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# Hippocampal and cortical communication around micro-arousals in slow-wave sleep

## Abstract

Sleep plays a crucial role in the regulation of body homeostasis and rhythmicity in mammals. Recently, a specific component of the sleep structure has been proposed as part of its homeostatic mechanism, named micro-arousal. Here, we studied the unique progression of the dynamic behavior of cortical and hippocampal local field potentials (LFPs) during slow-wave sleep-related to motor-bursts (micro-arousals) in mice. Our main results comprised: (i) an abrupt drop in hippocampal LFP amplitude preceding micro-arousals which persisted until the end of motor-bursts (we defined as t interval, around 4s) and a similar, but delayed amplitude reduction in cortical (S1/M1) LFP activity occurring at micro-arousal onset; (ii) two abrupt frequency jumps in hippocampal LFP activity: from Theta (6–12 Hz) to Delta (2–4 Hz), also t seconds before the micro-arousal onset, and followed by another frequency jump from Delta to Theta range (5–7 Hz), now occurring at micro-arousal onset; (iii) a pattern of cortico-hippocampal frequency communication precedes micro-arousals: the analysis between hippocampal and cortical LFP fluctuations reveal high coherence during τ interval in a broader frequency band (2–12 Hz), while at a lower frequency band (0.5–2 Hz) the coherence reaches its maximum after the onset of micro-arousals. In conclusion, these novel findings indicate that oscillatory dynamics pattern of cortical and hippocampal LFPs preceding micro-arousals could be part of the regulatory processes in sleep architecture.

## Introduction

Sleep has been described as a crucial behavioral state to physiological regulations, including hormone secretion1, learning processes2 and immune activity3. In mammals, sleep is divided into two main stages, rapid eye movement (REM)/Paradoxical sleep, and slow-wave sleep (SWS). While REM sleep is characterized by some physiological similarities to waking states in cortical electroencephalogram (EEG), such as low amplitude and high-frequency oscillations, SWS is defined by a high amplitude and low-frequency oscillations4. The sleep-wake cycle is controlled by periodic mechanisms within 24-hour periods, and the sleep phase itself oscillates between SWS and REM in an ultradian rhythm5,6. These two rhythms are fundamental to the homeostatic regulation of living organisms.

The coordinated activity among many encephalic structures promotes the transition from sleep to wake states. The arousal brain system is comprised by the ascending reticular activating system (ARAS) and a descending network from the cortex to the spinal cord. ARAS emerges from neuronal nuclei located in midbrain and pons and projects to sub-cortical structures that stimulate cerebral cortex and generate fast EEG oscillations7. In turn, a descending network is related to motor control allowing high amplitude movements, associated with increased electromyography amplitude (EMG), which corresponds to classical motor system pathways8.

In particular, sleep phases can be often interrupted by small, transient micro-arousal events that naturally emerge even during deep sleep stages. They are believed to recruit, at least partially, structures involved in the global arousal process9. The occurrence of micro-arousals was firstly interpreted as short wakefulness intrusions into sleep10. Pathological micro-arousals were also related to sleep disorders such as central apneas11,12,13, in which they were effective to restore ventilation overshoots14. Nevertheless, the role of physiological micro-arousals remains to be fully understood10,15. Micro-arousals are characterized by abrupt alterations of oscillatory patterns in the LFPs, in particular, of low-frequency bands5,16. In humans, micro-arousals are characterized by low-voltage fast-rhythm EEG oscillations, which can be accompanied by bursts of high-amplitude delta oscillations10. In rodents, micro-arousals during slow-wave sleep can encompass the emergence of hippocampal theta oscillations associated with increases in the frequency of ECG and in the amplitude of EMG17,18,19, which often include rapid and transient head movements20. Furthermore, physiological micro-arousals are quite common during SWS. They are associated with small irregular activity (SIA) in hippocampal LFPs, which are characterized by a sharp decrease in the oscillatory amplitude20,21,22,23. SIA has been previously proposed as a mechanism for maintaining context memory during sleep17. Recordings of single units in the rodent hippocampus indicate that SIA reflects a partial arousal in response to internal/external stimuli, which allows the animal to assess whether the full awakening is warranted, without disrupting the sleep cycle. Noteworthy, other studies suggested that the hippocampus may play a pivotal role in information processing during SIA events in sleep. Thus, the description of these electrical hippocampal phenomena during SIA events indicates that this brain structure could potentially act as part of the activating arousal brain system.

A recent work20 examined the temporal dynamics of micro-arousal events using head-accelerometer recordings. The authors reported that accelerometer signal (Acc) can predict motor-bursts typically observed in micro-arousals, through a specific technique called recurrence plot24. Furthermore, micro-arousals were preceded (about 4 seconds) by a drop in the determinism of Acc (a quantifier that computes how accurate are predictions based on past trajectories24), as well as by a simultaneous decrease in the standard deviation of the LFPs in the CA1 area of the hippocampus.

The interplay between cortical and sub-cortical structures of the brain contribute to the generation of oscillatory patterns found during SWS25. Recent studies have provided electrophysiological evidence of a robust coupling between hippocampal and cortical areas during SWS26,27,28. Lately, it was shown that micro-arousals were often preceded and succeeded by periods of low LFP amplitude and low firing rate activity, both in neocortical and hippocampal networks29. Despite such studies have shed some light on the understanding of physiology and physiopathology of micro-arousals in sleep architecture, little is known about the relationship involving hippocampal activities, transient cortico-hippocampal communication and micro-arousals episodes during SWS.

Here we investigated the following questions: Are micro-arousal events associated with cortical and hippocampal activities modulations? Do cortical and hippocampal activities change simultaneously at micro-arousals events? Is there a typical brainwave communication between cortex and hippocampus during SWS micro-arousals? Aiming to address these questions, we used micro-electrode arrays to record hippocampal and cortical LFP activities around micro-arousal events during deep SWS in mice.

## Methods

### Ethical compliance

This study complied with all ethical regulations regarding animal experimentation. All housing, surgical, and behavioral procedures presented in this work were in accordance with the National Institutes of Health guidelines, and approved by the Edmond and Lily Safra - International Institute of Neuroscience of Natal (ELS-IINN) Ethics Committee for Animal Experimentation (protocol number 08/2010).

### Experimental Procedure

#### Animals

Five adult male C57Bl-6 mice (2–5 months) were used in this series of experiments. After surgery, animals were housed in individual home cages under 12/12 h light/dark schedule (lights on at 7 am), with food and water freely available (Fig. 1).

#### Multielectrode Implantation Surgery

Animals were surgically implanted with chronic multielectrode arrays for intracranial local field potentials recordings. We used 13 teflon-coated tungsten microwires (diameter, 50 um; interelectrode spacing, 250 um; impedance, 0.5 MOhm at 1 KHz) attached to Omnetics 16-pin connector. The arrays were implanted through a rectangular craniotomy (Coordinates from Bregma: 0.55 mm to 1.65 mm to the right hemisphere, 0.0 mm to 2.2 mm posterior) under isoflurane anesthesia. For each animal, 5 electrodes were stereotaxically positioned in the dorsal CA1 area of the hippocampus, 4 electrodes in the M1 area of the motor cortex and 4 electrodes in the S1 area of the somatosensory cortex (1.5 deep into the brain). Three screws attached to the skull bone and soldered to a silver wire served as an electrical ground to the recording system. A three-axis accelerometer sensor attached to the headstage was used to record the X, Y, Z axes of movement of the head of the animals. Complementary, a 10-fold pre-amplification circuitry was located in association 4 cm distant from animals head, in order to reduce noise. The LFP signals were sampled at 1000 Hz, pre-amplified 500x and recorded in a Multichannel Acquisition Processor (MAP, Plexon System).

#### Open Field Apparatus

After one week of post-surgery recovery, animals were anesthetized with isoflurane (5%), for connecting the wires of the headstage with those of Plexon multi-electrode recording device. Around 5 minutes after that, animals were injected with saline (Nacl0.9%, 2 ml) and were placed in an open field apparatus (50 cm diameter and 30 cm high) at 10 am and recorded for 12 hours (Fig. 1(D)). During this time, animals were not disturbed in order to allow them to disclose a typical architecture of wake state, SWS, and REM sleep.

#### Identification of SWS episodes

Spectral analyzes of local field potentials were used to identify and quantify the occurrence of wake state, SWS, and REM sleep within the sleep-wake cycle (see Fig. 1(C)). State maps were employed for the characterization of waking and SWS/REM sleep states30, (ENeuFranca2015)31. We selected epochs of deep SWS within sleep clusters, located on the superior right border of the state map (see the shadow rectangle on the Fig. 1(C)). SWS states were behaviorally defined as presenting stillness with eyes closed at video recordings. Hippocampal LFPs obtained from deep sleep presented large-amplitude and slow-frequency oscillations and the head accelerometer signals were dominated by variations of very low amplitude. Only the first 4 recording hours were analyzed in this study. Sleep stages were grouped according to the following categories: WK (active exploration of the environment, large-amplitude variations on the accelerometer, and hippocampal theta (4–12 Hz) oscillations), SWS (stillness with eyes closed and large-amplitude slow (0.1–2 Hz) oscillations in the hippocampus) and REM sleep (stillness with prolonged whisking, no variations on the accelerometer signals, eyes closed and hippocampal theta oscillations).

#### Identification of micro-arousal with the accelerometer time series

Once we selected deep SWS episodes, we analyzed the occurrence of micro-arousal events on the accelerometer and LFPs signals [see Fig. 1(D) between dashed lines]10. Micro-arousal events were defined here as a fast motor-burst event during deep SWS on accelerometer signal with a minimum duration of 0.3 seconds, and an amplitude of more the two times the standard deviation of the basal signal (as previously described in20,31).

#### The τ (tau) event: the interval preceding the micro-arousal

The event τ interval was defined here as a duration of time beginning at the sharp drop in LPF amplitude at CA1 (>50%, using the Hilbert transformation envelope) [see Fig. (B) dashed line] and ends at the onset of the micro-arousal at Acc fluctuation (on average, the duration of τ is around 4 s). This pattern oscillatory CA1 LFP preceding motor explosions in Acc - during deep SWS - was persistently observed for all subjects (n = 176 in five animals). The statistic of the group for all 5 subjects who used ANOVA to test whether individuals present a distinct τ period did not find the significant difference (F(4,106) = 0.949, p = 0.44) [see Fig. 1 Supplementary Information].

#### Histological Preparations

In order to evaluate if electrodes were localized near the dorsal CA1 area of the hippocampus [see Fig. 1(B)], and at 1.5 mm deep in M1 and S1 cortical areas, brains were sliced and subjected to histological preparations. Briefly, after electrophysiological recordings, animals were euthanized in deep anesthesia with isoflurane (100%), and perfused with PBS (pH 7.4) and paraformaldehyde 4%; brains were removed, freezed at −80 °C and sliced at 30 um thickness. Nissl staining of coronal slices was performed in each brain of implanted animals, and in all cases brain tissular micro-lesions were located at 1.5 mm deep (cortex electrodes) or at CA1 hippocampal layer.

### Data Acquisition and Analysis

#### Data Treatment

The accelerometer output consists of three components of the acceleration vector a(i) = (ax, ay, az). To perform the mathematical analysis we combined the vector components in a single quantity similar to the vector module at the same time that we extract the average of each vector component. The new t(i) vector formed from the components of a(i) is defined as; $$Acc=({a}_{x}-{\hat{a}}_{x})+({a}_{y}-{\hat{a}}_{y})+({a}_{z}-{\hat{a}}_{z})$$ where $$\hat{a}$$ is the mean of a, shown in Fig. 1 - the second panel in(D), and with more detail in Fig. 2(A)20,31.

#### Envelope analysis using Hilbert Transform

Knowing the analytic signal is a complex signal. Mathematically the envelope e(t) of a signal x(t) is defined as the magnitude of the analytic signal as: $$e(t)=\sqrt{x{(t)}^{2}+\hat{x}{(t)}^{2}}$$; the $$\hat{x}{(t)}^{2}$$ is the Hilbert transform of the x(t). For each LFP signals from CA1 Hippocampus area [Fig. 2(B)] and S1/M1 Cortex region we calculated the envelope using Hilbert transform and smothering it calculates the average of each 200 time window and shifting it using 1 dot as step.

#### Spectrogram Analysis

The Spectrogram (Fast Fourier Transformed Windowed) was calculated for 1.2 sec windows (1200 dots) and shifted every each 100 dots = steps (overlapping = 1,100 dots = 1.1 sec). The normalization of the spectrogram matrix was computed by summing all value of frequency’s amplitudes for each time column and dividing it for each corresponding frequency amplitude. Next step was filtering the spectrogram’s matrix to make it smoother: The procedure was taking each 20 dots (window) and calculate the average of it. After that we shifted the windows for every each 1 dot (19 dots of overlapping). To guide the eye about the highest amplitude as a function of frequency and time of the spectrogram matrix we computed the maximum of it of each time column and plotted the red dot on it. As result we have plotted a red line throughout the entire time as shown in Fig. 2 panels (B) and (C). Following the sampling theorem (Nyquist theorem) and once our frequency bands of interest are delta (0.5–4.0 Hz) and theta (5.0–8.0 Hz) we built our spectrogram matrix choosing the frequency interval from 0.5 Hz to 11.5 Hz with the accurate (step) of 0.1 Hz.

#### All Event-related Spectrogram Analysis

Spectrograms were calculated using the “spectrogram” function of MatLab’s Signal Processing Toolbox (2015b). We first obtained CA1, M1 and S1 LFPs signals 20 sec around all micro-arousal events across all animals. Then we computed spectrograms in 2-s Hamming windows with 1.5-sec overlap to LFPs from each of the three brain areas. We then normalized each spectrogram from each brain area using “zscore” function and calculated the mean across all spectrograms from all animals to obtain the averaged event-related spectrogram for each brain area.

#### Power Spectral Density

We calculated the Power Spectral Density from CA1 Hippocampus and S1/M1 Cortex signals of all 5 mice of all deep sleep episodes during three epochs: [1] during the short time preceding τ interval; [2] during the τ interval; [3] during the micro-arousal interval as shown in Fig. 3 panel (C). The average (mean) and standard deviation of the mean for all event were calculated and plotted.

#### Event-related Coherogram Analysis

The coherence function is a frequency domain measure of the similarity of two signals x(t) and y(t). The magnitude-squared coherence function (MSC), which is a function of frequency, is given by formula: $${C}_{xy}(f)=\frac{|{S}_{xy}(f){|}^{2}}{{S}_{xx}(f){S}_{yy}(f)}$$. When Sxy(f ) is the cross-spectrum density between the x and y signals, and Sx(f ) or Sy(f ) are the individual power spectra of signals x and y respectively. The coherence of our physiological signals was calculated using the “mscohere” function (see above) of Matlab’s Signal Processing Toolbox (2015b). We first obtained CA1, M1 and S1 LFPs signals 20 s around all micro-arousal events across all animals [see Fig. 4]. Then we computed the coherence in 1-s Hamming windows with 750-ms overlap between pairs of LFPs from all the three brain areas as displayed in Figs 5 and 6. We then normalized each coherogram using zscore function (that returns the z-score for each element of a specific signal such that columns of this signal are centered to have mean 0 and scaled to have standard deviation 1) and calculated the mean across all coherograms from all animals to obtain the averaged event-related coherogram.

#### Statistical Analysis of the Mean Coherence across micro-arousal events

For comparisons of cross-structural LFP coherence among blocks of 5 s time intervals around micro-arousal events, we obtained the average normalized (zscored) coherence from all micro-arousal events (n = 167). We analyzed three bands of interest: Slow oscillations (0.5–2 Hz), Delta oscillations (2–4 Hz), and Theta oscillations (5–12 Hz) as shown in Fig. 7. We used one-way analysis of variance (ANOVA) followed by Tukey multiple comparison test, to find significant (p < 0.05) variations in the cross-structural LFP coherence across time (at pre-τ, τ and post-τ intervals) for each of these bands of interest. Data are expressed as mean ± SEM.

## Results

We studied hippocampal and cortical LFPs simultaneously recorded with head acceleration of 5 freely moving mice along the SWS stage (see Fig. 1). In this work we performed our analysis at periods of deep SWS presenting short bursts of motor-activity, which characterize micro-arousals as observed in Fig. 1(D). Moreover, we explored in detail the dynamic of LFP activity preceding micro-arousals across brain areas. Further, using coherence between hippocampal and cortical rhythms we detected a specific frequency-band communication preceding micro-arousal during SWS.

### Hippocampal dynamics organization preceding SWS micro-arousals

Our analysis revealed a multitude of motor-bursts on Acc activity (micro-arousals) that persistently occurred even during deep SWS stages. These micro-arousal events were consistently associated with periods of a significant drop in the CA1 LFP amplitude [see red rectangles in Fig. 1(D)], resembling patterns of brain activity typically observed during awake states. Panels (A) and (B) in Fig. 2 show examples of Acc and CA1 LFP activities around micro-arousal. Using Hilbert transform envelop, we observed a sharp drop in the CA1 LFP amplitude, which occurs a few seconds before the micro-arousal that we defined here as τ interval (2.2 seconds in this example). This CA1 LFP oscillatory pattern preceding motor-bursts in Acc during deep SWS was persistently observed for all individuals (n = 176 in five animals). In order to further investigate this LFP dynamics pattern around micro-arousals, we also analyzed the brain fluctuation frequencies over time. Figure 2(C), displays a spectrogram obtained from CA1 LFP. We firstly observed the rising of theta oscillations a few seconds before the onset of τ, followed by a frequency-band transition from 8.5 Hz to 2 Hz at τ onset (theta-delta). Hippocampal delta oscillations (2 Hz) remain stable throughout the τ interval. A second transition from 2 Hz to 6 Hz (delta-theta) occurs now precisely at the beginning of the micro-arousal. Please note that the CA1 low amplitude (starting at the τ onset) lasts until the end of the micro-arousal event. The CA1 LFP amplitude increases again, after the micro-arousal event. Further, in order to highlight the transition of dominant frequencies, in panel (D) we display a three-dimensional spectrogram of the CA1 signal shown in (C), and normalized by the maximum amplitude of the entire spectrogram matrix. A threshold at 70% of the maximum power spectrum was used to quantify the rate of frequency transition. We observed that 83% of all micro-arousal events were above the threshold for the two frequency-band transitions (theta-delta-theta). More examples can be found in Supplementary Material.

To confirm whether frequency jumps emerge in the group average, we performed the mean spectrogram and power-spectrum analysis around micro-arousal events (n = 174). Figure 3(A) depicts the Acc signal with a typical micro-arousal event, in which the average τ interval is 3.6 seconds [see Supplementary Information in Fig. 1]. Panel (B) shows the normalized mean spectrogram for a fixed time window (20 seconds) around the beginning of all micro-arousal events. The same outcome observed in Fig. 2(C) can be seen in the group results: a first frequency band transition (theta-delta) occurring at the onset of the average τ interval; and a second frequency band transition (delta-theta) occurring at the micro-arousal onset. Although the analysis here overlaps all events with different τ duration [see Supplementary Information], we observe a dominant power-spectrum for different intervals of time: (i) theta oscillations at pre-τ interval from −10 to −4 s; (ii) delta oscillations at the average τ interval itself, and (iii) a lower theta oscillations after τ interval (in average) that includes the micro-arousal/motor-burst activity itself (panel (A) related to Acc data).

To further investigate the precise frequencies that characterize each interval around micro-arousal events, we performed Fourier transform for three specific intervals: [1] pre-τ [2], τ, and [3] post-τ (see Fig. 3(C) inset and also Fig. 2(A,B)). Figure 3(C) shows the mean power spectrum of these three defined intervals for all micro-arousal events. We observe two main frequency bands in the pre-τ interval [1] (green): one between 5–12 Hz (maximum at 8.5 Hz) and the other as a lower peak whose local maximum is around 3 Hz. In its turn, the τ interval itself [2] (black) disclose only a slower oscillation (0.5–4.0 Hz, peak at 1.7 Hz), which corresponds to classical delta band. Finally, at post-τ interval [3] (red) we verified a sharp frequency range of 6–8 Hz, which peaked around 7 Hz. Noteworthy, pre-τ and post-τ intervals exhibit theta oscillations with different frequencies. While pre-τ is dominated by a wide theta frequency band (encompassing both type-I related to atropine resistant theta: 8–14 Hz; and type-II related to atropine sensitive theta: 4–8 Hz), the post-τ interval is related specifically to theta type-II32,33,34,35. Taken together, these results show a unique dynamic behavior in the hippocampal LFPs that rises seconds before micro-arousal events. Aiming to investigate sensory and motor cortical processing linked to micro-arousals during SWS, we next analyzed cortico-hippocampal communication around motor-bursts.

### Hippocampal and cortical frequency-band communication preceding micro-arousal

In order to investigate sensory and motor cortical processing linked to micro-arousals during SWS, we next analyzed cortical and hippocampal dynamics around motor-bursts. Figure 4 shows simultaneously recorded Acc signals (panel A), hippocampal (CA1, panel B), primary motor (M1, panel C) and primary somatosensory (S1, panel D) LFPs around a typical micro-arousal event. The sharp drop in hippocampal CA1 LFP amplitude observed during τ interval (indicated by τ in Fig. 4(B)) is not simultaneously accompanied by cortical M1 and S1 amplitude transition. The abrupt drop in the amplitude of S1 and M1 cortical LFPs often occurs simultaneously at the micro-arousal onset (see arrows in Fig. 4 panels C and D). This example represents the average LFPs amplitude and Acc signals around micro-arousals in the whole group of mice. Other examples of amplitude transitions can be verified at Supplementary Material at Fig. 2. In addition, using the same approach as Fig. 3(C), we performed Fourier transform from M1 and S1 cortical LFPs to investigate in more detail frequency ranges at intervals: pre-τ, τ, and post-τ. Differently, from hippocampal LFP that oscillates from 0.5 to 12 Hz, cortical LFPs oscillated basically only beneath 4.5 Hz, which corresponds to delta and slow rhythms [more details see Supplementary Information]. Specifically, we found a subtle peak frequency difference between M1 and S1 during the τ interval: while in S1 the peak is located around 2 Hz, the M1 peak is situated around 3.5 Hz [see Supplementary Material]. To further describe these temporal changes in multiple cerebral LFP fluctuations, we model four main intervals: pre-τ, τ, post-τ, and return to basal, where ellipses mean CA1 region, squares mean M1 region and triangles mean S1 region (Fig. 4(E)) and color represents changes in amplitude fluctuation characteristics.

Since previous works demonstrated cortico-hippocampal anatomical communication27 and once our results revealed that amplitude transitions in CA1 precedes those in M1 and S1, we performed frequency coherence analysis to evaluate a possible cross-talk among those areas around micro-arousal events. Figure 5 shows the coherence between cortical and hippocampal areas ±10 sec. around the micro-arousal events. Panel (A) shows the Acc, evidencing τ (in average) and the onset of the micro-arousal event (oval dashed line). Panels (B–D), display, respectively, the normalized coherogram between brain areas: CA1 × M1; CA1 × S1; and M1 × S1, in average overall micro-arousal events. In Fig. 5 we observed no prominent frequency coherence between brain areas during pre-τ [1] interval. We observed an increase of cortico-hippocampal coherences [panels (B) and (C)] that match exactly τ interval duration [2], which emerges in delta and theta frequency bands and peaks (2.5–5.5 Hz)(dashed ellipses). We also noted a remarkable high coherence transition from 2–6 Hz to 0.5–2 Hz frequency bands that occurs specifically at the onset of micro-arousal (post-τ interval [3]) [see oblique arrows in Fig. 5(B,C)]. Conversely, cortico-cortical coherence [panel D] only rises around micro-arousal onset, in a wide range of frequencies (0.5 Hz to 12 Hz).

To better investigate the coherence around micro-arousal [shown in Fig. 5], we measured the three frequency bands of interest as time function: Slow oscillation (0.5–2 Hz, in red), Delta range (2–4 Hz, in blue) and Theta range (5–10 Hz, in cyan), as observed in Fig. 6. Panel (A) displays accelerometer signal (Acc). The normalized coherences between CA1 × M1 are shown in panel (B), CA1 × S1 in panel (C) and S1 × M1 in panel (D). We verified: (i) no prominent coherence between cortico-hippocampal as well as cortico-cortical LFPs during the pre-τ interval (around −10 to −5 sec.) for all frequency bands at panels (B–D) [see Fig. 6]. (ii) a marked increase of the coherence between all brain areas during τ interval (in average from −4 to 0 sec.). However, in a closer view, Delta (blue) and theta (cyan) cortico-hippocampal coherences increase at τ onset and reaches its maximum positive value around half of the τ interval (about −2 sec.). On the other hand, slow oscillations (red) present slightly different temporal increase: at τ onset it reveals a maximum negative value and slowly gain positive values across τ interval, reaching its maximum positive value only at post-τ interval [see set of arrows in panels (B) and (C)]. Panel (D) shows that cortico-cortical coherence in these three frequency bands increase simultaneously during τ interval and reach its maximum values near the transition from τ to post-τ (around −1 to 0 sec.). (iii) that cortico-hippocampal coherences for slow oscillations (red) reveals its maximum positive values during post-τ interval (0 to 5 sec.). Conversely, coherences of delta and theta bands decrease returning to basal levels after micro-arousal [panels (B) and (C)]. Cortico-cortical coherence [panel (D)] at post-τ interval reveals a progressive decrease for all three analyzed band frequencies. Also, a slight coherence difference between delta and theta range rhythms can be perceived in detail: while slow and delta oscillations fall to negative values after τ, theta oscillation presents a faint fall but remains at positive values [see Fig. 6(D)].

Additionally, we checked whether frequency communication between brain areas displays statistical differences among pre-τ (from −10s to −5s), τ (−5s to 0 s) and post-τ (0 s to +5 s) intervals performing ANOVA comparisons (see Methods). Figure 7 depicts cortico-hippocampal and cortico-cortical coherences in boxes with 5 a seconds duration each. For all panels, the micro-arousal onset was used as reference (zero second).

The analysis of cortico-hippocampal frequency communication confirmed that coherence in Slow oscillations (panels A and D) increases significantly during the post-τ interval (CA1×S1: p = 0.002, F = 7.74, and CA1×M1: p < 0.001, F = 9.37; Tukey post-hoc p < 0.05, considering pre-τ and post-τ intervals in A and D). Also, statistical analysis of coherence in Delta revealed important significant differences (CA1×S1: p < 0.001, F = 15.51 and CA1 × M1: p < 0.001, F = 9.73; Tukey post hoc revealed in both cases an increase at τ interval, p < 0.05 compared to pre-τ and post -τ intervals). The analysis of coherence at Theta band considering cortico-hippocampal communication worked in the same direction of previous results at Delta, showing significant differences (HPxS1: p < 0.001, F = 17.13; HPxM1: p < 0.001, F = 15.67; Tukey post-hoc revealed in both cases an increase at τ interval, p < 0.05 compared to pre-τ and post -τ intervals) [see asterisks in panels (B, C, E, F)]. On the other hand, the analysis of cortico-cortical coherence shows no differences in slow (S1 × M1: p = 0.09, F = 2.5) or delta (S1 × M1: p = 0.49, F = 0.73) bands [see panels (G, H)], but theta (S1 × M1: p < 0.001, F = 9.82) band presents a significant increase from the pre-τ to τ intervals (p < 0.05 at post-hoc Tukey test) [see panel (I)]. Therefore, the foremost results here show that while cortico-hippocampal communication in delta and theta bands increase at τ onset [(panels B, C, E, F)], while cortico-cortical coherence presented only marginal differences (maximum p-value of 0.09, related to panel G) around the micro-arousal events. For all F values the degrees of freedom were (2, 29). In summary, here we described a distinct dynamic pattern around micro-arousals in the mice brain, and these results point to an activity shift occurring firstly at the hippocampus (CA1) and, in sequence, at primary cortical areas (S1 and M1).

## Discussion

According to the American Sleep Disorders Association definitions, physiological arousals could be considered events of sleep disruption, thus potentially signifying a risk of breaking a regular sleep pattern. However, recent studies pointed out that micro-arousals can be considered as an intrinsic part of sleep architecture, having a physiological role in its regulation and maintenance9,10,36,37,38. Results evidenced here corroborate those present-day views, in which micro-arousal events may be critical for the proper maintenance of SWS sleep state and disclose a complex but robust structure pattern. This idea is supported by our results, which gives the interpretation of neural processes providing a preparation (τ interval) for the emergence of motor-bursts (micro-arousal) in sub-cortical structures (such as the hippocampus). In detail, these processes are associated with LFP changes in frequency and amplitude parameters during SWS sleep. Furthermore, the increase of cortico-hippocampal coherence precedes the increase of this parameter in cortico-cortical networks [see Fig. 6]; these outcomes point out to the same direction of the previously described ascending reticular activating system (ARAS). In this sense, our coherence results indicate that cortical and motor activation are the culmination of a gradual processes, during the τ interval, performed to promote the rise of micro-arousals.

Other events in τ interval seem to be critical for the micro-arousal emergence. In the past related work, using the same acquisition methods, Lima et al.20 described that the accelerometer signal discloses a typical decrease in Determinism (DET, an output of recurrence plot analysis) before SWS micro-arousals, thus serving as an arousal predictor. The decrease in DET starts approximately 4 seconds before the micro-arousal onset, comprising an interval which coincides with τ interval as here described. This implies that, at τ interval, motor activity displays a gradual loss of entropy which coincides in time with a progressive increase in cortico-hippocampal frequency coherence. Also, as reported by Lima et al.20, around 4 seconds preceding micro-arousals, amplitude fluctuation, in the hippocampus, start to decrease, which coincides with the amplitude drop onset in this present paper. All these data also support the idea of a well-defined and gradual process occurring in sub-cortical, cortical and motor structures in the transition interval (τ) allowing micro-arousal emergence. However, these data do not yet really explain why micro-arousals emerge in SWS state, in terms of an adaptive physiological mechanisms.

So, from the descriptions evidenced here, we built two rationales to possibly situate micro-arousals as crucial events within SWS physiology. One, the micro-arousal may work as a regulatory process in deep sleep homeostasis, probably to avoid an excessive deepening of sleep that could lead to a pre-comatose sleep stage. The fact that both hippocampal and cortical LFP and accelerometer signal returns to basal states after motor-bursts provides some support for this consideration. In the second rationale, the micro-arousal may emerge as a result of an incomplete SWS transition for REM sleep. This last hypothesis is supported in our work by the occurrence of theta before and after a brief period of slow oscillation in CA1. Finally, these two rationales are not mutually excluding: a mouse could avoid an excessive deepening of SWS state through an incomplete (a micro-arousal with a return to the basal state) or even a complete transition to the REM stage.

Past works defined as SIA a small irregular activity that appears in hippocampus in varied phases of sleep/wake cycle17. Here, however, we specifically selected similar events related to SIA definition that were observed within SWS episodes. This was performed because recently, hippocampal LFP amplitude changes during SWS were associated with micro-arousal episodes in sleep20. In addition, theta rhythm was associated with the onset of the micro-arousal event revealed by accelerometer data (Acc), following similar results presented by Kang et al.39. Theta oscillations can be found in two sub-band frequencies described for hippocampal LFP: type-1, related to atropine-resistant (8–12 Hz) and type-2, related to atropine-sensitive (4–8 Hz)33,34. Our findings point to the occurrence of both high (8–12 Hz) and low theta (4–8 Hz) before shifting to slow oscillation/delta range (first frequency jump) and in sequence from slow oscillation to a lower theta range/type-2 theta (second frequency jump) [see Figs 2 and 3].

In detail, we describe frequency band transition associated with micro-arousals in a series of three main time intervals in both hippocampus and cortex: (i) Before micro-arousals (pre-τ interval), hippocampal oscillations are situated in both bands of type-1, atropine-resistant theta and in type-2 theta (atropine sensitive). (ii) In sequence, τ interval, there is a CA1 frequency band transition (jump) from theta to delta band (around 2.0 Hz), while in S1/M1, delta arises within a (1.5–4.0 Hz) band [see Supplementary Information Fig. 4]. (iii) Micro-arousal begins with the onset of a type-2 CA1 theta range (around 6.5 Hz). Such band was not found in the cortex in a high power-spectrum in this post-τ interval [see Supplementary Information Fig. 4]. However, we observe a cortical LFP amplitude decrease by the micro-arousal onset. Previous work described the existence of type-1 and type-2 theta bands during SWS sleep at the onset of arousals39 but, at wake state, type-2 theta is considered extremely fugacious. Type-1 theta (8–12 Hz) is associated with locomotion and some voluntary behavioral patterns, and, complementary, it is found in REM sleep, while type-2 theta (4–7 Hz) is typically a transient, short living rhythm, encountered when an animal is preparing to move or when it is frightened in a freezing state32. Conversely, our findings point to a different function of both theta types in SWS stage: around micro-arousals, type-1 theta appears when the animal does not present relevant motor activity (quiet behavior), while type-2 theta emerges when it performs a motor-burst. In light of the present data, we can not point to precise mechanisms related to this finding, and more research would be useful in this field. Nevertheless, because both theta-2 from REM and from micro-arousals may be related to the emergence of superficial sleep phases, they may arise through shared neurotransmitter networks, such as the cholinergic input from septal area40.

Concerning slow oscillation appearance in CA1, we detected delta-related frequencies immediately before micro-arousals. There are two types of slow rhythms in the hippocampus (0.5–4.5 Hz). One is the hippocampal slow oscillation, coupled to cortical LFPs, with a peak frequency of 0.64 ± 0.05 Hz (described in urethane-anesthetized rats)41. The other is the hippocampal respiration-induced rhythm (HRR), peak in 1.25 ± 0.10 Hz (described in urethane-anesthetized rats), and around 3.5 Hz (in mice)41,42, whose detection in hippocampus is dependent of airflow through mouse nostrils and the consequent respiratory rhythm generated in the olfactory bulb. This information enters in hippocampal networks through the entorhinal cortex42, which is also entrained by olfactory bulb respiratory pattern. According to Yanovsky and co-authors42, mouse respiration rate under urethane anesthesia (2.0–4.0 Hz) fulfills the range of HRR oscillations, but this rate may be different in a deep SWS physiological state. Usually, both brain-related olfactory bulb rhythm and respiration rate in mice appears in non-REM sleep stages (1.2–3.5 Hz), when detected with electrodes and simultaneous plethysmography43. This rate is consistent with our findings, which reveals that mice, in this work, disclose in CA1 peaks in the delta (1.0–3.5 Hz), which may reflect a respiratory pattern [see Fig. 3(C) - black circle/line during τ interval].

Taking a closer look at the results concerning frequency coherence analysis, the range of slow oscillations band (1.5–3.5 Hz), potentially associated with HRR, have a specific profile in cortico—hippocampal coherence, which is markedly different from the hippocampal slow oscillation (0.5–2 Hz): before micro-arousals, we have an increase of coherence of HRR-compatible oscillations and theta bands between hippocampus and cortex; immediately after the arousal this coherence is diminished. On the other hand, cortico-cortical coherence in all frequency bands increases during a micro-arousal event or until 2 seconds after this time point [see Fig. 5(D)]. From this observation, we may argue that, immediately before the micro-arousal, coherence between somatomotor cortex and sub-cortical structures (such as the hippocampus) in both HRR and theta rhythm, may be an important phenomenon linked to the onset of the arousal event. However, more studies are necessary to indicate a clear causal relationship between the increase of cortico-hippocampal communication related to these frequency ranges, in τ interval. These studies may include the analysis of coherence variance regarding respiratory-derived rhythms, associated with the micro-arousal event in SWS architecture. Considering that micro-arousals might appear as a result of oxygen and carbonic acid body fluctuations in SWS, future works also should focus on olfactory bulb patterns, associated with respiratory markers, O2, and C02 blood levels, and to hippocampal LFP oscillations, within the context of micro-arousals, in order to clarify this possible mechanism.

Additionally, concerning cortico-hippocampal communication, a previous study showed a robust correlation of neuronal discharges between the somatosensory cortex and hippocampus in different time scales in the mouse and rat27. Considering global neuro-anatomical and neurophysiological basis of arousal processes, we may interpret our results in terms of an increase in frequency-based communication considering hippocampus and cortical areas before the arousal onset. This makes sense, since hippocampus may work together with other sub-cortical structures, such as the thalamus in the ascending activating system, to activate the cortex and prepare for a possible arousal or another transitional state. Our results also suggest that, after the micro-arousal onset, hippocampal CA1 area decreases the coherence with the somatomotor cortical areas, while cortico-cortical communications between sensory and motor primary areas gradually become more relevant to process information required considering the imminence of a possible return to the waking state.

As a concluding remark, we described here a unique dynamical pattern of hippocampal and cortical activation associated with micro-arousals during sleep. Future questions raised by these descriptions could be addressed by a future similar detailed analysis comprising cortical, hippocampal and accelerometer signals, of SWS transition events to REM stage, in order to test if micro-arousal emergence in SWS shares common dynamical features with SWS-REM transitions. Complementary, the use of agonists and antagonists of sleep-related neurotransmitter pathways, such as acetylcholine, dopamine, and serotonin, could better illustrate their critical role in the onset of micro-arousals. The importance of these findings is related to research on SWS dynamical role and regulation, as well as to clinical studies involving pathological arousals associated with sleep disorders.

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## Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. This work has also financial support from CNPq (Grants 552148/2011–3, 470716/2013–4), CAPES (Grant: BEX: 11264/13–6) and FINEP (CT-Infra) (Brazilian agencies). FAPESP post-doc for financial support (2015/23487–8). INCT in Translational Medicine (Grant CNPQ-FAPESP 465458/2014–9 and 2014/50891). USA-Israel Binational Science Foundation (BSF) BSF Grant No. 2012219; National Institutes of Health (NIH)/National Heart Lung and Blood Institute (NHLBI) Grant No. (NIH Grant 1R01- HL098437); the Office of Naval Research (ONR Grant 000141010078) W. M. Keck Foundation, Medical Research Grant. We would like to thank Neuroscience Institute of Natal (IINN-ELS) allowing data collection in their laboratories and Miguel Nicolelis and Sidarta Ribeiro for the kind support.

## Author information

Authors

### Contributions

B.L.S., G.N. and J.F.-A. conceived the experiment(s), B.L.S., G.N. and A.C.F. conducted the experiment(s), G.Z.S.L., G.C., H.B., S.R.L., T.L.P. and P.C.I. analyzed the results. All authors reviewed the manuscript.

### Corresponding authors

Correspondence to Gustavo Zampier dos Santos Lima or Plamen Ch. Ivanov.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

dos Santos Lima, G.Z., Lobao-Soares, B., Corso, G. et al. Hippocampal and cortical communication around micro-arousals in slow-wave sleep. Sci Rep 9, 5876 (2019). https://doi.org/10.1038/s41598-019-42100-5

• Accepted:

• Published:

• ### Disruption of neocortical synchronisation during slow‐wave sleep in the rotenone model of Parkinson’s disease

• Gustavo Zampier Santos Lima
• , Adriano D. S. Targa
• , Samantha Freitas Cavalcante
• , Lais S. Rodrigues
• , John Fontenele‐Araújo
• , Pablo Torterolo
• , Monica L. Andersen
•  & Marcelo M. S. Lima

Journal of Sleep Research (2020)


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# Definition:Musical Interval/Historical Note

The Pythagoreans noticed that aesthetically pleasing musical intervals were obtained by sounding notes from strings whose lengths were in ratios of small natural numbers: $1 : 2$, $2 : 3$, $3 : 4$ and so on.

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## måndag 14 januari 2013

### Pyrgeometer Fiction of DLR and OLR

To understand the fiction of the DLR and OLR underlying CO2 alarmism supposedly evidenced by a pyrometer, as discussed in the previous sequence of posts, the following example can be helpful:
Suppose you check your bank account and discover that you have $1 as the net when you have received your net income and paid your bills. Not so bad. Suppose now you have forgotten how big your income was and you ask if from the present status of your bank account ($1), it is possible to figure out this number. If you can't, consider the following advice offered by the financial consultant of your bank:
This is relatively simple. Assume that your expenditures were $999, then your income was$1000 because you have $1 left. If you give it to me as consultant fee, then you will be even. But you are left with a feeling that something in the argument is not convincing: Suppose you instead assume that your expenditures were only$99, then your income would be only $100, which is way to little for someone with your credentials. On the other hand, if you assume that you have spent$9999, then you would have had the impressive income of $10000, which is more than you ever dreamed of. </ After some thought you understand that you have been fooled by the bank financial expert and you want your$1 back: Clearly, it is impossible to tell what A is from the knowledge that A - B = 1, unless you know B.
The pyrometer does the same trick as the bank consultant: It measures temperature difference as bank net balance and reports total DLR and OLR as total income. In both cases it is guess work which is not worth of \$1.
The guess work of the pyrgeometer is the outgoing radiation according to the formula sigma x T^4 which is Planck's law for radiation into surrounding space at 0 K. But the temperatur of the outside of the pyrgeometer is not 0 K, and so the outgoing radiation from the pyrgeometer as sigma x T^4 is only fiction. This is the myth of backradition discussed in detail on this blog.


text:
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Calculate online with matrix_product (product matrix calculator)
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text:
## How to Construct an Invertible Matrix? Just Choose Large Diagonals

Suppose that I asked you give me an example of an $n\times n$ complex matrix

$\displaystyle A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{bmatrix}$

which is invertible. You would probably give me the $n\times n$ identity matrix $I$, which is its own inverse (i.e. an involution). Now suppose I ask for an invertible matrix, which is not the identity. You may change one of the diagonal entries to a different nonzero complex number. Perhaps, you know a bit of linear algebra, and you give me an an upper-triangular matrix with no zeros on the diagonal.

So far, you have only given me matrices with some zero entries. Not surprisingly, matrices are easier to deal with when many of the entries are zero. To up the difficulty level, I now ask you for an invertible matrix, for which all the entries are nonzero. You scratch your head, mumble “um…”, and ask for a minute.

Before we dive into necessary and sufficient criteria for the invertibility of a matrix, I want to share an anecdote about Dennis Gaitsgory, who was my professor for Math 123 as an undergraduate student. Since the source for the anecdote is the Harvard College Math Review endpaper “How to Compute Determinants” written by Professor Gaitsgory himself, I have no reason to consider it apocryphal.

During his graduate student years, Professor Gaitsgory was a teaching fellow for an undergraduate linear algebra class. One early morning, he was explaining determinants to his students, and he made the claim that the determinant of a generic matrix is never zero. As you may know, having nonzero determinant is equivalent to a matrix being invertible. To backup his claim, he gave the pretty “generic looking” $3\times 3$ matrix

$\displaystyle A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$

as an example. The logic behind giving a single example to illustrate a broad claim is bit off, but in his defense, the students were non-math majors. Computing the determinant by Laplace expansion,

$\begin{array}{lcl}\displaystyle\det A&=&\displaystyle1\cdot\begin{vmatrix}5&6\\8&9\end{vmatrix}-2\cdot\begin{vmatrix}4&6\\7&9\end{vmatrix}+3\cdot\begin{vmatrix}4&5\\7&8\end{vmatrix}\\[1 em]&=&\displaystyle1\cdot\left(5\cdot9-6\cdot8\right)-2\cdot\left(4\cdot9-6\cdot7\right)+3\cdot\left(4\cdot8-5\cdot7\right)\\[1 em]&=&\displaystyle-3+12-9\\[1 em]&=&\displaystyle0\end{array}$

Professor–or rather, graduate student–Gaitsgory thought he made a mistake in his computation; otherwise, he had just given a nonexample! Well, he had. Even brilliant mathematicians struggle to give examples of invertible matrices early in the morning.

Gaitsgory’s example is not invertible because the columns are linearly dependent, and therefore the matrix is rank deficient. To see this, observe that

$\displaystyle\begin{bmatrix}2\\5\\8\end{bmatrix}-\begin{bmatrix}1\\4\\7\end{bmatrix}=\begin{bmatrix}1\\1\\1\end{bmatrix}\Longrightarrow\begin{bmatrix}3\\6\\9\end{bmatrix}=\begin{bmatrix}2\\5\\8\end{bmatrix}+\begin{bmatrix}1\\1\\1\end{bmatrix}=2\begin{bmatrix}2\\5\\8\end{bmatrix}-\begin{bmatrix}1\\4\\7\end{bmatrix}$

which shows that the third column is a linear combination of the first two columns.

Returning to my request for an invertible matrix with no zero entries, how would you, the reader, go about giving such a matrix. Choosing numbers more or less at random for the entries and then checking the determinant is slow and tedious, particularly if the size $n$ of the matrix is large. A better approach would be to start with a matrix whose entries are identical and nonzero and then start changing the entries so that the columns are linearly independent. For example, start with

$A_{0}=\begin{bmatrix}1&1&\cdots&1\\1&1&\cdots&1\\ \vdots&\vdots&\vdots&\vdots\\1&1&\cdots&1\end{bmatrix}$

Clearly, $A_{0}$ is not invertible; it has rank $1$. But say we change the second coordinate of the second column to $2$, the third coordinate of the third column to $3$, … , and the $n^{th}$ coordinate of the $n^{th}$ column to $n$ to obtain a matrix $A=(a_{ij})$. Symbolically,

$a_{ij}=\displaystyle\begin{cases} {1}&{i\neq j} \\ {j}&{i=j}\end{cases}$

I claim that $A$ is invertible. Suppose $\beta_{1},\ldots,\beta_{n}$ are scalars such that

$\displaystyle\beta_{1}\begin{bmatrix}1\\1\\ \vdots\\1\end{bmatrix}+\beta_{2}\begin{bmatrix}1\\2\\ \vdots\\1\end{bmatrix}+\cdots+\beta_{n}\begin{bmatrix}1\\1\\ \vdots\\n\end{bmatrix}=\begin{bmatrix}\beta_{1}+\beta_{2}+\cdots+\beta_{n}\\ \beta_{1}+2\beta_{2}+\cdots+\beta_{n}\\ \vdots\\ \beta_{1}+\beta_{2}+\cdots+n\beta_{n}\end{bmatrix}=\begin{bmatrix} 0\\ 0\\ \vdots\\ 0\end{bmatrix}$

Subtracting the first row from the second, we see that $\beta_{2}=0$. Subtracting the second row from the third row, we see that $2\beta_{3}=0\Leftrightarrow\beta_{3}=0$. Suppose that we have shown that $\beta_{2},\ldots,\beta_{k}=0$. Then subtracting the $k^{th}$ row from the $(k+1)^{th}$ row, we see that $(k+1)\beta_{k+1}=0\Leftrightarrow\beta_{k+1}=0$. By induction, we conclude that $\beta_{2},\ldots,\beta_{n}=0$. Applying this result to the first row, we conclude that all the $\beta_{i}$ are zero.

We have satisfied my request for an invertible matrix with all nonzero entries, but it took some effort. Now, I want to describe a method which focuses only on the magnitude of the diagonal entries of the matrix relative to the magnitude of the other entries of a given row. A matrix $A$ is diagonally dominant if

$\displaystyle\left|a_{i}\right|\geq\sum_{{j=1}\atop{j\neq i}}^{n}\left|a_{ij}\right|,\indent\forall i=1,\ldots,n$

$A$ is said to be strictly diagonally dominant if the inequality is strict. Define nonnegative quantities $R_{1},\ldots,R_{n}$ by

$\displaystyle R_{i}:=\sum_{{j=1}\atop{j\neq i}}^{n}\left|a_{ij}\right|,\indent\forall i=1,\ldots,n$

The closed disk $\overline{D}(a_{ii},R_{i})$ in the complex plane centered at $a_{ii}$ and of radius $R_{i}$ is an example of a Gershgorin disk. We can use Gershgorin’s circle theorem to construct invertible matrices with very little effort.

Theorem. Every eigenvalue of $A$ lies in at least one of the Gershgorin disks $\overline{D}(a_{ii},R_{i})$.

Proof. Let $\lambda$ be an eigenvalue of $A$, and let $x=(x_{1},\ldots,x_{n})\in\mathbb{C}^{n}$ be an associated eigenvector. Since $x$ is a nonzero vector by definition, there exists an index $i$ such that $\left|x_{i}\right|=\max_{j}\left|x_{j}\right|>0$. Applying the definition of the eigenvector,

$\displaystyle\sum_{j=1}^{n}a_{ij}x_{j}=\lambda x_{i},\indent\forall i=1,\ldots,n$

which we rewrite as

$\displaystyle\sum_{{j=1}\atop{j\neq i}}^{n}a_{ij}x_{j}=\lambda x_{i}-a_{ii}x_{i}=(\lambda-a_{iii})x_{i}$

Taking the magnitude of both sides and applying the triangle inequality to the sum, we obtain the estimate

$\displaystyle\left|\lambda-a_{ii}\right|\cdot\left|x_{i}\right|\leq\sum_{{j=1}\atop{j\neq i}}^{n}\left|a_{ij}\right|\cdot\left|x_{j}\right|\Longleftrightarrow\left|\lambda-a_{ii}\right|\leq\sum_{{j=1}\atop{j\neq i}}^{n}\left|a_{ij}\right|\cdot\left|\frac{x_{j}}{x_{i}}\right|$,

where division is possible since $\left|x_{i}\right|>0$ by construction. Moreover, since $x_{i}$ is taken to be the coordinate with the largest magnitude, we have that $\left|\frac{x_{j}}{x_{i}}\right|\leq1$, for all $j=1,\ldots,n$. We conclude that

$\displaystyle\left|\lambda-a_{ii}\right|\leq\sum_{{j=1}\atop{j\neq i}}^{n}\left|a_{ij}\right|=R_{i}$

$\Box$

Recall that a matrix is invertible if and only if it has trivial kernel; or equivalently, it does not have eigenvalue $\lambda=0$. If we choose the diagonal entries of $A$ sufficiently large so that $\left|a_{ii}\right|>R_{i}$ for $i=1,\ldots,n$, then the reverse triangle inequality implies that all the eigenvalues of $A$ have positive magnitude. Hence, $A$ is invertible.


text:
# libpysal.weights.w_local_cluster¶

libpysal.weights.w_local_cluster(w)[source]

Local clustering coefficients for each unit as a node in a graph.

Parameters
wW

spatial weights object

Returns
carray

(w.n,1) local clustering coefficients

Notes

The local clustering coefficient $$c_i$$ quantifies how close the neighbors of observation $$i$$ are to being a clique:

$c_i = | \{w_{j,k}\} |/ (k_i(k_i - 1)): j,k \in N_i$

where $$N_i$$ is the set of neighbors to $$i$$, $$k_i = |N_i|$$ and $$\{w_{j,k}\}$$ is the set of non-zero elements of the weights between pairs in $$N_i$$ [WS98].

Examples

>>> from libpysal.weights import lat2W, w_local_cluster
>>> w = lat2W(3,3, rook=False)
>>> w_local_cluster(w)
array([[1.        ],
[0.6       ],
[1.        ],
[0.6       ],
[0.42857143],
[0.6       ],
[1.        ],
[0.6       ],
[1.        ]])


text:
# Ask the name of a combinatorial theorem

It is a classical theorem. For given integer $n \ge 1$, among ${n\choose{n/2}} = 2^{(1-o(1)n)}$ strings in the cube $\{0, 1\}^n$ with weights $n/2$, i.e., $n/2$ indices are 1, there are at least $2^{cn}$ of these strings such that each pair has Hamming distance at least $n/4$, where $c$ is a constant between $0$ and 1.

This is for sure a known result. I hope to be aware of its name.

I don't have a name for the theorem, but I can give a quick proof in case you don't get hold of a name: Let $S$ be a maximal set of $n/4$-separated strings in the weight $n/2$ slice. Then the union of $n/4$-Hamming balls centred at elements of $S$ covers the entire slice. But each Hamming ball has $\binom{n}{0}+\ldots+\binom{n}{n/4}\sim 2^{an}$ elements, where $a=-\frac 14\log_2(\frac14)-\frac 34\log_2(\frac 34)<1$. Hence $S$ must consist of at least $2^{(1-a)n}$ elements.


text:
# \bibliography{} starting from any number

I'm aware of a previous question about changing bibliography starting number, but my problem is that I'm not using \begin{thebibliography} to build the bibliography, but directly reading it from a bibtex file. Is there any method of changing the default numbering, e.g. starting from 10 instead of 1 when compiling? The end of my file is as follows:

% REFERENCES AND NOTES
\bibliography{bibtextfile}
\bibliographystyle{style}

\end{document}

• Try \setcounter{enumiv}{9}, at least enumiv is the counter used by the article.cls version of \thebibliography for the numbering – user31729 Jan 26 '16 at 11:46
• Thanks but it doesn't work in my case. How can I check my counter type? – Fabio Lamanna Jan 26 '16 at 12:53

\setcounter{enumiv}{9} would work if the the \thebibliography wouldn't apply \usecounter{enumiv} which in turn resets the enumiv counter.

From article.cls

\newenvironment{thebibliography}[1]
{\section*{\refname}%
\@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%
\list{\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\@openbib@code
\usecounter{enumiv}%   This is the 'culprit'!!!!!
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\sloppy
\clubpenalty4000
\@clubpenalty \clubpenalty
\widowpenalty4000%
\sfcode\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty thebibliography' environment}}%
\endlist}

and from latex.ltx

\def\usecounter#1{\@nmbrlisttrue\def\@listctr{#1}\setcounter{#1}\z@}

i.e. \setcounter{#1}{0} effectively.

The following code uses a dummy counter mybibstartvalue and sets the enumiv counter after \usecounter{enumiv} using a patch.

However, this code is only meant for article.cls.

\documentclass{article}

\usepackage{xpatch}

\newcounter{mybibstartvalue}
\setcounter{mybibstartvalue}{9}

\xpatchcmd{\thebibliography}{%
\usecounter{enumiv}%
}{%
\usecounter{enumiv}%
\setcounter{enumiv}{\value{mybibstartvalue}}%
}{}{}

\begin{document}

\cite{Lam94}

\cite{GSM97}

\bibliography{biblio}
\bibliographystyle{unsrt}

\end{document}

• thanks for this elegant answer! – Libavius Oct 16 at 12:24


text:
# Katy MO

Kate draw triangle ABC. Middle of AB have mark as X and the center of the side AC as Y. On the side BC wants to find the point Z such that the content area of a 4gon AXZY was greatest. What part of the triangle ABC can maximally occupy 4-gon AXZY?

x =  0.5

### Step-by-step explanation:

$x=1\mathrm{/}2=\frac{1}{2}=0.5$

We will be pleased if You send us any improvements to this math problem. Thank you!

Tips to related online calculators

#### You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

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text:
# Filling circular segment using Tkz-Euclide

I tried to fill a circular segment but I'm having trouble doing it. Here's my MWE:

\documentclass[10pt]{scrartcl}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetikzlibrary{calc,backgrounds}
\usetkzobj{all}
\begin{document}

\begin{tikzpicture}
\tkzDefPoints{0/0/C, 4/0/B}
\tkzDrawTriangle[pythagore](C,B)
\tkzGetPoint{A}
\tkzDefCircle[circum](A,B,C)
\tkzGetPoint{O}
\tkzGetLength{cr}
\tkzDrawCircle[R](O,\cr pt)
\tkzDrawPoints(O)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](C)
\tkzLabelPoints[above left](B)
\tkzLabelPoints[below](O)
\tkzLabelSegment[above left](A,B){\Large 6 cm}
\tkzLabelSegment[above right](B,C){\Large 8 cm}
\begin{scope}[fill=gray, opacity=0.5]
\fill[clip] (B) -- (A) arc (90:-90:1.5cm) -- cycle;
\end{scope}
\end{tikzpicture}

\end{document}

I'll rotate the picture by 143 degrees.

• I added \usetkzobj{all} to make your code compile. – marmot Sep 24 '18 at 21:12

Really easy with the calc library. Perhaps even easier with tkz-euclide if you speak French, which I don't.

\documentclass[10pt]{scrartcl}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetikzlibrary{calc,backgrounds}
\usetkzobj{all}
\begin{document}

\begin{tikzpicture}
\tkzDefPoints{0/0/C, 4/0/B}
\tkzDrawTriangle[pythagore](C,B)
\tkzGetPoint{A}
\tkzDefCircle[circum](A,B,C)
\tkzGetPoint{O}
\tkzGetLength{cr}
\tkzDrawCircle[R](O,\cr pt)
\tkzDrawPoints(O)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](C)
\tkzLabelPoints[above left](B)
\tkzLabelPoints[below](O)
\tkzLabelSegment[above left](A,B){\Large 6 cm}
\tkzLabelSegment[above right](B,C){\Large 8 cm}
\begin{scope}[fill=gray, opacity=0.5]
\fill let \p1=($(A)-(O)$),\p2=($(B)-(O)$),
\n1={atan2(\y1,\x1)},\n2={atan2(\y2,\x2)},\n3={veclen(\x1,\y1)} in
(B) -- (A) arc (\n1:\n2:\n3) -- cycle;
\end{scope}
\end{tikzpicture}
\end{document}

Explanation: \p1=($(A)-(O)$) means that the coordinates of \p1, \x1 and \y1 will be the x- and y-coordinates of the vector O-A, likewise for \p2 and B. Correspondingly, \n1 and \n2 will be the angles of A and B, respectively, and \n3 the radius of the circle. These are the quantities needed to draw the arc.

• P.S. You can rotate the picture, of course, by saying \begin{tikzpicture}[rotate=143]. You can even compute the exact rotation angle by saying \path let \p1=($(A)-(O)$),\n1={180-atan2(\y1,\x1)} in \pgfextra{\typeout{\n1}};, after which you may decide to use \begin{tikzpicture}[rotate=143.13011]. ;-) – marmot Sep 24 '18 at 23:33

Also available in Metapost; here wrapped up in luamplib, so please compile with lualatex.

[I hope that I have correctly interpreted the OP "rotate the picture by 143 degrees".]

\documentclass[border=5mm]{standalone}
\usepackage{luatex85}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
path o;
o = fullcircle scaled 10 cm;

% "time" around the circle, 360° = 8 points
numeric a, b, c;
c = 0;                 % = 0°
b = 2/45 angle (3,4);  % = 106.26°
a = 4;                 % = 180°

pair A, B, C;
C = point c of o;
B = point b of o;
A = point a of o;

fill subpath (b,a) of o -- cycle withcolor 3/4 white;

draw o;
draw A--B--C--cycle;

dotlabel.lft("$A$", A);
dotlabel.top("$B$", B);
dotlabel.rt ("$C$", C);
dotlabel.bot("$O$", origin);

label.lrt (decimal round(abs(A-B)/cm) & "\thinspace cm", 1/2[A,B]);
label.llft(decimal round(abs(B-C)/cm) & "\thinspace cm", 1/2[B,C]);

endfig;
\end{mplibcode}
\end{document}

It is simply a calculation error of half the angle AOB which measures arcsin(3/5) or about 36.8699 degrees with a radius of 2.5 cm.

This give:

\fill[clip] (B) -- (A) arc (36.8699:-36.8699:2.5cm) -- cycle;

Full code:

\documentclass[10pt]{scrartcl}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetikzlibrary{calc,backgrounds}
\usetkzobj{all}
\begin{document}

\begin{tikzpicture}
\tkzDefPoints{0/0/C, 4/0/B}
\tkzDrawTriangle[pythagore](C,B)
\tkzGetPoint{A}
\tkzDefCircle[circum](A,B,C)
\tkzGetPoint{O}
\tkzGetLength{cr}
\tkzDrawCircle[R](O,\cr pt)
\tkzDrawPoints(O)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](C)
\tkzLabelPoints[above left](B)
\tkzLabelPoints[below](O)
\tkzLabelSegment[above left](A,B){\Large 6 cm}
\tkzLabelSegment[above right](B,C){\Large 8 cm}
\draw[fill=gray!50] (B) -- (A) arc (asin(3/5:-asin(3/5):2.5cm) -- cycle;
\end{tikzpicture}

\end{document}

• @marmot Yes, indeed, we can make these calculations by pgf. I never use and certainly mistakenly the computing capabilities of pgf.. You are right clip is useless, on the other hand it is necessary to make draw because otherwise the segment is not plotted. I edited my answers. I upvoted some time ago too on your answer :) – AndréC Sep 25 '18 at 4:33
• @marmot, I noticed that for some time now, you've been deleting your comments. It is not a good idea, reading the comments from the beginning often helps to understand the evolution of an answer. Moreover, this makes the other comments incomprehensible. :( – AndréC Sep 25 '18 at 5:29


text:
# Maximize money

1. Feb 17, 2007

### Loren Booda

Say you wanted to realize the greatest gain in wealth using past economic information, starting at a past investment date of your choosing. What is the most profit (e.g., in terms of GNP percentage) you could actualize before this historical data becomes unreliable (when return becomes relatively random) from the very (nonlinear?) effects following the introduction of your speculation?

2. Feb 18, 2007

### Staff: Mentor

Zero, of course. It says so right on the disclaimer for every investment product: past performance is no guarantee of future gain. Otherwise, everyone would make money on the stock market all the time.

3. Feb 18, 2007

### Loren Booda

So if I were able to decide, using knowledge of the past, to sell retroactively all my stocks in the summer of 1929 and invest in land, I would be no better off, on average, than if I held onto them?

4. Feb 18, 2007

### turbo

You would be better off, but land is a long-term investment and affords you limited liquidity - you can't move it around like money. Your best bet (given perfect hindsight) would be to invest in companies just before they boom, due to sudden investor interest, a major invention, or revolutionary development, then sell out and go on to invest in your next target company. This can make your investments grow very fast - lots of short-term gains.

Last edited: Feb 18, 2007
5. Feb 18, 2007

### Staff: Mentor

Ooooooh, I misread your initial post. You want to do an exercise in maximizing profit from a past date to the present using data for the period you are looking to generate profit in - not project it to the future. In that case, I don't understand the statement about unreliable data. A stock price is an exact number - take the historical price of a stock and you can calculate exactly how much money you'd have today given a purchase at a given time in the past.

Since this is just an exercise, why can't you just short-sell a company the day before it goes bankrupt and realize a near instant infinite return (no money invested for an arbitrary amount out)?

I guess I don't see the purpose/point of this question... Are you just asking what type of investments are best, historically?

Last edited: Feb 18, 2007
6. Feb 18, 2007

### Loren Booda

I am trying to understand

1. The effect an individual has on the market by participating or not, and if this choice would usually create a nonlinear disparity. Say I decide to invest in pork bellies (over some other investment) today. Is that choice likely to have a chaotic ripple effect eventually in the economy?

and similarly

2. How quickly any statistic would diverge from hypothetical hindsight of market activity, i. e., how radically and quickly projections become unreliable. Say I have a complete record of all financial transactions in history. If I were able to try maximizing profit (a la "time travel") using the record, how might my financial actions interfere with the eventual reliability of the data?

7. Feb 19, 2007

### Staff: Mentor

It depends a lot on how much you would invest and how diversified it would be. One thing though - once the purchase is made, the influence is over.

In the 20s, it was possible for one investor to influence the market if they were big enough. Today it is only really possible for an individual (with a few exceptions, like Bill Gates) to influence one individual stock. The market is just too big.

8. Mar 18, 2007

### Crosson

I knew that the year 2000 aquisition of AOL by TimeWarner was a disaster, and sure enough it cost TimeWarner about 100 billion dollars. I shhould have sold short

I like the part in Forest Gump where Forest invests in some apple company

9. Jul 24, 2007

### EnumaElish

I think one way to look at the OP's question is through the LTCM episode. They did not have future data, but they did have a mathematical model, which predicted the future. It's as if they had future data, only simulated. Acting on the model, they made profit-maximizing investment choices. But actual history turned out much different than what the model predicted.

Last edited: Jul 25, 2007
10. Jul 25, 2007

### AsianSensationK

This is quite an old thread, but really interesting. Although I'm not entirely sure if I am interpreting the question correctly.

I think the data starts to become unreliable as people begin to pick up on the fact that you're beating the market with your data. Expectations start to change. If people wind up following your lead, then you've affected the risk and return trade offs. What was once a risky bet is no longer risky anymore, and people start to expect better returns of your picks. This of course has an impact on the value of a company and ultimately its stock price.

I also don't think it'd be possible for you to go completely unnoticed in the marketplace, because if you had future data that helped you beat the market, you'd probably want to borrow to amplify your profits, right? That's where people would start to ask questions.

11. Jul 28, 2007

### Staff: Mentor

If one has a few thousand dollars or owns a few hundred or thousand shares, one will not have much of an impact on the stock market, and similarly for bonds. However, if one has tens or hundreds of millions or billions of dollars, one will have a significant impact on the markets, especially as AsianSensationK indicated others take notice of one's activities.

If one had records of rising stocks and falling stocks and one could buy low value stocks and sell high valued stocks one could make a lot of money certainly.

There have been extraordinary events in the past. For example, in the 1980's at least two companies, IIRC Santa Fe RR (or their holding company) and either UTC or GD declared one time huge dividends whereby they dumped huge amounts of cash ~$50/share on shareholders of record in order to avoid takeover. In contrast, other stocks have gone ex-dividend, and certainly if one knew that, one could dump the stock then repurchase it at a much lower price. Around 1986-87, ARCO stock went from$43/share to ~90$/share and the dividend went from$4/share to $4.50/share, which at a purchase price of$43/share was an attractive return. Even after the stock market crash in Oct 1987, ARCO did well. Also, there were other stockes like Quanex that went from ~$2/share to ~$27/share after the stock market crash. So it would have been a great investment if one had been able to anticipate its ascent.

In the mid 1980's, after Pons and Fleischman announced their cold fusion using palladium, the price of palladium (and platinum) soared, and then dropped.

In the mid 90's, there were settelements on some bonds of various defunct railroads (e.g. Erie Lackawanna which along with others had gone bankrupt with the collapse of the PennCentral RR), and those who were holding the bonds made huge amounts of money. Up to that point, those bonds were worthless, but some were able to purchase those bonds for cents on the dollar. The settlement was about 40% of the bonds' face value. The resolution of those bonds took about 20 years though.

There are numerous historical examples of great increase or decrease in the value of various financial instruments.

As they say, hindsight is 20/20.


text:
## HelloKitty17 one year ago MEDAL FOR RIGHT ANSWER HERE!!!!!!!!!!!!!!!! A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water: graph of y equals minus 2 times the square of x plus 20 times x plus 400 What do the x-intercepts of the graph represent? There were 20 larvae at 0 degrees Celsius. There were 10 larvae at 0 degrees Celsius. The water has no larvae at -10 degrees Celsius and 20 degrees Celsius. The water has maximum larvae at -10 degrees Celsius and 20 degrees Celsius.

1. HelloKitty17

@Mehek14

2. mathmath333

\large \color{black}{\begin{align} y=-2x^{2}+20x+400\hspace{.33em}\\~\\ \end{align}}

3. HelloKitty17

what does that mean

4. mathmath333

for $$x$$-intercepts set $$y=0$$ and solve \large \color{black}{\begin{align} 0=-2x^{2}+20x+400\hspace{.33em}\\~\\ \end{align}}

5. OregonDuck

6. mathmath333

do u know how to solve this \large \color{black}{\begin{align} 0=-2x^{2}+20x+400\hspace{.33em}\\~\\ \end{align}}

7. OregonDuck

no

8. OregonDuck

A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water: graph of y equals minus 2 times the square of x plus 20 times x plus 400 ..........y = -2x² + 20x + 400 What do the x-intercepts of the graph represent? ..........C) The water has no larvae at -10 degrees Celsius ................and 20 degrees Celsius.


text:
# Questions about capacitors in DC

1.Does the current flow through the capacitor when (a) it's not charged, (b) it's partially charged, (c) it's fully charged ? How does the flow changes in time (ex.: exponentially, logarithmically, linearly, etc.)?

2.If you charge a 16V capacitor from a 9V battery and then you disconnect the battery, what will be the capacitor voltage? 16 or 9?

-
Is this homework? –  starblue Apr 28 '12 at 11:55

If it's fully charged there's no current. When not or partly charged there will be current if you apply a voltage to it. The current curve depends on how your voltage source is connected. If you supply a constant current the voltage will rise linearly, if you charge over a resistor it will rise exponentially. That's because the rising voltage will cause the voltage drop over the resistor to become smaller and smaller, and hence the current through it will also decrease. A lower current means that charging goes slower.

How fast the capacitor charges depends on both the resistor's and the capacitor's value. The larger the resistor the slower the capacitor charges, and it also charges slower if the capacitance is greater. The exponential curve is said to have a time constant $RC$. After about $5 \times RC$ the capacitor is charged to 99%.

9V. The capacitor's rating has nothing to do with it, that only determines how high a voltage you can apply. Just look at what voltage is available, and that's 9V. Just take care that the capacitor is fit for the supplied voltage. You can safely charge a 16V capacitor to 9V, but not to 25V.

-
Thank you!!! Now I understand how it works. –  Cristi Apr 27 '12 at 12:22
yes son , there's no easy way to learn this without learning differential equations. –  sandun dhammika Apr 27 '12 at 13:34
No DC current ideally... :P –  Nick T Apr 27 '12 at 20:09

You seem to have some basic misconceptions about capacitors. The current thru a capacitor is proportional to the change in the voltage. In more mathematical terms:

current = K dV/dt

The proportionality constant K is the capacitance. When the units are Amperes, Volts, Farads, and seconds, it all works out with a scale factor of 1:

Amps = Farads * Volts / second

So if nothing is externally forcing the voltage of a capacitor to change, there will be no current thru it. Equivalently, if nothing is forcing current thru a capacitor, it's voltage won't change. If you charge up a capacitor to 9 V by connecting it accross a 9 V battery and waiting long enough for the current to go to zero (the voltage to stop changing), then disconnect it, the voltage on the capacitor will stay 9 V. That is for a perfect capacitor. Real capacitors have leakage, meaning a little current will leak thru them to discharge them, as if a large resistor was connected accross them.

The voltage rating of a capacitor only tells you how many volts you can charge it up to before it may break. So charging up a 16 V capacitor to 9 V is perfectly fine since 9 V is less than 16 V, but that 16 V rating has otherwise nothing to do with how many volts will be on the capacitor in a circuit.

-
Thank you!!! I'll keep that in mind. –  Cristi Apr 27 '12 at 12:27


text:
+1 vote

Hit by floods, the kharif (summer sown) crops in various parts of the country have been affected. Officials believe that the loss in production of the kharif crops can be recovered in the output of the rabi (winter sown) crops so that the country can achieve its food-grain production target of $291$ million tons in the crop year $2019-20$ (July – June). They are hopeful that good rains in July-August will help the soil retain moisture for a longer period, helping winter sown crops such as wheat and pulses during the November-February period.

Which of the following statements can be inferred from the given passage?

1. Officials declared that the food-grain production target will be met due to good rains.
2. Officials want the food-grain production target to be met by the November-February period.
3. Officials feel that the food-grain production target cannot be met due to floods.
4. Officials hope that the food-grain production target will be met due to a good rabi produce.

retagged

“Officials believe that the loss in production of the kharif crops can be recovered in the output of the rabi (winter sown) crops so that the country can achieve its food-grain production target of $291$ million tons”.


text:
Table 6.

Resting ventilation and heart rates reported in this and previous studies on S. officinalis

StudyTechniqueTemperature (°C)Cuttlefish size (mantle length, cm)Reported rateStandardized rate
Ventilation rate (breaths min-1
Present study Non-invasive (ultrasound) 15 15.5-18.5 35.0±9.2 35.0±9.2
Boal and Ni,1996  Non-invasive (visual observation) 19-21 7.5-10.5 45 32.1
Mislin,1966  Invasive (leads forced through mantle) 16-18 20-35 45±9 39.3
Bone et al.,1994  Invasive (electrodes in mantle tissue) 18.5 5.0-25.0 48-60 42.6
Heart rate (beats min-1
Present study Non-invasive (ultrasound) 15 15.5-18.5 24.0±5.1 24.0±5.1
Chichery and Chanelet, 1972  Invasive (EKG leads) 16 13.6-20.0 20-30 22.8
Mislin,1966  Invasive (leads forced through cuttlebone) 16-18 20-35 43±9 35.8
Chichery,1980  Invasive (EKG leads) 13 Not reported 45-50 57.1
StudyTechniqueTemperature (°C)Cuttlefish size (mantle length, cm)Reported rateStandardized rate
Ventilation rate (breaths min-1
Present study Non-invasive (ultrasound) 15 15.5-18.5 35.0±9.2 35.0±9.2
Boal and Ni,1996  Non-invasive (visual observation) 19-21 7.5-10.5 45 32.1
Mislin,1966  Invasive (leads forced through mantle) 16-18 20-35 45±9 39.3
Bone et al.,1994  Invasive (electrodes in mantle tissue) 18.5 5.0-25.0 48-60 42.6
Heart rate (beats min-1
Present study Non-invasive (ultrasound) 15 15.5-18.5 24.0±5.1 24.0±5.1
Chichery and Chanelet, 1972  Invasive (EKG leads) 16 13.6-20.0 20-30 22.8
Mislin,1966  Invasive (leads forced through cuttlebone) 16-18 20-35 43±9 35.8
Chichery,1980  Invasive (EKG leads) 13 Not reported 45-50 57.1

Rates are standardized to our temperature of 15°C using previously calculated Q10 values (King et al., 2005). If size was reported in mass instead of in mantle length, the equivalent mantle length was interpolated(Forsythe et al., 1994). We did not standardize rates for size. However, larger animals are expected to have slower heart rates (Chichery and Chanelet, 1972).

Close Modal


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# Morse theory of knot complements

Suppose $K^1 \subset S^3$ is a (connected) knot. First, I am wondering is there a Morse function $f$ on $S^3$ such that $f$ restricted to a neighborhood $D^2 \times S^1$ of $K^1$ have standard form, i.e the standard Morse function on $S^1$ with two critical points plus $r^2$, where $r$ is the radial coordinate on $D^2$. In particular, $f$ restricted to $D^2 \times S^1$ has two critical points of index 0,1 and $\nabla f$ is outward pointing along $\partial (D^2 \times S^1)$.

I am also wondering what the genus of a knot has to do with the number of critical points of such a Morse function. If the knot genus is $k$, is it true that there are at least $k+1$ critical points of $f$ of index $1$? If $K^1$ is the unknot, then $f$ can be a Morse function with 4 critical points of index $0,1,2,3$. Then by topology, there must be at least $k+1$ critical points of index 2 to cancel these critical points of index 1.

Finally, I am trying to understand the index 3 critical points, which are the most confusing to me. I think, in general, there need to be many index 3 critical points but I am not sure how they appear from the perspective of the genus/Seifert surface. It would be helpful if someone could explain or provide a reference to a concrete example, say the $(p,q)$-torus knot (which has genus $(p-1)(q-1)$). Maybe this has to do with Heegaard splittings of $S^3$. Essentially I want a concrete description of the knot complements in terms of Morse theory or cell decompositions.

• Your function is Morse in the neighborhood of the knot, so you can extend it arbitrarily and then perturb away from the knot to make it Morse. The number of necessary critical points has more to do with the topology of the knot complement (how many cells it needs) than anything about the knot itself (eg genus). – user98602 Nov 20 '16 at 20:25
• @MikeMiller The knot complement and the knot genus are both determined by the knot. One question I was asking is how they're related. – user39598 Nov 20 '16 at 20:33
• The index 3 critical points are uninteresting. You can always cancel all but exactly one of them off with index 2 points. – PVAL-inactive Nov 22 '16 at 19:54
• @PVAL-inactive Why is that? I agree that cancel algebraically but I'm not sure they cancel geometrically. The question is whether a 3-dimensional manifold with one boundary $S^1 \times S^1$ admits a Morse function with at most one critical point of index 3. In high dimensions, this is true but I am not sure this is true in dimension 3. Would the answer be different if I considered $K^1 \subset B^3$, the 3-dimensional ball? – user39598 Nov 22 '16 at 20:00
• @user39598 The attaching (ascending) sphere for a 3-handle is just a copy of $S^3$, and the the belt (descending) sphere for a 2-handle is an $S^0$. If they intersect algebraically once, they automatically intersect geometrically once. (This is also true for extraneous 0 and 1 handles (in this case not all of them are extraneous)). None of this answers your question it just means that all of the interesting part of your question relies on the interactions between the 1 and 2-handles. – PVAL-inactive Nov 22 '16 at 20:20