (Boekhoff et al., 1993; Nolte et al., 2016, 2013; Stengl, 2010; Stengl et al., 2001; Wicher and Miazzi, 2021; Ziegelberger et al., 1990). Thus, in hawkmoths, Orco is not part of the primary transduction channel but rather controls membrane potential oscillations and thereby the temporal resolution and general pheromone sensitivity of ORNs. The higher spontaneous burst frequency during the hawkmoth’s activity phase improves the resolution of the anticipated pheromone pulses during its mating flight in search for females.

In the mammalian suprachiasmatic nucleus (SCN), the clock that controls the sleep-wake cycle, cAMP levels oscillate at an ultradian frequency, and cAMP and cGMP induce opposing phase shifts in SCN spiking activity (Prosser et al., 1989; Prosser and Gillette, 1991). In M. sexta antennae, cAMP levels express daily oscillations and peak during the activity phase. In contrast, cGMP levels do not oscillate but are elevated by pheromone exposure, which adapts the spiking activity (Boekhoff et al., 1993; Dolzer et al., 2021; Flecke et al., 2006; Schendzielorz et al., 2015; Ziegelberger et al., 1990). This implies that cAMP and cGMP may have opposing effects on Orco conductivity, with results similar to what is seen in the SCN.

The upregulation of neural activity during wakefulness/alertness seems to be ubiquitous among animals. For example, the many synchronously recorded neurons in human EEG show high frequency oscillations in the beta and gamma ranges during alertness and low frequency oscillations in the delta and theta ranges during rest and sleep. Similarly, the spontaneous spiking activity of hawkmoth ORNs is upregulated during their activity phase, albeit only from below delta to the delta/theta range. The increased spontaneous spiking activity in the theta range during the activity phase of the hawkmoth matches the maximal temporal resolution of pheromone pulses of up to 10 Hz (Marion-Poll and Tobin, 1992), which supports a role for Orco as part of a membrane-bound PTFL clockwork that tunes the ORNs for active sensing. Even through oscillations in the field potential of electroantennogram responses to odor stimulation can reach 100 Hz in different insects (Szyszka et al., 2014), these frequencies most likely do not encode physiological odor responses. Apparently, they reflect the astounding adaptive ability of antennal ORNs to synchronize their endogenous membrane potential oscillations across the antenna to a broad range of stimulation frequencies. Pheromone pulse resolution of about 100 Hz was never reported in single olfactory sensilla recordings, and in behavioral experiments male moths stopped their upwind mate search if pheromone pulses exceeded 30 Hz (Baker et al., 1985; Bau et al., 2005, 2002; Tripathy et al., 2010).

Circadian oscillations in the ORN membrane potential would be ideally suited to tune sensitivity and temporal resolution of the pheromone-sensitive ORNs to the hawkmoth’s rest-activity cycle, synchronizing male and female mating behavior. Further experiments are needed to test our hypothesis that ORN signalosomes are highly plastic endogenous PTFL clocks comprising receptors for circadian and ultradian Zeitgebers that allow to tune into internal physiological and external environmental rhythms as basis for active sensing.

Stochastic ion channel dynamics and circadian modulation of Orco conductance explain modeled ORN firing patterns

Our modeling results provide mechanistic insight into the irregular spontaneous activity of pheromone-sensitive ORNs in M. sexta, highlighting how stochastic ion channel dynamics and circadian modulation of ion conductance together shape spiking patterns. By employing a Langevin approximation of the stochastic HH formalism, we demonstrate that the intrinsic noise arising from the random gating of ion channels is sufficient to generate both isolated spikes and bursts of spikes, as observed experimentally. Our model effectively recapitulates the spike distributions found in vivo and accounts for the role of channel noise as a critical determinant of ORN firing patterns.

A key contribution of our model is the incorporation of the Orco ion channel as a pacemaker component, with a conductance that varies linearly with cAMP concentration. We show that circadian modulation of Orco conductance alone is sufficient to produce the observed time-of-day dependent changes in spike distribution. This finding suggests that the rhythmic expression of cAMP levels across the circadian cycle may act as a primary driver of diurnal changes in ORN excitability, without requiring additional external inputs or modulation of other ion channels.

Importantly, our results reveal that the same set of ion channels can support the emergence of both single action potentials and complex bursting behaviors purely through the stochastic variability of their gating kinetics. This unifying explanation reduces the need for invoking separate biophysical mechanisms or circuit-level inputs to explain these distinct firing modes.

For simplicity, the circadian variation in cAMP concentration was modeled using a sinusoidal function. While this approximation is sufficient for capturing the qualitative features of the rhythmic modulation, future work could enhance the biological realism of the model by coupling Orco conductance dynamics to a molecular circadian oscillator, such as a Goodwin-type transcriptional-translational feedback loop. Incorporating this layer would allow for a more detailed exploration of how molecular clock components regulate cAMP synthesis and degradation. Additionally, further extensions could include downstream regulatory mechanisms such as phosphorylation and dephosphorylation events mediated by kinases and phosphatases under circadian control, potentially contributing to fine-tuned modulation of ion channel function over the daily cycle.

Together, these findings demonstrate the value of a biophysically grounded, noise-inclusive modeling framework for understanding complex temporal patterns in sensory neuron activity and open the door for future integrative models linking molecular circadian clocks to cellular excitability.

Acknowledgements

We thank Dr. Dieter Wicher (Max Planck Institute for Chemical Ecology, Jena, Germany) for generously providing OLC15, and Rishaban Radhakrishnan for help with Python codes. We thank Prof. Hülya Altuntaş for helpful discussions.

Additional information

Funding

AV, MF, YC, ACS, MEG, and MS were supported by Deutsche Forschungsgemeinschaft RTG 2749/1: “Biological Clocks on Multiple Time Scales”; KS and MS were supported by Deutsche Forschungsgemeinschaft STE531/20-2.

Author contributions

Conceptualization: AV, MF, MEG, MS; data curation: AV, MF, YC, PR, KS, ACS, MEG, MS; formal analysis: AV, MF, YC, PD, ACS; funding acquisition: MEG, MS; investigation: AV, MF, YC, KS; methodology: MF, PR, MEG, MS; project administration: ACS, MEG, MS; resources: MEG, MS; software: AV, MF, PR, MEG; supervision: MEG, MS; validation: AV, MF, YC, PR, KS, ACS, MEG, MS; visualization: AV, MF, YC, ACS, MS; writing – original draft: AV, MF, ACS, MEG, MS; writing –reviewing & editing: AV, MF, YC, ACS, MEG, MS.

Funding

Deutsche Forschungsgemeinschaft (RTG 2749/1)

Deutsche Forschungsgemeinschaft (STE531/20-2)

Supplementary Material

Mathematical Model

The mathematical model is a 22-dimensional system of stochastic differential equations designed to describe the behavior of ion channels and their effects on the membrane potential. I_LVA represents a low-voltage-activated calcium current, while I_Na and I_K are the transient sodium and potassium currents responsible for action potential generation. I_BK is a large-conductance potassium channel that is activated by both a depolarized membrane potential and an increased intracellular calcium concentration, and I_L is the leak current. Additionally, the Orco channel is a leaky, non-specific ion channel gated by cAMP, with the cAMP concentration oscillating in a circadian rhythm. To model the Orco conductance, we used a sinusoidal function of Zeitgeber time (ZT), independent of the membrane potential, as shown in Figure 2B, C. Each of these ion channels has been reported in vitro in previous studies (Dolzer et al., 2021).

The deterministic part of the 22D model is given by the mean field of the channel-based Langevin model. This approach was introduced by Fox and Lu’s (1994) for a 14D HH model. In their framework, one dimension tracks the membrane potential, while the other 13 dimensions correspond to the fraction of channels in each of the five states in the potassium (K⁺) Markov chain and the eight states in the sodium (Na⁺) Markov chain. Pu and Thomas (2020) demonstrated that the deterministic 14D model and the original 4D HH model are dynamically equivalent, meaning that each solution of the 4D model corresponds to a solution of the 14D model. Expanding upon Pu and Tomas’ methodology to represent out system, we added one dimension to track the calcium concentration, six dimensions for each possible state of the LVA channel, and two dimensions for each possible state of the BK channel. The state vectors for each ion channel are therefore defined as follows: a five-component vector NK for the K⁺ gates, an eight-component vector NNa for the Na⁺ gates, a six-component vector NLVA for the LVA gates, and a two-component vector NBK for the BK gates. The state vectors are represented as:

The K⁺ channel consists of four independent n (activation) gates, forming a five-vertex channel-state diagram with eight directed edges. The channel conducts a current only when it is in the rightmost state (Figure 2D). Similarly, the Na⁺ channel is composed of three identical n gates and one h (inactivation) gate, forming an 8-vertex diagram with 20 directed edges, one of which is conducting. The LVA channel includes two identical n gates and one h gate, resulting in a 6-vertex diagram with 14 directed edges, one of which is conducting. The BK channel consists of a single n gate, forming a 2-vertex diagram with two directed edges, one of which is conducting. The Markov chain for every channel is shown in Figure 2B.

For a sufficiently large number of channels, Schmidt and Thomas (2014) and Schmidt et al. (2018) demonstrated that, under voltage clamp, the equations governing the system can be approximated by a multidimensional Ornstein-Uhlenbeck (OU) process, or Langevin equation, as follows:

Here 𝜁^𝑖𝑜𝑛is the stoichiometry vector for the k^th directed edge. If we write i(k) for the source node and j(k) for the destination node of edge k, then . The voltage-dependent per capita transition rate along the k^th edge is and is the fractional occupancy of the source node for the k^th transition. The finite-time increment in the Poisson process is approximated by a gaussian process, namely, the increment in a Wiener (Brownian motion) process associated with each directed edge. These independent noise terms are scaled by , where represents the total number of that specific ion channel.

The membrane potential is governed by the current balance equation:

where the set 𝜒 contains the six ionic currents included in the model:

Each of the six ionic currents included in the model are described as:

[math11

Here, 𝑔_𝜒 is the maximal conductance, 𝑉 is the membrane potential, and 𝑉_𝜒 is the reversal potential for each ion current. The parameter values for each current can be found in Table 1.

Table 1:

Each transition rate is calculated in the following way:

where is the steady state value of the specific gate that switches states in the destination state and 𝜏_𝑝(𝑉) is the time constant of said gate. The steady state activation values and time constants for the LVA, Na, K, and BK channels were based on Viertel and Borisyuk (2019). The steady state functions for the activation gates are denoted with , and those for the inactivation gates with .

Each gate’s steady state activation function, except for the BK channel, takes the form:

The I_BK current is dependent on the intracellular calcium concentration and membrane potential as represented by the following equations:

The intracellular concentration of calcium ions is described by the following equation:

where 𝐶𝑎₀ is the steady state calcium concentration, 𝛾 describes calcium buffering, 𝑧 is the ionic valence of calcium, 𝐹 is Faraday’s constant and 𝑑 is the depth in microns at which calcium ions concentrate near the membrane.

The parameters of each steady state activation function as well as the time constant for each gating value can be found in Table 2.

Table 2: