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\begin{document}
\title{olddraft}
\author{test}
\affiliation{Affiliation not available}
\author{Chri}
\affiliation{Affiliation not available}
\selectlanguage{english}
\begin{abstract}
We propose that cerebral dynamics can be considered as a non-Hermitian
quantum system which during a short interval of the cardiac cycle has
real eigenvalues. We show that driving the system into PT symmetry is
only possible with the help of the active matter changing from a laminar
flow into a Ceilidh dance-like flow. During this change, the flow
experiences~ a transition from broken to unbroken PT symmetries which
results in a topological phase. The topological phase is then imparted
onto the topological defects which are an essential part of this Ceilidh
dance flow. This so-called topological braid could be the basis of
topological computing in the brain. Our recent experimental findings in
conscious humans have shown that the predicted PT symmetry changes may
exist. This is important because it is also related to consciousness
underpinning~ the computational aspect of the phenomenon.~%
\end{abstract}%
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In a recent brain study it has been shown that long-range quantum coherence emerges in the interim of arterial inflow and venous outflow, if and only if the brain is conscious \cite{Kerskens_2017a}. Such an observation may suggest that consciousness is based on quantum computing which, if true,
may only be realizable within a "protected" system. Taking into account that biology also contains diversities of topological defects \cite{Kumar_2018,Duclos_2016,Kawaguchi_2017,Saw_2017}, which may move also in a Ceilidh dance fashion \cite{Doostmohammadi_2018}, then it is intriguing to assume that such defects are "braided" in a topological phase with the aim to process information essential for consciousness.\\
%exceptional points
Topological phases exist among others at the exceptional point between broken and unbroken $\mathcal{PT}$ symmetry in non-Hermitian physics \cite{Heiss_2016,Yuce_2019}.
%pt symmetry in the brain
The brain may induce $\mathcal{PT}$ symmetry during the cardiac cycle in the following way:
Let us consider two density functions $\rho_{a,v}$
where the indices indicate the inflow and outflow, respectively. %The cardiac pressure wave oscillates the arterial and venous system together with the same frequency $a$ and with the same phase. This is due to their interweaved, highly symmetrical order, their close location and due the symmetrically induced pressure wave.
The arterial inflow acts as source $b \rho_{a}$ and the venous outflow as sink $-b \rho_{v}$. We can write
\begin{equation}
\imath \partial_t \begin{pmatrix} \rho_{a} \\ \rho_{v} \end{pmatrix} =\begin{bmatrix} -\imath b& 0 \\ 0 & +\imath b \end{bmatrix} \begin{pmatrix} \rho_{a} \\ \rho_{v} \end{pmatrix}.\label{lossgain}
\end{equation}
Both arteries and veins oscillate according to the cardiac frequency compensating each others volume changes, which results in a coupling $g$ which extends (\ref{lossgain}) to a loss-gain equation \cite{Bender_2002} as
\begin{equation}
\imath \partial_t \begin{pmatrix} \rho_{a} \\ \rho_{v} \end{pmatrix} =\begin{bmatrix} -\imath b& g \\ g & +\imath b \end{bmatrix} \begin{pmatrix} \rho_{a} \\ \rho_{v} \end{pmatrix}.\label{lossgainf}
\end{equation}
The system as described by equation (\ref{lossgain}) is not Hermitian but it has the eigenvalues
$E_\pm=\pm \sqrt{g^2-b^2}$ \cite{Bender_2002}. For a weak coupling $g^2 < b^2$, the eigenvalues are complex. The net inflow, which can be interpreted as cerebral blood flow, is then $\sqrt{b^2-g^2}$ which shows that cerebral blood flow is governed by a global parameter $g$ defined by the cardiac frequency and a tissue component $b$ which may be tunable by the tissue, locally. Both $b$ and $g$ aren't directly dependent on blood pressure which is known as cerebral autoregulation.
For real eigenvalues $g^2 \ge b^2$,
the equation (\ref{lossgainf}) is $\mathcal{PT}$ symmetric \cite{Bender_2002}. This can only happen if the probability flow is conserved. This may only occur at an interval between arterial inflow and venous outflow when in and outflow are compensating each other.
%Let us now assume that there is an interval during the cardiac cycle where inflow is equal to outflow. Then, the system as described by equation (\ref{lossgain}) has $\mathcal{PT}$ Symmetry with real eigenvalues
Then, the system is closed which allows a comparison with a non-linear Schr\"odinger equation which can be transferred into "classical physics" using the amplitude and phase decomposition \cite{Madelung_1927,Schleich_2013}. In our case the amplitude, which is associated with the density, depends only on time but not on any space coordinates. With $\rho_\pm =\rho_0 \exp[\imath E_\pm t]$ and $\Psi_\pm = \sqrt{\rho_\pm} \exp[\imath \phi t]$ we arrive at two non-linear Schr\"odinger equations
\begin{equation} \imath \partial_t\psi_\pm = \nabla^2 \psi_\pm + V \psi_\pm \pm E_\pm \psi_\pm, \label{2}
\end{equation}
where $V$ is the potential energy and the Planck constant $\hbar$ and mass was set to 1.
The Madelung potential $\nabla^2 \rho/(2\rho)$ is missing here because we assumed that $\rho$ is independent of space coordinates. It would, if present, relate pressure with flow \cite{Zhang_2007} which would violate the cerebral autoregulation. Equation (\ref{2}) has a corresponding classical Navier-Stokes equation which can be found again using the amplitude and phase decomposition \cite{Madelung_1927}. We arrive at
\begin{equation} \rho_0 (\partial_t u + u \nabla u) =\nabla \cdot \left( V\pm E_\pm \right).\label{euler1}
\end{equation}
On the other hand, the Navier-Stokes equation for active matter may have the following form
\begin{equation} \rho_0 (\partial_t u + u \nabla u )=\nabla \cdot \left( \Pi^{\mathrm{viscous}} + \Pi^{\mathrm{active}}\right),\label{euler2}
\end{equation}
where $\Pi^{\mathrm{viscous}}$ is the viscosity stress tensor and $\Pi^{\mathrm{active}}$ the active matter stress tensor \cite{Ramaswamy_2010}. The elasticity tensor $\Pi^{\mathrm{elastic}} $ is neglected because active stresses may dominate elastic stresses.
The active stress may result from a nematic-like tensor $\mathbf{Q}$ which comes from coarse-graining the dipolar flow fields that are generated by active cells \cite{Doostmohammadi_2018}. With $\Pi^{\mathrm{active}} = \zeta \mathbf{Q}$, the active matter tensor can exhibit two general modes depending on the sign of the activity coefficient $\zeta$. A comparison of (\ref{euler1}) and (\ref{euler2}) allows us to identify the stress tensors as $V \sim \Pi^{\mathrm{viscous}}$ and $E_\pm \sim \pm \mathbf{Q}$ because only active tensors $\Pi^{\mathrm{active}}$ may coexist with altered signs. As a consequence we find that the tensors $\pm \mathbf{Q}$ are also $\mathcal{PT}$ symmetric if the global system is $\mathcal{PT}$ symmetric and vise versa.
If active tensors $\pm \Pi^{\mathrm{active}}$ with altered signs coexist, active matter may show the Ceilidh dance behavior where topological defects are moved around. If $E_\pm$ are complex the flow must be laminar instead, which is coincide to greater $b$ and no defects present.\\
With the $\mathcal{PT}$ symmetry, which can only be reached and left by passing through the exceptional point, those defects impart a topological phase which results in so-called topological braiding.
That means only if the cells are active enough to influence $b$ in a way that $g^2 \ge b^2$ then topological computing is possible. But it also means only if the cardiac pulsation is sufficient that braiding becomes possible at all.
From observation, we know that on the one hand cardiac pulsation is a necessity for consciousness. On the other hand, recent experiments \cite{Kerskens_2017a} have shown that the $\mathcal{PT}$ symmetry may exist at the desired time interval and probably only during consciousness.
%In $\mathcal{PT}$ symmetry the angular momentum is conserved The global effect of $\mathcal{PT}$ symmetry extends into the microscopic world of nuclear spin via conversation of the angular momentum which includes the nuclear spin. the MRI may be explained by the flow dipoles orMoving an macroscopic entity like a topological defect in a flow field with its total angular moment $S_1$ involves a force which then alters $S_1$ by $\Delta S_1$. For a flow field with $\mathcal{PT}$ symmetry, where total angular $S_t$ moment must be conserved, a counterforce may apply to an other defect $S_2$ so that its alteration $\Delta S_2$ is $\Delta S_2=-\Delta S_1$. Then, both defects are entangled in $\mathcal{PT}$ symmetry.\\
%long-range entanglement
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