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\begin{document}

\title{%Complexity Changes of the Long-range Spin-spin Interactions in the Human Ageing Brain
Cardiac related Complexity Changes in zero-quantum coherence (ZQC) brain imaging compared between young and older healthy volunteers
%not happy yet}



\author[1,2]{David López Pérez}%
\author[2]{C K}%
\affil[1]{Institute of Psychology, Polish Academy of Sciences, Warsaw, Poland}%
\affil[2]{Institute of Neuroscience, Trinity College, Dublin, Ireland}%


\vspace{-1em}



  
  \date{May 10, 2024}


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\selectlanguage{english}
\begin{abstract}
Functional MRI (fMRI) is still one of the most popular methods to study the ageing brain. Getting older affects both the structure of the brain and our cognitive capabilities but there is still no solid evidence on how ageing influences the mechanisms underlying the fMRI signal. Here, we apply a recently developed fMRI-based sequence that was found to be sensitive to quantum fluctuations in the brain. We show that not only these fluctuations can be detected and measured, but also that they are affected by age. While comparing young and old participants, we found qualitative and quantitative evidence that the dynamics of these quantum fluctuations undergo strong changes with age.  Finally, we show how differences in these quantum fluctuations relate with measures from different cognitive batteries, suggesting that these quantum fluctuations may be key for cerebral dynamics and cognitive functioning. The profound sensitivity for dynamic changes shows the potential of this physiological effect with clinical relevance for all neuro-vascular diseases.%
\end{abstract}%



\sloppy


Keywords: Complexity, Quantum Coherence, Cognitive Ageing, MRI, Conciousness

\section{Introduction}

It is well established that ageing has cascading effects on many cognitive domains. Older adults have particular difficulties with episodic memory (REF), working memory (REF) or are slower processing different stimuli (REF). At the same time, some aspects of cognition are maintained, such as semantic memory (REF) or emotional regulation (REF). However, the cerebral mechanisms that underlie this better or worst performance are still poorly understood (REF). Additionally, ageing affects the brain at multiples levels ranging from sub- to macro-cellular (e.g., \hyperref[csl:1]{(Salat, 2004}; \hyperref[csl:2]{Allen et al., 2005)}) and a vast amount of studies have tried to link these structural changes to age differences in cognitive function. Magnetic resonance imaging (MRI) based methods have been particularly used to study changes in the ageing brain. Among the various MRI methods, functional MRI (fMRI) with more that published 10000 papers is probably the one more widely applied (e.g., REF). The blood oxygen level dependent (BOLD) signal obtained from fMRI is an indirect index of neural activity and reflects small metabolic changes in deoxyhaemoglobin concentrations that take place when a specific region of the brain is active (Ogawa et al., 1990-2?). These responses have been found, however, to be similar in young and older adults, (REFs) although there is some evidence that the magnitude of the BOLD response can be reduced in older adults (REF). 
Although still unknown, changes in cerebral vasculature with age are somehow expected to influence the mechanisms underlying the fMRI signal. Likewise, this should be related to differences in cognitive performance, but strong evidence has not been found so far.

This comes at a surprise because   heart functions also alter with age which then should have an effect an cerebral blood flow. Evenmore, it is well known that  several heartbeat-related effects  influence conscious perception (REF), e.g. it has been skown that the cardiac cycle may impact perception of visual or auditory stimuli(ref). As well, the existence of heartbeat-evoked potential (HEP) in general is a strong evidence that the heart beat influences neuronal functions (REF).
In a recent study, Kerskens and L\'{o}pez P\'{e}rez (2017) discovered phase transitions in the brain tissue stimulated by the heart beat which  may shed light into the underlying mechansism of HEPs and heart beat related conscious perceptions. Phase transition are accompanied with order changes  in the molecular structures of  brain tissue  which may play important roles for long-range signalling between brain regions.   MRI is generally sensitive for phase transition which are reflected in t1 and t2 changes but Kerskens and Lopez focussed on  long-range phase transition which can be detected with intermolecular multiple-quantum coherence (iMQC). Using a MRI time series acquisition, they found fluctuations in the brain, whose underlying mechanism was related to the intermolecular zero-quantum coherence (iZQC) properties of the magnetic resonance signal \hyperref[csl:3]{(Kerskens \& L{\'{o}}pez-P{\'{e}}rez, 2017)}. Those fluctuations were evoked  by the heart beat only during conscious awareness which underpins its importance for consciousness and cognition.
%t of  and whose strength, shape and variability were linked to the spontaneous character of the underlying brain physiological mechanisms. 
Quantum coherence 
%often hides in complex systems but researchers
have not traditionally considered  as a powerful tuning element for enhancing or explaining functions \hyperref[csl:4]{(Scholes et al., 2017)} in biology. However, a growing body of literature has recently demonstrated that quantum coherence in living organisms exists and it is itself important for their functioning (missing citation). With the recent rise of the field of quantum biology, it has been suggested that quantum phenomena might also influence brain activity and affect its cognitive abilites \hyperref[csl:6]{(Jedlicka, 2017)}. %Consequently, the dynamics of the long-range quantum coherence fluctuations, if linked to underlying brain mechanisms, should vary as we grow older. 

However, it is important to consider that the brain itself is a non-linear dynamical complex system, whose activity may vary if the system changes \hyperref[csl:6]{(Jedlicka, 2017)}. As a consequence, these quantum fluctuations that arise from the interaction between the brain and the heart would be high-dimensional, and thus, the complexity of the system (i.e., its ability to adapt and function in a ever-changing environment) will also be high-dimensional \hyperref[csl:7]{(C. K. Peng \& Goldberger, 2009)}. Recent studies have shown that high complexity is characteristic to healthy systems and that can degrade as consequence of disease or ageing \hyperref[csl:8]{(dos Santos et al., 2014)}. Thus, if this mechanism is important for cerebral dynamics, the complexity of these fluctuations will need to be kept high and any variation on the dynamics with age should affect the complexity of the system.

In this paper, we want to  study for the first time how the dynamical complexity of the long-range quantum coherence signal may vary with age. To characterise these fluctuations as complete as possible, we use a broad range of dynamical systems measures. First, we applied Recurrence Quantification Analysis (RQA; \hyperref[csl:9]{(Zbilut \& Webber, 1992)}), which is an increasingly popular method to analyse dynamical changes of behaviour in complex systems. This concept has been used to study physiological signals\unskip~\hyperref[csl:10]{(Marwan et al., 2002}; \hyperref[csl:11]{Wessel et al., 2003)}, heart rate variability\unskip~\hyperref[csl:12]{(Santos et al., 2014)} or the dynamics of heart rhythm modulation\unskip~\hyperref[csl:13]{(Censi et al., 2015)}. The main benefits of RQA in comparison to standard analysis resides in its sensitivity to small changes in the system dynamics\unskip~\hyperref[csl:12]{(Santos et al., 2014)}.  Secondly, we employed MultiFractal Detrended Fluctuation Analysis to extract the fractal properties of the signal (MFDFA; \unskip~\hyperref[csl:14]{(Ihlen, 2012)}). Multifractal analysis is another efficient chaos theory method to study the fractal scaling properties and long-range correlations of noisy signals \hyperref[csl:7]{(C. K. Peng \& Goldberger, 2009)}; for a review see\unskip~\hyperref[csl:15]{(Lopes \& Betrouni, 2009)}). Fractal differences as a consequence of ageing has been found between monofractal or multifractal signals in EEG\unskip~\hyperref[csl:16]{(Pereda et al., 1998)} or due to HRV changes\unskip~\hyperref[csl:17]{(Makowiec et al., 2011)}. Finally, we relate this measures with different cognitive batteries and show that those quantum fluctuations may be key for cerebral dynamics and cognitive functioning.

\section{Methods}

\subsection{\textit{Participants}}60 subjects (29 participants between 18 and 29 years old, and 31 participants over 65 years old) were scanned with the protocols approved by the St. James Hospital and the Adelaide and Meath Hospital, incorporating the National Children Hospital Research Ethics Committee. All participants were adults recruited for a larger study (REF) and came from the greater Dublin area.

All participants underwent the Cambridge Neuropsychological Test Automated Battery (CANTAB; \hyperref[csl:18]{(Robbins et al., 1994)}) which has been used to detect changes in neuropsychological performance and include tests of working memory, learning and executive function; visual, verbal and episodic memory; attention, information processing and reaction time; social and emotion recognition, decision making and response control. The CANTAB scores were normalised for age and IQ. Particularly, the following subtest were administered:

-The Paired Associate Learning Test is a measure of episodic memory where boxes are displayed on the screen and each one has a distinct pattern. The boxes are opened in random order, revealing the pattern behind the box. In the test phase, patterns are individually displayed in the centre of the screen, and participants must press the box that shields the respective pattern. 

- Pattern Recognition Memory is a test of visual pattern recognition memory in which the participant is presented with a series of visual patterns, one at a time, in the centre of the screen. In the recognition phase, the participant is required to choose between a pattern they have already seen and a novel pattern. In this phase, the test patterns are presented in the reverse order to the original order of presentation. This is then repeated, with new patterns. The second recognition phase can be given either immediately (immediate recall) or after a delay (delay recall).

-The Spatial Working Memory Test assesses spatial working memory in which boxes are presented on the computer screen and hidden behind one of the boxes is a yellow circle. Participants must find the box where the yellow circle is located. As the task progresses, the number of boxes on the screen increases. We analysed the spatial working memory strategies (i.e., the number of times participants begin a new search strategy from the same box). 

Moreover, participants performed the trail making test (TNT; \hyperref[csl:19]{(Reitan, 1958)}) which is neuropsychological test of visual attention and task switching. TNT test that can provide information about visual search speed, scanning, speed of processing, mental flexibility, as well as executive functioning \hyperref[csl:20]{(Arnett \& Labovitz, 1995)}.


\subsection{\textit{MRI data acquisition}}
Each participant was imaged in a 3.0 T Philips whole-body MRI scanner (Philips, The Netherlands) using a standard single-shot GE EPI sequence operating with a 8-channel array receiver coil in all cases. The parameters of the EPI time series sequence were as follows: Flip angle = 35\ensuremath{^\circ}, TR = 60 ms and the TE = 18 ms with a voxel size was 3.5 x 3.5 x 3.5 mm, matrix size was 64 x 64, SENSE factor 3, bandwidth readout direction was 2148 Hz and saturation pulse was 6 ms with 21 mT/m gradient strength. The imaging slice was set coronal above the ventricle to avoid pulsation effects (see Figure 1 for example). In addition, two saturation slices of 5 mm in thickness were placed parallel to the imaging slice (15 mm above and 20 mm below). These slabs were applied to introduce asymmetrical magnetic gradient scheme  for the  ZQC contrast (for a full description see \hyperref[csl:3]{(Kerskens \& L{\'{o}}pez-P{\'{e}}rez, 2017)}). Additional scans without the saturation slabs  using the same imaging parameters were carried out which had two effects on the ZQC; a change of the angulation between the asymmetric gradient field and the main magnet field towards the magic angle and a lengthening of the ZQC correlation distance. As a result ZQC were were strongly reduced leaving only higher order coherence in the time series. In this manuscript, we will refer to these scans as the SQC weighted signal to differentiate it from the ones that contain stronger ZQC effects (i.e., ZQC weighted signal).

Anatomical MRI images in all studies included a high-resolution sagittal, T1-weighted MP-RAGE (TR = 2.1 s, TE = 3.93 ms, flip angle = 7\ensuremath{^\circ})\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/e11e2bc7-2ccc-4964-a1da-cc1ba760a274-upicture1/e11e2bc7-2ccc-4964-a1da-cc1ba760a274-upicture1}
\caption{{\label{figure-72ae02e7fff2b4e6f2a0e044be117687} {\textbf{Figure 1.} Acquisition model which includes the image slice (central red line) and the REST slabs above and below the imaging slice both 5 mm thick and separated 15 mm and 20 mm respectively. }%
}}
\end{center}
\end{figure}


\subsection{\textit{Signal Preprocessing}}All calculations were developed in a Dell Optiplex 790 with 12 Gb RAM using Matlab 2017a (The MathWorks Inc., Natick, MA, 2017). Since motion correction could not be applied due to the single slice nature of the experiment, average time-series were visually inspected in search for irregularities which were manually removed from the analysis leaving the rest of the time-series unaltered. In addition, the data was not smoothed to avoid removing low frequencies which may lead to the loss of information\unskip~\hyperref[csl:21]{(Pignat et al., 2013)}. Manual segmentation was used to create a mask to remove cerebrospinal fluid (CSF) contributions which were later eroded to avoid partial volume effects at the edges. The first 100 scans were removed to avoid signal saturation effects.

\subsection{\textit{\textbf{Recurrence Quantification Analysis}}}We used Recurrence Quantification Analysis (RQA) to analyse the dynamical temporal characteristics of the MRI signals. RQA quantifies the repeated occurrences of a given state of a system (i.e., recurrences) by analysing the different structures present in a recurrence plot, which is a graphical representation of the recurrences in the dynamical system\unskip~\hyperref[csl:9]{(Zbilut \& Webber, 1992}; \hyperref[csl:12]{Santos et al., 2014)}. In our analysis, we considered the following RQA measures \hyperref[csl:22]{(Bosl et al., 2017)}:

\begin{itemize}
  \item \relax Determinism (Det): it represents a measure that quantifies repeating patterns in a system and it is a measure of its predictability. Regular, periodic signals, such as sine waves, will have higher DET values, while uncorrelated time series will cause low DET.
  \item \relax Mean Line (MeanL): it is the average length of repeating patterns in the system. It represents the mean prediction time of the signal, a measure of chaos or divergence from an initial point.
  \item \relax Entropy (Ent): it is computed as the Shannon entropy of the distribution of the repeating patterns of the system. If a signal has high entropy it exhibits diversity in short- and long-duration periodicities.
  \item \relax Laminarity (Lam): it determines the frequency of transitions from one state to another, without describing the length of these transition phases. It indexes the general level of persistence in some particular state of one of the time-series \hyperref[csl:23]{(Hirata \& Aihara, 2010)}.
  \item \relax Trapping Time (TT): it represents the average time the system remains on a given state and it is a measure of the stability of the system (REF Marwan). It was calculated here using the \textit{tt} function from the CRP Toolbox for Matlab\unskip~\hyperref[csl:10]{(Marwan et al., 2002)}.
  \item \relax Maximum Line (MaxL): it is the largest Lyapunov exponent of a chaotic signal, which gives the longest time spent in a single state by the system \hyperref[csl:24]{(G{\'{o}}mez \& Hornero, 2010)}.

\end{itemize}
  Three important parameters need to be set to calculate the recurrence plots. First, the smallest sufficient embedding dimension was determined by means of the \textit{fnn} function\unskip~\hyperref[csl:25]{(Kennel et al., 1992)} within the CRP Toolbox (Marwan, N.: Cross Recurrence Plot Toolbox\unskip~\hyperref[csl:26]{(Marwan et al., 2007)} for MATLAB, \textit{Ver}. 5.22 (R31.2), \url{http://tocsy.pik-potsdam.de/CRPtoolbox/).} This function estimates the minimum embedding dimension where the false nearest neighbours vanish. We applied the  \textit{fnn} to all time-series and obtained an average value of 15 which agrees with the typical values recommended for biological signals (see \hyperref[csl:27]{(Marwan \& Webber, 2014)}). The second parameter is the delay which we calculated using the mi function from the CRP Toolbox\unskip~\hyperref[csl:26]{(Marwan et al., 2007}; \hyperref[csl:28]{Roulston, 1999)}. This function finds the non-linear interrelations in the data and determines which delay fulfills the criterion of independence. In the same way as the embedding dimension, we applied the \textit{mi} function to all time-series and we obtained an average value of 3. Finally, several criteria have been suggested for the choice of the recurrence threshold\unskip~\hyperref[csl:29]{(Thiel et al., 2002)}. Here, we adapted the radius for each time-series using the embedding dimension and delay computed together with a recurrence rate sufficiently small (i.e., RR = 3\%)\unskip~\hyperref[csl:10]{(Marwan et al., 2002)}. Additional parameters in the RQA calculations were Euclidean normalisation of each time series and minimum line length equal to 2.

\subsection{\textit{\textbf{ Multifractal Detrended Fluctuation Analysis}}} In biological systems, coupling between different systems often exhibits different spatial and temporal scales and hence its complexity is also multi-scale and hierarchical \hyperref[csl:7]{(C. K. Peng \& Goldberger, 2009)}. Thus, to analyse the scale invariant properties of the MRI segments and its changes with age we used Multifractal Detrended Fluctuation Analysis (MFDFA). To do so, we first, calculated the multifractal spectrum, \textit{D,} of each time-series using the MFDFA  Matlab Toolbox \unskip~\hyperref[csl:14]{(Ihlen, 2012)}. The multifractal spectrum identifies the deviations in fractal structure within time periods with large and small fluctuations\unskip~\hyperref[csl:14]{(Ihlen, 2012)}. Each spectrum was computed using a window length with a minimum value of 2 and the maximum of half the length of the time-series. The \textit{q}-order statistical moments were chosen between -11 and 11 and divided into 21 steps (see further description in\unskip~\hyperref[csl:14]{(Ihlen, 2012)}).

From each fractal spectrum two parameters were calculated, i.e., width of the spectrum\textit{ W} and the position of the spectrum maxima \textit{H} . The width \textit{W} is calculated by subtracting the lower part of the spectrum, \textit{h\ensuremath{_{min}}} , to the upper part of the spectrum, h\ensuremath{_{max}}\unskip~\hyperref[csl:14]{(Ihlen, 2012}; \hyperref[csl:30]{Ma et al., 2006}; \hyperref[csl:17]{Makowiec et al., 2011)}:

$W\;=\;h_{max\;}-\;h_{min} $

a small width indicates that the time-series has fewer singularities and tends to be more monofractal. Finally, the\textit{H} parameter represents the value h in which the singularity spectra has its maximum \textit{h(D\ensuremath{_{max}})} \unskip~\hyperref[csl:17]{(Makowiec et al., 2011)} :

$H=h\left(D_{max}\right) $

the position of \textit{h\ensuremath{_{max}}} moves to higher values when the stronger singularities are present. Highly deterministic signals can often be explained by a lower number of fractal dimensions and are characterised by smaller \textit{W} and \textit{H} due to a decrease in the number of singularities (REF).

\subsection{\textit{Statistical Analysis}} Prior to any statistical analysis all variables were converted to z-scores. Those participants having z-scores larger 3 standard deviations in three non-linear parameters or more were rejected from the analysis. In total only 1 participant in the old group was removed. Independent \textit{t}-tests were computed to test differences between the RQA and fractal measures of the average MRI signals in both groups. Inspection of Q-Q Plots was carried out to all the measures to check if the data were normally distributed. Additionally, Levene's test for equality of variances was applied and in those cases where this assumption was violated, a \textit{t} statistic not assuming homogeneity of variance were computed on these measures. Finally, a linear regression between the non-linear measures and the participants age were performed.



\section{Results}

\subsection{\textit{The ZQC weighted signals}}Examples of the ZQC weighted signals and their Fourier transform of both age groups are shown in figure 2a-h. As it can be seen, strong cardiac signal fluctuations are resolved only in the average time-series of young subjects (Figure 2a and 2c) while in older subjects the strong cardiac signals are diminished (Figure 2e and 2g). The frequency spectra of the time-series (Figure 2b, 2d, 2f and 2h) showed the strongest cardiac frequencies for the young group, while in the older group, however, the spectra show stronger harmonics, envelope waves or both, in addition to the weaker cardiac frequencies. These envelope waves have a beat frequency of the (cardiac frequency)/n, where n takes values of (2,3,4,6,9 ...). These results are consistent over most subjects.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/fb3f0100-11d5-415a-9ff8-881e15a85253-uimage1-0/fb3f0100-11d5-415a-9ff8-881e15a85253-uimage1-0}
\caption{{\label{figure-c67d75493eda1dcc2b923f65a669b086} { \textbf{Figure 2.} Example of the time-series as well as its frequency spectra for two young (a-c and b-d) and two old healthy adults (e-g and f-h). These examples are representing the typical results in these groups.}%
}}
\end{center}
\end{figure}

\subsection{\textit{Non-linear dynamics of the iZQC weighted signal}}We also tested how the non-linear dynamics varied in the iZQC weighted time-series in all participants (see Supplementary Results for the SQC results). At a group level all the RQA parameters but the DET were statistically significantly higher in the old group in comparison to the young one (see Table 1 for group averages): MeanLine (\textit{t}(57) = 2.23, \textit{p} = .02; \textit{d} = .58), MaxLine (\textit{t}(57) = 2.81, \textit{p} = .007; \textit{d} = .73), Ent (\textit{t}(57) = 2.62, \textit{p} =.01; \textit{d} = .68), Lam (\textit{t}(57) = 3.68, \textit{p} = .001; \textit{d} = 0.96) and TT (\textit{t}(57) = 4.57, \textit{p} \textless .001; \textit{d} = 1.19) and Det (\textit{t}(57) = 1.23, \textit{p} = .22; \textit{d} = .32).

Likewise, the fractal properties of the iZQC weighted signal in the old group were statistically higher in \textit{W} (\textit{t} (57) = 5.44, \textit{p} \textless .001; \textit{d} = 1.41) and \textit{H} (\textit{t} (57) = 3.53, \textit{p} = .001; \textit{d} = .92) in comparison to the young group, suggesting a more chaotic behaviour in the old population.

Using regression analysis we analysed the relation between the non-linear features and the age of the participant (see Figure 4 and Supplementery Figure 1 for the remaining conditions). Very significant regression equations were found for W($R^{2}$ = .343 , F(1,57) = 19.70, \textit{p} \textless\ .001), H($R^{2}$ = .183, F(1,57) = 12.76, \textit{p} = .001), Lam($R^{2}$ = .228, F(1,57) = 21.6, \textit{p} \textless\ .001) or TT($R^{2}$ = .29, F(1,57) = 24.27, \textit{p} \textless\ .001).\selectlanguage{english}
\begin{table*}
\caption{{{\textbf{Table 1.} Group mean averages of the RQA and MFDFA parameters extracted from the iZQC weighted time-series for the young and old groups (\textit{p} \textless\ .05(*), \textit{p} \textless\ .01(**), \textit{p} {\textless}.001(***)).}}}
\label{table-wrap-6b1d1cf6c5c936aa400fd9759e2913ee} 

\ignorespaces
\centering

\begin{tabular}{ccc}
{\linewidth}
\hline
\textit{Parameter} & \textit{Young} & \textit{Old} \\
\hline
\textbf{\textit{Det}} &
  32.16 \ensuremath{\pm} 10.12 &
  35.72 \ensuremath{\pm} 11.99 \\
\textbf{\textit{MeanLine*}} &
  2.81 \ensuremath{\pm} .34 &
  3.07 \ensuremath{\pm} .51\\
\textit{\textbf{MaxLine**}} &
  99.10 \ensuremath{\pm} 38.35 &
  135.53 \ensuremath{\pm} 58.63\\
\textit{\textbf{Ent*}} &
  .77 \ensuremath{\pm} .19 &
  .92 \ensuremath{\pm} .24\\
\textbf{\textit{Lam**}} &
  35.42 \ensuremath{\pm} 8.24 &
  46.50 \ensuremath{\pm} 14.00\\
\textbf{\textit{TT***}} &
  2.24 \ensuremath{\pm} .11 &
  2.48 \ensuremath{\pm} .26\\
\textit{\textbf{W***}} &
  .15 \ensuremath{\pm} .08 &
  .27 \ensuremath{\pm} .07\\
\textbf{\textit{H***}} &
  .02 \ensuremath{\pm} .01 &
  .03 \ensuremath{\pm} .01\\
\hline
\end{tabular}
\end{table*}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Screenshot-2019-12-29-at-14-51-06/Screenshot-2019-12-29-at-14-51-06}
\caption{{Linear regressions between (a) fractal width, (b) fractal maxima, (c)
laminarity and (d) trapping time, and participants age.
{\label{962804}}%
}}
\end{center}
\end{figure}

\subsection{\textit{Are group differences coming from movement or cognitive differences?}}

Since the iZQC effects are sensitive to movement, we explored the relation between the non-linear parameters and motion quality control variables from a fMRI study within the same session. Although the information was not available for the young group, there were not significant correlations between these measures (see Table 2) or in other words, the non-linear dynamics in the old cohort were not worsened by motion.\selectlanguage{english}
\begin{table*}
\caption{{{\textbf{Table 2.} Spearman correlations between quality control movement measures and the non-linear parameters. In these correlations, n was equal to 27 since movement information was not available for one participant and another did not pass quality control (see section 2.5). \textit{p}-values are in parethensis.}}}

\label{table-wrap-6b1d1cf6c5c936aa400fd9759e2913ee} 

\ignorespaces
\centering

\begin{tabular}{ccc}
{\linewidth}
\hline
\textit{Parameter} & \textit{QC max movement} & \textit{QC mean movement} \\
\hline
\textbf{\textit{Det}} &
   -.36 (.07) &
   .13 (.50) \\
\textbf{\textit{MeanLine}} &
  -.10 (.60) &
   .00 (.97)\\
\textbf{\textit{MaxLine}} &
  -.30 (.13) &
   -.01 (.95)\\
\textit{\textbf{Ent}} &
  -.26 (.19) &
   .08 (.69)\\
\textbf{\textit{Lam}} &
  -.27 (.17) &
   .17 (.38)\\
\textbf{\textit{TT}} &
  -.13 (.52) &
   .22 (.27)\\
\textit{\textbf{W}} &
  .06 (.77) &
   .02 (.92)\\
\textbf{\textit{H}} &
  -.14 (.47) &
   -.06 (.77)\\
\hline
\end{tabular}
\end{table*}
\\

Finally, we explored any possible relation to cognitive measures and tests performed during the study (Table 3). CANTAB scores showed consistent negative correlations and trends between visual memory scores (pattern recognition memory and working memory) and the RQA parameters while no correlations arose with the TNT scores (see Figure 4). Consistently, young participants that showed higher complexity (i.e., smaller non-linear parameters) also had better cognitive scores.  Altogether, there were significant changes with ageing but even more with these cognitive scores which suggest that the iZQC signal may be related to aspects of cognitive abilities.\selectlanguage{english}
\begin{table*}
\caption{{{\textbf{Table 3.} Spearman correlations between the nonlinear parameters and the CANTAB and TNT scores. In these correlations, n varies between 59 (CANTAB scores) and 55 (TNT scores) since one participant did not pass quality control (see section 2.5) and the measures were not available to all of them. \textit{p}-values are in parethensis (trend \textit{(+)} \textit{p} \textless .05(*) and \textit{p} \textless .01(**)).}}}
\label{table-wrap-6b1d1cf6c5c936aa400fd9759e2913ee} 

\ignorespaces
\centering
\begin{tabular}{ccccccc}
{\linewidth}
\hline
\textit{Parameter} & \textit{Pat. Recognition Mem. (immediate recall)}& \textit{Paired Associates Learning} & \textit{Spat. working Mem. (strategy)} & \textit{Pat. Recognition Mem. (delayed recall)} &  \textit{Trial A} & \textit{Trial B} \\
\hline
\textbf{\textit{Det}} &
   \textbf{-.29 (.02)*} &
   .08 (.51) &
   -.20 (.12) &
   -.01 (.90) &
   .05 (.69) &
   -.02 (.87)\\
\textbf{\textit{MeanLine}} &
  \textbf{-.32 (.01)*} &
   -.03 (.82) &
   -.18 (.16) &
   -.13 (.32) &
   .00 (.98) &
   -.04 (.74)\\
\textbf{\textit{MaxLine}} &
  \textbf{-.34 (.008)**} &
   -.18 (.95) &
   \textbf{-.26 (.04)*} &
   -.20 (.12) &
   -.00 (.96) &
   -.02 (.86)\\
\textit{\textbf{Ent}} &
  \textbf{-.32 (.01)*} &
   .08 (.53) &
   -.21 (.09)+ &
   -.12 (.33) &
   .09 (.49) &
   -.04 (.76)\\
\textbf{\textit{Lam}} &
  \textbf{-.28 (.02)*} &
   .06 (.63) &
   \textbf{-.34 (.007)**}&
   -.09 (.48) &
   -.04 (.72) &
   -.05 (.68)\\
\textbf{\textit{TT}} &
    -.23 (.07)+ &
  -.06 (.62) &
  \textbf{-.30 (.01)*} &
  -.19 (.14) &
  .01 (.90) &
  -.07 (.57)\\
\textit{\textbf{W}} &
  -.17 (.19) &
   .02 (.84) &
   -.13 (.29) &
   -.18 (.16) &
   .25 (.06)+ &
   .13 (.31)\\
\textbf{\textit{H}} &
  -.22 (.08)+ &
   -.06 (.61) &
   -.22 (.09)+ &
   -.09 (.49)&
   .00 (.95) &
   .07 (.60)\\
\hline
\end{tabular}
\end{table*}
\\\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Picture-4/Picture-4}
\caption{{Examples of linear regressions between mean line (a) and entropy(b), and
Standardised CANTAB Pattern Recognition Memory Scores (Inmediate
Recall), and between Laminarity (c) and Trapping Time (d), and
standardised CANTAB working memory strategy scores.
{\label{820046}}{\label{820046}}%
}}
\end{center}
\end{figure}

\section{Discussion}

In this paper, we have analysed if the dynamical complexity of long-range quantum coherence fluctuations in the brain tissue varied with age. While comparing two populations, we presented qualitative evidence that the strong cardiac constant fluctuations are resolved only for the younger subjects while for the older ones the strong cardiac effect is diminished. Non linear analyses confirmed this effect and showed quantitative differences between both age groups, which were related to variations in complexity and chaos of the measured signals. However, more in-depth analysis showed that (1) the differences between both groups are not due to higher movement in the older group, (2) there were participants in the older group whose dynamics did not differ from the dynamics of the younger group and (3) higher complexity of this fMRI-based signal was related to better cognitive performance in some of the CANTAB scales. Altogether, this may suggest that the long-range quantum fluctuations may be sensitive to ageing, cognitive abilities or even differences in the conscious state.

The changes in the iZQC weighted signal were manifold, varying in shape, amplitude and frequency (see Figure 2). In particular, typical older participants possessed a higher number of frequencies as well as a decrease in the size and number of the cardiac bursts. On the other hand, typical younger volunteers were characterised by roughly constant bursts and clear cardiac frequencies with less harmonics. 
%I think this are harmonics. I know it varies with respiration but this is not visible in the data 
%Taking a closer look to the time-series it is noticeable, especially in the old group, that sometimes there are peaks at n (2, 3, 4...) times of the cardiac frequency (see Figure 2f and 2h). This is more evident in frequency domain, where the intensity of the fundamental cardiac frequency decreases and more harmonics arise. It could be hypothesised that the stronger signal fluctuations are correlated with the respiration cycle. In fact, some studies have reported the so-called respiratory pump, which is the increase in the cardiac output during inspiration\unskip~\hyperref[csl:31]{(Lin, 1999}; \hyperref[csl:32]{Hayen et al., 2013)}. This can be caused by mechanisms related to respiratory cycle changes in the main magnetic field\unskip~\hyperref[csl:33]{(Raj et al., 2001)} and changes in arterial CO2 pressure\unskip~\hyperref[csl:34]{(Wise et al., 2004)}. This could explain why more peaks arise in frequency domain, since the constant cardiac signal gets distorted due to respiration leading to a variation in the number of harmonics. However, this correlation with the respiration frequency could not entirely be confirmed because a respiration belt was not used in the study and future analysis using this sequence should consider including one to control for respiration changes.

To quantified the apparent differences between both groups, we  used non-linear time-series analyses to determine changes in the dynamics of the MRI signals. First, we applied Recurrence Quantification Analysis (RQA), which was proven to be a powerful discriminatory tool to detect significant differences in the dynamics of both groups (see Supplementary files for further discrimination analysis between both groups based on these measures). All the RQA measures (see Table 1 and Supplementary Table 1) were lower in the young group in comparison to the older group in both types of MRI signals, suggesting differences in the complexity of the signal dynamics in both populations. Second, fractal analysis showed an increase in the number of singularities with age, which is characterised by an increase in the width and position of the spectral maxima \unskip~\hyperref[csl:14]{(Ihlen, 2012}; \hyperref[csl:35]{Dick \& Svyatogor, 2012)}. These differences were supported by the RQA entropy which denotes the Shannon entropy of the histogram of the lengths of diagonal segments and thus indicates the complexity of the deterministic structure of the system \unskip~\hyperref[csl:12]{(Santos et al., 2014)}. This increase in the chaoticity of the signal is only visible when the quantum effects are measured, since no differences were found between the fractal parameters in the SQC time series (see Supplementary Table 1). %
Altogether, we have shown that evoked by the heart and under special conditions (for more details see \hyperref[csl:3]{(Kerskens \& L{\'{o}}pez-P{\'{e}}rez, 2017)}) long-range quantum fluctuations can be measured. 

Our results are in line with recent studies indicating that higher complexity in a system is a feature of healthy dynamics \hyperref[csl:8]{(dos Santos et al., 2014)} or greater degree of functional specialisation and integration in brain dynamics \unskip~\hyperref[csl:36]{(Ho et al., 2017)} and that this complexity declines with disease and age (e.g., \hyperref[csl:37]{(Manor \& Lipsitz, 2013)}). In fact, we observed some negative significant correlations between CANTAB scores, and RQA measures (see Tables 3), were lower scores (i.e., higher complexity) were related to better cognitive scores. Particularly significant were the relations with pattern recognition memory and working memory subscales, suggesting a link between the iZQC signal and short-term memory abilities. Research exploring the effect of age on neural changes has uncovered both structural and functional changes on short-memory related areas (e.g., Guttman et al., 1998; Raz, Gunning-Dixon, Head, Dupuis and Acker, 1998;Gunning-Dixon and Raz, 2003). In addition, fMRI studies have shown that healthy old adults present higher activity levels in some brain regions during the performance of cognitive tasks and these changes coexist with disrupted connectivity (for a review see Sala-Llonch, Bartr\selectlanguage{ngerman}és-Faz and Junqué, 2015). However, to the best of our knowledge, there are not fMRI-based signals that are able to predict consistently these CANTAB scores. This is specially surprising since the iZQC signal is a very rough and functional measurement that was placed without any particular anatomical selection. Thus we believe that these fluctuations, which had quantum origin, may be an important physiological effect for understanding cerebral regulation and have clinical relevance for all neurovascular diseases.


However, even though these results seemed consistent across groups, the regression plots showed that some young participants presented consistently higher non-linear parameters in comparison with the older group. Likewise, some participants in the old group showed very low parameters as participants in the young group. Following visual inspection of the individual signals (see Supplementary Figure X for examples), we confirmed that neither all participants in the younger group showed strong cardiac resolved signals nor all participants in the older group showed noisier signals. Initially, one may think that differences between both groups can be due to old participants moving more inside the scanner. In fact, Kerskens and L\'{o}pez P\'{e}rez (2017) reported that during hyperventilation inside the scanner the iZQC effect entirely vanished due to increased movement. Although the information was not available for the young group, motion quality control variables from a fMRI study within the same session did not correlate with any of the non-linear parameters (see Table 2). Thus, we can conclude that the dynamics in the old cohort were not worsened by motion. In spite of this, future studies using the sequence should try to minimise the effect of movement during the data acquisition (e.g, adding extra cushions to hole the head) which might help improve the intensity of the quantum effect \hyperref[csl:3]{(Kerskens \& L{\'{o}}pez-P{\'{e}}rez, 2017)}.

Another possibility is that these changes with age are due to the origin of the signal itself. The iZQC signal is accessible via multiple spin echoes (MSE) that induce a spontaneous symmetry breaking of the Landau\selectlanguage{english}'s order parameter \hyperref[csl:3]{(Kerskens \& L{\'{o}}pez-P{\'{e}}rez, 2017)}. These MSE oscillations appear in brain tissue simultaneously with the arterial pulse and might reveal the spontaneous character of the underlying physiological mechanism. Since ageing profoundly impacts not only the brain at multiple levels (e.g., \hyperref[csl:1]{(Salat, 2004}; \hyperref[csl:2]{Allen et al., 2005)}), but also the vascular and cardiac physiology (for a review see \hyperref[csl:38]{(Jackson \& Wenger, 2011)}), the dynamical interactions between the heart and the brain will be also altered. 

Altogether, these results provide further evidence of the existence of quantum fluctuations in the brain tissue. However, ageing and cognitive decline could be only one part of the explanation of the existing differences. Kerskens and Lopez Perez (2017) showed how a participant that reported falling sleep during testing presented a strong signal change during testing. Ageing is often accompanied by a variety of sleep complaints and sleep disorders (Cook and Ancoli-Israel, 2012), which leads to older participants falling asleep more easily during daytime. In this study, participants laid during testing in a supine position inside a dark room with no specific instructions but to remain still. Under these conditions, participants in the older group are more likely to feel sleepy thus affecting their conscious state. This might suggest that these quantum fluctuations in the brain tissue are sensitive not only to cognitive changes due to ageing but also depend on the conscious state of the participant. Further research is required to confirm and validate these findings.

Several limitations of this study arise from the acquisition protocol. In order to obtain the iZQC weighted signal, fast repetition times in combination with magnetisation transfer effects are required, which limits the number of imaging slices to just one. The use of one imaging slice complicates the study of particular areas and it could induce variability in the results across all the participants even when the position of the imaging slice is carefully planned. As a consequence, different slices should be acquired to study a larger region and improve the comparison between groups. There are some approaches that could be used to overcome this limitation. For example, we could excite three or more imaging slabs by means of Multi-band excitation\unskip~\hyperref[csl:39]{(Feinberg et al., 2010)}. It combines slice excitation at different off-resonance frequencies with demultiplexing based on spatial sensitivity differences of radiofrequency receiver coils which would allow covering larger brain areas. A second improvement can be achieved with the increase in the number of channels in the receiver coil. In this way, we could acquire higher number of slices in the same repetition time. Future research should focus on expanding the sequence protocol to be able to cover larger brain areas that would allow using the sequence in a wide range of studies. Finally, the group sizes in this study were small and the results need to be consider preliminary and therefore, further research is needed to confirm these findings.

\section{Conclusions}
 We have shown that evoked by the heart and under special conditions long-range quantum fluctuations can be detected and measured. Here, we provide further evidence not only of the existence of quantum fluctuations in the brain tissue, but also that they are sensitive to ageing and cognitive abilities. We showed qualitatively and quantitatively that these fluctuations worsen with age and that their decline is related to a decrease in the complexity of the quantum phenomenon. Consistently with the idea that higher complexity is related to healthier dynamics, our quantum phenomena showed higher complexity in younger population. Altogether, the long-range quantum coherence is a promising biomarker that need to be tested in larger and more diverse populations with clinical relevance for all neuro-vascular diseases.
 

\section*{Acknowledgements}We would like to thank S. Joseph for carrying out the imaging protocols in our participants and E. Stanaszek for reading the manuscript. This work was supported by Science Foundation Ireland (SFI-11/RFP.1/NES/3051), European Regional Development Fund through the Ireland Wales Programme 2007-13, Trinity College Institute of Neuroscience. This project has received additional funding from the Institute of Psychology, Polish Academy of Sciences.

\selectlanguage{english}
\FloatBarrier
\section*{References}\sloppy
\phantomsection
\label{csl:1}{Thinning of the Cerebral Cortex in Aging}. (2004). \textit{Cerebral Cortex}, \textit{14}(7), 721–730. \url{https://doi.org/10.1093/cercor/bhh032}

\phantomsection
\label{csl:2}{Normal neuroanatomical variation due to age: The major lobes and a parcellation of the temporal region}. (2005). \textit{Neurobiology of Aging}, \textit{26}(9), 1245–1260. \url{https://doi.org/10.1016/j.neurobiolaging.2005.05.023}

\phantomsection
\label{csl:3}\textit{{Cardiac pulsation evokes long-range quantum coherence in the conscious brain}}. (2017). \url{https://doi.org/10.1101/219931}

\phantomsection
\label{csl:4}{Using coherence to enhance function in chemical and biophysical systems}. (2017). \textit{Nature}, \textit{543}(7647), 647–656. \url{https://doi.org/10.1038/nature21425}

\phantomsection
\label{csl:6}{Revisiting the Quantum Brain Hypothesis: Toward Quantum (Neuro)biology?}. (2017). \textit{Frontiers in Molecular Neuroscience}, \textit{10}. \url{https://doi.org/10.3389/fnmol.2017.00366}

\phantomsection
\label{csl:7}{{ADAPTIVE} {DATA} {ANALYSIS} {OF} {COMPLEX} {FLUCTUATIONS} {IN} {PHYSIOLOGIC} {TIME} {SERIES}}. (2009). \textit{Advances in Adaptive Data Analysis}, \textit{01}(01), 61–70. \url{https://doi.org/10.1142/s1793536909000035}

\phantomsection
\label{csl:8}{Recurrence Quantification Analysis as a Tool for Discrimination Among Different Dynamics Classes: The Heart Rate Variability Associated to Different Age Groups}. (2014). In \textit{Springer Proceedings in Mathematics {\&} Statistics} (pp. 125–136). Springer International Publishing. \url{https://doi.org/10.1007/978-3-319-09531-8_8}

\phantomsection
\label{csl:9}{Embeddings and delays as derived from quantification of recurrence plots}. (1992). \textit{{Physics Letters}}, \textit{171}(3-4), 199–203.

\phantomsection
\label{csl:10}{Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data}. (2002). \textit{{Physical Review E}}, \textit{66}(2).

\phantomsection
\label{csl:11}\textit{{Beat-to-beat complexity analysis before the onset of ventricular tachycardia. Computers in Cardiology}} (pp. 477–480). (2003). (pp. ).

\phantomsection
\label{csl:12}\textit{{Recurrence Quantification Analysis as a Tool for Discrimination Among Different Dynamics Classes: The Heart Rate Variability Associated to Different Age Groups}}. (2014). 125–136.

\phantomsection
\label{csl:13}Webber, J. C., \& Marwan, N. (Eds.). (2015). \textit{{Dynamic Coupling Between Respiratory and Cardiovascular System}}. Springer.

\phantomsection
\label{csl:14}{Introduction to multifractal detrended fluctuation analysis in Matlab. Front}. (2012). \textit{{Physiol}}, \textit{141}(1{\textendash}18).

\phantomsection
\label{csl:15}{Fractal and multifractal analysis: A review}. (2009). \textit{{Medical Image Analysis}}, \textit{13}(4), 634–649.

\phantomsection
\label{csl:16}{Non-linear behaviour of human EEG: Fractal exponent versus correlation dimension in awake and sleep stages}. (1998). \textit{{Neuroscience}}, \textit{250}(2), 91–94.

\phantomsection
\label{csl:17}{Reading multifractal spectra: Aging by multifractal analysis of heart rate}. (2011). \textit{{EPL (Europhysics Letters)}}, \textit{94}(6), 68005.

\phantomsection
\label{csl:18}{Cambridge Neuropsychological Test Automated Battery (CANTAB): a factor analytic study of a large sample of normal elderly volunteers}. (1994). \textit{Dementia and Geriatric Cognitive Disorders}, \textit{5}(5), 266–281.

\phantomsection
\label{csl:19}{Validity of the Trail Making Test as an indicator of organic brain damage}. (1958). \textit{Perceptual and Motor Skills}, \textit{8}(3), 271–276.

\phantomsection
\label{csl:20}{Effect of physical layout in performance of the Trail Making Test.}. (1995). \textit{Psychological Assessment}, \textit{7}(2), 220.

\phantomsection
\label{csl:21}{The impact of denoising on independent component analysis of functional magnetic resonance imaging data}. (2013). \textit{{Journal of Neuroscience Methods}}, \textit{213}, 105–122.

\phantomsection
\label{csl:22}{Nonlinear {EEG} biomarker profiles for autism and absence epilepsy}. (2017). \textit{Neuropsychiatric Electrophysiology}, \textit{3}(1). \url{https://doi.org/10.1186/s40810-017-0023-x}

\phantomsection
\label{csl:23}{Devaney's chaos on recurrence plots}. (2010). \textit{Physical Review E}, \textit{82}(3). \url{https://doi.org/10.1103/physreve.82.036209}

\phantomsection
\label{csl:24}{Entropy and Complexity Analyses in Alzheimer's Disease: An {MEG} Study}. (2010). \textit{The Open Biomedical Engineering Journal}, \textit{4}(1), 223–235. \url{https://doi.org/10.2174/1874120701004010223}

\phantomsection
\label{csl:25}{Determining embedding dimension for phase-space reconstruction using a geometrical construction}. (1992). \textit{{Phys. Rev. A}}.

\phantomsection
\label{csl:26}{Recurrence Plots for the Analysis of Complex Systems}. (2007). \textit{{Physics Reports}}, \textit{438}(5-6), 237–329.

\phantomsection
\label{csl:27}{Mathematical and Computational Foundations of Recurrence Quantifications}. (2014). In \textit{Understanding Complex Systems} (pp. 3–43). Springer International Publishing. \url{https://doi.org/10.1007/978-3-319-07155-8_1}

\phantomsection
\label{csl:28}{Estimating the errors on measured entropy and mutual information}. (1999). \textit{{Physica D}}, \textit{125}.

\phantomsection
\label{csl:29}{Influence of observational noise on the recurrence quantification analysis}. (2002). \textit{{Physica D}}, \textit{171}, 138–152.

\phantomsection
\label{csl:30}{A new measure to characterize multifractality of sleep electroencephalogram}. (2006). \textit{{Chinese Science Bulletin}}, \textit{51}(24), 3059–3064.

\phantomsection
\label{csl:31}C., P., T., L., \& T, S. (Eds.). (1999). \textit{{Physiology of the circulation in Fundamentals of Anaesthesia}} (pp. 315–343). Medical Media.

\phantomsection
\label{csl:32}{The effects of altered intrathoracic pressure on resting cerebral blood flow and its response to visual stimulation}. (2013). \textit{{NeuroImage}}, \textit{66}, 479–488.

\phantomsection
\label{csl:33}{Respiratory effects in human functional magnetic resonance imaging due to bulk susceptibility changes}. (2001). \textit{{Physics in Medicine and Biology}}, \textit{46}(12), 3331–3340.

\phantomsection
\label{csl:34}{Resting fluctuations in arterial carbon dioxide induce significant low frequency variations in BOLD signal}. (2004). \textit{{NeuroImage}}, \textit{21}(4), 1652–1664.

\phantomsection
\label{csl:35}\textit{{Potentialities of the wavelet and multifractal techniques to evaluate changes in the functional state of the human brain}} (pp. 207–215). (2012). (pp. ).

\phantomsection
\label{csl:36}{Complexity analysis of resting state fMRI signals in depressive patients}. (2017). In \textit{{39th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC)}}.

\phantomsection
\label{csl:37}{Physiologic complexity and aging: Implications for physical function and rehabilitation}. (2013). \textit{Progress in Neuro-Psychopharmacology and Biological Psychiatry}, \textit{45}, 287–293. \url{https://doi.org/10.1016/j.pnpbp.2012.08.020}

\phantomsection
\label{csl:38}{Cardiovascular Disease in the Elderly}. (2011). \textit{Revista Espa{\~{n}}Ola De Cardiolog{\'{\i}}a (English Edition)}, \textit{64}(8), 697–712. \url{https://doi.org/10.1016/j.rec.2011.05.003}

\phantomsection
\label{csl:39}{Multiplexed Echo Planar Imaging for Sub-Second Whole Brain FMRI and Fast Diffusion Imaging}. (2010). \textit{{PLoS ONE}}, \textit{5}(12).

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